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A simulation study between 3+3, Rolling-6 and i3+3 design

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A simulation study between 3+3, Rolling-6 and i3+3 design

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A SIMULATION STUDY BETWEEN 3+3, ROLLING-6 AND
I3+3 DESIGN

by

Yuanliang Yang

A Thesis Presented to the
FACULTY OF THE USC KECK SCHOOL OF MEDICINE
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
MASTER OF SCIENCE
BIOSTATISTICS

August 2020

Copyright 2020 Yuanliang Yang

ii
Table of Contents
List of Tables ..................................................................................................................................................... iii
List of Figures .................................................................................................................................................... iv
Abstract .............................................................................................................................................................. v
1. Introduction .................................................................................................................................................... 1
2. Methods ........................................................................................................................................................... 3
2.1 Algorithm for designs ......................................................................................................................................... 3
2.1.1 3+3 design .................................................................................................................................................. 3
2.1.2 Rolling-6 design ......................................................................................................................................... 4
2.1.3 i3+3 design ................................................................................................................................................. 6
2.2 Evaluation metrics ............................................................................................................................................. 8
2.2.1 Reliability and safety .................................................................................................................................. 8
2.2.2 The duration of the trials ............................................................................................................................ 9
3. Simulations ...................................................................................................................................................... 9
3.1 Validating the algorithm .................................................................................................................................. 10
3.2 Setting up the parameters ................................................................................................................................. 12
3.3 Simulation result .............................................................................................................................................. 13
4. Discussion ...................................................................................................................................................... 19
REFERENCES ................................................................................................................................................. 22
Appendix ........................................................................................................................................................... 23

iii
List of Tables

Table 1 - Evaluation metrics ........................................................................................................... 8
Table 2 - Validation the algorithm ................................................................................................ 10
Table 3 - DLT probability scenarios ............................................................................................. 13
Table 4 - Result from probability scenarios #1 ............................................................................. 15
Table 5 - Result from probability scenarios #2 ............................................................................. 16
Table 6 - Result from probability scenarios #3 ............................................................................. 18

iv
List of Figures
Figure 1 - 3+3 design flowchart ...................................................................................................... 4
Figure 2 - decision properties of the 3+3 and Rolling-six designs ................................................. 5
Figure 3 - Pseudo code based on i3+3 rules ................................................................................... 7
Figure 4 - True toxicity probability of dose levels for each scenario ........................................... 13

v
Abstract
The Phase I clinical trial is the first step of new drug testing and aimed to estimate the true
Maximum tolerable dose (MTD) for human. In recent years, many new methods have been
proposed, but the classical 3+3 rule-based design is still most commonly applied since it is the
simplest to implement and requires no calculation during the trial (Hansen et al., 2014). This article
only focuses on the comparison among three rule-based designs, namely, the 3+3 design, the
Rolling-6 design and the i3+3 design. The reliability (i.e., the percentage of correct choices) and
safety (i.e., the number of patients receiving the treatment above the true MTD) are two main
criteria to evaluate a trial design, and the duration of the entire experiment is another consideration.
R code is used to implement the algorithms of these three designs, and the simulated data are
processed with these algorithms. The evaluation and comparison are made based on the results of
simulated data.

1
1. Introduction
The goal of clinical trials is to determine whether a treatment, prevention, or a behavior
approach is safe and effective. According to the research stage, a clinical trial can be divided into
phase Ⅰ-IV. Phase I clinical trials evaluate preliminary clinical pharmacology and human safety.
In recent decades, the human tolerance and pharmacokinetics of all new drugs should be initially
tested by phase I clinical trials before they are allowed to the market. ("NIH Clinical Research
Trials and You", 2020) The results achieved by the phase I trials are called MTD, meaning the
highest dose of a drug or treatment that does not cause any unacceptable side effects.
Different from normal drug trials, anti-tumor drug trials enroll cancer patients with no
effective treatment available as subjects, instead of healthy people. The ideal dose interval for each
individual should be lower than the MTD, and higher than the effective dose. In this case, if the
treatment is tested to be effective, more people can benefit from it.
The definition of toxicity may vary from drug to drug and can be long-term or short-term.
The dose-limiting toxicity (DLT) should be specified before the trial. Common Terminology
Criteria for Adverse Events (CTCAE) published by the National Cancer Institute provides
informative descriptions of AEs (adverse events).
Rule-based Designs: The dose-toxicity curve is not pre-estimated in rule-based designs.
Dose escalation or de-escalation is determined based on toxicity results at current dose levels. The
most significant advantages of rule-based designs are simple calculations and easy implementation.
The 3+3 design is the most popular phase I design (Hansen et al., 2014), which is simple, allowing
clinical researchers to explore MTD without complicated calculations. (Storer, 1989) The Rolling-

2
6 design was created by Skolnik et al. in 2008, aimed at shortening the duration of phase I clinical
trials while maintaining the same reliability and safety as the traditional 3+3 design. It was claimed
by Skolnik that their design could save about 60 days when compared with the traditional 3+3
design. (Skolnik et al., 2008) The i3+3 design was created by Meizi Liu et al. in 2019. They
asserted that their design not only had the accuracy as high as model-based designs (CRM,
titeCRM etc) but also maintained the simplicity of rule-based designs. The i3+3 design is as simple
as the 3+3 design but has more advanced rules. The decision table can be generated before the start
of the trial. Different from the other two rule-based designs, the i3+3 design has a flexible target
toxicity probability and the sample size can be enlarged to improve accuracy (Ji & Liu, 2019).
Model-based Design: The model-based design will establish a dose-toxicity curve model
before the subject is enrolled. During the test, use the toxicological data to modify the dose-toxicity
curve in real-time, which requires excellent biostatistical support to establish and modify the dose-
toxicity curve. Continual reassessment method (CRM) design is the first clinical trial design
method using the Bayesian theory. The toxic response information of the observed subjects was
used to gradually update the prior density parameter utilizing Bayesian theory during the test.
However, if the estimation is incorrect, the final result will be unreliable. Also, complex
calculations are required during the trial. (O’Quigley, 1990)
Model-assisted Design: In 2010, Ji Y et al. proposed the modified toxicity probability
interval (mTPI) design, which is an interval design, without the need to assume toxicity-dose
curves before the trial. (Ji Y et al., 2010) The Bayesian Optimum Interval (BOIN) design was
introduced by Liu and Yuan in 2014. BOIN is a relatively new class of phase I trial designs. The
observed toxicity rate at the current dose level is compared to a prespecified toxicity tolerance
interval, which is determined by the Bayesian approach. Compared to model-based design, the

3
BOIN design exhibits average performance when choosing MTD, but easier to implement and
greatly reduces the risk of assigning subtherapeutic or over-toxic doses to patients. (Liu and Yuan,
2014) (Fellman& Yuan, 2015)

2. Methods
2.1 Algorithm for designs
The algorithms of 3+3, i3+3, and Rolling-6 designs are written in R, and the code is shown
in Appendix-B. Three algorithms were applied to each set of simulated data simultaneously. On
account of the difference of three designs, the number of patients enrolled in each trial and the
dose level to each individual may vary.

2.1.1 3+3 design
A group of 3 patients is first tested at the lowest dose level, and their toxic reactions are
observed. If no DLT occurs in any of the three patients, a higher dose is given to the next group of
3 patients. If DLT occurs in 1 out of 3 patients, an additional group of patients would be tested at
this dose level. If no DLT occurs in any one of the second group of 3 patients, the next group of
patients would be tested at a higher dose. If more than one DLT occurs at a dose level, that dose
exceeds MTD. Those steps are repeated until the MTD is identified. If the highest level of the
tested dose exceeds the MTD and six patients have been tested at the lower dose, the dose before
the highest dose is defined as MTD; otherwise, more people would recur in the test at a lower dose
level. The rule of the 3+3 design is shown below.

4
Figure 1- 3+3 design flowchart

(from Luu)

2.1.2 Rolling-6 design
In the Rolling-6 design, 2 to 6 patients may be enrolled in the same cohort at a time. The
trial would be suspended only when waiting for the results from the six patients. Upon the arrival
of a new patient, the statistician needs to decide whether to engage this patient in and which dose
should be used. Dose level assignment depends on the number of patients with no DLT, the number
of DLTs, and the number of patients pending for results.
The comparison of decision properties between the 3+3 and Rolling-six designs is shown
below.

5

Figure 2- decision properties of the 3+3 and Rolling-six designs

(from Skolnik et al)

The number of patients tested in each dose level is flexible because the dose level is decided
according to both the number of DLTs and the number of pending patients. Therefore, the arrival
time of patients is significant for decision.
The algorithm takes the time when the DLT occurs as the benchmark time (arrival
time+length of treatment 21 days), and the decisions will be made by the time when the next
qualified patient comes. The result of this patient will be used to determine whether MTD is
reached or how many patients are needed at a certain dose level. It is noted that I assume that the
enrollments are before the report of DLT if they happen on the same day in this study.

6
Firstly, three arrays are created to record the numbers of patients with DLT (x[doselvl]),
patients without DLT (h[doselvl]), and patients enrolled (n[doselvl]) for each of 1-6 dose levels.
Patients enrolled are those arriving ahead of this time point and having been arranged with a dose
level. When new results are available (arrival time+length of treatment 21 days), n[doselvl],
h[doselvl], and x[doselvl] will be updated, and Rolling-6 rules can be applied. The dose level for
the next six patients will be decided.
The trial will end in four situations: 1) MTD is found; 2) More than 1 DLT is observed in
the 1st dose level; 3) the final DLT ratio is lower than the target interval in the highest dose level;
4) the total number of patients reach the upper limit preset before the test. In the last three cases,
the output of this cycle will be marked as -1, 99 and 9, respectively.

2.1.3 i3+3 design
According to the design principle of the i3+3, the total sample size is preset, and the cohort
size, target DLT rate, equivalence interval is adjustable. Three arrays are created to record the
number of patients with DLT(X[doselvl]), patients without DLT(H[doselvl]) and patients enrolled
(N[doselvl]) in dose level 1-6. A cohort of patients will be recruited into the trial each time, the
numbers of patients with DLT, patients without DLT and patients enrolled at the current dose level
will be updated. The scheduled dose for the next cohort of patients is determined by the two
quantities, X / N and (X-1) / N. The rule of 3+3 design is shown below.

7
Figure 3-Pseudo code based on i3+3 rules

(from Ji & Liu)

Pseudo code of two safety protection mechanisms in i3+3 design
(1) If the P(P[doselvl 1]>Ptarget | X[1], N[1])>0.95, terminate the trial due to excessive toxicity.
(2) If the P(P[dose]>Ptarget | X[dose], N[dose])>0.95, mark dose n as “DU”, and dose n and higher
doses will be removed from the trial.
The design assumes that the toxicity probability follows the beta distribution (P ∼
beta(X[dose] + 0.005, H[dose]+ 0.005)). A trial will terminate if the prespecified sample size is
reached or safety rule (1) is broken.
After the end of a trial, the posterior mean toxicity probability at each dose is given by
P=(X[dose]+0.005)/(N[dose]+0.01). The relationship between dose and toxicity should be
monotonic, and isotonic regression will be used to the posterior means probability using the pool
adjacent violators algorithm (PAVA function, in Iso package). The estimated value that violates
the monotonicity assumption will be replaced by a weighted average of its neighbors, where the
weight is the posterior variance of each dose level. The estimated MTD is the dose that minimizes
the difference between the target rate of DLT and the isotonic-transformed posterior means. The

8
MTD dose must have been tested and has an isotonic-transformed probability under the upper
boundary of the equivalence interval.

2.2 Evaluation metrics
The reliability (i.e., the probability of choosing the correct MTD) and safety (i.e., the
number of patients receiving a treatment above the true MTD and the number of DLTs) are two
main criteria for the evaluation of a trial design. The duration of the entire experiment is another
factor to be considered. The evaluation parameters are shown in the table 1.
Table 1-Evaluation metrics

2.2.1 Reliability and safety
The reliability of a design is represented by the percentage of correct choices. There is only
one correct dose in each probability scenario of six doses.
The safety of design is reflected by the number of patients receiving the treatment above
the true MTD and the number of patients experiencing the dose limiting toxicity (LTD).

9
2.2.2 The duration of the trials
In the Rolling-6 design, the total duration can be represented by the arrival time of last
patients adding 21 days. If we acquire MTD before we know the result of the last patient, the time
point we get the MTD is used as the duration. For the i3+3 design and 3+3 design, the duration of
the trial is much easier to get, because the cohort size is fixed. The total duration of the Rolling-6
design equals the arrival time of the last patient plus 21 days. The time point of getting the MTD
is taken as the duration if it precedes the achieving of the last patient’s result. The duration of i3+3
and 3+3 is the arrival time of the last patients add the product of the number of cohorts and
treatment cycle length.

3. Simulations
In 2014, Hansen et al. reviewed more than 1,200 trials from 1991 to 2006 and concluded
that 98% of Phase I clinical trials were conducted in a 3 + 3 design. Although the 3+3 design often
does not accurately estimate the MTD and exposes many patients to treatment below the effective
dose, it is still the most popular design for its simplicity and safety. This research centered on the
simple rule-based design, and the reliability and safety of the 3+3, Rolling-6, and i3+3 designs
were compared.
The evaluation and comparison are performed based on the results of simulated data. At
first, 100 patients’ data are generated in each iteration. Whether there is DLT or not in each patient
is generated from Bernoulli distributions with the probabilities of true toxicity risks. 1 and 0
represented the existence and absence of DLT, respectively. The arriving time is also produced in

10
this step with the Poisson distribution. After that, the data are analyzed under the rules of different
designs. One thousand simulations are performed for each set of parameters.

3.1 Validating the algorithm
The R code was validated with published articles or online simulators. The same
parameters and the probability scenario were used for comparison. The i3+3 and 3+3 algorithms
were validated with the result from the article (The i3+3 Design for Phase I Clinical Trials, Meizi
Liu, https://i3design.shinyapps.io/i3plus3/). The Rolling-6 algorithm was verified by the
simulation result of the U-design website (https://udesign.laiyaconsulting.com/my-home).
The validating results are shown in Table 2.1-2.6. For the four evaluation parameters of
percentage of correct selection, the number of patients, the number of patients with DLT and
treated with a dose higher than the true MTD, similar results confirmed the correctness of the code.
For the duration of the trial, articles using similar models were not found. However, this part of
the calculation was simple, so the validation was omitted.
Table 2.1 – Validation of 3+3design -1
Probability of selection
Dose level
(true P)
1(0.01) 2(0.05) 3(0.10) 4(0.60) 5(0.70) 6(0.90) # DLT No. of
Patients(sd)
#exceed
MTD
Author’s
result i=1000
0.025 0.126 0.813 0.032 0.001 0 3.12 16.34 3.74
My result
i=10000
0.026 0.111 0.826 0.033 0.001 0 3.12 16.35
(2.5)
3.76
Bold: Percentage of correct selection
i=Number of iterations
#DLT: The average number of patients with DLT
#exceed MTD: The average number of patients treated with a dose higher than the true MTD

11
Table 2.2 – Validation of 3+3design -2
Probability of selection
Dose level
(true P)
1(0.01) 2(0.17) 3(0.37) 4(0.57) 5(0.77) 6(0.92) # DLT No. of
Patients
#exceed
MTD
Author’s
result i=1000
0.255 0.535 0.186 0.013 0 0 4.52 14.29 5.12
My result
i=10000
0.247 0.528 0.207 0.015 0 0 4.47 14.28
(3.12)
5.30

Table 2.3 – Validation of Rolling-6 design -1(density=36 patients/year)
Probability of selection
Dose level 1(0.07) 2(0.14) 3(0.28) 4(0.37) 5(0.5) 6(0.7) # DLT No. of
Patients
Author’s
result i=300
0.18 0.40 0.26 0.08 0.01 0 3.18 17.1
My result1
i=10000
0.187 0.394 0.253 0.092 0.012 0 3.33 17.7
(6.0)

Table 2.4 – Validation of Rolling-6 design -2(density=36 patients/year)
Probability of selection
Dose level 1(0.01) 2(0.07) 3(0.14) 4(0.28) 5(0.7) 6(0.9) # DLT No. of
Patients
Author’s
result i=300
0.06 0.19 0.43 0.31 0 0 3.37 21.4
My result
i=10000
0.058 0.183 0.400 0.353 0.004 0 3.46 22.3
(5.0)

Table 2.5 – Validation of i3+3 design-1 (Target rate= 0.25(0.17-0.33) cohort size=3 sample size=30)
Probability of selection
Dose level 1(0.07) 2(0.14) 3(0.28) 4(0.37) 5(0.5) 6(0.7) # DLT #exceed MTD
Author’s
result i=10000
0.053 0.438 0.360 0.132 0.016 0 6.69 5.72
My result
i=10000
0.050 0.447 0.366 0.122 0.015 0 6.72 5.62

Table 2.6 – Validation of i3+3 design-2 (Target rate= 0.25(0.17-0.33) cohort size=3 sample size=18)
Probability of selection
Dose level 1(0.01) 2(0.07) 3(0.14) 4(0.28) 5(0.7) 6(0.9) #DLT #exceed MTD
Author’s
result i=10000
0.012 0.124 0.440 0.411 0.013 0 3.10 2.54
My result
i=10000
0.012 0.122 0.437 0.416 0.012 0 3.07 2.45

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3.2 Setting up the parameters
The three designs are all rule-based, and the dose levels are fixed and preset before the trial.
There are six dose levels (1-6) in total. The maximum number of patients is set to 21 and 30. The
cohort size of the i3+3 design is set to 3 and 5, and the target toxicity is 0.25(0.17-0.33), which is
the same as that of the 3+3 design. Both the length of the treatment cycle and the time interval
from the enrollment to the first DLT occurrence are set to 21 days which means even if the patients
come with a DLT but it will only be found after 21 days after their enrollment. The time interval
between two patients is assumed to obey the Exponential distribution, so the arrival time of each
patient follows the Poisson distribution, with the rate of 25 and 10 patients per year. The intervals
(days) are round to integers.
The comparison of the three designs is based on four different situations and three different
DLT probability scenarios. In the scenario chart, the x-axis presents the different dose level, and
the y-axis presents the probability of the toxicity rate.
The probability curves are shown in Table 3. Scenario#1 is the ideal condition, starting
with a suitable dose. The probability curve increases at a gentle slope. Scenario#2 starts with a low
dose. The probability curve increases at a steeper slope. Then Scenario#3 starts with a high dose.
The probability curve increases at a gentle slope.
The four situations are:
1. The maximum number of patients is 30, and the patient density is high.
2. The maximum number of patients is 30, and the patient density is low.
3. The maximum number of patients is 21, and the patient density is high.
4. The maximum number of patients is 21, and the patient density is low.

13
Table 3 – DLT probability scenarios
Scenario
No.
Probability of DLT at 6 Dose Levels
Dose 1 Dose 2 Dose 3 Dose 4 Dose 5 Dose 6
S1 0.07 0.14 0.28 0.37 0.5 0.7
S2 0.01 0.07 0.14 0.28 0.7 0.9
S3 0.3 0.44 0.55 0.66 0.77 0.88
Bold: target dose level

Figure 4 - True toxicity probability of dose levels for each scenario

3.3 Simulation result
Scenario#1(0.07, 0.14, 0.28, 0.37, 0.5, 0.7) is an ideal condition. The trials starts with a
suitable starting dose, and the probability curve increases at a gentle slope. The simulation results
for scenario#1 are shown in the table 4.
When the upper bound of recruits for the trial is limited to 30, basically all trials can be
completed. The two i3+3 design with the cohort size of 3 and 5 have more probability of choosing

14
the correct dose, which is 0.345 and 0.356, which is higher than the 3+3 and Rolling-6 design
(which is 0.253 and 0.257).
In terms of safety, with fewer patients recruited, 3+3 design and Rolling-6 design (which
is 3.3 and 3.1) exhibit better performance in reducing DLT occurrences during the trial. Within the
situation of low and high patient densities, Rolling-6 design has the shortest duration among all
designs and it also maintains a similar correct selection percentage compared to 3+3 design.
When the upper bound of recruits for the trial is limited to 21, the trial can be well
completed in both 3+3 and i3+3 designs, the probability of no select is less than 5%. The i3+3
design has a higher chance of selecting the correct MTD, which is 0.313. The average trial duration
under 3+3, Rolling-6, and i3+3 design is 332, 275, 455 days, i3+3 is time-consuming, to some
extent, the 3+3 and Rolling-6 run faster than i3+3, and they have more advantages in the people
number and safety (in the range of 2.4-3.1).
When patient density is high (25 patients per year), Rolling-6 even needs more patients
than it needs in the lower patient density. For example, in the high patient density, the Rolling-6
needs 16.2 patients, but in the lower patient density, it needs only 15.2 patients. Therefore, the
probability of not making a choice will also increase to 0.091 in high density and 0.171 in low
density which is the highest among the designs. The percentage of the right selection also drops
rashly to 0.173.

15
Table 4 - Result from probability scenarios #1
Max patient = 30, High patients’ density (25 patients/year)
Design type PCS
(dose3)
Prob. Of
No select
#exceed
MTD (SD)
#DLT
(SD)
#Average
patients (SD)
Trial duration
(SD) (days)
3+3 design 0.253 0.001 2.8(4.0) 3.3(1.2) 15.6(4.5) 338(115)
Rolling-6 0.247 0.005 2.4(3.7) 3.2(1.1) 17.0(5.6) 287(114)
i3+3 (cohort size=3) 0.345 0 5.5(6.9) 6.8(1.5) 30(NA) 650(82)
i3+3 (cohort size=5) 0.356 0 1.3(3.0) 5.1(1.3) 30(NA) 567(82)

Max patient = 30, Low patients’ density (10 patients/year)
Design type PCS
(dose3)
Prob. Of
No select
#exceed
MTD (SD)
#DLT
(SD)
#Average
patients (SD)
Trial duration
(SD) (days)
3+3 design 0.253 0.001 2.8(4.0) 3.3(1.2) 15.6(4.5) 680(247)
Rolling-6 0.264 0.001 2.3(3.5) 3.0(1.0) 15.5(4.9) 595(233)
i3+3 (cohort size=3) 0.345 0 5.5(6.9) 6.8(1.5) 30(NA) 1312(205)
i3+3 (cohort size=5) 0.356 0 1.3(3.0) 5.1(1.3) 30(NA) 1228(205)

Max patient = 21, High patients’ density (25 patients/year)
Design type PCS
(dose3)
Prob. Of
No select
#exceed
MTD (SD)
#DLT
(SD)
#Average
patients (SD)
Trial duration
(SD) (days)
3+3 design 0.237 0.05 2.7(3.6) 3.1(1.0) 15.3(4.1) 332(106)
Rolling-6 0.173 0.171 1.6(2.3) 2.8(0.9) 16.2(4.4) 275(100)
i3+3 (cohort size=3) 0.313 0 3.3(4.2) 4.4(1.1) 21(NA) 455(66)

Max patient = 21, Low patients’ density (10 patients/year)
Design type PCS
(dose3)
Prob. Of
No select
#exceed
MTD (SD)
#DLT
(SD)
#Average
patients (SD)
Trial duration
(SD) (days)
3+3 design 0.237 0.05 2.7(3.6) 3.1(1.0) 15.3(4.1) 670 (232)
Rolling-6 0.240 0.091 2.0(2.9) 2.9(0.9) 15.2(4.3) 583(127)
i3+3 (cohort size=3) 0.313 0 3.3(4.2) 4.4(1.1) 21(NA) 916(166)
PCS: Percentage of correct selection
#exceed MTD: The average number of patients treated with a dose higher than the true MTD
#DLT: The average number of patients with DLT
#Average patients: The average number of total patients
NA: Not available
SD: Standard deviation

Scenario#2 (0.01, 0.07, 0.14, 0.28, 0.7, 0.9) is the condition that the trial starts with a low
starting dose, and the probability curve increases at a steeper slope. The simulation results for this
scenario are shown in the table 5.
When the maximum number of recruits for the trial is limited to 30, all designs can
complete the trial, the i3+3 designs with a cohort size of 3 have more probability of choosing the

16
correct dose, which is 0.444, higher than the 3+3, Rolling-6 design and the i3+3 design with cohort
size 5 (which is 0.362, 0.251, 0.353). In terms of safety, with fewer patients recruited, the 3+3
design and Rolling-6 design have better performance on reducing DLT occurrences (3.3, 3.4)
during the trial.
When the maximum number of recruits for the trial is limited to 21, the trial can be well
completed under both 3+3 and i3+3 designs. The i3+3 design has a higher chance of selecting the
correct MTD, which is 0.424. But the 3+3 and Rolling-6 designs take less time and more
advantageous in the total number of people and safety (in range of 2.4-3.2).
When the density of patients is high (25 patients per year), the Rolling-6 will need more
patients than lower density(19.1 vs. 18.1), so the probability of not making a choice will also
increase (0.204,0.406), which is the highest among all the design, and the percentage of selecting
correct DLT will also significantly decline to 0.058, showing that the Rolling-6 design may not be
applicable when the maximum number of patients for the trial is limited to 21.
Table 5 - Result from probability scenarios #2
Max patient = 30, High patients’ density (25 patients/year)
Design type PCS
(dose4)
Prob. Of
No select
#exceed
MTD (SD)
#DLT
(SD)
#Average
patients (SD)
Trial duration
(SD) (days)
3+3 design 0.362 0 1.6(2.1) 3.3(1.1) 17.9(3.3) 386(95)
Rolling-6 0.351 0.001 1.5(2.1) 3.4(1.2) 20.9(4.5) 344(93)
i3+3 (cohort size=3) 0.444 0 2.5(3.2) 6.3(1.5) 30(NA) 650(82)
i3+3 (cohort size=5) 0.353 0 0.7(1.8) 4.1(1.2) 30(NA) 567(82)

Max patient = 30, Low patients’ density (10 patients/year)
Design type PCS
(dose4)
Prob. Of
No select
#exceed
MTD (SD)
#DLT
(SD)
#Average
patients (SD)
Trial duration
(SD) (days)
3+3 design 0.362 0 1.6(2.1) 3.3(1.1) 17.9(3.3) 778(214)
Rolling-6 0.358 0 1.4(2.0) 3.1(1.1) 18.7(3.8) 720(120)
i3+3 (cohort size=3) 0.444 0 2.5(3.2) 6.3(1.5) 30(NA) 1311(205)
i3+3 (cohort size=5) 0.353 0 0.7(1.8) 4.1(1.2) 30(NA) 1228(205)

17
Max patient = 21, High patients’ density (25 patients/year)
Design type PCS
(dose4)
Prob. Of
No select
#exceed
MTD (SD)
#DLT
(SD)
#Average
patients (SD)
Trial duration
(SD) (days)
3+3 design 0.345 0.01 1.5(2.0) 3.2(1.0) 17.6(2.9) 318(89)
Rolling-6 0.058 0.406 0.5(1.1) 2.4(1.0) 19.1(2.9) 321(78)
i3+3 (cohort size=3) 0.424 0 1.6(2.3) 4.0(1.0) 21(NA) 455(67)

Max patient = 21, Low patients’ density (10 patients/year)
Design type PCS
(dose4)
Prob. Of
No select
#exceed
MTD (SD)
#DLT
(SD)
#Average
patients (SD)
Trial duration
(SD) (days)
3+3 design 0.345 0.01 1.5(2.0) 3.2(1.0) 17.6(2.9) 768(203)
Rolling-6 0.209 0.204 1.0(1.5) 2.8(0.9) 18.1(3.1) 695(178)
i3+3 (cohort size=3) 0.424 0 1.6(2.3) 4.0(1.0) 21(NA) 916(166)

Scenario#3 (0.3, 0.44, 0.55, 0.66, 0.77, 0.88) is the condition that the trial starts with a high
starting dose, and the probability curve increases at a gentle slope. The simulation results for this
scenario are shown in table 6.
Because of the safety protection mechanism, if the starting dose is at a higher level, the
trials terminated earlier, which means the duration would be less than other experiments. Besides,
it needs fewer patients than Scenario#1 and Scenario#2. All of the experiments could conclude the
result under the limitation of patients' number which is 21. Therefore, only in the i3+3 design, the
result changes as the maximum number of people changes.
The i3+3 designs with a cohort size 3 and 5 have more probability of choosing the correct
dose, which is 0.696 and 0.715, higher than that of the 3+3(0.338) and Rolling-6 design (0.337).
In terms of safety, 3+3, and Rolling-6 designs, which have the same 3.0 DLT numbers, with fewer
recruited patients exhibit better performance in reducing DLT occurrences and duration of the trial.
In cases of both high and low patient densities, Rolling-6 requires the shortest time among
all designs while maintaining the percentage of correct selection and the number of DLTs similar

18
to 3+3 design. Compared to 3+3 and Rolling-6 designs, the safety protection mechanism of i3+3
design is more conservative and harder to trigger, resulting in a more extended trial cycle and more
DLT occurrence.
Table 6 - Result from probability scenarios #3
Max patient = 30, High patients’ density (25 patients/year)
Design type PCS
(dose1)
Prob. of early
termination
#exceed
MTD (SD)
#DLT
(SD)
#Average
patients (SD)
Trial duration
(SD) (days)
3+3 design 0.338 0.571 2.9(3.7) 3.0(1.1) 8.1(4.0) 174(94)
Rolling-6 0.337 0.586 2.6(3.5) 3.0(1.0) 8.0(4.0) 132(79)
i3+3 (cohort size=3) 0.696 0.258 6.3(7.7) 8.7(2.4) 25.7(8.4) 558(199)
i3+3 (cohort size=5) 0.715 0.250 3.1(5.5) 8.1(2.1) 26(8) 491(171)

Max patient = 30, Low patients’ density (10 patients/year)
Design type PCS
(dose1)
Prob. of early
termination
#exceed
MTD (SD)
#DLT
(SD)
#Average
patients (SD)
Trial duration
(SD) (days)
3+3 design 0.338 0.571 2.9(3.7) 3.0(1.1) 8.1(4.0) 352(200)
Rolling-6 0.335 0.576 2.4(3.2) 3.0(1.0) 7.5(3.8) 286(178)
i3+3 (cohort size=3) 0.696 0.258 6.3(7.7) 8.7(2.4) 25.7(8.4) 1127(417)
i3+3 (cohort size=5) 0.715 0.250 3.1(5.5) 8.1(2.1) 26(8) 1064(283)

Max patient = 21, High patients’ density (25 patients/year)
Design type PCS
(dose1)
Prob. of early
termination
#exceed
MTD (SD)
#DLT
(SD)
#Average
patients (SD)
Trial duration
(SD) (days)
3+3 design 0.338 0.571 2.9(3.7) 3.0(1.1) 8.1(4.0) 174(94)
Rolling-6 0.337 0.586 2.6(3.5) 3.0(1.0) 8.0(4.0) 132(79)
i3+3 (cohort size=3) 0.710 0.211 5.2(5.9) 6.4(1.5) 18.8(5.1) 406(127)

Max patient = 21, Low patients’ density (10 patients/year)
Design type PCS
(dose1)
Prob. of early
termination
#exceed
MTD (SD)
#DLT
(SD)
#Average
patients (SD)
Trial duration
(SD) (days)
3+3 design 0.338 0.571 2.9(3.7) 3.0(1.1) 8.1(4.0) 352(200)
Rolling-6 0.335 0.576 2.4(3.2) 3.0(1.0) 7.5(3.8) 286(178)
i3+3 (cohort size=3) 0.710 0.211 5.2(5.9) 6.4(1.5) 18.8(5.1) 767(203)

19
4. Discussion
Based on the above experiments, results may be valid only under specific test conditions.
If the maximum number of recruitable patients is preset to 21, Rolling-6 design may not be
applicable. The properties of Rolling-6 design make it require more patients than 3+3 design to
complete the trial. (Skolnik et al., 2008). Therefore, there is a high probability that the trial under
Rolling-6 design fails to make a choice because of the low number of patient’s limitation. The
failure to complete the trial is more pronounced when patients have a high density of visits. The
3+3 and i3+3 designs can complete the trial with less than 21 patients in most cases. The 3+3
design is advantageous in the test period, the number of patients and DLT, while the i3+3 design
has a high-level possibility of choosing the correct dose level. Detailed comparison results can be
found in appendix A1.
If the maximum number of patients that can be recruited reaches 30, all three designs could
be selected. They have their respective characteristics and advantages. The total number of patients
recruited in the 3+3 design is the lowest. Rolling-6 can get the result with the shortest duration
while keeping the same reliability and safety as the 3+3 design. The i3+3 design performs the best
in MTD selection, but it takes the most time, requires the largest number of recruits, and has more
patients with DLT. Detailed comparison results can be found in the appendix A2.
This study also finds some characteristics of different experimental designs. The i3+3
design shows better reliability than the other two designs. Basically, in each case and scenario, it
can select the MTD with the highest probability. Additionally, the targeted MTD rate and the total
sample size are adjustable, which provide a new way for experimental designers to design trial
according to their own needs. When the total sample size is 21, which is slightly higher than the

20
average number of patients in 3+3 and Rolling-6 designs, the correct selection percentage of i3+3
is higher than that of 3+3 and Rolling-6 designs in all scenarios. However, more patients are
recruited and treated with a dose higher than MTD than 3+3 and Rolling-6 designs. When the total
sample size is enlarged to 30, the result is more reliable, but more patients are treated with a dose
over MTD. The i3+3 design with a cohort size of 3 performs better in reliability but worse in safety
than that with a cohort size of 5.
Depending on the characteristics of Rolling-6 design, the actual arrangement of the trial
will vary according to patient density. When the interval between patients is short, the rule tends
to enroll patients at a dose level before any results are known. If the interval is long, dosage
schedule will be more dependent on information that is already known. However, the simulation
results show that the percentage of correct selection does not significantly change with the interval.
Besides, when the interval is long, the average number of the patients treated over the true MTDs
decreases, so dose the average number of patients recruited in the trial and the number of DLTs.
The experiment period in Rolling-6 design is the shortest. Compared with 3+3 design, the
Rolling-6 design can indeed shorten the total duration of the trial while maintaining the same
reliability and safety, thus achieving its design purpose. The Rolling 6 takes less time than 3+3
and i3+3 design in each scenario.
The significance of this thesis is mainly to provide a reference for choosing among different
rule-based designs and verify the conclusions by the trial design author. The i3+3 design is a new
rule-based design proposed in 2019 and has not been compared with the Rolling-6.
There are some limitations in this thesis. In reality, investigators may have very little
information about new drugs, so any dose-toxicity relationship is possible. Although the three dose

21
toxicity scenarios selected in this study can cover a wider range, they cannot cover all situations.
In my simulation, the possibility of a patient leaving the experiment in the middle of the experiment
and becoming unobserved was not considered. In addition, this thesis only simulated the rule-
based designs and did not compare them with model-based designs and model-assisted design.

22
REFERENCES
Hansen, A., Graham, D., Pond, G., & Siu, L. (2014). Phase 1 Trial Design: Is 3 + 3 the
Best?. Cancer Control, 21(3), 200-208.
NIH Clinical Research Trials and You. National Institutes of Health (NIH). (2020). Retrieved 22
June 2020, from https://www.nih.gov/health-information/nih-clinical-research-trials-you.
O’Quigley J, Pepe M, Fisher L (1990). Continual reassessment method: a practical design for
phase I clinical trials in cancer. Biometrics; 46 (1): 33-48.
Storer, B. E. (1989). Design and Analysis of Phase I Clinical Trials. Biometrics, 45(3), 925. doi:
10.2307/2531693
Skolnik, J., Barrett, J., Jayaraman, B., Patel, D., & Adamson, P. (2008). Shortening the Timeline
of Pediatric Phase I Trials: The Rolling Six Design. Journal Of Clinical Oncology, 26(2),
190-195.
Liu, S., & Yuan, Y. (2014). Bayesian optimal interval designs for phase I clinical trials. Journal
Of The Royal Statistical Society: Series C (Applied Statistics), 64(3), 507-523.
Yuan J., Ping L., Yisheng L., & Nebiyou Bekele, B. (2010). A modified toxicity probability
interval method for dose-finding trials. Clinical Trials: Journal Of The Society For
Clinical Trials, 7(6), 653-663.
Wang, T, (2012) A simulation study of the standard design, the rolling six design, the CRM, and
the modified CRM in Phase I clinical trials. Texas Medical Center Dissertations (via
ProQuest).
Fellman, B., & Yuan, Y. (2015). Bayesian Optimal Interval Design for Phase I Oncology Clinical
Trials. The Stata Journal: Promoting Communications On Statistics And Stata, 15(1), 110-
120.
Morgensztern, D., & LoRusso, P. (2014). The Molecular Basis of Cancer(Fourth Edition), Phase
I Trial Today (4th ed., pp. 661-676).
Luu, J. (2019). A Simulation Evaluation of the Effectiveness and Usability of the 3+3 Rules-based
Design for Phase I Clinical Trials.

23
Appendix
A1 Table- Comparison of different designs with a maximum number of 21
Max patient = 21, High patients’ density (25 patients/year)
Design type PCS Prob. Of No
select
Number of
DLTs
Trial duration
Scenario1 3+3 design 0.237 0.05 3.1 332
Rolling-6 0.173 0.171 2.8 275
i3+3 (cohort size=3) 0.313 0 4.4 455
Scenario2 3+3 design 0.345 0.01 3.2 318
Rolling-6 0.058 0.406 2.4 321
i3+3 (cohort size=3) 0.424 0 4.0 455
Scenario3 3+3 design 0.338 0.571 3.0 174
Rolling-6 0.337 0.586 3.0 132
i3+3 (cohort size=3) 0.710 0.211 6.4 406

Max patient = 21, High patients’ density (10 patients/year)
Design type PCS Prob. Of No
select
Number of
DLTs
Trial duration
Scenario1 3+3 design 0.237 0.05 3.1 670
Rolling-6 0.240 0.091 2.9 583
i3+3 (cohort size=3) 0.313 0 4.4 916
Scenario2 3+3 design 0.345 0.01 3.2 768
Rolling-6 0.209 0.204 2.8 695
i3+3 (cohort size=3) 0.424 0 4.0 916
Scenario3 3+3 design 0.338 0.571 3.0 352
Rolling-6 0.335 0.576 3.0 286
i3+3 (cohort size=3) 0.710 0.211 6.4 767

A2 Table - Comparison of different designs with a maximum number of 30
Max patient = 30, High patients’ density (25 patients/year)
Design type PCS Prob. Of No
select
Number of
DLTs
Trial duration
Scenario1 3+3 design 0.253 0.001 3.3 338
Rolling-6 0.247 0.005 3.2 287
i3+3 (cohort size=3) 0.345 0 6.8 650
i3+3 (cohort size=5) 0.356 0 5.1 567
Scenario2 3+3 design 0.362 0 3.3 386
Rolling-6 0.351 0.001 3.4 344
i3+3 (cohort size=3) 0.444 0 6.3 650
i3+3 (cohort size=5) 0.353 0 4.1 567
Scenario3 3+3 design 0.338 0.571 3.0 174
Rolling-6 0.337 0.586 3.0 132
i3+3 (cohort size=3) 0.696 0.258 8.7 558
i3+3 (cohort size=5) 0.715 0.250 8.1 491

24
Max patient = 30, low patients’ density (10 patients/year)
Design type PCS Prob. Of No
select
Number of
DLTs
Trial duration
Scenario1 3+3 design 0.253 0.001 3.3 680
Rolling-6 0.264 0.001 3.0 595
i3+3 (cohort size=3) 0.345 0 6.8 1312
i3+3 (cohort size=5) 0.356 0 5.1 1228
Scenario2 3+3 design 0.362 0 3.3 778
Rolling-6 0.358 0 3.1 720
i3+3 (cohort size=3) 0.444 0 6.3 1311
i3+3 (cohort size=5) 0.353 0 4.1 1228
Scenario3 3+3 design 0.338 0.571 3.0 352
Rolling-6 0.335 0.576 3.0 286
i3+3 (cohort size=3) 0.696 0.258 8.7 1127
i3+3 (cohort size=5) 0.715 0.250 8.1 1064

25
Appendix B.
Codes

scenario1<-c(0.07, 0.14, 0.28, 0.37,0.5, 0.7)
scenario3<-c(0.01,0.07, 0.14, 0.28, 0.7, 0.9)
scenario5<-c(0.3, 0.44, 0.55, 0.66, 0.77, 0.88)

library(Iso)
set.seed(2020)
iterations=10000
resulttime<-totaldlt<-totalpatient<-safety<-reroll<-rep(0,iterations)

for(re in 1:iterations){
p.dlt<-scenario3

data<-array(-1,dim=c(60,9))
t9=0

for(num in 1:60){
data[num,7]=num
for(doselvl in 1:6){
data[num,doselvl]<- rbinom(1,1,prob=p.dlt[doselvl])}
data[num,8]<-round(rexp(1, rate=10/365))
t9<-t9+data[num,8]
data[num,9] =t9 #arrive time
}

cohortsize=3
numofgroup=10
pt=0.25
ei=0.08
numofdose=6
p<-rep(1,6)
var<-du<-h<-n<-x<-rep(0,6)#dose lvl1
t=0
doselvl=1
result<-rep(-1,60)
dose<-rep(0,60)
dose[1:cohortsize]=rep(1,cohortsize)

26
narranged=0

for(num in 1:numofgroup){ #number of groups
doselvl=dose[(num*cohortsize)]

for(haha in 1:cohortsize){
result[((num-1)*cohortsize+haha)]=data[((num-
1)*cohortsize+haha),doselvl] #test

if (data[((num-1)*cohortsize+haha),doselvl]==1){
x[doselvl]=x[doselvl]+1
}else {h[doselvl]=h[doselvl]+1}
}

n[doselvl]=x[doselvl]+h[doselvl]
p[doselvl]=(x[doselvl]+0.005)/(n[doselvl]+0.01) #beta

if (pbeta(pt, (x[doselvl]+0.005), (h[doselvl]+0.005))<0.05){
du[doselvl]=1
} #du

if(p[doselvl]<(pt-ei)){
if
(doselvl==6){dose[(num*cohortsize+1):(num*cohortsize+cohortsize)]=rep((doselvl
),cohortsize)}
else {if
(du[(doselvl+1)]==1){dose[(num*cohortsize+1):(num*cohortsize+cohortsize)]=rep(
(doselvl),cohortsize)}
else
{dose[(num*cohortsize+1):(num*cohortsize+cohortsize)]=rep((doselvl+1),cohortsi
ze) }}}

if(p[doselvl]>(pt-ei) && p[doselvl]<(pt+ei) ){

dose[(num*cohortsize+1):(num*cohortsize+cohortsize)]=rep((doselvl),cohortsize)
}

if(p[doselvl]>(pt+ei)){

27
if (doselvl==1){ if (du[1]==1){ doselvl=-1;break; }else
{dose[(num*cohortsize+1):(num*cohortsize+cohortsize)]=rep((doselvl),cohortsize
)}
} else if ((p[doselvl]-1/n[doselvl])<(pt-
ei)){dose[(num*cohortsize+1):(num*cohortsize+cohortsize)]=rep((doselvl),cohort
size)}
else
{dose[(num*cohortsize+1):(num*cohortsize+cohortsize)]=rep((doselvl-
1),cohortsize)}}

}

var<- ((x+0.005)*(h+0.005)/(((x+h+0.01)^2)*(x+h+1.01)))
pa<-pava(p, var, decreasing=FALSE, long.out=FALSE, stepfun=FALSE)
for(hahaha in 1:6) {if(p[hahaha]>(pt+ei)) {pa[hahaha]=1}}
for(hahaha in 1:6) {if(n[hahaha]==0) {pa[hahaha]=1}}

whp<-which((abs(pa-pt))==min(abs(pa-pt)))
if(pa[whp[1]]>pt){
flvl<-whp[1]} else {flvl<-whp[length(whp)]}
if(doselvl==-1){flvl<- -1}

reroll[re]=flvl

resulttime[re]=data[cohortsize*num,9]+21*num

totalpatient[re]=sum(n)

totaldlt[re]=sum(x)

safety[re]=sum(n)- sum(n[1:4]) #####

}
table(reroll)
mean(safety);sd(safety);
mean(totaldlt);sd(totaldlt);
mean(totalpatient);sd(totalpatient)

28
mean(resulttime);sd(resulttime)

set.seed(2020)
iterations<-1000
maxp<-30
mtdlvl=1
resulttime<-totaldlt<-totalpatient<-safety<-reroll<-rep(0,iterations)
nd1<-nd2<-nd3<-nd4<-nd5<-nd6<-rep(0,iterations)
for(re in 1:iterations){

p.dlt<-scenario5
data<-array(-1,dim=c(100,9))
t9=0

for(num in 1:100){
data[num,7]=num
for(doselvl in 1:6){
data[num,doselvl]<- rbinom(1,1,prob=p.dlt[doselvl])}
data[num,8]<-round(rexp(1, rate=10/365))
t9<-t9+data[num,8]
data[num,9] =t9 #arrive time
}
#data
#the toxicity probability of the MTD was estimated in a range from 16% to
33%.

h<-n<-x<-rep(0,6)#dose lvl1
mtdvisit=0
t=0
doselvl=1
dose<-rep(0,100)
th<-tx<-tn<-result<-rep(-1,100) #test
time<-rep(0,100) #test
dose[1:6]=c(1,1,1,1,1,1)
narranged=0
change=-2

for(num in 1:100){
time[num]=data[num,9] #test

29
if(dose[num]==0){next;}
result[num]=data[num,doselvl] #test

doselvl=dose[num]
t=data[num,9]+21
if (data[num,doselvl]==1){
x[doselvl]=x[doselvl]+1
}else {h[doselvl]=h[doselvl]+1}
for(pending in 1:100){ #arranged
if (data[(pending+1),9]>t){break;}
}

n<-rep(0,6)
for(hahaha in 1:pending){
for(lvl in 1:6){
if (dose[hahaha]==lvl){ n[lvl]=n[lvl]+1;}
}}

tn[num]=n[doselvl] #test
th[num]=h[doselvl]
tx[num]=x[doselvl]
c<-data.frame(num,dose,result,time,tn,th,tx)

if(x[1]>1){doselvl=-1;break;}#toaxity too high
if(h[6]>4){doselvl=99;break;}#toaxity too low

if(mtdvisit==0 && (n[doselvl]>2) && h[doselvl]==n[doselvl]) #doselvl+1
{ change=1
if(doselvl==6){doselvl=99;break;
}else{
dose[(pending+1):(pending+6)]=rep(c((doselvl+1),0),times=c((6-
n[doselvl+1]),n[doselvl+1]))}}

if(mtdvisit==0 && (n[doselvl]==6) && h[doselvl]==5) #doselvl+1
{ change=1
if(doselvl==6){doselvl=99;break;
} else {dose[(pending+1):(pending+6)]=rep(c((doselvl+1),0),times=c((6-
n[doselvl+1]),n[doselvl+1]))}

30
}

if(x[doselvl]>=2) #doselvl-1
{ change=-1
dose[(pending+1):(pending+6)]=rep(c((doselvl-1),0),times=c((6-n[doselvl-
1]),n[doselvl-1]))
mtdvisit=1
if (n[doselvl-1]==6){doselvl=doselvl-1; break;}
}

if(n[doselvl]==6 && x[doselvl]<2 && h[doselvl]<5) #suspended
{ change=0
dose[(pending+1):(pending+6)]=c(0,0,0,0,0,0)}

if(mtdvisit==1 && n[doselvl]==6 && h[doselvl]>4)
{ change=0
break} #MTD

if(change==-2)
{
dose[(pending+1):(pending+6)]=rep(c((doselvl),0),times=c((6-
n[doselvl]),n[doselvl]))} #same doselvl

}
reroll[re]=doselvl
# result time
resulttime[re]=data[num,9]+21
# resultdlt[re]=sum(x)
totalpatient[re]=sum(n)
res=rep(0,100)

for (xxx in 1:100){
if (dose[xxx]!=0){res[xxx]=data[xxx,dose[xxx]]
last=xxx
} }
#totaltime[re]=data[last,9]+21
totaldlt[re]=sum(res)
safety[re]=sum(n)- sum(n[1:mtdlvl])

tll=0

31
mnummax<-mmmmm<-99
for (xxxxx in 1:num){
if (dose[xxxxx]!=0){
tll=tll+1
if (tll> maxp){
mnummax<-mmmmm
resulttime[re]=data[mnummax,9]+21
reroll[re]=9; break}
mmmmm<-xxxxx

}
}

litmax=0
nummax<-mmmm<-99
for (xxxx in 1:99){
if (dose[xxxx]!=0){
litmax=litmax+1
if (litmax> maxp){ nummax<-mmmm; break}
mmmm<-xxxx
}
}

if (litmax> maxp){
totalpatient[re]=maxp
totaldlt[re]=sum(res[1:nummax])
dose<-dose[1:nummax]
safety[re]<-sum(dose>mtdlvl)
}

}
table(reroll)
mean(safety);sd(safety);
mean(totaldlt);sd(totaldlt);
mean(totalpatient);sd(totalpatient)
mean(resulttime);sd(resulttime)
#The paramaters I want to compare in my thesis

32
set.seed(2020)

iterations<-1000
resulttime<-totaldlt<-totalpatient<-safety<-reroll<-rep(0,iterations)
nd1<-nd2<-nd3<-nd4<-nd5<-nd6<-rep(0,iterations)
for(re in 1:iterations){

p.dlt<-validating2
data<-array(-1,dim=c(100,9))
t9=0

for(num in 1:100){
data[num,7]=num
for(doselvl in 1:6){
data[num,doselvl]<- rbinom(1,1,prob=p.dlt[doselvl])}
data[num,8]<-round(rexp(1, rate=10/365))
t9<-t9+data[num,8]
data[num,9] =t9 #arrive time
}
#data

cohortsize=3
numofgroup=7

h<-n<-x<-rep(0,6)
doselvl=1
result<-rep(-1,60)
dose<-rep(0,60)
dose[1:cohortsize]=rep(1,cohortsize)

for(num in 1:numofgroup){ #number of groups
doselvl=dose[(num*cohortsize)]

for(haha in 1:cohortsize){
result[((num-1)*cohortsize+haha)]=data[((num-1)*cohortsize+haha),doselvl]

if (data[((num-1)*cohortsize+haha),doselvl]==1){
x[doselvl]=x[doselvl]+1
}else {h[doselvl]=h[doselvl]+1}
}

33

n[doselvl]=x[doselvl]+h[doselvl]

if(x[doselvl]==0 && doselvl==6){ doselvl=99;break}
if(x[doselvl]==1 && doselvl==6 && n[doselvl]==6){ doselvl=99;break}
if(x[doselvl]==0 && n[doselvl]==3){

dose[(num*cohortsize+1):(num*cohortsize+cohortsize)]=rep((doselvl+1),cohortsiz
e) }

if(x[doselvl]==1 && n[doselvl]==3){

dose[(num*cohortsize+1):(num*cohortsize+cohortsize)]=rep((doselvl),cohortsize)
}

if(x[doselvl]==1 && n[doselvl]==6){
if(n[doselvl+1]==0){

dose[(num*cohortsize+1):(num*cohortsize+cohortsize)]=rep((doselvl+1),cohortsiz
e)
} else{break}}

if(x[doselvl]>1 && doselvl==1){ doselvl=-1;break}
if(x[doselvl]>1 ){
if(n[doselvl-1]==6){ doselvl=doselvl-1;break
}else {dose[(num*cohortsize+1):(num*cohortsize+cohortsize)]=rep((doselvl-
1),cohortsize) }
}
if(x[doselvl]==0 && n[doselvl]==6){break}
}

reroll[re]=doselvl
resulttime[re]=data[3*num,9]+21*num

totalpatient[re]=sum(n)

totaldlt[re]=sum(x)

safety[re]=sum(n)- sum(n[1:2]) #####

34

}
table(reroll)
mean(safety);sd(safety);
mean(totaldlt);sd(totaldlt);
mean(totalpatient);sd(totalpatient)
mean(resulttime);sd(resulttime)

#The paramaters I want to compare in my thesis

Abstract (if available)

Abstract The Phase I clinical trial is the first step of new drug testing and aimed to estimate the true Maximum tolerable dose (MTD) for human. In recent years, many new methods have been proposed, but the classical 3+3 rule-based design is still most commonly applied since it is the simplest to implement and requires no calculation during the trial (Hansen et al., 2014). This article only focuses on the comparison among three rule-based designs, namely, the 3+3 design, the Rolling-6 design and the i3+3 design. The reliability (i.e., the percentage of correct choices) and safety (i.e., the number of patients receiving the treatment above the true MTD) are two main criteria to evaluate a trial design, and the duration of the entire experiment is another consideration. R code is used to implement the algorithms of these three designs, and the simulated data are processed with these algorithms. The evaluation and comparison are made based on the results of simulated data.

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University of Southern California Dissertations and Theses

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Asset Metadata

Creator Yang, Yuanliang (author)

Core Title A simulation study between 3+3, Rolling-6 and i3+3 design

School Keck School of Medicine

Degree Master of Science

Degree Program Biostatistics

Publication Date 07/26/2020

Defense Date 07/22/2020

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

Tag 3 3 design,i3 3 design,OAI-PMH Harvest,phase I clinical trial,Rolling-6 design,simulation study

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Piao, Jin (committee chair), Barkauskas, Donald (committee member), Dang, Ha (committee member)

Creator Email 1211546036@qq.com,yangyuan@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c89-339821

Unique identifier UC11664203

Identifier etd-YangYuanli-8765.pdf (filename),usctheses-c89-339821 (legacy record id)

Legacy Identifier etd-YangYuanli-8765.pdf

Dmrecord 339821

Document Type Thesis

Rights Yang, Yuanliang

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 a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA