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How Many Stratification Factors Is Too Many" to Use in a Randomization

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How Many Stratification Factors Is Too Many How many Strati cation Factors is \Too Many" to Use in a Randomization Plan? Terry M. Therneau, PhD Abstract The issue of strati cation and its role in patient assignment has gen- erated much discussion, mostly fo cused on its imp ortance to a study [1,2] or lack thereof [3,4]. This rep ort fo cuses on a much narrower prob- lem: assuming that strati ed assignment is desired, how many factors can b e accomo dated? This is investigated for two metho ds of balanced patient assignment, the rst is based on the minimization metho d of Taves [5] and the second on the commonly used metho d of strati ed assignment [6,7]. Simulation results show that the former metho d can accomo date a large number of factors 10-20 without diculty, but that the latter b egins to fail if the total numb er of distinct combinations of factor levels is greater than approximately n=2. The two metho ds are related to a linear discriminant mo del, which helps to explain the results. 1 1 Intro duction This work arose out of my involvement with lung cancer studies within the North Central Cancer Treatment Group. These studies are small, typically 100 to 200 patients, and there are several imp ortant prognostic factors on whichwe wish to balance. Some of the factors, such as p erformance or Karnofsky score, are likely to contribute a larger survival impact than the treatment under study; hazards ratios of 3{4 are not uncommon. In this setting it b ecomes very dicult to interpret study results if any of the factors are signi cantly out of balance. With these small samples, however, there is concern that including to o many strati cation factors will place us in the situation of the old proverb that \He that attempts to please everyone pleases no one," i.e., that in trying to balance on all of the factors there may be balance on none. To investigate this simulations were p erformed using two metho ds of treatment assignment, dynamic allo cation and strati ed assignment. The imbalance of a single factor was examined in cases where the chosen pro ceedure is attempting to balance on the relevant factor concurrently with from 0 to 11 others. It must b e p ointed out that the issue of strati cation is a controversial one. Peto et. al. [3] hold the view that it is unnecessary, esp ecially in large trials where complete randomization is fairly ecient, and also b ecause retrosp ective strati cation by means of an analysis involving covariates takes care of p ossible imbalances. The opp osing view, articulated in Begg and Iglewicz [2], among others, is that balancing provides a more ecient comparison of treatments for trials of the typical size and, more imp ortant, that trials in which the 2 prognostic factors are well balanced are far more convincing to a scienti c audience than sophisticated covariate analysis alone. This rep ort addresses a more narrow problem: assuming that balance is desired, how many factors can be managed? 2 Metho ds The North Central Cancer Treatment Group and the Mayo Comprehensive Cancer Center use a metho d of dynamic allo cation to assign patients. This is based on a technique found in Pocock and Simon [7] and in Taves [5]. Minor changes have b een made that allow easier computation by the randomization p ersonnel. The numer k of categorical randomization variables or factors on which balance is desired usually ranges from 1 to 5, a factor commonly has between 2 and 4 levels. The use of the word al location rather than randomiza- tion is purp oseful: with this and other minimization techniques the treatment assignment of nearly every patient, given the prior assignments, is fully deter- mined. A coin or other randomizing device is rarely used. As an example, with three factors the metho d works as follows: A new sub ject arrives with values for the three factors of F =1, F = 0, and F =0. 1 2 3 A table like the one shown in Table 1 is constructed and lled in with the numb er of patients who fall into each category. The row lab els of the table are based on the new patient's characteristics, the counts within it are based on the patients randomized to this p oint. The columns of the table corresp ond to the treatment group. A given patient may count multiple times in a column. The column totals T , T , etc. are compared, and the sub ject is assigned 1 2 3 Table 1: Example of a Working Table to the treatment with the smallest total. If there is a tie between two or more treatment groups, then the treatment with the smallest total number of patients on study ignoring the factors is chosen. If there is still a tie, a random choice is made b ewteen the tied groups. Note that the construction and contents of the table did not dep end on the number of levels for each factor, but only on the realized values of them for the patient ab out to b e assigned. If the additional rule to \break ties based on the treatment totals" is dropp ed this has little e ect on the p erformance of dynamic allo cation. It turns out that the column totals in table 1 rarely are tied unless the overall treatment totals are also tied. To see this, assume that treatment group A has more total patients than B. Then treating each row of the table as a sample from the total study, in exp ectation the total sum for A will be larger than that for B. Another common way to control balance is by blo cked randomization within cells. The sample space is divided into the L L ::: L unique cells formed 1 2 k 4 by the k strati cation factors, where L is the number of levels for factor i number i. Balance is maintained separately within each cell using random- ized blo cks or some other simple metho d. This metho d will be refered to as strati ed assignment. For simplicity, the simulation exp eriments were evaluated with two treat- ment groups, and the k strati cation factors had either 2 or 3 levels with corre- sp onding probabilities of 1=2, 1=2or1=3, 1=3, 1=3. An exp eriment consisted of randomizing either 100, 200 or 400 patients to a `trial' using k =1; 2;:::;or 12 factors. The imbalance I for any exp eriment is computed as the number of patients with factor F = 0 on treatment A minus the numb er with F =0 on 1 1 treatment B. By symmetry, I will b e p ositive or negative with equal likeliho o d and E I = 0. If only factor F is considered, the b est concievable assignment scheme 1 would achieve an imbalance of 0 whenever p ossible, i.e., whenever the total numb er of patients with F =0was even. When there was an o dd numb er with 1 F = 0 the imbalance would be 1. The least e ective assignment metho d 1 is represented by a randomization scheme that paid no attention whatever to factor 1. Of course, a maliciously designed metho d could do even worse by striving for imbalance, but this is not of interest here. Thus, the simulation exp eriments quantify how the balance on one partic- ular factor will be degraded by addition of other \extraneous" factors to the balancing scheme. The relative placement of the exp ected imbalance b etween the b est case and worst case values provides a measure of randomization \ef- ciency" with resp ect to that factor. A strategy whose exp ected imbalance for F is no b etter than it would be if that factor had b een ignored has by 1 5 de nition an eciency of 0 | there is no return b ene t for the investigator's time and e ort to collect and use the strati cation data. A strategy, if any, that matches the ideal imbalance would have an eciency of 1. Exp erience with dynamic allo cation has revealed that when there are two treatment groups, the total number on treatment is nearly always balanced after every second patient. To make randomization within cells not app ear worse only b ecause of a global treatment imbalance, the metho d used within a cell was a randomized blo ck of length 2. 3 Results 3.1 Best case imbalance Let p = P F = 0 and q = 1 p. By adding the binomial expansions for 1 n n p + q and p q , we see that the probability that the study will contain n an even numb er of patients with F =0 is :5+q p =2=:5+, where n is 1 the number of sub jects. Thus the imbalance I takes on values of 0, 1, and -1 with probabilities :5+, :25 =2 and :25 =2, resp ectively. For p = :5 or n !1 we have =0 exactly. For p = :05, an extreme case that might arise if F had several levels, and n = 100 the probability of an 1 even total is .500027. For any realistic n and p, then, the b est case will have 2 E jI j=E I 0:5. 6 Table 2: Simple Randomization 3.2 Worst case imbalance If the chosen factor F were ignored in the randomization scheme then there 1 are two imp ortant p ossibilities for it's imbalance, dep ending on whether the overall assignmentscheme is based on simple randomization or restricted ran- domization.
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