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A joint model for Poisson and normal data for analyzing tumor response in cancer studies

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A joint model for Poisson and normal data for analyzing tumor response in cancer studies

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Content A Joint Model for Poisson and Normal Data for Analyzing Tumor Response in Cancer Studies Copyright 2001 by Xiaoguang Steve Cai A Thesis Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree MASTER OF SCIENCE (APPLIED BIOMETRY AND EPIDEMIOLOGY) May 2001 Xiaoguang Steve Cai R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. UMI Number: 1 4 06439 ___ < g > UMI UMI Microform 1406439 Copyright 2001 by ProQuest Information and Learning Company. Ail rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. UNIVERSITY OF SOUTHERN CALIFORNIA The Graduate School University Park LOS ANGELES, CALIFORNIA 90089-1695 This thesis, written b y X'Ao Q , a G S ''T £ | / £ Under the direction o f h J L . T > Thesis C om m ittee, and approved b y a ll its members, has been presented to and accepted b y The Graduate School in partial fu lfillm en t o f requirem ents fo r th e degree o f ' ± £L 0 ~ v /1 Dean o f Graduate Studies Date May 11, 2001__________ THESIS COMMITTEE I R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Acknowledgement I am very grateful to Dr. Susan Groshen, my thesis committee chair, for her helpful guidance, providing this research topic and furnishing the tumor response data. I am also very grateful for my thesis committee members, Dr. Sather and Dr. Siegmund, for their academic advice and kindly support. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Contents 1. Acknowledgment ii 2. Contents iii 3. List of table iv 4. Abstract v 5. Introduction 1 6. Statistical model 7 7. Study 1: description, results, discussion 13 8. Study 2: description, results, discussion 14 9. Discussions and future research 15 7. References 17 8. Appendix 1: MLE 20 9. Appendix 2: LRT 23 10. Appendix 3: SAS program 24 iii R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. List of Tables 1. Tablet: Tumor response study 1. 2. Table2: Tumor response study 2. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Abstract The number of events such as tumor lesions, seizures, or other indices in diseases is a common measure in clinical trials and epidemiological studies. Usually the measure of severity for each event is available but not incorporated in the analysis. This paper proposes a methodology for jointly modeling the number of events and a measure of severity of the events. The parameter of the Poisson distribution for the counts and parameters of the normal distribution for the measure of severity are functionally linked by Bayes’ rule. The maximum likelihood method is used to estimate the parameters. Maximum likelihood ratio tests are used to test hypotheses. SAS programs are written for the model to test whether the number of events or severity of events is different for treatment groups. The results of two case studies are presented in this paper followed by analysis and discussion. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Introduction In cancer clinical trials and in many animal tumor models, disease burden is measured as the number of tumor lesions or sum of the volume or cross-sectional area of each of the measurable lesions or even the sum of the longest diameters of all measurable lesions. One of these is then selected as the primary endpoint and univariate models are used to analyze the results of the studies. (Bradford, 1962; Finney, 1978; Lee, 1981). Recently, statistical models have been proposed which will allow the analysis of the number of lesions and a measure of the size of the lesions jointly. (Albert, 1997). One simple and natural model considers the number of lesions to follow a Poisson distribution, with parameter X, and the size (total volume or area or length) to follow a normal distribution. The dependence between the number of lesions and the size exists only in the parameterization of the normal mean and variance. The Poisson distribution, which is named after a French mathematician, S. D. Poisson, provides a realistic model for many random phenomena, especially for some rare events. (Larson, 1973). Although not all rare phenomenon can be considered to be Poisson distributed, many can. If certain assumptions regarding the phenomenon under observation are satisfied, the Poisson model is the correct model. The three fundamental assumptions regarding Poisson distribution are: 1 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. (1) The probability that exactly one happening will occur in a specific length of time interval is approximately constant from interval to interval, as long as the length of the interval is fixed. (2) The probability of more than one happening in a small specific length of time interval is negligible when compared to the probability of just one happening in the same interval. (3) The numbers of happenings in non-overlapping time intervals are independent. If the above three assumptions are satisfied, then the number of occurrences in a period of time has a Poisson distribution with parameter X. X is the expected number of random events in that time period. The Poisson probability distribution for M, the number of events in an interval of fixed length is defined as e * * X m ) = ----------- — . m ! Much use is made of the Poisson distribution in biological sciences such as growth of bacteria and other organism or abnormal cell growth. (Davis, 1958). In bacteriology, the knowledge of the Poisson distribution permits one to measure the precision of the estimate for the density of live organisms in a suspension. In a similar fashion, number of cancer lesions, which are colonies of abnormal cells, will be investigated in this article by assuming a Poisson distribution. 2 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. The most important continuous probability distribution is the normal distribution, which is named after the German mathematician, C. F. Gauss, or, as it’s frequently called, the Gaussian distribution. Larson (Harold J. Larson, 1974) described the normal distribution as: “The distribution is basically symmetrical about the middle and exhibits a shape rather like a bell, with the pronounced peak in the middle and a gradual falling off of the frequency in the two tails.” The probability density, f { x ) , of a normally distributed random variable, X, is given by the expression Here p is the expectation or mean value of X and a 2 is the standard deviation of X (Note that 7t is the mathematical constant 3.14159....). The symmetry of the distribution about p may be inferred from the above mathematical expression, since the change in sign but not the magnitude of x-p leaves f( x ) unchanged. In our study, the severity of disease, which is the size of all the lesions, follows the normal distribution. We let the variable X, be the sum of the sizes of all (m) lesions; it will have parameters mean m p and variance m e 2. f x (*) = 3 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Based on Bayes’ rule, the joint distribution for M=m lesions with total size X=x is While the maximum likelihood method of estimating unknown parameters was employed by C. F. Gauss 170 years ago for isolated problems, it was described as a generally appropriate method, and extensively used, by Sir Ronald A. Fisher. It has proven to be a very powerful technique and is widely used today. Although for some problems both the method of moments and maximum likelihood lead to exactly the same estimator, and for others they do not; when the two methods do not agree the maximum likelihood estimators are generally to be preferred. (Mood, The maximum likelihood method was described by Larson as follows. Suppose that X is a random variable with distribution function f x (x) which is indexed by (i.e depends on) an unknown parameter 0. Fisher proposed that the 4 1974) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. likelihood function, Lx (x ), be defined as the joint density function of the sample if X is continuous, or as the joint probability function of the sample if X is discrete. In either case, it is evaluated at the observed sample values, x ,of variable X. Lx (x) then is a function of the (single or more than one) unknown parameter 0. Choosing the value of 0 that maximizes the Lx (x) yields the maximum likelihood estimate of 0 since Lx (x) has been evaluated at the observed sample. Values of 0 which maximize value Lx (x) will be a function of the observed sample values. For the discrete random variable X whose probability function px (x,0) depends on an unknown parameter 0, the likelihood function of the sample is defined to be Px (x1,9)px (X2,0)px (x3,0)px(x4,0)...px (xn,0) = (0). For the continuous random variable, X, whose probability function f x (x,0) depends on an unknown parameter 0. The likelihood function of the sample is defined to be f x O l . 9 ) f x ( x 2 • Q ) f x ( * 3 ’ ( * 4 ’ 0 ) - f x i X n ’0 ) = ( # )• R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. In most situations we are able to take the derivative of the log of the maximum likelihood function and equate it to zero to find the value of 0 which maximizes the likelihood function and thus becomes the maximum likelihood estimator (MLE). (Mood, 1974). It is possible, however, for the likelihood function to have more than one point at which the derivative is zero. It is also possible that sometimes the derivative is not zero. In these cases, other techniques must be relied upon to find the estimator. If Lx (x) is indexed by more than one parameter, the likelihood function is the function of more than one parameters. The generalized likelihood ratio test criterion is a method which can be employed to derive a critical region for testing Ho versus Ha. It generally leads to a good test in the sense of giving small probabilities of type II error for a given a. (Mood, 1974). The likelihood ratio test statistics is defined as UfQ) ^ A L(Q) Here to indicates that in the numerator, the MLE has been chosen to maximize the likelihood, but subject to the restriction that 0 e to where to represents values of 0 compatible with the null hypothesis; and Q indicates that the MLE of 0 has been 6 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. chosen subject to the restriction that 0e£2 where Q represents those values of 0 compatible with the alternative hypothesis. The closer A is to 1, the more the data are compatible with the null hypothesis; the smaller A is, the more the data are incompatible with the null hypothesis. Under certain regulatory conditions, which are frequently met, it can be shown that -21og of likelihood ratio (i.e. -21og(A)) is approximately a chi-square random variable with r degree of freedom if Ho is true, for large values of n. Here r is the difference in the dimensions of £2 and to. In summary, we assume that the number of lesions follows a Poisson distribution and the size of total lesions follows a normal distribution, and we will use the joint Poisson and normal distribution model to analyze our tumor response data. We will show how we can compare 2 groups of animals (or patients) with the model by using maximum likelihood estimator and likelihood ratio test. Statistical Model 1. The Model. As we discussed earlier, we assume that the our data for number of cancer lesions have the following distribution: 7 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. where Mj, the random variable representing the number of lesions in the ith patient or animal, follows a Poisson distribution. mj is the observed number of lesions for patient or animal i. X is the Poisson distribution parameter and represents the average number of tumor lesions in the animals or patients being studied. We also assume that our data for total size of the cancer lesions for the specific animal have the following distribution: Where X t is the random variable representing the sum of the size of m; lesions in the ith animal or patient. xt is the observed sum of sizes of the lesions for subject i. mifi is the mean of Xi. m p 2 is the variance of X j. The joint model is given by 1 e 8 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. P{Mi = m ,)a n d /(MX)(m. *,.) = In this model, the maximum likelihood estimators for parameters, k, p, and a, are: i= l See appendix 1 for details of how these were derived. 2. Comparison of two groups Suppose we have two independent samples (possibly a control group and a treatment group), both following the Poisson-Normal distribution under consideration. In this situation we have (Mu, Xu) 1=1, 2, 3, ..n with parameters A,i, H\ > and o f and (M 2 j, X 2 j) j= l, 2, 3, ..k with parameters k 2, n 2 and a \ . R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. If these samples represent different treatments, then we would be interested in the testing the null hypothesis that the parameters of the two distributions are equal against the alternative that not all the parameters are equal. That is we have H0: Ai = X2, //j = M2 ’ anc* a \ ~ a l vs Ha: Xi + X2, * fi 2, and cr2 # (global test). Details of calculation by maximum likelihood estimates and likelihood ratio test are given in Appendices 1 and 2, but the key points are summarized below. Under Ho hypothesis the maximum likelihood estimates for the three common parameters are: '2 j A = f i 2 = f i = 1 = 1 ix+z m 2j i=l J= 1 +E m 2 j = X2 = X — 1=1 j= 1 n + k and 10 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. <72 = [E (xii - mii fi )2 /mii + S (x2 j - m2 j )2 /m2j]/(n+k) Under alternative hypothesis the MLE’s of the six parameters are: n x * . (= i Y j n , i= i k X * i & = Jr — ' L m , 7=1 for p i and ?il the x ’s and m ’s should have subscripts li (i.e xh) and i should run from 1 to n. for p2 and X I the subscripts should be 2j and j should run from 1 to k. Please fix these up. The denominators for the lambda hats are switched. Em, A =■" n k and 11 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. If we reject Ho in favor of Ha hypothesis then there are other hypothesis that we like to test: He: Xi * X2, pi * P2 and cr2 = Hd: X\ * X2, pi = P2, and a 2 = He: X\ = X2, pi * p2, and a 2 = o \ Vs Ha: X\ * X2, pi * p2 and a 2 -£ a \ . MLE estimates of the parameters in these models are all straight forward except the estimate of o \ = a \ = o 2 in hypothesis Hg. In this case a 2 = [S (xh - m u/*! )2 /mii + 2 (x2 j - m2 j fi2 )2 /m2 j]/(n+k). 3. SAS program R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. SAS programs have been provided to perform the maximum likelihood estimators and likelihood ratio tests. Details of SAS program are in Appendix 3. Nude Mouse with Liver Metastasis Model The data were provided by Dr. Maria Gordon at U. S. C.. Tumors known to metastasize to the liver were injected into nude mice. One week after tumor cell injection, mice were treated with placebo or an experimental gene therapy. By day 14. Mice were sacrificed and liver lesion numbers and total size were measured. 1. Study 1 In study 1, tumor response data are listed in table 1. (See table list). Following results were obtained by using maximum likelihood estimators and the likelihood ratio test: Group Total animals Average number of lesion per animal (parameter /I ) Average mean size of lesion per subject per X lesion (-----) m Variance Null Hypothesis Ho Approxim ate X 2 (-21ogA) df=3 P value Treatment K=5 =10.5 (parameter p) f a =0.043 o f = 2.35 A = «2- 0.75184 P=0.6866 Do not Control N=2 4 = 3 8 .2 f a =0.079 0-2 =4.21 - X^ and reject Ho Under Ho N=7 X =15.5 p. = 0.065 <r2 =3.33 _ 2 _ _ 2 (7 i — <T2 13 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Therefore, the null hypothesis that average number of lesions and total size of lesions are not different between control and treatment group can not be rejected. (LR=0.75184 on 2 degree of freedom and p=0.69). 2. Study 2 In study 2, tumor response data are listed in table2. (See table list). Testing results: Group Total Average Average mean animals number of lesion per animal (paramete size of lesion per subject per lesion X r A) (— ) (paramet m erp) Treatm ent K=4 4=9.25 > ~l O o Contro 1 N=4 4=56.5 f i 2 = 0 .1 1 Under Ho N=8 A =33.4 f l = 0.075 Null Hypothesis Ho a t = 0.129 a 2 = 0.001 a 2 =0.065 A = «2- \ = A ^ and a : Approxi mate z 2 (-21ogA) df=3 59.6049 P value P=0.0 01 Rejec tHo Hypothesis test results suggest that at least one of the parameters is significantly different following the two treatments. (P=0.001). With the result of this overall, global test, we would like to know whether the differences are due to the differences in numbers of lesions, or due to differences in the sizes of the lesions. 14 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Results from further testing show us both number and size of tumor lesions for treatment group are significantly less and smaller than the control group. Testl -21ogA2 He: o f = a 2 Ha: o f * < X 2 Df=l x 2 =4.9675 P=0.0257 Do not reject Ho < T i — <r2 Test2 -21ogAl Hd: A = fi2 He: A Df=l =27.9877 P=0.0001 Reject Ho A = A Test3 -21ogA2 He: \ = \ He: \ ^ A Df=l Z2 =26.6474 P=0.0001 Reject Ho A - A R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Discussions and further study plan The model and the hypothesis testing we present can be conveniently applied to clinical trials and in vivo tumor response studies. The SAS program written for this model can produce other maximum likelihood estimators and hypothesis tests. The sample size, on which our analysis based, is small. Although the examples we provided above gave us the results we expected, in general this model requires relatively good sample sizes since the likelihood ratio is approximately chi-square distribution when sample size is big enough. With large clinical trials and animal tumor response studies, this the large sample approximation should be fairly good . For our study, we assumed that the variables for the number of lesions and for the size of the lesions are independent. However, in the situation where there is correlation among variables, we would approach the problem using the multivariate normal distribution. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Bibliography Albert, P., Follmann, D. and Barnhart, H. (1997). A Generalized Estimating Equation Approach for Modeling Random Length Binary Vector Data. Biometrics 53,1116-1124. Bradford, H. A. (1962). Statistical Methods in clinical and Preventive Medicine. Livingstone, Edinburgh. Cox, D. R. and Hinkley, J. J. (1985). Theoretical Statistics. Chapman and Hall, London Davies, O. L. (1958). The designs of screening tests in the pharmaceutical industry. Bulletin of the International Statistical Institute 36, 226-241. Finney, D J. (1978). Statistical Method in Biological Assay, 3rd edition. The University Press, Cambridge. Larson, H. J. (1973) Introduction to the Theory of Statistics. (1973). John Wiley, New York. Lee, Y. J. and Wesley, R. A. (1981). Statistical contributions to Phase II trials in cancer: Interpretation, Analysis and Design. Siminar in Oncology 8. 403-417. Mood, A. M., Graybill, F. A. and Boes, D. C. (1974). Introduction to the theory of statistics. John Wiley, New York. Pearson E. S. and Hartley H. O. (1966) Biometrika Tables for Statistician, Vol. 1, 3rd edition. Cambridge Univesity Press, Cambridge. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Table 1 Mouse tumor study 1 results. Mouse ID s.a. Tumor #Foci Mean of Foci PBS Control 39012a 0.0406 6 0.0068 39012b 0.5317 2 0.2659 39012c 0.0227 5 0.0045 39014a 0.0286 1 0.0286 39014b 1.0507 4 0.2627 39014c 0.024 3 0.0080 Gene Therapy 39005a 0.3709 3 0.1236 39005b 0.0732 13 0.0056 39005c 4.7225 22 0.2147 39007a 1.1878 46 0.0258 39007b 0.2236 7 0.0319 39007c 1.4349 40 0.0359 39008a 0.2675 33 0.0081 39008b 0.0031 1 0.0031 39008c 0.0702 6 0.0117 39010a 0.0042 2 0.0021 39010b 0.002 2 0.0010 39010c 0.0016 5 0.0003 39013a 0.0003 1 0.0003 39013b 0.0211 9 0.0023 39013c 0.0008 1 0.0008 The above table lists liver tumor sizes and numbers of a control group and a retroviral vector treated group, a, b, and c are different liver sections, right Caudate lobes, left lobe and median lobe respectively for one mouse. Sum of total size (a+b+c) for one mouse is defined as total size of tumor and is the number used in the analyses described. The number of lesion is the sum of lesion from the a, b and c sections for each mouse. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Table 2 Mouse tumor study 2 results. Mouse ID s.a. Tumor # Foci %Tum/Liv Mean of Foci GeneTherapy n=4 26a 0.023 1 0.028 0.023 26b 0.150 4 0.065 0.038 26c 0.015 4 0.009 0.004 27a 0.001 1 0.001 0.001 27b 0.001 1 0.006 0.001 27c 0.008 3 0.005 0.003 28a 0.018 3 0.038 0.006 28b 0.139 9 0.071 0.015 28c 0.047 4 0.025 0.012 29a 0.015 2 0.038 0.008 29b 0.016 1 0.020 0.016 29c 0.018 4 0.012 0.005 PBS Control n=4 30a 1.107 9 2.218 0.123 30b 0.723 5 0.450 0.145 30c 0.723 7 0.653 0.103 31a 0.036 1 0.070 0.036 31b 1.059 5 0.648 0.212 31c 2.401 5 2.044 0.480 32a 0.232 1 0.270 0.232 32b 0.079 5 0.043 0.016 32c 0.643 6 0.435 0.107 33a 4.263 49 10.350 0.087 33b 8.028 79 8.057 0.102 33c 6.366 54 26.005 0.118 The entire notation in Table 2 is essentially the same as table 1. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Appendix 1 Maximum Likelihood Estimates of Parameters Simple Model-Single sample: = f r e-'' ♦A ,-' , 1 1 1 m;! i=l ^2 m n i cr, The logarithm of the likelihood function is: n n i----- L* = - n ;lj + i m,*log Aj - X log( /n,!)- n log( ^Jl7t m .a i = 1 i = 1 £ - y -*(JC « -|” «-^i)2 <=i 2 m ia { To derive the MLE’s, the derivative is taken of the log of the likelihood, and set to zero. i.e. ... and now give the formula for dL/dp... 20 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Same reasoning for dL * n 1 = —n + i m ,— =0 d A { i=1 Ay n R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix 2 Likelihood Ratio Test Likelihood ratio A = Slip L(A, ju,a m^rrij, xf ,Xj)l s u p L(AX ,X2,px,p2,ai, < J 2mi , ms , , x , ) - 2 log A is approximately X 1 distribution . R estricted . ■ 2 log A = - 2 log(— ) unrestricted , Unrestricted. : 2 log(—------ — -) Re stncted = 2 log s u p ,A2,px,p2,<Jx, (7 2mt, M j , x u , X j ) l s u p U.A, p, a , m,., m j , x x , x}) n n ft £ 1 ~"VSd7( :ti''m '*A )2 » I I " ' 1 — e /1 ~ 1 l- I 1 I» 2 I fa " 1 * y____ * __________ 1 fa 1 (mi\)n (-v[27tmia l)n (mj\)k (42nmj&2)k ■Hog y m , " i - i i e A * 1 ^ 2m ,d i * g A * 1 „ 2mi & 2 (m;!)" {42nmiax)n (mj\)k {42amj&2)k MLE’ s as described previously can now be substituted. Reject Ho if and only if - 21og A > Zi-a(3 ) ■ 23 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Appendix 3 SAS Program j ********************************* Program f o r t e s t i n g c an ce r le s io n number, le s io n s iz e and v a ria n c e . By S te ve C a i, May 2 3 , 2000. Last re v is e d 0 3 /2 0 /0 1 . i i ' k 'k 'k i f k i f k - k 'k i t ' k i c k ' k ' k 'k - k ' k i t 'k ' k i i ' k ' k i t 'k - k 'k 'k i e l /* i n p u t d ata f o r c o n t r o l g ro u p */ d a ta a; in p u t y $ t s; cards; a 1 .107 9 b 0 .7 2 3 5 c 0 .7 2 3 7 a 0 .0 3 6 1 b 1 .059 5 c 2.401 5 a 0 .2 3 2 1 b 0 .0 7 9 5 c 0 .6 4 3 6 a 4 .2 6 3 49 b 8 .0 2 8 79 c 6 .3 6 6 54 run; d a ta a 1 ; s e t a; i f y = ' a ' then group1+1; run; proc s o r t; by g r o u p l; /* d a t a m a n ip u la tio n to g e t s iz e o f le s io n and number o f le s io n f o r each c o n t r o l * / R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. d ata a 2 ;m e rg e r= 1 ; s e t a 1 ; by g r o u p l; i f f i r s t . g r o u p l then do; c o u n t t l= 0 ; c o u n ts 1 = 0 ;end ; c o u n t t l + t ; c o u n ts l+ s ; i f la s t .g r o u p l then o u tp u t; run; /* p r o c p r i n t ; r u n ; * / /*a d d t o t a l le s io n s iz e and d evide by t o t a l le s io n number to g e t average u, t h i s i s u1 f o r c o n t r o l * / d ata a 3 ;m e rg e r= 1 ; s e t a2 end=z; t t 1 + c o u n t t 1 ; ts 1 + c o u n ts 1 ; u 1 = tt 1 / t s l ; i f z; *p ro c p r i n t ; run; /*m erge d ata a3 and d a ta a2 to c a c u la te lam dal and s ig m a l* / d a ta a4; merge a2 a 3 ;b y m erger; Ia m d a 1 = ts 1 /g ro u p l; p r e s ig 1 = ( (c o u n t t l-u 1 * c o u n t s 1 ) * * 2 ) / c o u n t s l ; s ig m a l+ p re s ig 1 ; r u n ;/* p r o c p r i n t ; r u n ; * / /♦ o u tp u t o n ly s ig m a l, v a ria n c e f o r c o n t r o l g ro u p */ d a ta s s s 1 ; s e t a4 e n d = z;m e rg er= 1 ; i f z ;o u tp u t; *p ro c p r i n t ; r u n ; d ata a5; s e t a4 end=z; i f z ; o u t p u t; * p r o c p r i n t ; r u n ; proc s o r t d a ta = a 5 ;b y m e rg er;ru n ; /* * * * * * s a m e p rocess f o r d a ta o f t r e a t m e n t * * * * * * * * / d ata b; in p u t y $ t s; R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. card s ; a 0 .0 2 3 1 b 0 .1 5 0 4 c 0 .0 1 5 4 a 0.001 1 b 0.001 1 c 0 .0 0 8 3 a 0 .0 1 8 3 b 0 .1 3 9 9 c 0 .0 4 7 4 a 0 .0 1 5 2 b 0 .0 1 6 1 c 0 .0 1 8 4 run; d a ta b l ; s e t b; i f y = 'a ' then gro u p 2+1; run; proc s o r t; by group2; 3 d a ta b 2;m e rg er= 1 ; s e t b 1 ; by group2; i f f i r s t . g r o u p 2 then do; /* c o u n t+ 1 ? * / c o u n tt2 = 0 ;c o u n ts 2 = 0 ;e n d ; c o u n tt2 + t;c o u n ts 2 + s ; i f la s t.g r o u p 2 then o u tp u t; run; /* p r o c p r i n t ; r u n ; * / d a ta b 3;m e rg er= 1 ; s e t b2 end=z; tt2 + c o u n tt2 ; ts2 + c o u n ts2 ; u 2 = t t 2 / t s 2 ; i f z; *p ro c p r i n t ; run; d a ta b4; merge b2 (drop=group2) b3;by m erger; lam d a2= ts2/g ro u p 2; p r e s ig 2 = ((c o u n tt2 -u 2 * c o u n ts 2 )* * 2 )/c o u n ts 2 ; sig m a2+p resig 2; R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. run;/*proc print;run;*/ /* o u tp u t o n ly sigma2, v a ria n c e f o r tre a tm e n t g ro u p */ d ata sss2; s e t b4 e n d = z;m erg er= 1; i f z ;o u tp u t; *p ro c p r in t ; r u n ; d ata b5; s e t b4 end=z; i f z ;o u tp u t;* p r o c p r in t ; r u n ; proc s o r t d a ta = b 5;b y m e rg er;ru n ; / * * * * * * * * * * * * c a l c u l a t e uu and H am d a, which is common u and common lamda f o r r e s t r i c t e d l i k e l y h o o d * * * / d ata ab; merge a5 b5 sss1 sss2; by m erger; Ila m d a = (t s 1 + t s 2 ) / (g ro u p 1 + g ro u p 2 ); s ig m a l1 = s ig m a 1 /g ro u p l; sig m a22=sigm a2/group2; sigm a_s=( sig m al+sigm a2) / (g r o u p l+ g r o u p 2 ); u u = (s ig m a 1 * tt2 + s ig m a 2 *tt1 ) / (s ig m a l* ts 2 + s ig m a 2 * ts 1 ); proc p r in t ; r u n ; / * * * * * * * * * T h e fo llo w in g i s t e s t o f h y p o th es is: Ho:Lamda1<=lamda2 and u1<=u2 Ha:Lamda2<lamda1 o r u2<u1 The c a lc u la t io n have been tr u n c a te d in t o th r e e p a r ts , 1. e**lam d a p o r tio n (lo g LL L ) 2 . la m d a * * M i/j p o r tio n (lo g rL L L ) 3 . p a r t o f norm al d i s t r i b u t i o n p o r tio n (logKKK) C o n tro l and tre a tm e n t group a re c a lc u la te d s e p a r a te ly P lease see a tta c e d fo rm u la o f lik e ly h o o d r a t i o * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * j / * * * * * * * * * * * * * * * * j d i S s te p is t o c a lc u la t e * * * * 3 . p a r t o f normal d i s t r i b u t i o n p o r t i o n * * * * * / d a ta ab_a4; merge ab a4 SSS1 end=z; by m erger; k 1 = ( ( c o u n t t l- c o u n t s l* u u ) * * 2 ) / (2 *c o u n ts 1 *s ig m a 1 1 ); R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. k k 1 + k 1 ; uk1 = ( ( c o u n t t l - c o u n t s l * u 1 ) * * 2 ) / (2 *c o u n ts 1 *s ig m a 1 1 ); ukk1+uk1; i f z ;r u n ;p r o c p r in t ; r u n ; d a ta ab_b4; merge ab b4 SSS2 end=z; by m erger; k 2 = ( ( c o u n tt 2 - c o u n t s 2 * u u )* * 2 ) / (2 *c o u n ts 2 *s ig m a 2 2 ); kk2+k2; u k 2 = ( (c o u n tt 2 - c o u n t s 2 * u 2 ) * * 2 ) / (2 *c o u n ts 2 *s ig m a 2 2 ); ukk2+uk2; i f z ;r u n ;p r o c p r in t ; r u n ; d a ta ab_ab; merge ab_a4 ab_b4;by merger; lo g r L L L = -t t 1 * lo g ( lla m d a /la m d a l) - t t 2 * lo g ( l la m d a /la m d a 2 ) ; lo g kk k = kk 1+ kk 2-u kk 1-u kk 2 ; lo g S IG M A = lo g (s ig m a _ s )-0 .5 *lo g (s ig m a 1 1 *s ig m a 2 2 ); lo g L L L = lla m d a *(g ro u p l+ g ro u p 2 )-g ro u p l*la m d a 1 -g ro u p 2 *la m d a 2 ; ru n ;p ro c p r i n t ; r u n ; / * * * * * * * * * c o n b i n e a l l th r e e p o rtio n s to g e th e r and t e s t h yp o th es is 3 t t h i s s t e p * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * / d a ta x; s e t ab_ab; x x = 2 * ( lo g r lll+ lo g k k k + lo g S IG M A + lo g lll) ; p r o b = 1 -p r o b c h i( x x ,3 ) ; i f p ro b < 0.05 then p = 'u2 a n d /o r lamda2 a n d /o r sigma2 i s / a r e s i g n i f i c a n t l y d i f f . . . 1; e ls e p = 'u 2 , lamda2 and sigma2 a re not s i g n i f i c a n t l y d i f f e r ru n ;p ro c p r i n t ; r u n ; / * * * * * * * * t e s t h yp o th es is i f u1>=u2 vs Ha u 2 > u 1 * * * * * * * * * * * A d a ta xu; s e t ab_ab; xuu = 2*lo g kkk; p r o b = 1 -p ro b c h i(x u u ,1 ); i f p ro b < 0.05 then p = 'u 2 is s i g n i f . d i f f e r e n t l y from u1'; e ls e p='u2 i s not s i g n i f . d i f f . from u1'; ru n ;p ro c p r i n t ; r u n ; R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. / * * * * * * * * t e s t h yp o th es is i f Iamda1>=lamda2 vs Ha Iamda2>lamda1* * * * * * * * * * * * * * * * * * * / d a ta x l; s e t ab_ab; x l = 2 * ( l o g r l l l + l o g l l l ) ; p r o b = 1 -p r o b c h i( x l,1 ); i f p ro b < 0.05 then p ='lam d a2 i s s i g n i f i c a n t l y d i f f e r e n t l y from la m d a l1; e ls e p='lam da2 is not s i g n i f i c a n t l y d i f f e r e n t l y from la m d a l'; ru n ;p ro c p r in t ; r u n ; / * * * * * * * * t e s t h yp o th es is i f sigma1>=sigma2 vs Ha sigma2>sigma1* * * * * * * * * * * * * * * * * * * / d ata x l; s e t ab_ab; x l = 2 * ( logSIG M A ); p r o b = 1 -p r o b c h i( x l,1 ); i f p ro b < 0.05 then p ='sig m a2 i s s i g n i f i c a n t l y d i f f e r e n t l y from s ig m a l'; e ls e p='Sigma2 is not s i g n i f i c a n t l y d i f f e r e n t l y from s ig m a l'; ru n ;p ro c p r in t ; r u n ; R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.

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Asset Metadata

Creator Cai, Xiaoguang Steve (author)

Core Title A joint model for Poisson and normal data for analyzing tumor response in cancer studies

School Graduate School

Degree Master of Science

Degree Program Applied Biometry

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

Tag biology, biostatistics,health sciences, oncology,OAI-PMH Harvest

Language English

Contributor Digitized by ProQuest (provenance)

Advisor Groshen, Susan (committee chair), Sather, Harland (committee member), Siegmund, Kimberly (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c16-36838

Unique identifier UC11336841

Identifier 1406439.pdf (filename),usctheses-c16-36838 (legacy record id)

Legacy Identifier 1406439.pdf

Dmrecord 36838

Document Type Thesis

Rights Cai, Xiaoguang Steve

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