New re-randomization tests

The insights we gained from the behavior of in Section 4.3 prompted us to propose a weighted version of the fixed-entry-order re-randomization test for minimization with unequal allocation. An obvious way is to correct the fluctuation and use ̃ ∑ ( ̅) as a test statistic. Even though explicit expressions for , are unavailable, they can be well approximated through Monte Carlo simulations because the re-randomization mechanism is known. Therefore S_ can be calculated after have been evaluated by simulation. We propose 10,000 re-randomizations to estimate which can then be utilised in the calculation of the test statistic for each re-randomization. Ideally the same starting seed for running the re-randomizations to estimate should be used when the re-randomization program is run again with the addition that the test statistic is calculated.

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However we also need to consider the impact of the behavior of on the variance.

Note that in re-randomization tests, the responses are considered as fixed quantities. The observed test statistic is calculated as

̅ ̅

∑

So each observation contributes equally in the sense that the ‘weights’ or coefficients for the observed responses are the same. Therefore we want also equal weights of the observed responses in re-randomized tests.

First consider the behavior of the re-randomization test under the complete randomization. The variance of the re-randomization test given by formula (4.3) can be computed as (4.6) ( ) { ∑ ̅ ∑ ∑ ̅ ̅ }

Here the subscript  in and indicates that these quantities are evaluated under the re-randomization distribution. Note that is a Bernoulli random variable with the success probability_j. For minimization, is not zero although the exact form is hard to derive.

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Now assume that the trial was conducted using complete randomization. In this case, { ,T j_j 1,..., }n can be considered as independent and identically distributed binary random variables. Therefore the second term of formula (4.6) is zero and the variance is

(4.7) ( ) ∑ ̅ ( ∑ ̅ )

In (4.7) each ̅ carries the same weight so that is proportional to the sample variance _{∑ ̅}

. This is obviously a desired property that makes comparison with observed test statistic valid.

Now consider a variant of the complete randomization procedure. Assume subjects are independently randomized to two treatments with predefined but unequal allocation probabilities . In this case, , j=1,…,n} are independent but non-identical. For trials randomized with this procedure,

( ) ∑ ̅ ( ) (∑ ̅ )

We see that each

(y

_j

y)

2 is not equally weighted to the calculation of the variance unless is constant. When is close to 0, subject j is severely down-weighted. When , subject has the largest weight. Obviously, in order for each subject response to have equal influence to the conditional variance, ̅ should be re- weighted.

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Therefore we define a mean-centered and information-weighted re-randomization test as, (4.8) _{∑ ̅} ∑ ̅ √ ( ) √

Under the general minimization procedure with no covariates, the weighted randomization test S_wt is centered around zero. Its variance can be calculated using (4.6), which comprise two terms. The first term now becomes the sample variance and the second term is a linear combination of the terms ̅ ̅

, which is generally intractable. Nevertheless, the second term has an expectation of 0 under the null hypothesis. In our simulations, we indeed observed small values for the second term. Therefore the variance of _{is dominated by the first term in many cases.}

Finally for a minimization procedure with covariates and where the responses follow model (4.1), the ANCOVA is a valid test when a correct model is specified between the response and covariates [112], and the simple -test, without any covariate, is conservative in terms of type I error rates. As the covariate imbalance is minimized in covariate-adaptive minimization, the weighted randomization test without adjusting for covariates may still yield valid results. Alternatively, we propose to perform the re- randomization test on covariate-adjusted residuals, which can be obtained by fitting a regression model on baseline covariates, but without using the treatment indicator [95]. Note that in theory the covariate-adjusted re-randomization test remains valid even the fitted model is misspecified. Frequently, covariate-adjusted re-randomization inference

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can reduce bias and increase efficiency by accounting for imbalanced influential covariates due to finite samples.

4.4.2 Alternative re-randomization test using random entry order

The weighted re-randomization test described in Section 4.4.1 keeps the original subject entry order during the re-randomization process. When the subject entry order does not convey any information, random entry order re-randomization test can be performed and we expect this re-randomization test to center at 0 and perform well. However, when subject entry conveys certain information due to the temporal trend, this test may be invalid as this information is lost after the permutation of the entry order. We mainly use this random-entry-order test for numerical comparisons.

In document Statistical analysis of clinical trial data using Monte Carlo methods (Page 98-102)