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A Longitudinal Model Of Residence Change

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A Longitudinal Model Of Residence Change

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Xsrox University M icrofilms
300 N orth Z M b R oad
Ann A/toor, M leM esn 4S10S
7 4 - 1 1 , 7 2 0
YEE, W illiam Jan k , 1935-
A LCffclTUDBttL M O D E L OF RESIDENCE CH A N G E.
U n iv ersity o f Southern C a lifo rn ia , P h.D ., 1973
Sociology, demography
University Microfilms, A X ER O X Com pany, Ann Arbor, Michigan
THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED.
A LONGITUDINAL MODEL OF
RESIDENCE CHANGE
by
William Yee
A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In P artial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(Sociology)
June 1973
UNIVERSITY O F S O U T H E R N CA LIFO RN IA
T H C O R A O U A T * SC H O O L .
U N IV U S IT Y PARK
U W ANOCLCS, CALIFO RNIA »OOOY
This dissertation, w ritten by
W illla m Y ss
under the direction of i l l . . Dissertation C om
m ittee, and approved by all its members, has
been presented to and accepted by T he Graduate
Schoolt in partial fu lfillm en t of requirements of
the degree of
D O C T O R O F P H IL O S O P H Y
..................................................................................... { / £>«m
D ate f l J Z j t ” .......................
FORWARD
Some papers begin with p a r t i a l l y conceived Ideas and
t h e i r progress is f i l l e d with guesses and questions. At
the same time there 1s th is I n t u i t iv e feeling th at the
approach 1s 1n the "correct" d ire c tio n . The re s u lt --
"Not communicated." Such was th is paper. Why the Im
passe? Where does I t not communicate? The p a tie n t-
impatient quest with others . . . the rew riting , the r e
phrasing, and re-th1nk1ng . . . and f i n a l l y , a disq uieting
understanding. Not "eureka" but "how simple . . . should
have s t a r te d with the d ir e c t route . . . now th a t I know
what I am doing, I should do I t over." Instead, I sub
mit th is paper as a “f i n a l i t y " of a search with the
hopes th at i t 1s a beginning.
With a simple pride, mention is made that part of
this paper received the Bogardus Award at the University
of Southern C alifornia and was read at the Pacific Socio
logical Association Meeting. The re tro sp e c tiv e "burden"
of hindsight on e a r l i e r versions Is tempered by the
moments of lev ity 1n a s c i e n t i f i c community.
The 11st of c re d its to "old" teachers and friends
is burled 1n the realm of fond memories. Thanks Is ex
pressed to members of my committee over the years: Alan
11
Acock, Judith Friedman, LaMar Empey, Malcolm Klein, San
ford Labovltz, Robert Newman, Rodger Rice, and Maurice
Van Arsdol. "Survivors of the ordeal" were Alan Acock,
LaMar Empey, Judith Friedman, and Robert Newman. A debt
Is acknowledged to Maurice D. Van Arsdol for his conveying
in te r e s t which resulted 1n the I n itia l paper. "Contin
uity of e f f o r t 1 1 was provided by Rodger Rice. Gerald
Butler, Nancy Edwards, David Heer, Steven Lubeck, and
Robert Yeaman provided many comments that were incorpor
ated .
Appreciation 1s expressed to Robert McGinnis where
his "preliminary paper" 1s cited without permission. A
recent paper "substantiates" the approach of his model.
In this paper, "free uses" of his Ideas are made.
The University Computing Center provided f a c i l i t i e s
on the Honeywell H-600 and the IBM 360,
Most important was the atmosphere 1n the Sociology
Department at the University of Southern California.
Wisdom of change, 1 1 Ignorance" of e rro rs , and a knowledge
beyond the technical to the sim plicity of human beings.
As human beings there Is disagreement, and the responsi
b i l i t y of the final paper is mine.
William Yee
February 1973
1 1 1
TABLE OF CONTENTS
Page
FORWARD....................................................................................................... 11
LIST OF TABLES........................................................................................ v1
LIST OF FIGURES................................................................................... v11
Chapter
I A LONGITUDINAL M ODEL OF RESIDENCE CHANGE . . 1
II THEORY AND LITERATURE............................................... 6
Life Cycle Explanation ................................................. 6
Cornell Model................................................................. 9
Mobile-Stable (Mover-Stayer)
Population Explanation ............................................ 10
S Longitudinal Model ...................................................... 11
III METHODOLOGY FOR CONSTRUCTING
THE LONGITUDINAL MODEL ................................................. 15
Empirical and Mathematical Problems
In Model C onstruction............................. 15
Life Table Cohort Analysis ......... 20
Methodology of the Model and the
Use of Theory................................................ 21
Defining One Longitudinal Move . . . . . . . 33
Comb 1 n a tio n a l-P ro b a b l11ty Formulation. . . . 38
Age and the Series of Moving
and S taying............................................................ 39
Life H istories of Residences ......................... 41
Average Life History of Residence Change . 44
Stayers and Last Past M o v e.......................... 45
Comparison to the Cornell Model................. 53
IV THE DATA............................................................................ 56
V RESULTS................................................................................. 59
Age and Staying at the Same House................. 61
Life H istories of R e s i d e n c e s .......................... 63
1 v
TABLE OF CONTENTS - continued
C hapter Page
Average Number of Hoves In Life H istory. . . 67
P ast-R etrospective Hovers........................ 67
Comparison to Cornell Model. .. . . . . . . 69
VI DISCUSSION............................................................................... 72
Empirical V alidity ........................................................... 74
Methodology—Longltud1nal P ro b a b ilitie s . . . 75
Independent or Dependent P ro b a b ilitie s . . . 76
Life Cycle Explanation ................................... 83
Cornell Model.......................................................................... 84
Mobile-Stable Population Explanation . . . . 85
The Longitudinal Model's Life
Cycle Stage E xplanation............................................ 88
Problems and E valuation................................................. 93
VII SIGNIFICANCE OF THE MOOEL AND
GENERAL CONCLUSIONS........................................................... 98
REFERENCES.................................................................................................. 101
v
LIST OF TABLES
Table
1
2
3
4
5
6
7
P age
Comparison of the Featuras of t h e I_
Cycle Explanation, the Cornell M o d « l »
and the Mobile-Stable Population E x
planation .................................................. . _ _ . . . 12
Illustration of the P a s t - R e tr o s p e c t. "I v «
Mover Probabilities by Single Y e a r s o Y
Age and Single Years to Last M ove U s ~f r e g
Three Series.............................................. _ _ _ . . . 50
Probabilities of a Subsequent L o n g l ' t e a —
dlnal M ove Using Age-Related Mob1 T 1 'fc.y'
Rates for 1959 and Five E s t im a te s o * F t h e
Length of Time Between Moves by A g e D s —
trlbutlon Used In the Model. . . . _ . .. 58
Probability of Experiencing a G i v e n N u m
ber of Moves by Age and D i f f e r e n c e s { y
in Moving Probabilities Between A g e
Categories - Life Cycle E stim a te - _ . . . . 60
Per Cent Stayers-Always at the S a m e l-l o u s e
by Age, United States, 1952 and 1 9 5 9 —
Actual and Estimated Results U s i n g F" ■! e r «
Distributions - ...................................... . _ * . . 62
Life History of Number of R e s i d e n c e s
Age and Per Cent Within C a t e g o r i e s » U r e -f ' f e e d
States, 1959 - Actual and E s t i m a t e d —
suits Using Five Distributions _ . . . 64
Past-Retrospective Movers--Per C e n t : o Y
the Total Population who were S t a y e r s
During the Survey Year by Age a n d b y Y « e r
of Last Mover,* United States, 1 9 5 9 —
Actual and Estimated Using Five D I s t r * i —
butlons - ................................................ _ _ . . . 68
The Modified S Residence Change M o d e 1
Compared to the Cornell Model s S 1 m u 1 « a " t *1 o n
Estimates..................................................... . _ _ . . . 70
vl
LIST OF FIGURES
F1qure Page
1 I llu s tra tio n of Partitioning the Age
Related Mobility Rate Distribution into
P robabilities of One Longitudinal Move
for the S Longitudinal M o d el................................. 22
v 11
CHAPTER I
A LONGITUDINAL MODEL OF RESIDENCE CHANGE
For each year of the past two decades» approxi
mately 20 per cent of the United States population
changed Its residence (U.S. Census, P-20 S e rie s ) . This
r a t e , applied to the 1970 population, Indicates th a t 41
m illion persons moved In th a t year. What are the causes
for such a volume of movers? Do these causes change over
the person's l i f e h istory or are the causes re la te d to
one another? How many times will a person move In his
l i f e time?
Of I n t e r e s t here are the causes of moving and the
number of residence changes made by persons 1n t h e i r l i f e
h isto ry . Three th e o re tic a l formulations have focused on
the causes of moving for an aggregate population. These
are: (1) the l i f e cycle explanation (Lee, 1966), (2) the
Cornell Model (McGinnis, 1968), and (3) the m obile-stable
population (or m over-stayer) explanation (Goldstein,
1965; Morrison, 1969). A major d i s t i n c t i o n among them 1s
whether they assume th at a given move Is dependent on
past moves or h i s t o r i e s or th a t the moves are Independent
of each o th e r, being re la te d only to the person's current
age and decision making s i t u a t i o n . This d i s t i n c t i o n Is
1
re fle c te d in both conceptual and p ro b a b ility d iffe re n c e s .
A mathematical model1 will be constructed for th is
research 1n order to analyze the extent to which current
p r o b a b ilitie s of moving at a given age are Independent of
or dependent on e a r l i e r residence change.2 This approach
to analyzing the Influence of past moving h is t o r i e s 1s
u t i li z e d because longitudinal data 1s necessary. However,
there 1s no longitudinal re g is try of persons by address
1n the United S tates. There are annual sample surveys
(U.S. Census, P-20 s e rie s ) giving the per cent who moved
(a change of house) In the previous year, and decennlal
census (U.S. Census, PC (1)-A1). There are only a few
studies of 11fe hi s to ri es of moving (U.S. Census, P-23
s e r i e s , No. 25; Taeuber, Chlazze. and Haenszel, 1968) and
these are limited to changes between areas of r e s i d e n c e .’
1 The term "model" 1s used In a sense sim ilar to
H ille r ( 1 9 6 7 : 1 5 ) - - " . ..a conceptualization of a group of
phenomena constructed by means of a r a ti o n a l e , where the
ultimate purpose Is to furnish the terms and r e la ti o n s ,
the propositions of a formal system which 1f validated ,
becomes theory." Here, unlike H i l l e r , a system of
ra tio n a le 1s considered theory.
2 The co n sisten t finding of h i s t o r i c a l streams and
counter-streams of migrants between areas at the macro
level Is not at Issue.
3 Th* annual survey (P-20 s e r i e s , No. 188: 2) asks,
"Has . . . 11vTng Tn th is house (date) a year ago?" If the
answer was "No," the question 1s asked, "Has __ living
In th is same county on (date) a year ago?" If the answer
was, "No," the question 1s asked, "Hhat State (or foreign
country) was . . . liv in g 1n on (date) a year ago?" In the
l i f e history data (P-23 S eries, No. 25:4-5), the question
3
How can the cross sectional data on moving be mathematic
a lly re la te d 1n order to construct a longj t u d l n a l . l i f e
history sequence of moving and of areas of residences for
an aggregate population 1n order to answer the th e o re tic a l
p ro b a b ility question? How can the conceptual theory on
causes for moving be combined with th is empirical data
Into a more complete mathematical formulation of the pro
cess of residence change and on residence h is to r ie s ?
The purpose of th is study 1s to construct a lo n g i
tu d in a l, l i f e h isto ry mathematical model of residence
change which r e f l e c t s "theory" in Its formulation, Is
based on empirical data, can be rew ritten to account for
d i f f e r e n t types of data, and 1s validated against actual
data. The model computes estimates of l i f e h is t o r i e s of
residence change from annual data. These longitudinal
l i f e history estimates are computed using an Independent,
1s asked, "How many years have you lived 1n 7" An
"Always" response terminates the Interview. Otfierwlse, he
1s asked to s t a t e the number of years of continuous r e s i
dence there and the location , siz e -ty p e , and duration of
residence of each of his previous residences 1n order, up
to a maximum of th re e , and f i n a l l y , his place of b i r t h .
Thus, the residence h isto ry covered is at most five r e s i
dences (p re sen t, 3 p ast, and b i r t h ) . The respondent was
also asked whether he had lived for at l e a s t a year In
any residence In addition to the five about which Informaa-
tion was obtained on the questio nn aire. This question
provided the c l a s s i f i c a t i o n of the population by number
of residences up through six or more. In ad d itio n, an
area of residence was defined as a p o l i ti c a l u n it, an In
corporated c ity or county, 1n which the person had lived
continuously for at l e a s t a year. Thus, residence implies
an area, ra th e r than a house.
4
*ge context, probability approach (a model cannot s t a r t
with both an Independent and a dependent probability
formulation). The estimates are compared to actual data
to determine how close a " fit" an Independent probability
approach has to a wide variety of data. Comparisons are
also made to dependent probability estim ates. This 1s
done 1n order to consider the theo retical question of in
dependent or dependent p ro b a b ilitie s for a given age and
th e ir relation to causes of residence change.
The model Incorporates the causal residence change
theory 1n an elementary probability approach which In
cludes adding, multiplying, and combining p ro b a b ilitie s .
Here, these probability rules are called "combinational
mathematical theory1 ' and assume Independent p ro b a b ilitie s .
In u t i li z i n g combinational theory (Harkov Chain Processes)
in developing the model, two methods -- l i f e table cohort
analysis ( l a t e r referred to as "cohort analysis") with
Independent p ro b a b ilitie s (rates) and Stochastic Pro
cesses with dependent p ro b a b ilitie s -- are also examined.
For sake of presentation, the proposed model is considered
as a life table cohort analysis procedure which has been
modified by combinational mathematical theory. In this
presentation, d istrib u tio n s or function are used to con
s tru c t p ro b a b ilitie s . In referring to d istrib u tio n s or
functions, no convention 1s followed whether the graph or
equation should be referred to as "x by y" or "y by x."
No th e o re tic a l causal re la tio n s h ip 1s Implied. In e ith e r
case, the methods used are part of the th e o re tic a l causal
formulations considered.
Although a comparison Is made of three th e o rie s ,
the emphasis Is on Independent stages 1n the l i f e cycle
in the model's development. This emphasis on the stages
1n the l i f e cycle 1s the basis for callin g the metho
dology the S Longitudinal Residence Change Model.
CHAPTER II
THEORY AND LITERATURE
A central th e o re tic a l Issue 1s whether the causes
and the asso ciated p r o b a b il i t ie s of moving at a given age
are independent of or dependent on e a r l i e r residence
change. Three theories focusing on th is Issue are the
l i f e cycle explanation, the Cornell Model, and the mobile-
stab le population explanation. These theories of r e s i
dence change p o sit an an a ly sis somewhere between large
populatlon aggregates, e . g . , the Interchange of migrants
between c i t i e s (S to u ffe r, 1962) and reasons why an 1nd1-
vldual moves (Rossi, 1955). These explanations are sum
marized here.
Life Cycle Explanation
The l i f e cycle explanation Is part of the "push-
pull" theory (Lee, 1966). The push-pull theory considers
both the macro population level and the interm ediate l i f e
cycle explanation. At the macro le v e l, push factors such
as a high unemployment ra te are crucial causes for r e s i
dence change. In addition to these macro factors are the
"normative" 11fe cycle fa c to rs such as marriage among the
young and the expectation th a t they will live away from
6
7
t h e i r parent's home. In the push-pull theory, both these
macro and normative f a c t o r s Impinge on the I n d i v i d u a l 's
decision and result I n stay in g or 1n a residence change.
This discussion focuses on the normative r a t h e r
than the macro f a c t o r s 1n Its use of the U f a c y c le ex
planation as theory. These normative factors a r e th eo re
t i c a l l y posited as I n f l u e n c i n g behavior at d i f f e r e n t ages.
For many persons ( a g g r e g a t e ) these behaviors o ccur at
sim ilar ages, and t h e r e is an aggregate co n tin u ity over
time 1n when they a r e Im portant. Thus, age Is an e f f e c
tiv e predictor v a r i a b l e regarding residence change
(Bogue, 1959; Shryock, 1964; Eldrldge, 1965) 1n the l i f e
cycle explanation ( R o s s i , 1955; Taeuber et al . , 1968;
Sabagh, Van Arsdol, a n d B u tle r, 1968; Yee and Van Arsdol,
1968). These normative behaviors within age c a te g o r i e s
are considered as c a u s e s fo r moving 1n the l i f e cycle
explanatlon.
These causes In t h e l i f e cycle explanation of r e s i
dence change result In d e fin in g a set of p r o b a b i l i t i e s
(the concept of plus o r minus "valences” was used 1n Lee's
[1966] article) which vary according to the sta g e 1n that
cycle. These s ta g e - c y c le s are demarked by the causes of
moving by age for an a g g r e g a te population1 and Include:
1 In d i f f e r e n t i a t i n g the "aggregate" p o p u latio n
from the "Individual" l e v e l , two processes 1n a n a ly s is
are used. The f i r s t p r o c e s s 1s to aggregate "upward"
e
c e s s a tio n of education, marriage, d iv o r c e , labor force
p a r t i c i p a t i o n , retirement from work, and death. G11ck
( 1947, 1955) provides median ages for some of these
e v e n ts based on Census data. As these events occur 1n
t h e l i f e cycle, the behavior changes and the p ro b a b ili
t i e s of residence change are a lte re d u n t i l another event
o c c u r s .
These behavlorlal changes r e f l e c t movement from
on e stage to another and the associated p r o b a b ilitie s
a r e Independent of past residence change h is t o r ie s .
From th eir analysis of empirical data, Taeuber, at al.
( 1968:23) suggested that current c h a r a c t e r i s t i c s are more
Im portant than background. The p r o b a b i l it y that a per
son will move Is dependent upon the Immediate context
1n which he makes his decision to move o r not move, upon
h i s knowledge of his present place of re s id e n c e , and upon
h i s possible destin ation s. This In dlv ld ual decision
making aspect 1s not d ire c tly "tested" in th is study.
from the Individual to larger groupings based on common
dim ensions. The second process 1s to "d isagg regate
downward" from the larger population to the Individual.
I f enough dimensions are u til i z e d , an in d iv id u a l can be
I d e n t i f i e d from a larger grouping. Here, In the l if e
c y c l e , a homogeneous aggregate Is being analyzed, I . e . ,
t h e Individuals In the larger grouping have commonality
on many dimensions. In the examples, t h i s homogeneous
a g g re g a te Is discussed as the unit of a n a l y s i s . The pro
b a b i l i t i e s calculated from this aggregate re fe r to a
h y p o th e tic a l unit where all Individuals a re Id en tica l,
and only vary on the specified dimension, e . g . , by age
o r length of time at a residence.
9
Rattier, the push-pull theory's explanation 1s accepted
with regards to decision making within an age context.
The portion that will be "tested" 1s the age ( l i f e cycle)
factor where the p ro b a b ilitie s vary by age aggregate and
are Independent of one another.
Cornell Model
In co n trast, the Cornell Model (McGinnis, 1968}
uses Stochastic Process mathematics and theo retical as
sumptions about past mobility are posited within th is
formulation. The main assumption 1n the Cornell Model 1s
the concept of "cumulative In e rtia ." This concept In d i
cates that the person's past history Is a crucial factor:
"The longer a person resides In a community, the more
likely he will remain there" (McGinnis, 1968:716).2 This
theorem (assumption) 1s used 1n Stochastic Process theory
which emphasizes the Idea of dependent p ro b a b ilitie s in a
tra n s itio n matrix of conditional p ro b a b ilitie s . Implied
In this theorem Is a continuous, Increasing stayer prob
a b ility for those persons who remain longer times at a
residence. Also, stayer time rather than mover time 1s
2 More precisely, cumulative In e rtia 1s: "The pro
b a b ility of remaining 1n any s ta te of nature Increases as
a s t r i c t monotone function of duration of prior residence
1n that state" (McGinnis, 1968:716). McGinnis applies
this concept to social mobility, emotional disturbances,
and crim inality. The mechanism he suggests to explain
cumulative In e rtia Is that "ties" develop to a position.
considered. Therefore, In terms of the push-pull theory,
only the pull factors for staying are considered by the
Cornell Model.
A summary of the comparison between the l i f e cycle
explanation and the Cornell Model 1s given l a t e r in
Table 1.
Mobile-Stable (Moyer-Staver)
Population Explanation
The 11fe cycle and the Cornell Model are explana-
tlons directed towards "predicting" the behavior of an
aggregate population. Instead of a theoretical o r ie n ta
tio n , the mover-stayer explanation Is an empirical
analys1s from which Inferences about residence change
are made according to data d i f f e r e n t i a l s . The mover-
stayer explanation divides the population Into two groups
(1) those who previously moved, and (2) those who stayed.
The obtained data d iffe re n tia l* e . g . , background v a r i
ables, are used to predict future behavior. Sometimes
this explanation 1s extended to Include the concept of
repeated mover. This 1s an Individual who Is tra n sie n t
and reports a larger number of moves. When this concept
1s Included (Goldstein, 1955), the mover-stayer explana
tion suggests that there are two populations: (1) those
who stay and are "stable," and (2) those who move and are
repeated movers ("mobile"). Morrison (1969) reformulates
11
the mover-stayer explanation and suggests Including Ideas
from both the l i f e cycle and the Cornell Model. However,
this formulation Is not considered here* since Morrison
does not Indicate how these two theories should be com
bi ned.
The Cornell Model and the mobile-stable population
explanation have common features. The comparison of
these two theories and the l i f e cycle explanation are
glven In Table 1.
The major s im ila r itie s of the Cornell Model and
the mobile-stable population explanation are the predis
position to move or to stay based on sodal-psycholog1cal
factors occurring In the past histo ry . For these
reasons, the Cornell Model and the mobile-stable popula
tion explanation are considered sim ilar formulations,
especially with regards to the central Issue that past
mobility history Is the crucial factor.
S Longitudinal Model
This discussion will u t i l i z e some of the theore
tic a l features from the l i f e cycle explanation, the
Cornell Model and the mobile-stable population explana
tion 1n a mathematical formulation. The emphasis Is on
the l i f e cycle stages in the Longitudinal Model's con
stru ctio n . However, features of the other two frameworks
are also Incorporated J_n the construction. In addition,
TABLE 1
Comparison of tho Features of the Life Cycle Explanation, the Cornell
Model, and the Nobile-Stable Population Explanation
Dimension
Life Cycle
Explanation Cornell Model*
Mobhe-Stable Popu
lation (Nover-Staycr)
Explanation
Phenomenon
Explained Hovers and Stayers Stayers Only Movers and Stayers
Causal Factor Life cycle stages ac
cording to age. Norma
tive factors Including
those related to both
"push" and "pull" at
the macro level.
Continuous time at
residence by stayers
results In "Inertia,"
and neighboring "ties."
Only "Pull* factors
for staying are
studied.
Predisposition to
move or to stay - a
psychological factor
(Goldstein, 1955).
Probability
Implication
as Input Into
the Model
Direct probabilities
(Valences) are used for
each age stage In the
li f e cycle. These prob-
111t1es are treated as
dependent only on the
stage 1n the cycle.
However, stages are In
dependent of each other.
Conditional probabili
ties are used which are
dependent on past h is
tory of time at a re si
dence.
Persons who moved In
the past have a higher
probability of moving
In the future and per
sons who stayed In the
past are likely to
stay 1n the future.
FormalIzatlon
of Nodal
No methodological form
alization has been done.
This will be done here.
Independence of stages
suggests a Markov Chain
Process would be appro
priate.
Formalization using
Stochastic Process
theory.
No mathematical form
alization.
* The Cornell Model (McGinnis, 1960) does not discount age as a factor; however, the
■ajor emphasis 1s on tine at a residence. How age enters Into the Cornell Model's
concept of cumulative Inertia Is not specified 1n detail (only that there Is an age
system).
13
a f te r the model 1s constructed based on l i f e cycle stages,
It will attempt to derive features (see Table 1) of the
other two frameworks. If this derivation Is possible,
then the l i f e cycle explanation Is suggested as more
"general" than the other models.
In constructing the model, the concept of a pro
b ab ility formulation and a time between moves factor 1s
taken from the Cornell Model and u tiliz e d In the l i f e
cycle context. Instead of the Cornell Model's Stochastic
Processes, a special case -- Markov Chain Process -- 1s
used 1n this context. For purposes of studying the
theoretical Implications of the l i f e cycle stages and
the idea that only normative l i f e cycle events a l t e r
the probability of moving, a step function9 Is u tiliz e d
(Yee and Van Arsdol, 1968). Here, the sim plification Is
made that the p ro b a b ilitie s are constant for a stage and
are only altered at age points where a l i f e cycle event
(step) occurs which a lte rs the time between moves which
In turn a lte rs the probability. Also, for sim p licity ,
3 An I l l u s t r a t i o n of a step function Is as follows:
Step 2
Age
A homogeneous age aggregate Is assumed where Identical
behaviors are enacted until a tra n s itio n event occurs at
a specified age and resu lts 1n a behavior change.
14
assumption Is made that the normative event occurs at one
l i f e cycle age point. The assumption Implies a homogen
eous rather than a heterogenous aggregate population.
Associated with each age stage 1s one probability. With
regards to using a probability formulation and the time
between moves, the Cornell Model and the S Longitudinal
Model are conceptually p a ra lle l. Here, age and the l i f e
cycle event are considered as Influencing both the length
of time at a residence and residence change.
Therefore, 1n this discussion, a Longitudinal Model
Is constructed which u tiliz e s theoretical features from
the Cornell Model 1n a life cycle context of independent
p ro b a b ilitie s . These p ro b a b ilitie s are used in combina
tional mathematical theory. Rewriting the equations of
the model, the estimates are compared to sets of actual
data. Also, the dependent probability features of the
Cornell Model and the mobile-stable population explana
tion are analyzed by calculating estimates using the S
Model 1n order to answer the theoretical question.
CHAPTER I I I
METHODOLOGY FOR CONSTRUCTING THE
LONGITUDINAL M ODEL
The methodology for constructing the S Longitudinal
Mathematical Model Incorporates the previous theo ries.
However, several problems are encountered 1n constructing
the model. These problems will be addressed by Indi
cating the specific Issue Involved, then showing how the
methodology will resolve them.
Emp1r1cal and Mathematical Problems
In Model Construction
In the construction of the Longitudinal Model, com
binational mathematical theory Is u tiliz e d . Once having
chosen this Independent probability framework, r e s t r i c
tions are placed on the model's framework. These r e s t r i c
tions concern the Incorporation of sociological theory
with data for the mathematical formulation. The Inclu
sion of theory will be considered 1n the next section.
Here, the central problem 1s how to determine the prob
a b i l i t i e s for combinational theory from empirical data.
This choice of using empirical data was made as opposed to
using a theoretical probability d istrib u tio n (Spllerman,
15
16
1972) or simulation equations (McGinnis, 1968). Combina
tional theory within the l i f e cycle explanation using
empirically derived p ro b a b ilitie s 1s the approach u t i l
ized.
However, the problems with using empirical data 1n
combinational theory are numerous. These problems are
not unique to constructing a model of residence change
and are present whenever empirical data are used 1n an
Independent combinational probability framework. Here
the problems are presented In general terms, and the de
t a i l s of the solution u tiliz e d 1n constructing the Longi
tudinal Model are given In the next three sections.
F i r s t , empirical data on residence change are sur
vey data which contain three sources of variatio n; these
are sample, within year, and between years v ariation .
Each of these types may present problems 1n defining pro
b a b ilitie s for combinational theory and are considered
here.
Sample v a r ia tio n : The data on residence change are
sample surveys where "retrospective" recall of the previous
y ear's moving Is recorded. Since these are surveys,
sampling variation 1s Involved. While this error 1s min
imal when estimating the proportion of people who moved
ov erall, 1t 1s larger for sub-samples by age aggregates.
For example, a sample of 2,000 persons may be adequate for
estimating the mover rate for the United States. However,
1 7
divided Into 70 age categories (or about 30 persons per
age category), the age rate would produce less re lia b le
estimates of the p ro b ab ilities required for combinational
theory. Given this problem of sample variation and r e
lating 1t to p ro b a b ilitie s , the solution given l a t e r Is
to use "smooth" curves and to examine h isto rical data and
studies for consistency of re su lts .
Within Year V ariation: Within year, or seasonal,
variation In residence change may occur; however, this
variation is not considered, since one year rates are
analyzed. Using one year survey periods, within year
variation Is like sample variation and has the problems
previously discussed.
Between Years V ariation: The concept of probability
Involves a parameter which sample data only approximates.
Since the residence change pattern of the early 1900's
might be d iffe ren t from the 1970's, combining samples
(or between years) for a long h isto rical period would
not yield the concept of a probability. Here, the "true"
residence change effect Is "confounded" with the sampling
error and departs from the concept of a probability.
Each of these three sources of variation are likely
minimal under the condition that across h isto rical time
there has been l i t t l e variation. Without such s t a b i l i t y ,
the concept of a probability or parameter becomes more
d i f f i c u l t to define.
Second, In addition to survey v ariatio n problems,
combinational theory requires the defining of the
p ro b a b ility of one event which 1s Independent of previous
events. Two problems are I n t e r - r e l a t e d In attempting to
meet data requirements for combinational theory. The
f i r s t problem 1s Inherent In the counting of the events
of moving or staying during a one year survey period and
the second Is 1n using annual data (sy n th etic cohort) to
define the p r o b a b il i t ie s for longitudinal h is to r ie s (real
cohort). When the event of a move or a stay 1s counted
during the survey, persons who moved twice or more are
counted as having moved once. In add ition , variable
times during which the move occurred between the survey
periods are counted as equiv alent, while 1n f a c t , 1f the
concept of cumulative I n e r tia Is tru e , a d i f f e r e n t i a l
p otential of fu rth e r moves may be estab lish ed. This d i f
ference In potential for fu rth e r moves would have less of
an e f f e c t on a real cohort of the same persons over th e ir
l i f e time than on a syn th etic cohort or cro ss-section al
data on d lf f e r e n t persons. Since a re g is try 1s not
available to e s ta b lis h data for a real cohort, cross-
sectional data are u t i l i z e d by n ecessity . Thus, 1n
cro ss-sectio n al data the two problems of counting the
actual event and the p o ten tial events are I n te r r e la te d
1n defining p r o b a b ilitie s for combinational theory.
19
The solution to th is problem, discussed In d etail
l a t e r , 1s to use a second d i s t r i b u t i o n 1n order to define
the longitudinal p ro b a b ility of moving using data obtained
1n the survey year. By using both the cro ss-sectio n al
mover p ro b a b ility and the length of time between moves
d i s t r i b u t i o n , according to age, the longitudinal poten tial
p ro b a b ility of moving will be Incorporated. These d e t a i l s
are discussed l a t e r In the section on the methodology of
Defining One Longitudinal Hove.
Third, the p o s s i b i l i t y ex is ts th at a dependent
p ro b a b ility fa cto r Is present In the cro ss-sectio n al data.
Here, combinational theory u t i l i z e s an independent pro
b a b i l i ty framework which may be contrary to the actual
data. The problem then becomes one of maintaining the
two possible answers to the th e o re tic a l question of the
paper. The solution to th is problem and discussed l a t e r
1s determining how well the mathematically computed e s t i
mates approximate other sets of actual data on residence
change. In ad d itio n, the features or expected re s u lts of
the dependent p ro b a b ility models will be derived by the
Longitudinal Model.
The problems j u s t discussed have bearing on the
problem of defining the p r o b a b ilitie s of a longitudinal
move used 1n combinational theory. The approach towards
so lu tio n , discussed 1n d e ta il In the next se c tio n , 1s to:
(1) use empirical data d i s tr ib u tio n s and to provide
2 0
empirical evidence where possible that these data will
give an approximation to the mathematical concept of
p ro bability, (2) use a second data d is trib u tio n -- age
by the time between moves -- to obtain an approximation
of the potential time at a given residence and re la te It
to the actual moving data d is tr ib u tio n , and (3} compute
the mathematically expected re su lts and compare these to
other sets of data or to the features of other theories.
In order to re la te combinational theory (Elementary
Probability Theory), theory on residence change and em
pirical data, the methodology of the model 1s presented
1n relationship to 11 fe table cohort analysis which uses
empirical rates. The next section begins with cohort
analysis, introduces a modification that 1s designed to
Incorporate residence change theory and then provides a
second modification to Include combinational theory.
Life Table Cohort Analysis
A review of constructing a simple synthetic l i f e
table cohort analysis has the following steps: (1) Em
p ir i c a l l y determined rates for a chosen series of age
categories are obtained, e . g . , five year rates (r^).
Implied In this f i r s t step are two d is trib u tio n s : (a) an
age category d is trib u tio n with fixed age In terv als, for
example, constant five years, and (b) a frequency d i s t r i
bution by which the age categories serve as a basis for
21
grouping the data 1n order to obtain age related ra te s ,
e . g . , mortality (gx) or survivor rates (Px)* (2) These
rates are multiplied by a radix (N), usually 100,000, and
the continuous l i f e history of this cohort Is obtained,
e . g . . the number of persons who survive to age 5 (N"*lx *
Nr0_5). (3) The number of persons at the end of the l i f e
history cohort or from a given age (j) and older (k) Is
k
obtained ( I N. * L * N"). (4) This number (N"-Lv) Is
1 - j 1 x x
used In a "new ra te ," e .g ., li f e expectancy (expectancy -
N"/N*L /I ). A sim ilar method (Double Decrement Tables)
A A
1s used by Wilber (1963) to construct expected life
h isto ries of moves and migrations. Wilber includes both
mortality and residence change; however, he does not sug
gest a theo retical rationale for the method.
Methodology of the Model
and the Use of Theory
In constructing the S Longitudinal Model, the r e s i
dence change theory and the evidence for using rates as
p ro b a b ilitie s are Incorporated 1n the above l i f e table
cohort procedure step (1) where a series of longitudinal
p ro b a b ilitie s (ra tes) are obtained. Three parts are pre
sented which p a r a l le l, 1n concept, this f i r s t step 1n
cohort analysis. Two d istrib u tio n s are discussed and are
combined to obtain longitudinal p ro b a b ilitie s .
22
The f i r s t d is trib u tio n discussed 1s one year age
related mobility rates (U.S. Census. P-20 series where
an example Is given la te r In Figure 1). This 1s similar
to a cohort analysis' mortality or survivor d istrib u tio n
except graphed or plotted as a continuous rather than a
d iscre te age d is trib u tio n . The second d istrib u tio n d i s
cussed 1s age by time between residence change (also
shown 1n Figure 1).
The following paragraphs are devoted to these two
d is trib u tio n s and the combining of them. This Is done to
In terp ret the theoretical frameworks of the l i f e cycle
and the Cornell Model Into combinational methods. Also,
the evidence for using these two d istrib u tio n s as proba
b i l i t y d is trib u tio n s is discussed.
a) Age Related Mobility Rate Distribution: In
constructing the Longitudinal Model, the assumption 1s
made that one year empirical mover rates are appropriate
as a probability d istrib u tio n of moving. In an e a r l ie r
section, the problems In using sample data were discussed.
Here, the evidence th at this assumption 1s tenable 1s pre
sented.
These age related mobility rates (U.S. Census, P-20
series) are sample data th at have been collected annually
for more than 20 years. The rates are compiled on whether
the person moved or not during the previous year and show
that age Is related to residence change. Although these
23
rates flu c tu a te by each h is to ric a l year (Lowry, 1966:29).
the general form of the d i s t r i b u ti o n has been within a
narrow lim it. The narrow range of flu c tu a tio n over t h is
20 year h i s t o r i c a l period Indicates empirical support
th a t these are r e l i a b l e estimates of a p ro b a b ility d i s
tr ib u tio n .
b) Age by Length of Time at a Residence: In a d d i
tion to the fa cto r of age Influencing residence change
(above a), age is assumed to be a facto r 1n the potential
length of time at a residence. Here, the assumption Is
made th at there ex is ts both an age by length of time at
residence d i s t r i b u ti o n and a corresponding age by mobi
l i t y ra te d i s t r i b u ti o n for each h i s t o r i c a l year. If such
an assumption were true then both d i s tr ib u tio n s could be
sy stem atically related to each other 1n order to define
p otential longitudinal moves.
However, United States data on length of time at a
residence by age 1s minimal (Taeuber, 1961; Taeuber,
Haenszel , and Slrken, 1961). Since the data are lim ited,
th is assumption about an age by length of residence d i s
t r ib u tio n cannot be stated as validated by evidence.
Therefore, several approximations to th is d i s tr ib u tio n
are made using th e o re tic a l Ideas from the U f a cycle and
Cornell Model.
The Cornell Model's basic concept th at the length
of time at a residence Is a crucial fa c to r Is u t i l i z e d
24
In developing this assumption. The problem with using
age related mobility rate data alone (above a) 1s that
these are d istrib u tio n s of persons who had moved in the
previous year but with no indication of how long they
stayed prior to the move. The use of this length of
time d is trib u tio n 1s discussed next, and the d e ta ils for
approximating this d is trib u tio n will be discussed la te r
when Defining One Longitudinal Move.
c) Combining the Age Related Mobility Rate D i s t r i
bution and the Age by Length of Time at a Residence Dis
trib u tio n : Since 1t was assumed previously that age
Influences both the length of time at a residence and
the prob ability of a residence change In the near future,
the questions then become (1) how to combine these factors
and (2) why they can represent longitudinal p ro b a b ilitie s .
How to combine: The Importance of the d istrib u tio n
of age by length of time at a residence Is to obtain var
iable age categories for "grouping" the age related
mobility rate d is trib u tio n Into longitudinal p ro b a b ili
t ie s . This method of grouping attempts to Incorporate
the potential for future moves as well as the actual past
moves. Also, this concept of grouping p aralle ls l i f e
table cohort analysis. However, 1n cohort analysis, the
age categories are usually a constant In terv al, e .g .,
five years. In this case, variable age categories will
be used. These variable age categories will be
25
determined by the estimated length of time at a residence.
For example, at age 20, assuming 2 s ta y e r y ears, the next
p ro b a b ility of moving would be at age 22 years. While
at age 64, assuming 8 stay er y e a rs , the next p ro b a b ility
of moving would be at age 72. In th is manner the length
of time concept of the Cornell Model 1s p a r t i a l l y Incor
porated Into the Longitudinal Model.
Why can they be combined: Since 1t Is known th at
age re la te d mobility rates have been r e l a t i v e l y stab le for
at l e a s t 20 y e a rs , and i f age Influences the length of
time at a residence with a sim ilar s t a b i l i t y , then they
can be combined lo n g itu d in a lly . Imagine a three dimen
sional figure or graph where h is to r ic a l time is assumed
not to be a fa c to r. The three axes of t h is figure are
age, length of time at a residence, and residence change.
Since the two above mentioned d i s t r i b u ti o n s have age as
a common f a c t o r , then such a three-dimensional figure can
be constructed. The three two-dimensional figures 1n th is
construction are: age-moving, age-time, and t1me-mov1ng.
If the assumption 1s made th a t the three-dimensional
figure 1s constant and applies over h i s t o r i c a l time, then
a two dimensional figure u t i l i z i n g age can apply lo n g i
tu d in a lly across l i f e h isto ry . The two-dimensional f i g
ure Implies the re la tio n s h ip of t1me-age and age-moving.
Thus, I t uses age re la te d mobility rates (age-moving)
26
where the age catego ries for grouping the data are d e t e r
mined by the length of residence (tim e-age).
Thus* in th is l i f e cycle study, the p ro b a b ility
d i s t r i b u t i o n of moving (above a) 1s p a rtitio n e d Into
*
rates which correspond to age categories of v ariable
widths. These v a ria b le widths correspond to the e s t i
mated length of residence time for the age category.
The assumption Is made th a t the length of time by age
d i s t r i b u t i o n and the mobility ra te d i s t r i b u ti o n by age
have been constant over h i s to r ic a l time and th is p a r t i
tioning will provide an estimate of the longitudinal
p ro b a b ility of moving. The adequacy of th is assumption
will be considered by comparing the calculated estimates
to actual data. These data and th is comparison are given
In the DATA and RESULTS s e c tio n s , re sp e ctiv ely .
These three steps 1n co nstructing the model p a ra lle l
the f i r s t step of the previously outlined l i f e table co
hort analysis and provide longitudinal p ro b a b ility e s t i
mates of one move or one stay within an age In te rv a l.
In cohort a n a ly s is , constant age categories are usually
used to define ra te s ( p r o b a b i l i t i e s ) . Such a procedure
Is ap p licab le 1n constructing l i f e tab les since m ortality
1s a single event whereas residence change can occur
numerous times 1n a person's l i f e h is to ry . Therefore,
ra th e r than using constant age categ ories 1n the Longitu
dinal Model, age categories which p a ra lle l the length
27
of time at a residence d i s tr ib u tio n are used to define
one event. Assuming a homogeneous a g g reg ate» these
moving or staying events can then be combined as 1f they
are Independent events In combinational theory In order
to obtain the sequence of moves or stays by d iffe rin g
ag es.
d) The use of theory In defining variable length of
time ca te g o ries: In the previous section on constructing
the Longitudinal Hodel , the l i f e cycle explanation and the
Cornell Model's theory were discussed. These two theories
will enter conceptually into the model's construction of
the age by length of time at a residence d i s t r i b u ti o n
(above b), esp e cially since the data are lim ited. Here,
the stages In the l i f e cycle are posited as a step func
tion over age. while the Cornell Model's cumulative
I n e r tia Is posited as a continuous function where the
length of time at a residence increases with Increased
past length of time over age.
The concept of cumulative I n e rtia Is Incorporated
by using past length of time; however. It will be modi
fied by assuming 1n addition th at with Increased age the
expected length of time at a residence Increases. The
original concept of cumulative I n e r tia (McGinnis. 1968)
assumes t h a t with increased length of time at a residence,
without regards to age, there will be a gre ater Increase
1n the future p o ten tial for staying. An age function
26
with Increasing length of time at a residence Is p a r t i
ally su b s titu te d for past length of time a t a residence.
The Increasing length of time 1n the future Is retained
from the concept of cumulative I n e r t ia . This re p resen ta
tion re su lts In a continuous d i s t r i b u t i o n , and along with
the step function of the l i f e cycle, Is Incorporated as a
separate age by length of time at a residence d i s t r i b u
tion. More d e t a i l s on these two represen tation s of
theory are discussed l a t e r 1n Defining One Longitudinal
Move.
Since d ir e c t Independent p r o b a b ilitie s are defined
above, they will not e x p l i c i t l y t e s t the Cornell Model.
The th e o re tic a l Issue of the discussion 1s whether the
p r o b a b ilitie s at a given age are Independent o f, or are
dependent on past h is to rie s of residence change. In the
above d e f i n i t i o n s , the p r o b a b il i t ie s were defined using
varying age In te r v a ls . Although these p r o b a b ilitie s are
Independent of each other, the concept of cumulative
In e r tia was p a r t i a l l y Incorporated In the varying age
In te rv a ls . These varying age In terv als use a continuous
d i s tr ib u tio n of past length of time which Increases, and
th is d i s t r i b u ti o n Is superimposed on an age continuum.
In a l a t e r section on s ta tin g the Longitudinal Model 1n
formal terms, however, a method of estimating the Cornell
Model's re s u lts Is shown and Is used as a " te s t" of
29
adequacy of the Independent probability formulation com
pared to the dependent framework.
Viewed In terms of empirical data, the step func
tio n 's length of time at a residence framework has less
conceptual v a lid ity than a continuous function (whether
constructed from age or past stayer time). The step
function posited that events occur at a single age point
and the probability Is only altered at the next age
stage. On the other hand, the continuous function would
correspond to the empirical expectation that the d i s t r i
bution of ages at which events occur varies rather than
occurs at a single age point. The rationale for use of
the step function Is to provide a theoretical connector
to events occurring at a single age point In the l i f e
history 1n order to determine which events are crucial
as causes. If such events by age can be Id e n tifie d , then
theoretical connectors can be made to causes and explana
tion of residence change. Thus, rather than only emphasi
zing graphs or d is tr ib u tio n s , the causal events by age
are considered Important. Here, the balance Is made be
tween theory and precision.
Thus, In this study, the shape of the d istrib u tio n
Involving age by time at a residence Is considered as both
a step or continuous function to define p ro b a b ilitie s
which are used and tested 1n the model. These p ro b a b ili
tie s using the step or continuous function once estimated
30
are treated as Independent, which f i t combinational
theory. Therefore, the t e s t of the Cornell Model's de
pendent. tra n s itio n p ro b a b ilitie s will only be by In fe r
ence on how well the continuous, Increasing function of
time at a residence by age " f i t s " the c r i t e r i a data as
opposed to the step function of the l i f e cycle. Xn addi
tion. the Longitudinal Model's formulation, using inde
pendent p ro b a b ilitie s will compute estimates of the
Cornell Model's estimates (McGinnis, preliminary, 1968).
In summary, the f l r s t step In the outlined l i f e
table cohort analysis was modified from Its standard form
to Include Ideas from theory and obtained Independent
longitudinal p ro b a b ilitie s . This modification was made
by Including an age d istrib u tio n which had variable
lengths of time at a residence. Residence change theory
and assumptions about h is to ric a l time were Incorporated
Into this modification. The residence change theory
resulted 1n positing a step or a continuous function of
length of time at a residence by age. These two func
tions were used as the variable age d istrib u tio n to
"group" the age related mobility rate d is trib u tio n Into
longitudinal p ro b a b ilitie s .
The second step In l i f e table cohort analysis 1s
d ire c tly related to the m ultiplication of p ro b a b ilitie s
used 1n constructing the Longitudinal Model's estimates
of c r i t e r i a data. The difference Is that l i f e table
n
cohort analysis uses a radix of 100.000 while the Longitu
dinal Model uses p ro b a b ilitie s d i r e c tl y and 1npl1es a
radix of l . 1 The re s u lts of the model are unlike cohort
analysis 1n th at a variety of combinations of mover-stayer
series Is obtained ra th e r than a single s e r ie s . For
example, 1n the model a series of moves at age 2 and 22
or at ages 3. 7, and 15 and staying at the other ages
could be obtained using combinational theory. In th is
study, the sequence of events over the l i f e h isto ry 1s
considered, and the p r o b a b il i t ie s defined In the f i r s t
step are used In combination to account for the m ultiple
residence change events. The d e t a i l s for using combina
tional theory In providing estimates of a c tu a l, c r i t e r i a
data are given l a t e r .
The th1rd step and fourth step In l i f e tab le cohort
analysis where "new" rates are obtained, has no counter
part In constructing the Longitudinal Model. The model
uses combinational theory to group the d i f f e r e n t sequences
1 The p a ra lle l between m u ltip lic a tio n of probabi
l i t i e s and l i f e table cohort analysis Is as follows;
Cohort M ultip licatio n
N1 fTj - fr-j )N-- - - -
N2 * N1*r 2* N2 * *r l^ (r 2*N
N3 - N2( r 3) N3 * (r-j) ( r 2) ( r 3)N
If N, the radix Is equal to 1, then N3 Is a rate or pro
b a b i l i ty of the m ultiple occurrence of the three Indepen
dent events, r 1 , r 2 , and r 3.
32
of age ( U f a h i s t o r i e s ) and moves into combinations which
can be compared to actual data or to theory.
Since a wide range of data ex is ts on residence
change, the mathematical formula o u t ! 1ned above 1n terms
of combinational theory are then rew ritten such th a t they
meet the tab u latio n format of the actual data. Rewriting
the equations of combinational theory Is lo g ic a l. For
example, persons who moved five years ago rath er than one
year ago (annual survey data) should have the same explan
atio n , unless h i s t o r i c a l factors have a lte re d the r a te s .
Since l i f e h is t o r i e s are being considered and assuming
the h i s to r ic a l conditions are the same, then the only d i f
ference is 1n the procedure of tabu lation of data. Or If
all moves of a person are recorded In a r e g is t r y or per
sons had "perfect r e c a l l , " then from a given date, r e t r o
spective moves are those moves In the past and should be
derivable from one year ra te s .
Thus, major assumptions 1n constructing the model
are: (1) the age mobility (residence change) rates have
been In v ariant for the l i f e h isto ry (72 y e a rs ), and "p ar
titio n e d " or age-grouped r a te s , based on v ariab le age
c a te g o rie s , can apply for the actual and p otential moves.
(2) The procedure used here 1s a Markov Chain Process
(see McGinnis' (1968) c r i t i c i s m s ) , 1n c o n tra s t to the
Cornell Model which uses dependent p r o b a b ilitie s in a
Stochastic Process. (3) Distance moved Is not a f a c t o r ,
whereas I t Is a crucial fa c to r 1n defining the d iffe ren ce
between re s id e n tia l mobility and migration (Shryock,
1964).
This above ou tlin e Is an overview of the procedure
and Its problems. Details of the procedure are given
1n the following sectio n s.
Defining One Longitudinal Move
The model's procedure for defining one longitudinal
mover or stay er period p ro b a b ility , using annual data,
was as follows: F i r s t , a d i s tr ib u tio n of age related
mobility rates (U.S. Census, 1960, P-20 Series) was
posited to represent the p ro b a b ility d i s t r i b u ti o n of
moving. Secondly, the d i s t r i b u ti o n of age by time at a
residence was posited. However, th is d i s t r i b u ti o n had
less documentation for the United States (Taeuber, 1961,
provides a set of d ata). Therefore, several a l t e r n a ti v e
d i s tr ib u tio n s of the age by time between moves were
posited and were used 1n the previously outlined method
of p a r titio n in g or grouping rates by variable age c a t e
g ories. These a lte r n a ti v e d i s t r i b u ti o n s are sp ecified
a f t e r the "p a r t i t l o n l n g " of the age related mobility
ra te d i s t r i b u t i o n according to the length of time at a
residence by age d i s tr ib u tio n Is b rie f ly reviewed.
Like l i f e table cohort analysis which uses the
frequency ra tio (or r a t e ) , e . g . , deaths/population for
34
the given age category, the age ra te as shown on the
continuous age re la te d mobility ra te d i s t r i b u t i o n was
u t i l i z e d In the model. Figure 1 I l l u s t r a t e s the defining
of the p ro b a b ility of one longitudinal move where the
assumption Is made th at a constant six years 1s the
length of residence time between moves for all ages.
Instead of a constant six year I n te r v a l, the f o l
lowing five a l t e r n a t i v e estim ates were used to obtain
p r o b a b il i t ie s for t e s tin g the model. These a l t e r n a ti v e
d i s t r i b u ti o n s of age by length of time at a residence
posited as approximations were: Two constant fu nction s,
two step fu n ctio n s, and a continuous function. The two
constant functions (estim ate 1 and 2) were considered
as baseline standards or an “age only effect* to determine
1f the following step or continuous function had any
merit above "no data or explanatory d iffe re n c e ." The step
fu n c tio n s 2 represented the l i f e cycle explanation ( e s t i
mates 3 and 4). The f i r s t of the two step functions ( e s t i
mate 3) was used as a "baseline standard" to compare to
the second (estim ate 4) for l i f e cycle p ro b a b ility changes
2 The ( l i f e cycle) step function of age by time at
a residence of g re a te s t I n t e r e s t (which had a sequence
of 2, 4, and 8 years; ca lled estim ate 4) was a guess based
on a paper by Yee and Van Arsdol (1968) for the age d i
visions of 0.05-32.5, 32.5-64.5, and 64.5+ which was con
sidered an "employment stage" and on a study by Kayne,
1963, which Included duration of employment. An aposter-
1or1 analysis of Taauber's (1961) data suggests th at
sequence 2, 5.5, and 8 years might give a b e tte r " f i t " to
the c r i t e r i a data.
Figure 1. Illustration of Partitioning the Age Related nobility Rate
Distribution Into Probabilities of O ne Longitudinal novo
for the S Longitudinal nodel
Rate
.40
Age Related nobility Rate Distribution
(Total Movers)
.35
.30
.25
- 2 1
.20
.10
-.10
.05
.00
TIm
c c
Distri
bution
A
Distri
bution
I
•o V i V i V l *3+I1 V i V i V i *7+11 V ri V i *10+,1 *11+11
Length of Use Between M oves (Distribution B-exanpla • 5 years) Converted to Age Categories
of#One°Move * HrtHHy Rates (Distribution A) Into Longitudinal Probabilities
1^ equals the length of tine at a residence (tlee between mvos) at age a^
P| equals the probability of one longitudinal stove.
ui
< 7 1
36
at the older ages. The continuous Increasing function
with Increasing past time at a residence superimposed on
age represented the Cornell Model (estimate 5). Each of
these five d is trib u tio n s Is based on a combination of 2,
4, and 8 years between moves sequence for the co rres
ponding ages of 0.5-32.5, 32.5-64.5, and 64.5+. These age
points represented an employment stage (Yee and Van
Arsdol, 1968).
1. Constant 2 Years A constant interval of 2
years between moves for all ages:
Estimate 1: 1^ * I ■ 2 years for ages
0.0-80.0+
2. Constant 4 Years A constant Interval of 4
years between moves for all ages:
Estimate 2: I^ * 1 * 4 years for ages
0.0-80.0+
3. Step Function - Young Ages A constant interval
of 2 years between moves for persons 32.5 years
of age or younger and a step up to 4 years for
persons over 32.5 years of age:
Estimate 3: 1^- 2 years for ages 00.0-32.5
» 4 years for ages 32.5-80.0+
4. Step Function - Young and Elderly Ages A con
s ta n t Interval of two years between moves for
persons 32.5 years of age or younger, a step
up to 4 years for persons over 32.5 years up
37
to 64.5 years of age, then a step up to eight
years from 64.5 years old and oldar: This e s t i
mate 1s considered to represent the l i f e cycle
step function explanation. (This step function
1s not the only possible a lte r n a ti v e - see
estim ate 3). This step function was based on
a p rior study (Yee and Van A rsdol, 1968).
Estimate 4: 1^- 2 years for ages 00.0-32.5
- 4 years for ages 32.5-64.5
« 8 years for ages 64.5-80.0+
5. Continuous The continuous, Increasing function
of the form ea where a is age which has been determined
from the past time at a residence: This represents the
Cornell Model's cumulative In e rtia which uses oast tim e,
or the empirical expectation th at ages at which events
occur v a rie s. Unlike a completelv homogeneous population
where every person experiences the event at the same age,
the stages are continuous and the length of time between
moves increases with age. The following continuous func
tion passes through the mid-points of the 2, 4, and 8
year step function (estim ate 4) so th a t only the question
of a step or a continuous function can be analyzed using
the same seale of sequence time. The amount of time (or
height of the d i s t r i b u t i o n ) was not a lte re d .
Estimate 5: ^ * a 1+1 + e*5 + *021
38
n
where 1 - 0 to n and z I. < 72
1-1 1 “
These five estimates of time between moves were
used to p a r t i t i o n the age re la te d mobility ra te d i s t r i
bution Into longitudinal p r o b a b il i t ie s of moving. The
actual r a te s , which defined one longitudinal move proba
b i l i t y for the five estim ates of the age by time between
moves used In the model are given In the DATA section .
The five sets of mover p r o b a b il i t ie s were u t i l i z e d
1n the following combinational formulation In order to
r e la te and to compare the variety of sets of actual data
used as the c r i t e r i a for determining the adequacy of the
Longitudinal Model. Also, formulae, based on the use of
Independent p r o b a b il i t ie s and the l i f e cycle, were used
to derive the features (see Table 1) of the two other
theories (Cornell Model and m obile-stable population ex-
pl a n a tlo n ).
Combi n a tio n a l-P ro b a b l11 tv FormulatIon
The formulae 1n th is section are divided Into five
parts which correspond to five sets of c r i t e r i a data used
to t e s t the model. These formulae are based on combina
tional theory and on the two previously mentioned d i s t r i
butions which defined the longitudinal p r o b a b i l i t i e s .
They are mathematically rew ritten such th a t the model
corresponds to the actual data. The five formulations In
the following sections estim ate actual data for: (1) age
39
and the series of moving and staying with focus on stayers
at the same house all of th e ir liv e s , (2) l i f e h isto ries
of residence, (3) average number of moves 1n the l i f e
h istory, (4) stayers and th e ir l a s t move, and (5) compar
ison to the Cornell Model.
Age and the Series of Moving and Staving
From the previous section a set of longitudinal
p ro b a b ilitie s (Pj) were defined for one move for each age
interval (a^). Since these p ro b a b ilitie s were assumed
to be Independent of one another, the m u ltiplication rule
of probability applies. The probability of moving (P^)
1n two consecutive (age) time periods -- at age a-j and
at a2 -- Is equal to the product P-jPg- In th1s context,
the probability of staying at age 1 Is defined as (1 - P^)
or Qj. Thus, the probability of staying at both a^ and
a2 1s equal to ( 1 - P ^ (1 - P2) or Also, a variety
of combinations of movers and stayer periods can be m ulti
plied as above for a given age category. The number of
p ro b a b ilitie s mutllplled 1s an Important consideration.
These p ro b a b ilitie s are then added according to the
number of moves 1n the series for a given age. Using
this procedure (Combination Theory), the table of these
p ro b a b ilitie s using each of the five estimates Is denoted
as .
In formal notatlon consider a matrix S containing
three sets of n age data points: (1) mover p ro b a b ilitie s
40
P^l, (2) s ta y e r p r o b a b il i t ie s (1 -P j), and (3) a s e rie s of
time In terv als I j , where 1 ■ 1 to n. These p ro b a b ilitie s
are m utH plled to obtain al 1 series of combinations of
movers and stayers over the n age In terv als or l i f e
h isto ry data p o in ts. These combinations could range from
a move In every time period to staying In the same r e s i
dence (place of b ir th ) for the e n tir e l i f e h isto ry .
Intermediate to these two extremes are the variety of
combinations of the number of moves or stay s. This com
plete set of combinations Is called matrix S^. For
sim p licity of no tatio n , when the s e rie s of p r o b a b ilitie s
are grouped (added) according to the number of moves for
an age category ac » a matrix Sj Is obtained. The formula
and grouping of the number of combinations for which the
p r o b a b il i t ie s are added are as follows:
Equation 1 : Number
of Combinations
for a Given Number
of Moves bv Age
Event
Period Age Formula
1
Moves
0___ 1 2 3
1 1
2
(2) S, ■ SqPj , S0 (l-P.j) 1 2 1
3
^ S2 * S1P2* Sl ^ * p2^
3 3
Note: S.j has one possible outcome of no moves,
(1-P0 ) O - P j) ; two possible outcomes of one move, PqO -P j )
and 0 * po)P'|; and one possible outcome of two moves, Pj Pq ;
these possible outcomes are given to the rig h t of the
eq u a tio n .
Using combinational theory, the m atrix, called Sj
1s three dimensional since 1t shows age, by the number of
moves, by the p ro b a b ility 1n th a t category. This matrix
or table can be Interpreted as the estimated p ro b a b ility
of persons of a p a r ti c u l a r age having m moves over a given
number of age-t1me periods. The p ro b ab ility In the "no
move" category for the d i f f e r e n t ages will provide e s t i
mates for comparison with the U.S. Census data (P-20
Series) on the per cent of non-movers -- always at the
same house. Since the age categories In the actual data
on stayers at the same house were not equivalent to those
given by this Information, the estimates for corresponding
age categories 1n the model were summed and the average
compared to the actual data.
Life H isto ries of Residences
These serie s of mover-stayer p r o b a b ilitie s by age
In matrix Sj are mutually exclusive and can be added to
obtain aggregate l i f e h is to r ie s up to a given age. For
example, given two years between moves, s t a r t i n g with the
42
f i r s t age Interval the p ro b a b ilitie s are added according
to the number of moves up to the t h i r ty - s ix th In terv al,
giving the aggregate probability of a l i f e history of a
given number of moves. Or stated d if f e r e n tly , the sum-
c
matlon of the mover p ro b a b ilitie s (L. - Z P.) less than
c 1 1-1 1
or equal to an age category (a_ - Z I.) gives an
c 1*1 1
estimate of the aggregate l i f e history probability of the
number of moves to that age.
In formal notation the regrouped series of m ulti
plied p ro b a b ilitie s (P^) Into the number of moves (m) for
age (1) Is denoted as Sjm and the relationship 1s:
Equation 2:
S \ * grouped Sf - SJm - S1Q, Sn , S12 . . . S1m
where 1 1s the age Index and m Is the
number of moves Index.
The l i f e history of m moves by age category a Is ob-
c
talned by summing the p ro b a b ilitie s v e rtic a lly (along
the age 1 dimension) according to the number of moves up
c
to and Including age category c where (a„ - Z I . ) .
c 1*1 1
Then an adjustment for the number of time Intervals (c)
1s made, I . e . , at each variable age Interval 1^, the pro
b a b ility of all combinations Is equal to 1 and the a d ju s t
ment 1s made to maintain a l i f e history probability equal
to 1. The three dimensional array of p ro b ab ilities of
l i f e h isto rie s of m moves by age category c Is denoted
as and Is equal to:
l m
43
Equation 3 :
1
where c - the number of age
categories from j-1 to 1.
m (moves) Is held constant.
Using the average p ro b a b ility for age Interval by
number of moves categories which correspond to the actual
d ata, the estim ates obtained were compared to the l i f e
h is to rie s of residence change data (U.S. Census, P-23
s e r i e s . No. 25). However, p rio r to th is comparison, an
adjustment had to be made.
In the actual l i f e h isto ry d ata, the term “area of
residences" was used, and Includes more than persons who
had moved from or stayed at the same house. Included
are local moves within the area. I t was noted In the
Census data (P-23, No. 25, Table 2 ), used as the l i f e
history c r i t e r i a th at a co rrection for local moves had to
be made 1n made 1n defining " res 1dences•" when considering
d is c r e te mover ca te g o ries. For example, in 1958, at age
65 and over, 19.1 per cent only had one residence (U.S.
Census, P. 23, No. 25; Table 2) while 2.2 per cent had
always lived at the same house (U.S. Census, P-20, No.
104, Table 11). In order for the estim ate obtained by the
model's formulation to correspond to the actual d ata, a
conversion of moves to residences was made by:
One residence * no moves + .5 (one move)
Two residences - .5 (one move + two moves)
44
Three or more residences * .5 (etc. + e tc .)
The factor of .5 for local moves was based on an approx-
Imatlon that 1t had to be less than about .67 (within
County moves). Four-ninths or two-th1rds of two-thirds
might have been used as an approximate factor for local
moves, however, .5 was selected.
In formal notation the l i f e history moves -- the
L1m Probat> 111 ty estimate (see equation 3) — requires
the following local mover adjustment 1n order for the
definitio ns of moves (m) and residences (Rj^) to corres
pond. The formulae used were:
Equation 4 :
■ one residence * g + .5L^ ^
R .j k * two to m-2 residences - . 5 (L ^ ^ (<+])
where k ■ 1 to m - 1
^1k * residences « .5L^ ^
where k * m - 1
The l i f e history of residences estimates were compared to
actual data (U.S. Census, P-23 Series, No. 25, Table 2).
Average Life History of Residence Change
The average number of moves 1n a person's l i f e
history was obtained from the l i f e history of residence
estimate given In (Equation 4) above. Assume, that a
person's l i f e time Is 72 years. At age 72, a l i f e history
45
estimate Is given from Equation 4 and Indicates the number
of residences (m^) and a probability of having that number
of residences (R ^)* If the number of residences 1s
multiplied by the probability of that number of residences
and the resu lts are summed* the expected (average) number
of residences experienced by all persons aged 72 Is ob
tained. Finally, If the f i r s t residence 1s subtracted,
then the average number of moves (migrations - see Defini
tion of Residence) 1n a person's l i f e time 1s calculated
using the model's formulation.
In formal n o ta tio n , the average number of l i f e time
moves 1s equal to:
Equation 5:
c
average number of ■ L (m.R,, .) - 1
l i f e time moves k*l /«•*
where mk Is the number of residences; RJ2 k Is the pro
b ab ility obtained from Equation 4 at age 72 for each of
the mk residences; and c 1s the number of RJ2 k residence
categories at age 72.
This estimate of the average number of l i f e time
moves (migrations) was compared to Shryock (1965) and
Wilber's (1963) estimates.
Stayers and Last Past Hove
Retrospective movers Imply recall of past residence
change behavior. Therefore, the mover events, length
of time and associated p ro b a b ilitie s should correspond
46
to events from older towards younger ages ra th e r than
younger towards older as 1n the l i f e h i s t o r i e s . Of prime
concern 1s the p ro b a b ility of having had a past move k
years ago, given th a t the person Is age a. For example,
given that the person 1s age 29, what is his pro b a b ility
of having made his l a s t move 9 years ago when he was age
2 0 .
The form of the past mover equation 1s to multiply
p r o b a b ilitie s back for k stayer events which represent a
given non-mover period and then multiply by a mover event.
Rather than all combinations of the series of movers and
s ta y e r s , as 1n Equation 1, only one or a series of stayer
events and one mover event are of concern.
The problem here 1s converting the combinational
Equation 1 to f i t the c r i t e r i a data. The c r i t e r i a data
are stayers In the survey year by age categories by number
of years ago when the l a s t move occurred. For a given
age, the Equation 1 can c a lc u la te p ro b a b ilitie s for the
series of stayer periods back to a move. However, these
formulae do not give single years when the l a s t move
occurred nor are they conditional p r o b a b il i t ie s for an
age aggregate, i . e . , given the person Is age 29 and had
not moved for 10 years what 1s his p ro b a b ility of having
moved 1n his 11th year? Rather, the equ ation's probabi
l i t i e s are a series of v ariable stay er periods back to a
move for a given age category, e . g . , at age 29, the
p r o b a b ilitie s of a l a s t move occurring exactly at 2, 4,
6, and 8 years ago. This variable length of time at a
residence when the move occurred becomes a problem when
determining moves by sing le years. When events are
counted as 1n Equation 1. th is problem Is not of great
concern since the d e f in itio n of one mover period was used
In the c r i t e r i a data.
The following procedure was used to convert Equation
1 to f i t conceptually the c r i t e r i a data on past movers.
Here, only the l i f e cycle step function (Estimate 4) was
used ra th e r than all five a l t e r n a ti v e d i s tr ib u tio n s of
age by length of time at a residence. The mathematics
for the continuous function (Estimate 5) were considered
too complicated and the other a lte r n a tiv e s were not of
central issue 1n this discussion.
1. The f i r s t modification Is to assume that the
move event occurs 1n the time Interval half-way Into the
mover time period. This modification Is made since It
would be more lik e ly th at a move occurred half-way ra th e r
than at the beginning or end of the time In te rv a l.
Equation l ' s section on the length of time between moves
or stays was rew ritten to Incorporate th is modification
for: (1) considering only stayer events and one mover
event, and (2) using half a time interval for the mover
period. This modified length of time at a residence by
age d i s t r i b u ti o n was used to p a r ti t i o n the age related
48
mobility ra te d i s t r i b u t i o n Into p r o b a b il i t ie s from older
towards younger ages (see p r io r section on Defining One
Longitudinal Hove).
In formal notatlon the equation on the length of
stayer time back to a move was:
Equation 6 :
Tj,1+1 ’ *5 I 1+1 +kkJ 1I kk * Tm
where 1^ 1s the time d i s t r i b u t i o n ; I kk
are stayer periods; kk * 1 to 1; and
I 1s a mover p ro b a b ility time; and
1+1
j Is the age.
2. These longitudinal mover-stayer p ro b a b ilitie s
obtained by p a r ti t i o n in g , using Equation 6, were used
1n Equation 1 to multiply and obtain the p ro b a b ility of
the serie s of stay er period back to a move for a given
age. For example, for age 72 years and two years between
moves, the se rie s would Include the p r o b a b ilitie s of a
l a s t move at age 69, 67, 65, 63 . . . 3. These serie s were
calcu lated on single years of age as a s ta r tin g p oint,
e . g . , 72, 71, 70 . . . 3. Table 2 given l a t e r I l l u s t r a t e s
th is matrix.
3. Given the 1onqest serie s at age 72, a d ire c t
procedure Is to assume th a t the p r o b a b il i t ie s of a l a s t
move one year ago would be the p ro b a b ility at age 71 or
2 years ago at age 70. These p r o b a b il i t ie s by length of
time (In terms of age) were obtained by step 2 above.
49
This r e s u lts 1n a matrix of age by year of l a s t move by
p ro b a b ility for each year by age.
However, two problems s t i l l e x i s t -- f i r s t , only
one stayer se rie s (step 3) Is used ra th e r than all pos
s ib le calcu lated stayer combinations up to a move;
second, I t 1s d i f f i c u l t to obtain categorlzed conditional
p r o b a b il i t ie s which f i t the c r i t e r i a data.
4. Rather than using only the one continuous age
s e r i e s , s t a r t i n g with the o ld e s t age category as In step
3 above, a "new" ca lc u la te d series using the step func
tion was obtained. This re s u lta n t p ro b a b ility was used
and the age determined p ro b a b ility was excluded. Here,
the supposition was made th at the ca lc u la te d p ro b a b ility
using the step function was more accurate than the age
defined p ro b a b ility from step 3. An example of th is
m odification 1s the series of single years of age proba
b i l i t i e s from 72-64 being used, then a newly calculated
s e rie s used from 64-60 to complete the 12-year span where
the assumption was made th a t there were 8 and then 4 years
between moves for the respective age ca te g o ries.
A table of p r o b a b il i t ie s was constructed where a
portion was obtained by the age re la tio n s h ip (step 2) and
by the c a lc u la tio n of new series (step 4). Here, R_ , Is
S
the re tro s p e c tiv e p ro b a b ility of stay in g, then moving,
where a 1s the age and s Is the age defined s e rie s . An
I l l u s t r a t i o n of th is procedure Is given In Table 2. The
TABLE 2
50
I l l u s t r a t i o n of the Past-R etrospective Mover
P ro b a b ilitie s by Single Years of Age and
Single Years to Last Move Using
Three Series
Age
Years ago to la s t i move
1 2 4 5 6 7 B 9 H o Move
1
R11
S1
2
R1 2 R11
S2
3
R1 3 R12 R11
S3
4
R14 R1 3 R1 2 R21
S4
5
R1 5 R14 R1 3 R22 R21 S5
6
R1 6 R1 5 R1 4 R23 R22 R21 S6
7
R1 7 R1 6 R1 5 R1 4 R1 3 R32 R21 S7
8
R18 R1 7 R16 R1 5 R1 4 R33 R32 R21 S8
9
R19 R1 8 R1 7 R1 6 R1 5 R14 R33 R31 R21 S9
fts e r i es age ts the p a s t - r e t r o s p e c t 1ve p ro b a b ility of
moving 1n that single year.
Note: In this example, for 9 years* assuming 2 years be
tween moves or sta y s, 3 age categories are used and th is
r e s u lts In 3 s e r ie s . Series number 1 1s by sin g le years
of age s t a r t i n g with the o ld est age. Series number 2-3
are calcu lated p r o b a b ilitie s from older towards younger
ages using the concept of 2 years between moves.
51
serie s numbered 1 in Table 2 and the single years of age
or younger are d i r e c tl y ca lc u la te d p r o b a b i l i t i e s . Series
numbered 2 and 3 are the newly ca lc u la te d series of pro
b a b i l i t i e s where the assumption Is made th at there are 2
years between moves. The age Index (a) decreases towards
the younger ages.
5. In order to categorize these p r o b a b il i t ie s by
single years of age and single years to l a s t move and to
conceptually f i t the c r i t e r i a d ata, aggregate categorized
conditional p r o b a b il i t ie s are required. Having moved
during the f i f t h through seventh year, for example, 1s
conditional on not having moved zero to four years ago.
Also, more than one move could have occurred during the
f i f t h through seventh year. Here, the problem 1s to
determine the p ro b a b ility of a move beginning b years ago
to the end e of a category (b to e). The methodology for
th is ca lc u la tio n was based on Equations 6 and 1 and used
a p ro b a b ility matrix like the one I l l u s t r a t e d 1n Table 2.
These p r o b a b ilitie s represent a move in the p a s t. In terms
of single years ago, for each age, and were used to ap
proximate conditional p r o b a b i l i t i e s .
A "better" mathematical procedure could be developed
for obtaining categorical conditional p r o b a b il i t ie s of
stayer periods to a move, given p r o b a b ilitie s of moving
1n the past by sin g le years and for each age. However,
a simple sub traction procedure (of mutually exclusive
52
events) was used here to compute the conditional pro
b a b ility estimate. Using subtraction, for example, the
probability of moving at. time 2, given the person stayed
at time 1 would be obtained by subtracting this proba
b i l i t y from the probability of moving at. time 1 or
(past move 1) - (past move 2) or (Rj) - ( (1 - R1) R2) where
age Is held constant. This procedure does not a n a ly tic
ally " f it" the Idea of conditional probability; however,
i t was used as an estimate. (The use of categories
creates the problem. For example, using age categories
20-25 and moving 4-7 years ago, the one year condition
of staying or moving both by age and years age 1s a
problem.)
In formal notation the probability of having a move
a specified number of years ago given a specific age 1s
obtained by:
Equation 7:
Probability of l a s t move
s ta rtin g at b to the end - R .
year e for single age c,o ,e
category c
e
■ h + 1 I j -R- ui I
c.b 1 c ,1 c ,1+1'
In order to r e la te the single year age categories
to a range of ages, e . g . , 25-44 years of age, the average
probability from (7) was calculated within the age range.
In formal notatlon the age p ast-retro spective pro
b a b ilitie s were:
53
Equation 8 :
P robability of last
move, sta rtin g at
year b to the end
year e for age
category r to s
Rr , s i b,e
1 e
s-r-1
These average p ro b a b ilitie s (R„ . . . were summated
r |S | D | f
across the number of years to la s t move categories plus
stayers at the same house all of t h e i r 11fe estimates
(using equation 3) and adjusted to yield a row sum pro
b ab ility equal to 1.00. These estimates were compared to
the U.S. Census. P-20 series data on stayers at the same
house 1n the survey year, by age and year of l a s t move
c a te g o rie s .
Comparison to the Cornell Model
The Cornell Model focuses on stayers and 1s based
on the concept of cumulative In e rtia of stayer time. The
l i f e cycle step function 1s not considered 1n the Cornell
Model as an a lte rn a tiv e explanation. Therefore, a sup
position was made to determine 1f the use of a constant
time between moves function for each amount of past time
at a residence In the S Longitudinal Model would provide
an estimate to the "simulation" data generated by the Cor
nell Model. This supposition was made, because the "prob
lem" was considered as centered In defining longitudinal
p ro b a b ilitie s of movers, not stayers. Therefore, a series
54
of constant functions to p a r titio n the age re la te d mobi
l i t y ra te would s u ffic e and would represent Independent
mover and s ta y e r p r o b a b i l i t i e s . These p r o b a b ilitie s were
used to estim ate the dependent, simulation p ro b a b ilitie s
(equations) obtained by the Cornell Model (g e n e ra to r^ P j,
type B. McGinnis, Preliminary (1968).*
The procedure for obtaining comparable p ro b a b ilitie s
using the Longitudinal Model and the Cornell Model was
f i r s t to e s ta b lis h a correspondence between time between
moves by age and duration age. (Duration age Is the past
time of staying at a residence a f t e r having stayed with
out regards to actual chronological age). The adjustment
for correspondence (time between moves - duration age - 1
year) was made since the f i r s t duration age was assumed
by the Cornell Model as one year. A constant function
corresponding to the time between moves (duration age)
was used to p a r t i t i o n the age related mobility rate d i s
tr ib u tio n Into p r o b a b i l i t i e s . These p r o b a b ilitie s were
used 1n the l i f e history of residence formulation for
each time between moves (duration age).
3 McGinnis reported the re s u lts of four simulation
d i s t r i b u t i o n s . His type B generator "looked reason
able" and the t r a n s i t i o n matrix had p r o b a b il i t ie s th a t
appeared " c o r r e c t," and was selected a p rio ri for compar
ison. The supposition of a correspondence between the
s erie s of constant functions and "duration age" was a
guess.
55
Then for each of the Longitudinal Model's computed
tables of l i f e h is to rie s of residence change (Equation 3)
by age and by time between moves (duration age), only
the p ro b a b ilitie s of one residence (stay ers) was con
sidered. In the computed t a b l e s , the average" for all
age categories (as defined by the time between moves or
duration age) was used as the estimate for the Cornell
Model, generator ^Pj. type B.
In formal notation the comparison of the Cornell
and modified S Model u t i l i z e d :
Equation 9 :
PJ ‘ •!?! PJ . 1 . 1 /C
where c is the number of age c a te g o rie s .
Pj I ^ is the p ro b a b ility of one residence
for age category 1, and constant function
time between moves j (which corresponds to
duration age and 1s used In Equation 3 to
ca lc u la te the p ro b a b ilitie s of , , ) .
j • * 11
In summary, the five above sections of equations
were developed to r e la te d i f f e r e n t types of actual or
th e o re tic a l simulation data. All of these equations stem
from the same combinational conception and when combined
with l i f e cycle theory Is called the S Longitudinal
Model of Residence Change. The data used In these
equations are given 1n the next section and the re s u lts
of the methodology and data follows.
" The summation may cancel out the age e f fe c t and
only duration age remains.
CHAPTER IV
THE DATA
Four major sets of data were related In the S
Longitudinal Residence Change Model. Three of these
sets were used as c r i t e r i a to t e s t the accuracy of the
model. These data were for the United States and con
sisted of: (1) age related mobility rates (U.S. Census,
P-20 Series, No. 104) for 1959, 1 (2) stayers at the
same house all of th e ir lives (U.S. Census, P-20 Series,
No. 47 and 104) for 19S2 and 1959, (3) the 1959 study of
the l i f e h isto ries of mobility (U.S. Census, P-23 Series,
No. 25), and (4) stayers 1n the survey year by year of
la s t move (U.S. Census, P-20 Series, No. 104) for 1959.
In addition, estimates of (5), the average number of
moves in a person's l i f e history were examined using
Shryock (1965) and Wilber's (1963) estimates. Shryock
(1965:591) estimates that 1n a person's l i f e history 2.1
1 Under a h is to ric a l " s ta b ility " condition, a
"better" choice than using 1959 rates a$ the age related
mobility probability d is trib u tio n would have been the
20 year average. Lowry (1966:29) gives the average from
1953 to 1964. Of the ten age categories by Lowry, the
maximum difference was 4 per cent in the 30-34 age c a te
gory when 1959 and the average are compared. Two age
categories had 2 per cent difference; four had one per
cent difference; and three had no difference.
56
57
migrations are made. Wilber (1963) estim ates 13.0 ex
pected moves and 4.2 expected m igrations. Using W ilber's
assumptions. Shryock (1965) obtained an estim ate of 3.15
migrations and considered th is a maximal estim ate. In
a d d itio n , (6) McGinnis' (prelim inary 1968) estim ates from
his g en erato r, type B in the Cornell Model were com
pared to the S Longitudinal Model.
Using the 1959 age re la te d mobility ra te d i s t r i
bution of to ta l movers, and the fiv e estim ates of the
length of time between moves (see Section on Defining
One Longitudinal Move), the p r o b a b il i t ie s used 1n the
longitudinal model's formula are given 1n Table 3.
TABLE 3
P ro b a b l1I t l e s of a Subsequent L on g itu d in al Nova
Using Age-Relatad M o b ility Ratos f o r 1959 and
Five E stim ates of th a Langth of Tina
Between Moves by Ago D is tr ib u tio n
Usad In tha Modal
E s tln a ta
1 % M *2.4.8
Ago Yaar Yaar Yaar Yaar Continuous
00-01.99 .31 .31 .31 ,31 .31 0 . 0 0 )
02-03.99 .29 .29 .29 .28 1.65
04-05.99 .26 .26 .26 .26 .25 3. 35
06-07.99 .21 .21 .21 .21 5.11
08-09.99 .18 . 18 .18 .18 .20 6.94
10-11.99 .17 .17 .17 .18 8.83
12-13.99 .16 .16 . 16 .16 .17 10.89
14-15.99 .15 .15 .15 .16 12.83
16-17.99 .15 .15 .15 .15 .15 15.00
18-19.99 . 23 .23 .23 .18 17.20
20-21.99 .37 .37 .37 .37 .39 19.50
22-23.99 .42 .42 .42 .43 21.90
24-25.99 .43 .43 .43 .43 .42 24.50
26-27.99 .41 .41 .41 .36 27.20;
28-29.99 . 32 .32 .32 .32
30-31.99 .27 • 27 j[30.00)
32-33.99 .23 .23 .23 .23 .221 33.00)
34-35.99 .20
36-37.99 .18 . 18 .18 .18 .181[36.20)
38-39.99 .16 .141*39.60)
40-41.99 .15 .15 .15 .15
42-43.99 .14 .1 3 (4 3 .3 0 )
44-45.99 .13 .13 .13 .13
46-47.99 .12 .12(47.20)
48-49.99 .12 .12 .12 .12
50-51.99 .11 .1 2 (5 1 .4 0 )
52-53.99 .11 .11 .11 .11
54-55.99 .11
56-57.99 . 10 .10 . 10 .10 .1 1 (5 6 .1 0 )
58-59.99 .10
60-61.99 .10 .10 .10 .10 .1 0 (6 1 .1 0 )
62-63.99 .09
64-65.99 .09 .09 .09
66-67.99 .09 .1 0 (6 6 .7 0 )
68-69.99 .09 .09 .09 .09
70-71.99 .09
72-73.99 .10 .10 .10 .1 1 (7 3 .0 0 )
74-75.99 .11
Nota: *2 year" assunas a c o n sta n t In ta rv a l fo r a l l a g a s .
This 1s a ls o tru a fo r tha "4 year" I n t a r v a l . Tha "2*4
yaar" has a s ta p fron 2 to 4 yaars a t ago 32.5. In a d d i
tio n to tha 2*4 y a a r s to p s , tha "2*4*8 y a a r ” has a s ta p
fron 4 to 8 y a a rs a t ago 64.5 . For tha "continuous"
fu n c tio n th a ago c a ta g o r la s shown a ra a p p ro x ln a tlo n s (and
ara shown 1n th a p a ra n th a sa s - saa E s tln a ta 5 1n tha
sa c tlo n on Doflnlng Ona Nova). Tha nunbar of d ata p ro
b a b i l i t i e s 1s a f a c t o r 1n tha a s t l n a t a o b tain ed by con-
b ln a tlo n a l th e o ry .
CHAPTER V
RESULTS
The five p ro b a b ility estim ates re s u ltin g from the
d i s t r i b u ti o n of age by the time in terv al O ^ ) between
moves and age re la te d mobility rates were te ste d 1n the
combinational p ro b a b ility formulation of the Longitudinal
Residence Change Model. The d i s t r i b u t i o n re ferred to as
estimate 4 was considered the l i f e cycle step fun ctio n,
and estimate 5 was considered the continuous function,
representing the complementary explanation of the Cornell
Model and m obile-stable population explanation. In add i
tio n , the continuous function represented the expected
empirical re s u lt where age events do not occur at a single
age point as posited in the l i f e cycle. The three other
estim ates, referred to as estimates 1, 2, and 3 were used
as baseline estim ates. These five estimates were used
In the formulae to compute tables for comparison to actual
d a t a .
Table 4 used estimate 4 and gives the l i f e cycle
step fu n c tio n 's estimate of the p ro b a b ilitie s of moving
by age category and the number of past moves corresponding
to Equation 1. This basic ta b le will be referred to
l a t e r and 1n the DISCUSSION section that follows.
59
TABLE 4
Probability of Experiencing a Given Umber of Moves by Age and Differences (d)
In Moving Probabilities Between Age Categories
-U fa Cycle Estloate-
One
Humber of Past Moves
Two Three Four
3 p— d p
Five
■ 0 -4-
00.00-01.99 .690 .310
02.00-03.99 .490 -.200 .420 .110 .090
04.00-05.99 .363 -.127 .438 .018 .176 .086 .023
06.00-07.99 .286 -.077 .422 -.016 .231 .055 .055 .032 .005
08.00-09.99 .235 -.051 .398 -.024 .265 .034 .087 .032 .014 .009 .001
10.00-11.99 .195 -.040 .370 -.028 .288 .023 .117 .030 .026 .012 .003 .002
12.00-13.99 .164 -.031 .342 -.028 .301 .013 .145 .028 .041 .015 .007 .004 .001
14.00-15.99 .139 -.025 .315 -.027 .307 .006 .168 .023 .056 .015 .012 .005 .002 .001
.118 -.021 .289 -.026 .308 .001 .189 .021 .073 .017 .019 .007 .003 .001
.091 -.027 ' .758 -.839 .304 -.004 .216 .027 .100 .927 .031 .012 .998 .995
20.00-21.99 .057 -.034 .191 -.059 .284 -.020 .249 .033 .143 .043 .057 .026 .019 .011
22.00-23.99 .033 -.024 .135 -.056 .245 -.039 .264 .015 .187 .044 .093 .036 .043 .024
24.00-25.99 .019 -.014 .091 -.044 .198 -.047 .256 -.008 .220 .033 .134 .041 .083 .040
26.00-27.99 .011 -.008 .062 -.029 .154 -.044 .232 -.024 .235 .015 .169 .035 .138 .055
28.00-29.99 .008 -.003 .045 -.017 .124 -.030 .207 -.025 .234 -.001 .190 .021 .192 .054
30.00-31.99 .006 -.002 .035 -.010 .103 -.021 .185 -.022 .226 -.008 .202 .012 .243 .051
32.00-35.99 .004 -.002 .028 -.007 .087 -.016 .166 -.017 .217 -.009 .208 .006 .290 .047
36.00-39.99 .004 -.000 .024 -.004 .077 -.010 .152 --014 .206 -.009
■222 •991
• 327
•22Z
40.00-43.99 • 003 -.001 .021 -.893 .069 - .008 .141 -.011 .199 -7905 .209 .086 .358 .931
44.00-47.99 .003 .000 .019 -.002 .063 -.006 .131 -.010 .192 -.007 .208 -.001 .385 .027
48.00-51.99 .002 -.001 .017 -.002 .057 -.006 .123 -.008 .184 -.008 .206 -.002 .410 .025
52.00-55.99 .002 .000 .015 -.002 .053 -.004 .116 -.007 .178 -.006 .203 -.003 .433 .023
56.00-59.99 .002 .000 .014 -.001 .049 -.004 .109 -.007 .171 -.007 .201 -.002 .453 .020
60.00-63.99 .002 .000 .013 -.001 .046 -.003 .103 -.006 .165 -.006 .198 -.003 .473 .020
64.00-71.99 .001 -.001 .012 -.001 .043 -.003 .098 -.005 .160 -.005 .195 -.003 .491 .018
72.00+ .001 .000 .011 -.001 .040 -.003 .093 -.005 .154 -.006 .192 -.003 .509 .018
Rote: d - >9*44.1
- *9*1
probability LIFE CY CLE *
?
years for ages (00.0-32.5
-d: probability of Mvlng Is decreasing
+d: probability of sovlng Is Increasing
betwoon age intervals a^ and a ^
4 years for ages (32.5-64.5
8 years for ages (64.5-80.0
61
Age and Staving at the Same House
Table 5 was obtained by using the "stay" column
in Table 4 or a sim ilar column from tab les of the other
four estim ates not presented. This table represents
estim ates of stayers at the same house a ll of t h e i r
l i v e s . These estimates were compared to actual data
{U.S. Census, P-20 S eries, Nos. 47 and 104) using Robin
son's (1957) A to determine the degree of agreement.
A11 five estimates tended to follow the actual data
and to confirm the a n a ly t i c a l l y expected re la tio n s h ip
between the l i f e h isto ry of staying and age. For
example, children have a sh o rter l i f e h is to ry , th e re fo re ,
have a higher p ro b a b ility of living at the same house all
of t h e i r l i f e . Robinson's A for both the l i f e cycle step
function and the continuous function was .99. Both ex
planations closely f i t the actual data. The maximum
absolute departure of the estimate from the actual data
for the step function was 3.2 per cent In 1952 and 9.5
per cent 1n 1959. The maximum absolute departure for
the continuous function was 3.7 per cent In 1952 and
7.2 per cent In 1959. These departures were all 1n the
age category 1-13 y ears. I t might be expected th at part
of the d iffe ren ce between the estimate and the actual
data can be accounted for by the wide v aria tio n In the
rates at the younger ages.
TA4LE 5
Per Cent Stayers-Aluays at tha Saaa House by Aft. United States, 1952 and 1959
- Actual and Estimated laiu lti Using Five Distributions* *
Actual Estl n { i t Estimate 2 Estimate 3 Estli
l l f l
■ate 4
Cjfcl*
Estiaata 5
Continuous
Ago I 1 “ 7 " I i I i 1 1 T
1552 Data A>.99 A-.42 A-,99 A*.99 A*. 99
1-13 27.0 30.2 -3.2 42.9 -15.9 30.2 -3.2 30.2 -3.2 30.7 -3.7
14-17 11.1 14.5 -3,0 30.2 19.4 14.4 -3,0 14.4 -3.0 10.7 1.1
14-15 7.9 5.3 •0.4 23.0 -15,1 4.3 -0.4 4.3 -0.4 5.5 1.3
20-24 3.5 3.4 •0.1 13.9 -10.4 3.5 -0,1 3.5 •0.1 3.5 -0,1
25-25 2.0 1.0 1.0 4.1 -4.1 1.0 1.0 1.0 1.0 1.0 1.0
30-34 1.1 0.4 0,7 5.4 -4.5 0.4 0.7 0.4 0.7 0.7 0-4
35-35 1.1 0.3 0.5 4.3 -3.2 0.4 0.7 0.4 0.7 0.4 0.4
40-44 1.1 0.2 0.9 3.4 -2.5 0.4 0.7 0.4 0.7 0.4 0.4
45-54 1.1 0.1 1.0 2.4 -1.7 0.3 0.4 0.3 0.4 0.3 0.4
55-44 1.4 0.1 1.5 2.2 •0.4 0.3 1.3 0.3 1.3 0.3 1.3
45-75+ 2.2 0.0 2.2 1.7 -0.5 0.1 2.1 0.1 2.1 0.2 2.1
1951 Data A-.99 A-.93 A».99 A-.99 A-.99
1-4 57.5 45.0 9.5 42.1 -4.4 44.0 9.5 44.0 9.5 50.3 7.2
5-4 33.9 30.5 3.4 49.4 -15.7 30.5 3.4 30.5 3.4 24.5 5.4
7-13 21.9 20.0 1.9 35.4 -14.7 20.0 1.9 20.0 1.9 17.4 4.5
14-17 15.2 14.4 0.4 30.2 -15.0 14.4 0.4 14.4 0.4 10.7 1.1
14-15 12.2 9.3 3.9 23.0 -10.4 4.3 3.9 4.3 3.9 4.4 1.3
24-21 7.1 5.1 2.0 17.4 -10.1 5.1 2.0 5.1 2.0 4.5 2.4
22-24 3.7 2.4 1,1 12.7 • 9.0 2.4 1.1 2.5 1.1 2.4 1.1
24-29 2.0 1.0 1.0 4.1 • 4.1 1.0 1.0 1.0 1.0 1.0 1.0
tt - 34 1.4 0.4 1.2 5.4 • 4.0 0.4 1.2 0.4 1.2 0.7 0.4
35-44 0.9 0.2 0.7 3.9 - 3.0 0.4 0.5 0.4 0.5 0.5 0.4
41-44 1.1 0.1 1.0 2.5 - 1.4 0.3 0.4 0.3 0.4 0.3 0.4
45-74 1.1 0.0 1.1 1.7 • 0.4 0.1 1.0 0.1 1.0 0.2 0.9
75+ 1,1 0.0 1.1 1.4 - 0.3 0.1 1.0 0.1 1.0 0.2 0.9
Estiaete-Tin intarval Assumptions: " "
1. 1| * I yaars far all egos. 2. » A yaars for all agas.
1. I, * 2 yaars for sgts 0.0-32.5; 4 yaars. agas 32.5-90.
4. 1, - 2 yaars for agas 32.5*. 4 yaars, agas 32.5*54.5; I yaars, agas 44.5-10.
». ij “ I
* Tbasa gar cants do not add up to 1001 across ago bacausa tba n e a r portion bat boon ts-
cladad. Across botb savers and stayors, tba par cants agaal 1005.
•♦Source: U.S. lornau of tba Cansus, Currant Population Reports, Sorias P-20, bos. 47,
Tabla 5, and 104, Tabln 11, "Papulation Characteristics,- U.S. Oavaranat Printing Office,
basblngtan, D.C., 1553 and 1540. w
ro
63
The f i t of all five estim ates tended to confirm
th a t age re la te d mobility rates can be used to approximate
the actual l i f e h is t o r i e s of staying at the same house
using the model. For all p ra c tic a l purposes, e i t h e r the
step or the continuous function yields the same r e s u l t .
The continuous function f i t s the actual data a l i t t l e
b e t t e r ; however, the step function f i t s almost as well and
was an Ideal th e o re tic a l formulation. The Cornell Model's
explanation 1s oriented towards p redicting s ta y e r s , thus
the continuous function should have f i t b e t t e r . W e note
that the l i f e cycle step function f i t s almost as well
when used to define one longitudinal mover p ro b a b ility for
combinational theory.
Life H isto ries of Residences
The sum of the p r o b a b il i t ie s In Table 4, or In
four other tables not presented, by the number of moves
up to and Including age a^ (and ad justing to equal to
1.0 and accounting for "residence") gives the l i f e
h isto ry of number of residences p r o b a b ilitie s by age.
Table 6 gives the actual re su lts (U.S. Census, P-23
S e rie s, No. 25) and the five estimates of the l i f e
h is to r ie s of the number of residences p r o b a b il i t ie s by
age.
Here, the a n a ly t i c a l l y expected r e s u l t was th at
with Increased age there would be an Increasing number
64
TAIL! •
L lfa Mlitory of Nunfear of IIo i K oocoi by Am and f a r Caat Within C atag o rlas.
Uni tad S ta ta a . 1111 - Actual and C stlnataa Raoulto Using Flva D ls trfb u tla n s -
1 — — j j j —
I f - 14 1 1 -1 1 1 1 -1 1 41! - ¥ f ” 66-64 65*
n1 d ■t
21 30 40 60 60 _ 70
Nunbar of
Ratldancas I d S d f d
s d
f 4
I .
Actual Data
■ *
1 40.6 26.0 24.7 23.3 21.6 19.1
2 30.6 29.2 27.9 26.3 30.1 26.7
3 15.3 19.6 19.9 19.5 21.4 21.9
4 7.2 10.2 11 .4 11.5 11.1 12.3
S 2.9 S .3 6.0 6.5 6.5 7.1
6+
3.1
7.4 10.2 10.6 9.4 10-7
E stln ata 1:
1 41 .3 -0.7 31.0 -3 .0 23.9 0.6 19.5 3.6 16.4 5.2 14.2 4.9
2 26.2 2.6 23.4 S, 6 16.9 9.0 15.7 12.6 13.5 16.6 11,7 17.0
3 , 17.6 -2.3 16.0 1.6 16.2 3.7 14.4 5.1 12.6 8.6 11.6 10.4
4* 13.4 -0.2 2 7 .S -4.1 40.9 -13.3
N M
-21.6 67.4 -30.4 62.6 -32.6
f s t l n i t a !:
1 59. 3 • 19.7 49.0 - 21.0 40.6 •16.1 35.0 -11.7 30.7 -9.1 27.4 -6.3
2 27.4 3.4 27.6 1.6 26.4 1.5 24.9 3.4 23.5 6.6 22.1 6.6
3 10.1 5.1 14.6 5.2 17.4 2.5 16.6 0.7 19.5 1.9 19.6 2.1
4 2.6 4.6 6.2 4.0 9.6 1.6 12.0 •0.5 13.6 -2.7 15.0 -2.7
S 0.4 2.5 2.0 3. 3 4.1 1.9 6.0 0.5 7.7 -1.2 9.0 -2.9
6> 0.1 3.0 0.6 1.7 6.6 3.1 7.5 4.6 4.6 6.6 4.1
E stln a ta 3:
1 41.3 -0.7 31.0 -3.0 26.4 -1.7 23.4 -0.1 21 .1 0.5 19.1 0.0
2 26.2 2.6 23.4 S . 6 20.5 7.4 16.6 9.7 17.0 13. 1 15.6 13.1
3 17.6 -2.3 16.0 1.6 17.0 2.9 16.1 3.4 15.2 6.2 14.5 7.4
4 6.4 • 1.2 12.4 -2.2 13.3 -1 .9 13.5 -2.0 13.6 -2.5 13.4 -1,1
S 3.2 -0.3 7.7 -2.4 9.6 -3 .6 10.9 -4.4 11.7 -5.2 12.3 -5.2
6+ 1 1.9 7.4 0.0 13.0 - 2 . 1 17,4 -6.6 21.4 -12.0 2 8 J -14.4
E stln ata 4: Llfa Cyclo Stap A" .69
1 41.3 -0.7 31 .0 -3.0 26.4 -1.7 23.4 -0.1 21.1 0.5 19.7 - 0.6
2 26.2 2.6 23.4 5.6 20.5 7.4 16.6 9.7 17.0 13.1 16.0 12.7
3 17.6 -2.3 16.0 1.6 17.0 2.9 16.1 3.4 15.2 6.2 14.7 7.2
4 6.4 -1.2 12.4 -2.2 13.3 -1 .9 13.5 • 2.0 13.6 -2.5 13.5 -1.3
S 3.2 • 0.3 7.7 -2.4 9.6 -3 .6 10.9 -4 .4 11.7 -5.2 12.1 -6.0
6a 1.2 1.9 7.4 0.0 13.0 -2 .6 17.4 -6 .6 21.4 •12.0 24.0 -13.3
E stln a ta 5: A-.93
1 41 .0 -0.4 33.0 -5.0 27.9 -3.2 24.5 -1.2 22.4 -0.6 20.9 • 1.6
2 26.0 2.6 24. S 4. 7 21.6 6.3 19.5 8.8 18.0 12.1 17.0 11.7
3 17.7 -2.4 16.3 1.5 17.7 2.2 16.9 2.6 16.3 5.1 15.6 6.1
4 9.0 -1 .6 11.9 -1 . 7 13.3 -1 .9 13.6 -2 .3 14.1 -3.0 14.2 -1.9
5 3. S -0.6 6.6 -1.5 9.2 -3.2 10.6 -4.1 11.5 -5.0 12.1 -6.0
1.6 1.5 5.5 1.9 19-4 -0 .2 14,3 -3 .5 17.9 -8.5 2Q.1 _ _ -J.4
j
E stln ata 1:
*1 "
2 yaars for a n agas E stln ata 2: I. ■ 4 yaars fa r a ll agas.
E s tln a ta 3;
>1 '
2 yaars for agat 0.0 -32.5 yaars for agat 32 .5-60 yaars
E s t l n a t a 4: I
E stln a ta S: I
2 yaar* for ago* 0 .0 - 3 2 .Si 4 yaar* for a«o« 32.S-C4.Si •
ago* C4.S-IO yaar*.
♦ .-5 ♦ o h
actual a* tln a ta d
Only 4+ ratld an cas c a tc u la ta d *1nco 1-6 would raqulra 3 hour* coaputar tin * on N-600,
and th is 1* only a C atalin a o t t l a a t o . Thi* a s tln a ta wat conparad to tha tun of par
cant for 4-6 ro tld a n e a t In tha actual data.
•Saurea: U.S. Buraaa of tha Cantu*. Curront Copulation R aports. S arlo t f-2 3 . No. 2S»
Tabla 2. " L ift Tina Migration H ls to rla t of tha Anarlcan Paapla," U.S. Oatarnnant
P rin tin g Off lea. Maihlngton, D.C., 1966.
65
of residences. Estimates 1 and 2 (constant time between
moves) departed more than the l i f e cycle step function
or the continuous function from the actual re su lt. Thus*
a constant function of age by time between residences
used 1n combinational theory does not provide as close
a f i t to actual data as does a step or continuous func-
t1 on.
A comparison of the two step functions Indicated
that Inclusion of the older age stage (estimate 4 com
pared to estimate 3 which does not have an elderly stage)
Improves the estimate s l ig h tly . Although the data on the
elderly 1s limited 1n terms of recall and being " tru n
cated." 1n further discussion, only reference to e s t i
mate 4 as the "step function" Is Intended.
Robinson's A for the step function (estimate 4) was
.89 and for the continuous function was .94. For all
practical purposes, the step and continuous function
yield the same r e su lt. The maximum absolute departure
from the actual data for the step function was 13.3 per
cent, for the continuous function was 12.1 per cent.
These departures were In the older ages and 1n the
estimate for the higher number of residences. These
departures were expected since the assumption 1n con
structin g the model was that the age related mobility
rate and the time Interval d is trib u tio n were Invariant
66
for the past 72 years. Also, the actual data are tru n
cated and limited.
Since the age related mobility rate d is trib u tio n
1s known to have been r e la tiv e ly stable for the past 20
years (U.S. Census, P-20 series) rather than 72 years,
the expected re su lt (Table 6) for the 18-24 year old
category should be equivalent to the actual. This expec
tation was confirmed for both the step and continuous
function. Robinson's A for both estimates was .99. The
maximum absolute departure for the step function was 2.6
per cent and 2.8 per cent for the continuous function.
Since the previous re su lt indicated that stayers at
the same house (U.S. Census, P-20 Series) was closely
approximated by the model, the model's estimate of the
probability of persons with one residence In th e ir l i f e
history (Table 6) should closely f i t the actual data (U.S.
Census, P-23 Series, No. 25). This expectation was v e r i
fied by both step and continuous function having a Robin
son's A of .99 for one residence by age. The maximum ab
solute departure from the actual re s u lt was 3.0 per cent
for the step function and 5.0 for the continuous function.
The two estimates for all practical purposes yield the
same r e s u lt.
67
Average Number of Moves in L ift History
Using the estimate of the l i f e history of r e s i
dences In Table 6 for persons age 72 years for the l i f e
cycle step function and Equation 5, an estimate of 2.77
migrations was obtained. ( I t should be noted that the
U.S. Census, P-23 Series data approximate the d efin itio n
of "migration." Also the shape of the rate d istrib u tio n s
of "resid en tial mobility" and "migration" are similar
(Eldrldge, 1965), and the rates -- not the numbers -- were
used 1n the model). This 2.77 estimate was within the
range specified by Shryock (1965), which was 2.1 to 3.15
migration. This estimate was less than Wilber's (1963)
estimate of 4.2 migrations. The model's estimate was
probably s lig h tly high due to the assumption of Invariant
rates for 72 years.
Past-Retrospect1ve Movers
Table 7 gives the actual data (U.S. Census, P-20
Series, No. 104) and the estimates of stayers 1n the
current year by age and by year of l a s t p ast-retro sp ectiv e
move. Table 7 was only calculated on Estimate 4 (the
2, 4, and 8 year) -- the l i f e cycle step function. The
continuous function (Estimate 5) was not tested for past-
retrospective movers so no comparison with the step func
tion was possible.
TABLE 7
Past-Retrospect1ve Novers--Per Cent of the Totel Population Mho Mere Stayers
During the Survey Year by Age and by Year of Last Mover,* United S tates, 19S9
• Actual and Estimated Using Five Distributions -
Tears Ago to Last Move
1 Tear 2-3 Tears 4-7 Years 8-17 Years 18+ Years
Age i ” e d a** e d — a** e d a** e d a** e d
1-4 .191 .201 - .011 .155 .208 .053 .033 .155 -.122
_ .
5-6 .173 .223 - .050 .263 .233 .031 .207 .217 -.010
- . . . -
7-13 .131 .160 - .049 .212 .202 .010 .348 .227 .121 .081 .162 -.081
-
14-17 .110 .178 - .068 .157 .189 .032 .317 .213 .104 .259 .217 .042
-
18-19 .127 .170 - .043 .152 .171 .018 .256 .210 .047 .305 .261 .044 .031 .075 -.044
20-21 .214 .224 - .010 .174 .211 .037 .202 .190 .022 ,248 .227 .021 .084 .078 .006
22-24 .337 .277 .061 .287 .287 .002 .159 .224 -.064 .116 .125 -.010 .059 .056 .002
25-29 .296 .287 .009 .344 .295 .049 .234 .304 .069 .073 .079 -.006 .031 .025 .006
30-34 .80 .227 - .048 .292 .261 .031 .369 .315 .054 .110 .178 -.069 .033 .014 .019
35-44 .129 .167 - .037 .210 .183 .027 .362 .258 ,104 .240 .308 -.068 .050 .082 -.032
45-64 .083 .141 - .059 .145 .147 -.003 .248 .184 .064 .285 .209 .076 .228 .315 -.087
65-74 .079 .138 - .060 .112 .142 -.030 .187 .160 ,027 .224 .191 .033 .387 .367 .020
75+ .064 .141 - .077 .118 .140 -.022 .178 .142 .036 .211 .175 .036 .418 .401 .018
Ave .1^3 .197 .634 .202 .205 -.004 .238 .215 .023 .165 .164 .001 .102 .109 - .ffW
a * actual e * estimate ave ■ average per age category
Note; In the actual data, the f i r s t coaiplete y ear's data Mere used as the f i r s t past
year In order to exclude partial years ( e .g ., 3 Months). The estimates of the actual
(a ), above was obtained by sumlng the f i r s t to the 18+ year category plus the stayers
at the sawe house a ll of th e ir l i f e and obtaining the per cent of each category
(adjusting to a probability equal to 1.00). Robinson's A * .92.
* Per cents do not add up to 100 according to age because stayers are excluded.
In addition, rather than the year of la s t wove, the nuaber of years ago 1s
u tiliz e d to abridge the original table.
** Source; U.S. Bureau of the Census, Current Population Reports, Series P-20, No. 104,
Table 11, 'Population C haracteristics,* U.S. Governnent Printing Office,
Washington, D.C., 1960.
69
The maximum absolute difference between the actual
and the step function estimate was 12.2 per cent. This
age category only contains persons who are currently 4
years old and moved at b ir th , even though the category
Is defined m o r e broadly. The Robinson's A for the
agreement between the actual and step function estimate
was (A ■ ) .92.
Comparison to Cornell Hodel
Table 8 gives the model's estimate of the compar
ison to the Cornell Hodel (generator j Pj » type B; McGin
nis. preliminary, 1968). The model's estimate to the
Cornell Model was based on a series of constant functions
corresponding to duration age (previously defined in
METHODOLOGY section) rather than a step or a continuous
function. For each constant function, the model calcu
lated the probability for that duration age and repeated
the calculation for each duration age. The results are
given In Table 8.
Numerical differences were noted. The maximum ab
solute difference 1n estimate between the two models for
the f i r s t 12 years was 11 per cent at duration age of
six. From the 12th or older duration age. the two models
d iffe red . At duration age of 36 years, the difference
was 17 per cent. On the other hand, the numerical
sim ila rity of estimates between both models suggests a
70
TABLE 8
The Modified S Residence Change Model Compared
to the Cornell Model's Simulation Estimates
Tom e 11 Hodel {THodal
Duratl on*
Age (years) Probabl11ty a
"Time"
Between
Moves
(years) Probabl11 tv
1
2 0
-
3 .27 2 .24
4 .34 3 .38
5 .40 4 .40
6 .43 5 .54
12 .62 11 .68b
36 .90 35 . 73
* The Cornell Model has the f i r s t duration age as 1 year;
th erefo re, the time between moves equals the duration
age - one year. I . e . . time between moves * duration age-
1. Duration age 1s the time of staying at a residence.
a These numerical values were approximated from a graph by
McGinnis (preliminary. 1968). A smooth approximation
curve was used.
b Taeuber. et al . , (1968:37) show 28.5 per cent having one
area of residence from age 25-34 years and 25.2 per cent
having one residence from age 35-44 years. An average
per cent at age 35 years for staying would be (1.00 -
average movers) * (1.00 - 26.8) or 73.2 per cent. For
65 years and older, Taeuber shows one residence for 19.6
per cent or an approximate asymtope of (1.00 - 20 per
cent) « 80.0 per cent. Sample survey data (U.S. Census,
P-20, 1960, Table 11) Indicates approximately 2 per cent
of the population, age 65 or older, who have "always
lived In the present house." These two sets of data
are for 1958-59.
71
commonality of concept although the formulations are
d iffe re n t. It should be noted, however, th at the modified
$ model which used a series of constant functions to ap
proximate the Cornell Model would not estimate actual
data as closely as the continuous or step function.
CHAPTER VI
DISCUSSION
The S Longitudinal Hodel of Residence Change was de
veloped using combinational theory and then compared to
actual data. Causal theory was Included In the procedure
for specifying the p r o b a b ilitie s of one longitudinal move
used 1n the model. In evaluating the model, the c r i t e r i a
used were Its a b i l i t y to account for a wide range of
actual d ata, using the same formulation. This an aly tic
framework was used to consider the th e o re tic a l question
of whether the p r o b a b ilitie s of moving at a given age
were Independent of or dependent on past residence change
h i s t o r i e s . After considering the p ro b a b ility question,
the causal reasons for moving are examined 1n terms of an
Independent or dependent conceptual theory.
The comparison of the estim ates by the model to
actual data suggests that an Independent, combinational
p ro b a b ility approach has p o tential for fu rth e r develop
ment. Both the l i f e cycle step function and the co n tinu
ous function of (age by) Increasing length of time at a
residence according to past amount of time closely ap
proximated the data. However, the l i f e cycle step
7 2
73
function had th e o re tic a l advantages which will be d i s
cussed l a t e r .
Of empirical Importance 1s that the model's e s t i
mates clo sely approximated the actual data and suggested
th a t the four sets of data used in evaluation were r e
la te d . The estimates suggest th a t age re la te d mobility
rates approximate the p ro b a b ility d i s t r i b u t i o n of moving
and are re la te d to stayers at the same house a ll of t h e i r
l i v e s , to l i f e h is to r ie s of moving, and to stayers by age
and t h e i r year of l a s t move. These four sets of data
have been tre a te d as separate aspects of residence change.
The model confirms the logical expectation th a t the data
are re la te d and are focused on a single phenomenon, the
Internal residence change pattern of the population of
the United S tates. The Implication Is th a t although these
sets of data are tabulated d i f f e r e n t l y or focus on a d i f
ferent aspect of the residence change, they should be
considered as a single phenomenon. Constructing a model
1s an approach for combining these d i f f e r e n t aspects. In
a d d itio n , knowing portions of the d ata, other aspects can
be derived.
The model supported the th e o re tic a l re la tio n s h ip
between age, length of time at a residence and residence
change as noted In the l i t e r a t u r e . In a d d itio n , the
model suggested a method for combining these r e l a t i o n
ships Into a sin g le formulation. This formulation
74
suggests a re la tio n s h ip between: 1) the l i f e cycle ex
p lan atio n , 2) the Cornell Model, and 3) the m obile-stable
population explanation. These formulations are discussed
1a te r .
Thus, model construction on residence change has
em pirical, methodological, and th e o re tic a l Importance.
Each of these three aspects 1s considered In the following
d ls c u s s 1 on.
Empirical V alidity
The computer estim ates, using the step function
representing the l i f e cycle and the continuous function --
representing both the complementary explanation of the
Cornell Model and the m obile-stable population explana
tion - - yield approximately the same r e s u l t s . The most
s a l i e n t evidence for the accuracy of the approach used
was the Longitudinal Model's estimates of the cross tabu
latio n s of age by l i f e history of residence and per cent
within the c a te g o rie s , and p a s t- r e tr o s p e c tiv e movers by
age and by year of l a s t move. F ittin g data to a single
dimension, e . g . , age and stayers at the same house all of
t h e i r l i v e s , 1s “e a s i e r 1 1 than f i t t i n g cross tabulated
data. The model was mathematically rew ritten to f i t two
cross ta b u la tio n s .
In the development of the methodology of the S
Model, the emphasis was on l i f e cycle stages and
75
Independent p r o b a b il i t ie s which estimated diverse sets of
data. This model was of methodological and th e o re tic a l
I n t e r e s t as well as having empirical v a l i d i ty .
Methodology
Longitudinal P ro b a b ilitie s
The model's construction of longitudinal p ro b a b il
i t i e s using annual ra te data was accomplished by using
age categories with v aria b le widths which corresponded to
the length of time at a residence at those ages. Also,
combinational theory was used In order to obtain l i f e
h is t o r i e s of residence change. The e f f e c t of these two
methodological procedures can be best I l l u s t r a t e d by a
comparison of the model's estim ate and Wilber's cohort
analysis procedure (Double Decrement Tables) which does
not Incorporate these two m odifications.
Wilber's (1963) cohort analysis approach used e ith e r
five or ten year in te rv a ls between moves (depending on the
age c a te g o riz a tio n of data) 1n order to estimate the
average number of "expected" l i f e time migrations.
Shryock (1965) used a more d i r e c t approach to obtain his
estim ate of th is average. The model's construction sug
gested th a t the length of time at a residence Is re la te d
to age, and the p ro b a b ility of one move -- migration or
mobility -- 1s not best estimated year by year (or equal
time Interval for all ages). This fa c to r appears to
76
explain why Wilber's estimate was higher than Shryock's.
The estimate by this model was within Shryock's range.
Thus, a closer correspondence between Wilber's and
Shryock's estimates was suggested by the model.
Since the model's estimates used length of time at
a residence within age 1n order to obtain the variable
age categories, then the Issue of Independent or depen
dent p ro b a b ilitie s based on past residence change needs
to be considered. Is a combined (or jo in t) fa cto r pre
sent or Is past residence change crucial?
Independent or Dependent P rob ab ilities
Two d istrib u tio n s were used to construct the Inde
pendent p ro b a b ilitie s used 1n the longitudinal model.
These d istrib u tio n s were age related mobility rates which
were partitio ned by an age by length of time at a r e s i
dence d is trib u tio n which had variable widths of time be
tween moves by age categories. A theoretical question
can be raised whether these p ro b ab ilities are Independent
only "by definitio n" or "by construction." Is there a
t a c i t dependency assumption?
When considering how the p ro b a b ilitie s were con
stru cted , 1t can be argued that the p ro b a b ilitie s are
conceptual 1v dependent on the length of time Interval
for the given age. This argument poses a paradox 1n that
as probabl11 ties per se they are Independent; however.
77
1n the conceptualization these are conditional probabi
l i t i e s (dependent) on the length of time at a residence
for a given age. Since this question of Independence
or dependence of past moving history (e ith er as time or
behavior) Is the theo retical question of this paper, an
answer to this paradox Is requ lred.
F i r s t , i t can be noted th at the probability formu
lations of the Markov Chain Process (Independent) d if f e r
from the Stochastic Processes (dependent) 1n terms of an
additional h is to ric a l dependency assumption. The question
then becomes one of determining the meaning of the term
"dependency" In a conceptual sense and with regards to
the l i f e h istory. Should the dependency assumption be
considered 1n terms of the recent stayer period, the
sequence of stayer periods, or over the total l i f e
history? In Stochastic Processes, the usual condition
specified (McGinnis, 1968) Is that the tran sitio n matrix
applies for any sta te . Thus, for the Cornell Model the
prior length of stayer time 1s the crucial factor 1n the
conceptual d e fin itio n of the dependent p ro b a b ilitie s .
Second, It can be noted that the p ro b a b ilitie s used
1n the Longitudinal Model were conceptually defined using
length of residence within age, therefore, were condi
tional on this factor. In this way, the Longitudinal
Model Is conceptually sim ilar to the Cornell Model even
though the tra n s itio n probability matrix for each would
78
d i f f e r . Conceptual differences* namely, which Is more
crucial -- age or length of time -- need to be examined
1n order to make a c le a r e r determination of Independence
of or dependence on past h is to ry .
I t 1s argued here th at even though the length of
residence was used In defining the p r o b a b i l i t i e s , the
Longitudinal Model used an age sequence of time at a r e s i
dence. Therefore, age Is the c r i t i c a l v ariab le and length
of time 15 secondary. In ad d itio n , the age stages were
Important In determining the length of time between
residences; the model posited a sequential length of time
series and not a random s e r ie s . Therefore, these age
stages are Independent and the use of the step function
Is suggestive of th is Independence. By th is approach,
the conceptual and p ro b a b ility frameworks of the Longitu
dinal Model are reconciled 1n terms of an Independent
framework. The meaning of th is Independent framework
In terms of a s p e d f 1c explanatory theory will be d i s
cussed l a t e r . Here, general p ro b a b ility and th e o re tic a l
considerations are discussed 1n re la tio n to the concept
of Independence or dependence and with regards to s t a t i s
t i c a l data and reasons for moving.
Explanatory theory goes beyond s t a t i s t i c a l analysis
of Independence or dependence of p r o b a b i l i t i e s . The
re la tio n s h ip between the conceptual construction of the
p r o b a b il i t ie s and the p r o b a b il i t ie s themselves poses many
79
In terestin g questions and suggests a need for a guide
1n re la tin g theory. For this limited study, only five
questions are raised rather than answered when considering
theory and Its relationship to the p ro b a b ilitie s .
Accounting for the d a t a : The s t a t i s t i c a l question
of which framework, the Independent or dependent, accounts
for more data is only a preliminary analysis and will not
arrive at theory. Relating the conception and p ro b a b ili
ties 1s necessary.
Reasons and Data: After the preliminary analysis
of s t a t i s t i c a l data above, the relation ship between data
and reasons needs to be examined. The crucial question
1s then which framework, the Independent or dependent
probability , accounts for more of the numerical data
according to the Independent or dependent reasons given
for moving. The aggregated data and reasons are In te r
related 1n the f i r s t analysis, and when disaggregated
according to conceptual reasons, the data should "best
f it" according to the respective formulations. I . e . ,
Independent or dependent. This type of theoretical
analysis presupposes that Independent or dependent causal
reasons can be determined prior to the analysis.
Mover and Stayer Reasons: Prior to making the
determination of whether a reason 1s Independent or de
pendent, both mover and stayer reasons must be considered.
It 1s suggested, that both dimensions of mover-stayer
80
reasons and Independence-dependence need to be considered.
In a d d itio n , within each of these four c e l l s , defined
by these two dimensions, d i f f e r e n t types of explanations
need to be considered, e . g . . psychological or social
psychological, or l i f e cycle.
If cu rren t l i t e r a t u r e 1s examined, then an Inde
pendent framework can be readily claimed since most
studies are on reasons for moving (not for staying ).
Rossi (1955:459) l i s t s "reasons for moving" categories
none of which Include more than the present or the l a s t
residence. Reasons mentioned Include: b e t t e r .quarters
or b e t t e r lo c a tio n , more space, rent too high, or house
too la rg e , house In need of r e p a i r s , clo ser to location
where employed, to build or purchase home, house sold,
re p aired , renovated, occupied by owner, house burnt or
torn down, and marriage. There Is not l i s t e d any reason
why, for example, an event occurring three residences ago
was a fa cto r In the perso n’s current move. These reasons
give a "clear case" for the Independent p r o b a b ility , based
on conceptual con sideratio ns.
The p o s s i b i l i t y ex is ts th at the dependent or the
stayer fa cto rs have not been studied. What types of
reasons might be considered as psycho!oalcal propens 1 t v ?
Perhaps psychological reasons for moving might be. "I
like to move," "I like to liv e in d i f f e r e n t places,"
"I moved around a lot as a child and s t i l l maintain th is
81
h ab it." It appears unlikely th a t these reasons for
moving are very prevalent or a r t i c u l a t e d . However,
a lte re d with regards to staying they may be more meaning*
f u l. Examples might be, "I like the neighborhood," "It
Is near stores I go to ." Considered in terms of 1nde-
pendence or dependence, long term dependency does not
seem to be a fa c to r In view of the age pattern which 1s
re la te d to residence change. This area requires more
re s e a rc h .
From a social psycho!oglcal point of view {assuming
that the psychological perspective can be separated, 1n
view of the unlikelihood or small e f f e c t of long term
psychological reasons), what types of responses would
be expected? McGinnis (1968) suggests the broad c a t e
gory of " t i e s to a neighborhood" as reasons for staying.
Lee (1966) suggests that these neighborhood or present
place fa c to rs can be e i t h e r po sitiv e or negative In terms
of moving or staying. Long (1972) argues th a t age of
children (a l i f e cycle fa c to r) lik e ly represent " tie s"
to the neighborhood and 1s a fa c to r most closely related
to residence change, a f t e r age of parents Is considered.
Long’s conclusion will be discussed l a t e r In terms of the
Longitudinal Model's explanation. Here, the question is
raised I f Long's re s u lts support the Cornell Model or
Stochastic Processes. The response to th is question sug
gests th a t the meaning of the term dependency must be
82
defined both conceptually as well as mathematically.*
This question Is considered 1n terms of the following
mathematical discussion.
Any State or an Age S t a t e : In Stochastic Processes,
the usual proposition Is that any s ta te or length of
time 1s being considered 1n the t r a n s i t i o n matrix. Long's
r e s u l t s c le a rly indicate an age s t a t e . In order for
Stochastic Processes mathematics to be ap p lic a b le , a study
of f i t t i n g data to the same t r a n s i t i o n matrix across ages
Is necessary. Without such a study, here. Long's re su lts
are claimed as belonging more to an age stage rath er than
to any s t a t e .
Spllerman (1972) uses Stochastic Processes 1n
order to estimate the actual longitudinal residence
hi s to r ie s of populations according to areal units (not
age). He compares both the Stochastic and the Markov
Processes and concludes th a t Stochastic Processes produce
a superior f i t , although Markov processes f i t almost as
well. It 1s argued here th at the In te rp re ta tio n of
Spllerman's re s u lts should be th a t over-and-above the
independent framework, the dependent framework can only
add explanatory power. Although Spllerman argues for Sto
chastic Processes, here It Is argued th at the Independent
framework takes precedence over the dependent fa c to rs .
This In te r p r e ta tio n leads to the question of theory or
models.
83
Theory or Hodels: This paper considers theory or
explanation as being more Important than models. The
l i f e cycle explanation re su ltin g from the analysis of the
Longitudinal Model 1s given a f t e r the previously mentioned
theories are discussed.
Life Cycle Explanation
The l i f e cycle explanation has the advantage of
being readily understood, and the p o ten tial for d e t e r
mining age points at which behavior occurs. These be
haviors can be em pirically v e rifie d by age and fu rth e r
analyzed. The disadvantage Is Its lack of precision In
defining “the l i f e cycle." Here, a mathematical formula
tion was developed to extend the l i f e cycle explanation
In the area of residence change.
The Longitudinal Model's Independent p ro b a b ility
framework 1s conceptually "simpler" (Markov Chain Process)
than Stochastic Processes where dependent p r o b a b il i t ie s
are used. This mathematical framework permits elabo ration
of the model.
In a d d itio n , the l i f e cycle explanation o ffers the
p oten tial for including normative events 1n the explan
ation of residence change. This explanatory p otential
of the U f a cycle re s u lts In Its lik eliho od as a major
th e o re tic a l framework. For example. Incorporating
mobility plans and choices and possibly values, e ith e r
84
1n terms of age or age and areal d istrib utio n * would
strengthen the link between demography and sociology. In
this discussion, a methodology for extending the l i f e
cycle explanation was suggested and represents one of a
variety of possible approaches.
Cornell Hodel
The Cornell Model suggested the relationship that
was Incorporated Into the Longitudinal Model. The pro
b a b ility assumption of "cumulative Inertia" was modified
to an age by length of time (depending on pest time) at
a residence d is trib u tio n used 1n the model. Also. 1t was
shown In the results that the l i f e cycle step function
and the continuous function suggestive of the Cornell
Model were similar and closely approximated the actual
r e s u l t s .
The d irec t comparison of the Longitudinal Model to
the Cornell Model (-|Pj» type B) according to "duration
age" were similar. However, the modification of the
Longitudinal Model Into constant functions according to
duration age that was used to approximate the Cornell
Model's estimates would less lik ely re su lt In accurate
approximations of actual data. Rather than a series of
constant functions, the model suggests that the age by
length of time between residence should be considered
1n terms of the l i f e cycle step or continuous function
85
approximating the 2, 4, and 8 years between moves with
tra n s itio n or step ages of 32.5 and 64.5.
On the other hand, the Cornell Model has greater
potential for precision and has correspondence with the
empirical expectation of a continuous (rather than a step)
function. This model could be readily modified and could
provide " b e tte r ” estimates.
Mobile-Stable Population Explanation
In the mobile-stable population explanation, and In
the concept of cumulative I n e r tia , the assumption Is made
that past experience Influences future behavior. The
mobile-stable population explanation dlchotlmlzes the
empirical data by an a rb itra ry number of past moves, e .g .,
none, or one or more, to Infer about future behavior.
It will be argued that the Inference about individuals
and the Influence of th e ir past residence behavior on
th e ir future 1s age related and the mobile-stable popu
lation explanation may need c l a r i f ic a t i o n .
The relation sh ip which forms the basis for the
mobile-stable population explanation can be noted in
Table 4. The probability of moving by those with more
prior moves Is greater than those with fewer moves. How
ever, the age relatio nship Is presently where the younger
persons have a higher probability of moving 1n the future
(Table 4, column labeled d to denote p ro b a b ilitie s of
86
moving 1n the next age category). The basic p ro b a b ilitie s
(Pm) used to compute this table were constant for an age
category. That 1s, there was not an assumption that
within an age category the probability would increase,
the more moves a person makes (Stochastic Processes). Yet
the estimates closely approximated the actual stayer
results (movers are only a dichotomy 1n the mover-stayer
explanation, I . e . , movers - 1.0 - sta y e rs). Although an
assumption about dependent p ro b a b ilitie s might result
1n a more accurate estimate, the age, Independent pro*
b a b l l i tl e s will account for a large portion of the e s t i
mate to actual data.
The explanation offered to account for the equal
probability for an age category and the Increased pro
bab ility of moving by the mobile population In Table 4
Is based on combination theory. With Increasing numbers
of past moves (p) for a given number of mover periods or
years (y), there are Increasing numbers of combinations
(J). These combinations are of the form y !/( (y-p)!p*) ).
The summation of the sets of p ro b ab ilities for the com
bination of p past moves, especially when truncated to
six plus past moves, results In an Increased probability
of moving as the number of past moves Increases for an
aggregate population. In addition, h isto rical time 1s
truncated. I . e . , only a b rie f period of time 1s considered
rather than the l i f e history of persons.
87
This analysis does not Indicate that the mobile-
stable population explanation should be discarded. It
only Indicates that age is a factor In this explanation.
The younger the person, the more apt he 1s to move and
to move repeatedly while the converse Is true for the
eld erly. Also care must be taken In specifying the type
of population and the way data are categorized 1n an
analysis. In the grouping of data by the number of moves
at the aggregate level, the Longitudinal Model Indicates
that the mobile-stable population differences can be
noted empirically within an age context. Also, at the
Individual lev el, an age facto r may enter 1n the decision
to move. This decision appears to be dependent on the
Immediate situation rather than on past learning or
psychological propensity. The question raised Is not
that the person's past experience has no bearing on his
decision, but the question must be asked, M Wh1ch portion
of a person's past experience Is Important?" (The d i f
ference between the Independent and dependent models might
be "chronic movers.") Also, I t might be observed that
persons may become more e f f i c i e n t In the process of
moving; however, do they become more e f f ic ie n t In the
decision to move?
88
The Longitudinal Model’s Life
Cycle Stage Explanation"
In the THEORY Section presented e a rlie r* the l i f e
cycle stage explanation within the Push-Pull theory
(Lee, 1966) was used as a frame of reference for con
stru c tin g the Longitudinal Model. The rep resen tatio n of
the l i f e cycle stage 1n the model was through the use of
a step function of the length of time between moves by
age. Invariance was assumed across h is to r ic a l time for
the length of time by age and for the cro ss -s e c tio n a l age
related mobility rate d i s t r i b u t i o n ; Independent, lo n g i
tudinal p r o b a b ilitie s of one move were obtained. These
p r o b a b ilitie s were used In combinational theory In order
to obtain the series of l i f e h is t o r i e s of residence
change. The Important p ro b a b ility fa c to rs 1n the model
were the length of time between moves within age c a t e
gories and the annual mobility ra te . The Important ex
planatory facto rs were the l i f e cycle step function and
the use of age.
What 1s I t about the l i f e cycle which would explain
these two d i s t r i b u t i o n s , t h e i r combining and the accuracy
of t h e i r re su lts ? The following l i f e cycle explanation
uses age and two age generations to explain residence
change with regards to the two d i s t r i b u t i o n s . A step age
at approximately 32 years was used 1n the age by time
between moves d i s t r i b u t i o n In co nstructing the model.
89
This step age was previously noted by Yee and Van Arsdol
(1968) and was called an employment age. The following
Is a more sp ec ific argument of why there should be a
step occurring at th is age. Since age 1s only a scale
Index, the sequential causal factors 1n explaining r e s i
dence change need to be sp ec ifie d .
It Is argued th a t at age approximately 32 and for
approximately the following 10 y ears, the composition
of the family 1s crucial 1n the step (function) upwards
towards a more stable residence p attern (from 2 to 4
years between moves). Two explanations are suggested,
both of which are i n t e r - r e l a t e d and r e s u l t 1n residence
change d i f f e r e n t i a l s . The two explanations are called
"married and early completed family size" and "married
and la te completed family s iz e ." The common feature of
both 1s the age of the children variable and the d i f
f e r e n t i a t i n g featu re may be education.
Long (1972) notes that a f t e r the age of parents,
the age of children 1s crucial In explaining the r e s i
dence change pattern ( r e s id e n tia l mobility and migration).
For "short moves" ( re s id e n tia l m o b ility ), Long notes
th at p rio r to and a f t e r 45 the mover pattern changes.
For migration, Long notes age 35 as c r u c ia l. Using the
two types of fa m ilie s , Long's re s u lts are explained as
f o l 1ows:
90
Early completed Faml1l e s : Using Long's re s u lts
plus an approximate marriage age of 21. the age of c h i l d
ren at age 32 to 45 would be from 11 to 24 years of age.
These ages on the age re la te d mobility ra te d i s t r i b u t i o n
(see Figure 1) shows the lowest point at about age 14
y e a r s . and t h e r e a f t e r the ra te rapidly Increases as young
persons leave home to a peak ra te at about age 20 y ears.
At these younger ages, there 1s a sex d iffe ren ce where
females have an e a r l i e r Increase 1n residence change than
males. In ad d itio n , females marry at a s l i g h t l y younger
age than males. Therefore, 1t Is argued that young
persons are leaving home to form "new" fa m ilie s , and there
4 decrease 1n the "older" family size In con trast to
the e a r l i e r period where the family size was In c re a s ln o .
Moving to a larger place ceases to be as Important as 1n
the early y e a rs , or the parents are moving to a sm aller
place (re s id e n tia l m obility) to age 45 years. This space
reason 1s among the most frequently noted 1n survey data
(Rossi, 1955).
Late Completed Famlly S i z e : Since the peak child
bearing ages for women are 1n t h e i r l a te 2 0 's , th is cycle
has passed when they are 1n t h e i r 30's and fam ilies have
a tta in e d most or a ll of t h e i r completed family size.
Therefore, the space fa c to r also ceases to be an Important
determinant of residence change.
91
The age of children which Influences space consid
eratio ns In residence change also 1s a determinant of
women's labor force s t a t u s , which 1n turn fu rth e r I n f l u
ences residence s t a b i l i t y . Women 1n t h e i r 30's have com
pleted t h e i r family size or the size 1s decreasing (early
completed family s i z e ) . Thus, married women can return
to the labor force at about age 35 (Hauser, 1964: 181-
182). Thus, i t Is argued that the age of children 1s
crucial as they enter grade school ( l a t e completed family
size) or with the development of the second generation of
new fam ilies and potential movers (early completed family
s iz e ) . In both cases, the crucial age is approximately
35-45 fo r the older family. Thus, over the l i f e cycle,
the early facto rs (child bearing) which resu lted In
moving are now a lte re d (p otential labor force s ta tu s ) a c
cording to the conditions In the l a t e r years when a new
generation 1s formed. This female labor force statu s adds
s t a b i l i t y to the residence change pattern -- the larg er
the female labor force the more lik e ly r e s id e n tia l mobil
ity ra th e r than migration will occur.
Marital statu s and employment s ta tu s are Important
1n the H f e cycle pattern for women with regards to
residence change. In the above, married women were con
sidered. The U.S. Census (1969:1) Indicates th a t a f t e r
age 35, married and single persons have the same r e s i
dence change ra te which 1s somewhat below those who are
separated* divorced* or widowed. In a d d itio n , marital
statu s 1s more of a fa c to r than having young children as
a determinant of labor force s t a t u s . Being married and
having young children re s u lts In less of a likelihood
of being 1n the labor force. Thus, as children become
older or the peak child bearing ages are passed, the
lik eliho od of married women re -e n te rin g the labor force
Increases (Mott, 1972:178-181). For married women, t h e i r
r e s id e n tia l change pattern would be more apt to follow
that of t h e i r husbands. Thus, Long's r e s u l t which In
dicates th a t female heads of fam ilies are more re s id e n
t i a l ly mobile than male heads may be due to employment
d i f f e r e n t i a l s which have been Inter-tw ined with family
s t r u c tu r e . On the other hand, the employment and r e s i
dential p attern of single women would be more like th a t of
married women due to the s i m il a r it y to the married hus
band's employment p attern . Thus, 1n the l i f e cy cle,
marital s t a t u s , age of children (family size and s t r u c
t u r e ) , and employment statu s are I n t e r - r e l a t e d Influences
on residence change. In ad d itio n . Income and residence
change are I n te r r e la te d .
If th is l i f e cycle employment context 1s accu rate,
then the Inverted U shaped d i s t r i b u t i o n of family income
might be explained by the Increase In position (social
mobility) of men (and have a migration e f fe c t to age 35
years old) and the entrance of married women back Into
93
the labor force at about the age of 35 years. The In
crease in wage earners in th e ir 30’s re su lts 1n the peak
Income ages in the f o r t i e s . The decrease In family Income
a f te r age 40 Income might be due to mortality of males
beginning at these ages or women leaving the labor force.
Both persons (husband and wife 1n th e ir fo rtie s ) might be
employed In the same geographic location and re s u lt In
the "shorter" move (re sid en tia l mobility to age 45), how
ever, to a "smaller," "better" place. Thus, using this
li f e cycle explanation of residence change, employment,
and family Income, the larger macro-economic analysis
( e .g ., Lowry, 1966) might be Incorporated into a more
complete theory of residence change. Then the explanatory
factors of the l i f e cycle would be Incorporated with the
push-pull factors.
In summary, the preceding sections discussed the
empirical v a lid ity , the methodology, and the theoretical
explanation of the Longitudinal Model. Problems with the
model are discussed In the following section.
Problems and Evaluation
There were many problems with the Longitudinal
Model. F i r s t , the c r i t e r i a data were based on survey r e
sults and the "true" re su lt can only be approximated.
This approximation 1s especially serious when the sample
data are disaggregated Into slngle-years-of-age rates.
94
More accurate data on rates by single years of age 1s
needed. S im ilarly , the model s ta rte d with empirical rates
to estim ate Ht r u e N p ro b a b ility d i s t r i b u ti o n s and these
I n i t i a l data are subject to v a r ia tio n . Since a lim ited
amount of data were ava ilab le for the age by length
of time at a residence d i s t r i b u t i o n , the use of five
estim ates to approximate the NformM or shape of the
d i s t r i b u ti o n only represents a "guess1 1 about the actual
d i s t r i b u t i o n . A fu rth e r study Is needed In th is area In
order to b e t t e r define the longitudinal p ro b a b ility of
one move.
The use of cross sectional data to approximate
actual longitudinal data needs fu rth e r study, e s p e c ia lly
of the adequacy of p a r ti t i o n in g the d ata, and of the
fa c to rs which Influence the p a r ti t i o n in g . Here, the
h i s t o r i c a l facto r of changing rates was not taken Into
account 1n the Longitudinal Model. Over h i s to r ic a l time,
the d i s t r i b u ti o n s of age re la te d mobility r a t e s , and age
by time between moves should vary due to social and
technological d iffe re n c e s. The p a r titio n in g procedure's
Inadequacy and the h i s t o r i c a l facto r re s u lts 1n ra te s or
per cents being used as long-term p r o b a b ilitie s In the
model.
The two d i s t r i b u t i o n s th at were used In the model
to construct p r o b a b ilitie s were very limited 1n terms of
t h e i r elaboration with demographic methods and in terms
95
of the few fa c to rs used In the model's explanation. The
model was lim ited to estim ates of per cents r a th e r than
numbers of population. Therefore, many app licatio n s would
require the model to be expanded to the macro level 1n
terms of: (1) the re la tio n s h ip between the age s tru c tu re
and the p ro b a b ility d i s t r i b u t i o n (ra te s ) of moving and
(2) the macro reasons for moving. If a solution could be
found for Incorporating the age s t r u c t u r e , then the sum
of the population by age would give the to ta l population
(P) and f a c i l i t a t e the Incorporation of macro population
models ( e . g . , S to u ffer. 1962). Stated d i f f e r e n t l y , the
problem with the model 1s a t a c i t assumption about a con
stan t age s tru c tu re where the variable Is the age by
residence change d i s t r i b u t i o n .
A sim ila r d e f in itio n a l problem Is noted In the use
of "to ta l movers" ra te s as the p ro b a b ility d i s t r i b u ti o n
used 1n the model. This d i s t r i b u ti o n does not take into
account the areal differences In d e f in itio n between r e s i
dential m obility, m igration, and residence change. This
fa c to r was only approximated 1n r e la tin g moves from a
house to area of residence and 1n the re la tio n s h ip between
the model's estim ate and Shryock's and Wilber's estim ates.
This areal problem needs to be incorporated more r i g o r
ously.
96
The reasons persons have for moving or staying need
to be analyzed 1n g re ater d e ta il to determine 1f 1n fact
they are "Independent" of past h is to ry . I t may be that
the r e s u lts of the model are a r t i f a c t s of numerical
analysis ra th e r than of l i f e cycle fa c to rs . Here, only
a single age point was used to demark events 1n the l i f e
cycle step function. The model has to be modified and
Incorporate the v ariab le age d i s t r i b u ti o n of an event 1n
order to more closely approximate the empirical expecta
tion th a t behavior events c l u s t e r . Such a modification
would bring the model closer to the Cornell Model In terms
of a p ro b a b ility formulation.
In order to d i r e c tl y compare the S model and the
Cornell Model, d ir e c t estimates of actual data on r e s i
dence change need to be calcu lated using Stochastic
Process theory. Only then can It be determined 1f the
l i f e cycle or the concept of cumulative In e rtia Is a
b e tte r p ro b a b ility background from which to determine
the most Important types of reasons for moving to consider
In theory co nstruction .
Prior to comparison of estim ates, a b e tte r s t a t i s t i c
needs to be developed to measure the " f i t " between the
estimates and the actual data. This s t a t i s t i c would com
pare the agreement of Independent (or possibly dependent)
columns of events 1n a cross tab u latio n form. Robinson's
97
A's were very close to 1.0 end yet departures were noted.
A more sen sitiv e s t a t i s t i c Is needed.
In addition, higher mathematics are required than
were used to obtain b e tte r estim ates, especially with
regard to the p ast-retro sp ectiv e mover conditional
p r o b a b ilitie s . Also, only a simple step function was
used. The use of a continuous function 1n an elaborated
model which Includes variable ages by behavior event d i s
trib u tio n s around the step would be more appropriate.
Also, higher mathematics could te s t out the properties
of the model.
CHAPTER V II
SIGNIFICANCE OF THE M ODEL AND
GENERAL CONCLUSIONS
For all the numerous problems Indicated e a r l i e r ,
the consistency of the estimates to actual data and to
the variety of data suggests the model's potential for
further development. The most Important consideration
Is that these d iffe re n t tables of actual data should
have a single explanation (not an explanation for each
t a b le ) , since they represent the same underlying pheno
menon but tabulated d iffe re n tly .
The re su lts of the model suggest that the current
age situ atio n rather than past mover history 1s the
crucial factor 1n residence change. The model's resu lts
indicate that residence change p ro b a b ilitie s may be
treated as Independent of past history. Age and the l i f e
cycle stage events (here posited as employment or an "un
known event" occurring about age 35-64) are Important
f a c t o r s .
Unlike other analyses, this model Is not a basic
set of data for analysis, but uses other data to derive
the expected re s u lts . Sets of data were related by the
model rath er than the data being analyzed. Here a model
98
99
Is evaluated according to Its a b i l i t y to estimate actual
d ata, constrained by Its own rules of mathematical logic*
If th is model 1s adequate, then previous a n a l y t i c
a lly obtained data r e s u lts and findings could be re la te d
by the model. Life h isto ry and p a s t-r e tro s p e c tiv e mover
estim ates were connected to annual data where re g is tr y
data were not a v a ila b le . With these annual, p a st, and
l i f e h isto ry mathematical connectors, theory might be
f a c i l i t a t e d using the p a r a lle l verbal re s u lts of studies
as connectors. In addition to re la tin g analyses, d e riv a
tion of untabulated re s u lts (deductions 1n a limited
sense of the word) Is possible using the mathematical
framework.
The model 1s based on previous r e s u lts which In d i
cate th at age was a major fa c to r 1n moving, ra th e r than
on a few mathematical prop o sition s. If age 1s the major
f a c t o r , and the p r o b a b ilitie s of moving for an aggregate
are Independent of past l i f e h is to ry , then the age-context
or s itu a tio n 1s the major facto r ra th e r than a s o c la l-
psychology propensity such as learning 1n residence
change.
If such an Independent p ro b a b ility approach 1s
accepted as a th e o re tic a l base, the estim ates can be used
as baseline data for comparison to survey r e s u l t s , I . e . ,
an expected r e s u l t could be ca lc u la te d . In a d d itio n ,
fu rth e r elaboration of the model Is possible. For
100
example, h is to ric a l sh ifts In the longitudinal
p ro b a b ilitie s could be Included where decennial censuses
since 1890 are examined. Or the Income or educational
factor a lte rin g the l i f e cycle can be studied. Here the
Longitudinal Model's elaboration of l i f e table cohort
analysis suggests its potential as a research methodology.
REFERENCES
Bogue, Donald
1959 "Internal Migration." In The Study of Popula
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Duncan. Chicago: University of Chicago Press.
Eldrldge, Hope T.
1965 "Primary, Secondary, and Return Migration 1n
the United S tates, 1955-1960." Demography 2:
444-456.
GUck, Paul C.
1947 "The Family Cycle." American Sociological
Review 12 (April): 16T-17 4 .
1955 "The Life of the Family." Marriage and Family
Living 17 (February): 3-9.
Goldstein, Sidney
1955 "Repeated Migration as a Factor In High Mobi
l i t y Rates." American Sociological Review 19:
536-541.
Hauser, Phi 111p M .
1964 "Labor Force." 1n Handbook of Modern Soc1-
ology . edited by Robert E. L. Farls. Chicago:
Aand McNally, 161-190.
Lee, Everett
1966 "A Theory of Migration." Demography 3(1):
47-57.
Long, Larry H.
1972 "The Influence of Number and Age of Children
on Residential Mobility." Demography 9
(August):371-382.
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Asset Metadata

Creator Yee, William Jank (author)

Core Title A Longitudinal Model Of Residence Change

Degree Doctor of Philosophy

Degree Program Sociology

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag OAI-PMH Harvest,sociology, demography

Language English

Contributor Digitized by ProQuest (provenance)

Advisor Acock, Alan C. (committee chair), Empey, Lamar T. (committee member), Friedman, Judith J. (committee member), Newman, J. Robert (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c18-833288

Unique identifier UC11355801

Identifier 7411720.pdf (filename),usctheses-c18-833288 (legacy record id)

Legacy Identifier 7411720.pdf

Dmrecord 833288

Document Type Dissertation

Rights Yee, William Jank

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the au...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus, Los Angeles, California 90089, USA