DescriptionThis web application calculates the sample size for comparing twostage adaptive treatment strategies in a SMART trial using weighted log rank test. In these trials, all subjects are randomized to one of two initial treatments, denoted by A_{1} and A_{2}; the probability that a subject being assigned to A_{1} is denoted by p_{1}. All subjects with insufficient response to the firststage treatment (nonresponders) are rerandomized to one of two secondstage treatments: those who initially receive A_{1} and do not respond are further randomized to B_{11} or B_{12}; those who initially receive A_{2} and do not respond are further randomized to B_{21} or B_{22}. For nonresponders to A_{1}, the probability of being randomized to B_{11} is p_{21}, and for nonresponders to A_{2}, the probability of being randomized to B_{21} is p_{22}. The responding subjects are offered the same treatment (no randomization; usually a continuation of the firststage treatment or a maintenance/relapse prevention treatment). The timing for observation of nonresponse along with the criteria for nonresponse should be defined in the study protocol. Nonresponse may be assessed at a fixed point in time (e.g. 2 months after study entry) or at regular intervals, or may be assessed at a nonresponse timetoevent such as the time until two unexcused counseling sessions are missed or time until a second drug positive urine is collected. The trial design is depicted in the graph below. The primary outcome of interest is a failure time. There are 4 adaptive treatment strategies in this trial, denoted by 11, 12, 21, 22. Strategy jk is the strategy in which A_{j} is offered first, then B_{jk} is offered to nonresponding subjects and C_{j} is offered to responders, where j, k can be either 1 or 2 (see graph below). InstructionsTo use the sample size calculator, enter the following information in the appropriate boxes to the right:
Output

Sample size calculation 
In this example, suppose children with ADHD are first randomized to either a behavioral modification therapy (BT) or a medication (Med), with equal probability. Beginning at 2 months and every month thereafter each child’s classroom behavior is assessed and compared to a prespecified criterion. Exceeding the criterion is interpreted as a sign of nonresponse; the nonresponding children are then rerandomized to either intensification of current treatment (more intensive behavioral therapy (BT+) or higher dose medication (Med+)), or a combined treatment (behavioral therapy and medication (BT+Med)), with equal probability. Children who do not show signs of nonresponse continue on their initial treatment (see the figure below). Suppose the outcome of interest is time until a major school disciplinary event.
For the sample size calculation, you need to first specify A_{j}, B_{jk} and C_{j}, j = 1,2. In this example, the first and second stage treatments are: A_{1}=BT, A_{2}=Med, B_{11}=BT+,B_{12}=BT&Med, C_{1}=BT, B_{21}=Med, B_{22} =BT&Med, and C_{2}=Med.
After this, you can determine p_{1}, p_{21} and p_{22}. In this example, p_{1} is the probability that a subject is initially randomized to behavioral treatment, since treatment A_{1} is the behavioral therapy. Also, p_{21} is the probability that a subject not responding to behavioral therapy is randomized to more intensive behavioral therapy, and p_{22} is the probability that a subject not responding to medicine is randomized to higher dose medication. In this example, we have p_{1} =p_{21} =p_{22} =0.5.
Then you need to determine which two strategies you want to compare. For example, suppose you want to compare strategies 11 and 22. Strategy 11 is: offer behavioral therapy first, and offer more intensive behavioral therapy if the subject does not respond; and stay on the original behavioral therapy if the subject responds. Strategy 22 is: offer medicine first, then offer combined treatment (add behavioral therapy) to nonresponding subjects; responding subjects stay on the medication.
Finally, specify the probability that you observe a major school disciplinary event during the study period among children assigned strategy 11, where “observe an event” on a subject implies that the subject has not dropped out of the study when the event occurs.
This web applet calculates the sample size necessary to detect a meaningful difference between two adaptive treatment strategies (also called adaptive interventions or dynamic treatment regimens) in a SMART trial [1] with two stages. The primary outcome is a failure time and the sample size calculator is based on the weighted log rank test with time independent weights given in [2] (also see [3]).
This sample size calculator can be used to size a SMART trial for comparing two strategies beginning with different firststage treatments (e.g. 11 versus 21 or 11 versus 22 or 12 versus 21, or 12 versus 22). Often investigators compare the two strategies that can be viewed as most extreme in terms of intensity and burden or two strategies that represent opposing clinical approaches to treatment. The primary outcome, a failure time, may be censored before or at the end of study. The failure time under strategy jk is denoted by T_{jk}. This is the failure time if the subject had followed strategy jk.
In deriving the sample size formula, we make the following working assumptions:
Assumption 1 is common in trials in metal health and substance abuse, etc. Assumption 2, the independent censoring assumption, is the usual assumption in standard failure time studies. Assumption 3 permits the specification of an effect size. Under Assumption 3, the effect size for the comparison of the two strategies is taken to be the hazard ratio. However the use of the weighted log rank test in the data analysis does not require Assumptions 1 and 3 (these assumptions are only used to size the study).
This sample size calculator usually results in conservative sample sizes. This occurs because, in deriving the sample size formula, the variances involved in the test statistics are replaced by their upper bounds. We find that it is easier to elicit necessary information to size the study when we use these upper bounds [2]. Moreover, the degree of conservatism depends on the percentage of subjects rerandomized in the second stage. The higher percentage of subjects rerandomized, the less conservative the sample sizes [2].
To improve power in the data analysis, we recommend using the weighted log rank test with time dependent weights [2]. This test is more powerful than the weighted log rank test with time independent weights (see [2] and [4] for details).
Although we assumed only nonresponders are rerandomized in the above, this sample size calculator can also be used if responding instead of nonresponding subjects are re randomized. Then strategy jk is: offer treatment A_{j} in the first stage, offer second stage treatment B_{jk} if the subject has met the response criteria. Offer a fixed treatment C_{j} (e.g. continue on current treatment or provide a salvage treatment) if the subject does not meet the response criteria. Everything else is the same as above.
[1]. Murphy S.A. An experimental design for the development of adaptive treatment strategies. Statistics in Medicine 2005; 24:14551481.
[2]. Z. Li and S.A. Murphy, Sample Size Formulae for TwoStage Randomized Trials with Survival Outcomes. Biometrika 2011; 98(3):503518. Click here to view paper Click here to obtain the simulation code in Matlab Click here to obtain the supplementary material.
[3]. Guo X. Statistical analysis in two stage randomization designs in clinical trials. unpublished PhD thesis, Department of Statistics, North Carolina State University, 2005. Click here to view thesis.
[4]. Guo X. and Tsiatis A.A., A weighted risk set estimator for survival distributions in twostage randomization designs with censored survival data. The International Journal of Biostatistics 2005; 1(1):115.
If you use this applet in your own research, we would greatly appreciate it if you cite one or more of the articles listed in the references section of this web page. Thank you very much!