Global Statistical Test

 The Global Statistical Test (GST), also known as the Wei-Lachin test, is a nonparametric method designed to simultaneously evaluate multiple outcomes (e.g., clinical endpoints) in a multivariate fashion. It is especially useful in clinical trials where patients have several related outcome measures and you wish to assess an overall treatment effect.

 Key Concepts

1. Purpose

• To combine several endpoints (continuous, ordinal, or binary) into a single global test statistic.

• More powerful than univariate tests when treatment effects are consistent across outcomes.

2. Typical Applications

• Neurology trials (e.g., stroke studies)

• Ophthalmology trials (e.g., visual acuity and field)

• Rare disease trials with multiple small sample endpoints

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🧪 Test Mechanics

Step-by-Step:

1. Rank-Based:

   • Each endpoint is converted into a rank across subjects.

   • Ties are handled via average ranks.

2. Global Rank Vector:

   • For each patient, construct a vector of their ranks for each outcome.

3. Statistical Model:

   • Use the generalized estimating equations (GEE) or Hotelling’s T²-type statistic on the rank vectors.

   • Null hypothesis: No overall treatment effect across all outcomes.

4. Test Statistic:

   •  General Form of the Test Statistic

     A common structure for the global test statistic is:

     $$T_{global} = \sum_{j=1}^{K} w_j T_j$$

     Where:

     • $K$: number of endpoints

     • $T_j$: test statistic for the $j^{th}$ endpoint (usually a t-statistic, Z-score, or rank-based statistic)

     • $w_j$: weight for the $j^{th}$ endpoint (e.g., equal weights, inverse-variance weights, or based on clinical importance)

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     🔹 Example: Wei-Lachin Procedure (1984)

     If you're using the Wei-Lachin test, the global test statistic may take the form:

     $$T_{WL} = \mathbf{c}^T \mathbf{S}^{-1} \mathbf{U}$$
     

     Where:

     • $\mathbf{U}$: vector of test statistics (e.g., means or differences between treatment groups for each endpoint)

     • $\mathbf{S}$: covariance matrix of the test statistics

     • $\mathbf{c}$: contrast vector (often $[1,1,\dots ,1]\; if\; testing\; for\; overall\; superiority)$

     This leads to a quadratic form, often compared to a chi-squared distribution with degrees of freedom equal to the number of endpoints.

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     🔹 Permutation-Based Global Test Statistic

     In permutation-based methods (e.g., Li et al., 2020):

     1. For each permutation of treatment assignment:

        • Recalculate the LS-estimate-based t-statistics for each endpoint:

          $$t_j = \frac{\hat{\mu}_{1j} - \hat{\mu}_{0j}}{SE_j}$$

          where $\hat{\mu}_{1j}, \hat{\mu}_{0j}$ are LS means for treated and control.

     2. Sum or combine these:

        $$T_{sum} = \sum_{j=1}^{K} t_j^{*}$$

        (Reversing the sign for some endpoints if higher values mean worse outcomes)

     3. Compare the observed test statistic to the permutation distribution to obtain a 2-sided p-value:

        $$p = 2 \times \frac{\#(T_{perm} \geq |T_{obs}|)}{N}$$

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     🔹 When to Use What

     Scenario	Test Statistic
     Multivariate Normal Assumption	Wei-Lachin Quadratic Form
     Non-parametric or ranks	Sum of ranks or Wilcoxon-type statistics
     Mixed-type endpoints or small sample size	Permutation-based test statistic

5. Inference:

   • Under the null hypothesis, $T$ follows a chi-square distribution with degrees of freedom equal to the number of endpoints.

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📊 Advantages

• Handles multiple endpoints simultaneously.

• Robust to distributional assumptions (nonparametric).

• Can accommodate missing data under MCAR or MAR conditions (if implemented carefully).

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⚠️ Limitations

• Assumes all outcomes are measured on the same patients.

• Cannot easily interpret individual endpoint contributions.

• May lose power if treatment effects are in opposite directions.

Example:

The primary analysis is investigating the totality of evidence of  Drug A over active comparator through global statistical test on multiple endpoints consist of Endpoint1, Endpoint2, Endpoint3, and Endpoint4 on Functional Assessment Analysis Set. The Wei-Lachin procedure will be used to perform a global statistical test on multiple endpoints through permutations.

The analysis will be conducted using permutation test, where 10,000 re-randomizations will be performed according to a randomization procedure. For each re-randomized dataset, the LS-estimate of t statistic for each endpoint, defined as the LS estimate of mean over standard error of treatment difference between the treatment group, will be calculated using the analysis method described in xxx. The sum of t statistics will be computed. The signs of the t statistic for some endpoints will be reversed to align the direction across all endpoints, if needed. The 2‑sided nominal p-value will be calculated as twice as much as the proportion of the permutated treatment differences that are at least as extreme as the observed sum of z-scores as calculated from the propensity score weighting method of the above listed endpoints.

The global statistical test will include test results on the Endpoint1, Endpoint2, Endpoint3, and Endpoint4 change from baseline at 1 year.

 

Hypothesis

For the primary analysis comparing Drug A treated subjects and active comparator treated subjects:

·       Null hypothesis: there is no treatment difference in any included endpoints (ie, Endpoint1, Endpoint2, Endpoint3, and Endpoint4 from baseline at 1 year).

·       Alternative hypothesis: there is treatment difference in included endpoints.

Even though the alternative hypothesis is 2-sided, only superiority of Drug A over active comparator will be of interest.