Leveraging multiple endpoints in small clinical trials Robin Ristl Section for Medical Statistics, Center for Medical Statistics, Informatics and Intelligent Systems, Medical University of Vienna EMA FP7 small-populations workshop, March 2017 1 / 15
Reasons to use multiple endpoints in clinical trials in rare diseases Setting 1: Co-primary endpoints Some diseases are multi-faceted and we need to adress multiple endpoints for full characterization Main goal: Show efficacy in all co-primary endpoints Challenge: With small sample size, the probability to miss the main goal can be large and we need some fallback strategy Setting 2: Global tests Aim for conclusion of some overall treatment effect Use information from multiple endpoints to counteract low information from small sample Increased power compared to single endpoint test Challenge: Optimize global test for defined alternative 2 / 15
Hypothesis tests with multiple endpoints E.g. for two endpoints, we consider the following null hypothesis ◮ No effect in any endpoint (global null hypothesis) ◮ No effect in endpoint 1 ( H 1 ) ◮ No effect in endpoint 2 ( H 2 ) The probability for any false positive decision must be ≤ α The probabiltiy to identify a true effect (power) should be large To find optimized test procedures, we consider general multidimensional rejection regions. 3 / 15
How to find a multidimensional rejection region? Rejection region 1 ≤ 2.5% T 2 0 ≥ 97.5% −1 −1 0 1 T 1 T 1 and T 2 are the test statistics for endpoints 1 and 2 4 / 15
Setting 1: Co-primary endpoints Two endpoints are co-primary, if the main aim is to show an effect for both endpoints. Standard co-primary endpoint test: ◮ Perform a separate test for each endpoint ◮ If both tests are significant at level α , conclude effect in both endpoints ◮ Else, no conclusion Co−primary endpoint test Most powerful for main aim, but reduces to all or nothing reject both decision. z 1 −α ◮ E.g.: Test for H 1 may be highly significant while the T 2 0 test for H 2 is not significant. ◮ Then the standard test does z α not allow for rejection of any null hypothesis. z α z 1 −α 0 T 1 5 / 15
New method: Fallback tests for co-primary endpoints What can we learn if a co-primary endpoint trial does not achieve the main goal to reject all null hypothesis? We extend the standard co-primary endpoint test with a fallback option: Even if the main goal is not reached, there is the option to reject some null hypothesis. Important in small sample situtation, as the probability to miss the main aim may be high 6 / 15
A fallback test for two co-primary endpoints Diagonally trimmed Simes test reject H 2 reject both z 1 −α 2 z 1 −α T 2 0 reject H 1 z α z α 2 z α 2 z α z 1 −α z 1 −α 2 0 T 1 Retains power of standard test Adds decision rules to claim partial success Strict type I error rate control for arbitrarily correlated test statistics (if bivariate normal or t-distributed). 7 / 15
Further results A Fallback tests for three endpoints Adjusted p-values for the fallback tests General approach to combine simple fallback test to a more complex testing procedure R. Ristl, F. Frommlet, A. Koch, M. Posch, “Fallback tests for co-primary endpoints”, Statistics in Medicine, 2016, 35:2669-2686 8 / 15
Setting 2: Global tests for multiple endpoints The power to reject a global null hypothesis can be large compared to the power to show some endpoint-specific effect Conclusion on endpoint-specific effects requires extension to multiple testing procedure Challenge: Find powerful rejection region for global test, control type I error rate Proposed solution: Exact tests through multivariate permutation, optimization algorithms to find rejection regions We studied in particular optimal exact tests for multiple binary endpoint. 9 / 15
Example: Non-24-hour sleep-wake disorder Assume a study similar to Lockley et al., Lancet 2015, 386:1754-64 EP 1: Entrainment (synchronization of the master body clock to the 24-hour day) EP 2: Clinical response Assumed success rates for planning Treatment Control Endpoint 1&2 0.35 0.03 Endpoint 1 0.1 0.03 Endpoint 2 0.1 0.03 None 0.45 0.91 Observed blinded frequencies in the example Treatment Control Total Endpoint 1&2 blinded blinded 16 Endpoint 1 blinded blinded 4 Endpoint 2 blinded blinded 6 None blinded blinded 4 Total 15 15 30 10 / 15
Discrete null distribution found through permutation Null distribution 0 0.2 2.2 9.7 22.7 30 22.7 9.7 2.2 0.2 0 15 0.1 14 1.6 13 9 23.3 12 T 2 11 31.8 10 23.3 9 9 8 1.6 7 0.1 5 6 7 8 9 10 11 12 13 14 15 T 1 Test statistics T 1 , T 2 are the number of successes for endpoints 1 and 2 in the treatment group. Dark fields correspond to larger probability under the global null hypothesis. 11 / 15
Rejection region with maximal power under the assumed alternative Optimal power 0 0 0 0 0.5 3.6 14.9 31.2 32.4 15.1 2.4 8.7 15 14 31.2 13 37 12 18.5 T 2 11 4.2 10 0.4 9 0 8 0 0 7 5 6 7 8 9 10 11 12 13 14 15 T 1 Power to reject global null hypothesis in the example ◮ Optimal joint permutation test: 81% ◮ Fisher exact tests with Bonferroni correction: 61% ◮ Single endpoint Fisher exact test: 59% 12 / 15
Further results Optimally weighted Bonferroni tests Construction of multiple testing procedures Adjusted p-values Fast greedy algorithm for approximate solution R. Ristl, X. Dong, E. Glimm, M. Posch, “Optimal exact tests for binary endpoints”, arXiv:1612.07561 13 / 15
Points for discussion 1 In co-primary endpoint trials, what is the impact of a partial claim of success on regulatory decision making? 2 In which situation is it sufficient to show a global treatment effect on multiple endpoints? 14 / 15
References R. Ristl, F. Frommlet, A. Koch, M. Posch, “Fallback tests for co-primary endpoints”, Statistics in Medicine , 2016, 35:2669-2686 R. Ristl, X. Dong, E. Glimm, M. Posch, “Optimal exact tests for binary endpoints”, arXiv:1612.07561 R. Ristl, S. Urach, G. Rosenkranz, M. Posch, “Methods for the analysis of multiple endpoints in small populations: A review”, submitted to Statistical Methods in Medical Research 15 / 15
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