A Unified Approach to the Statistical Evaluation of Differential Vaccine Efficacy

Transcription

1 A Unified Approach to the Statistical Evaluation of Differential Vaccine Efficacy Erin E Gabriel Department of Medical Epidemiology and Biostatistics, Karolinska Institutet, Stockholm, Sweden Dean Follmann Biostatistics Research Branch, NIAID, NIH, USA JSM Vancouver 2018: July 30, 2018

2 1 Background 2 Unified 3 Mechanism (maybe) 4 Conclusions

3 Notation We are considering a randomized clinical trial where we can observe an intermediate outcome and a clinical outcome which are matched, the same as the vaccine, or mismatched. z: randomized treatment Y m (z): potential outcome under treatment z for antigen m S(z): potential candidate surrogate outcome under treatment z for a matched antigen m, realized level s 1. z: vaccine and placebo m and m for matched or mismatched (M = 1 and M = 0) We are going to ignore the problem of observing both the potential outcomes!

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5 Sieve A very simplified version of a Sieve analysis: Y = β 0 + β 1 z + β 2 M + β 3 M z + ɛ assuming this structural model and under the standard consistency and SUTVA and ignoring competing risk β 3 0 is evidence of a sieve effect. Gilbert et al. (1998,2001), Juraska and Gilbert (2013), Follmann and Huang (2017)

6 CoP A very simplified version of a Correlates of Protection analysis: Y = γ 0 + γ 1 z + γ 2 s 1 + γ 3 s 1 z + ɛ assuming this structural model and under the standard consistency and SUTVA γ 1 + γ 3 s 1 = E[Y (1) s(1) = s 1 ] E[Y (0) s(1) = s 1 ] γ 3 0 provides evidence of at least some value as a CoP. Follmann (2006), Gilbert and Hudgens (2008), Gabriel et al. (2014) Gabriel et al. (2017)

7 CEP(s 1 ) Causal effect predictiveness curve, RD(s 1 ) = E[Y (1) s(1) = s 1 ] E[Y (0) s(1) = s 1 ] For a good target of vaccine improvement we want RD(s 1 ) to vary in s 1, but one could also see how you would want RD(0) = 0 and RD(s 1 ) > 0 for all s 1 > 0. (Gilbert and Hudgens 2008)

8 Unified Setting Parasitemia X(1) = Actual/counterfactual immune response to vaccine

9 Unified Model Three way interaction model: Y m = γ 0m + γ 1m z + γ 2m s 1 + γ 3m s 1 z + ɛ Y m = γ 0m + γ 1m z + γ 2m s 1 + γ 3m s 1 z + ɛ

10 Sieved CoP or Differential Sieve if γ 3m γ 3m then there is a different modification of the CEP by the CoP for matched and mismatched infecting agents. Looked at differently: if γ 3m γ 3m the sieve effect varies with S(1). A sieve effect for subjects with (or that would have had) a measurable immune response S(1) > 0, but no sieve effect when when S(1) = 0.

11 Efficiency versus standard CoP When there is the same CoP for both matched and mismatch infections, we can lose efficiency using the unified models. However, when there is a CoP for matched alone we can have large gains in efficiency Assume exactly half match and half mismatch infections and constant variance. The expected Wald statistic for the standard CoP when there is a CoP only for match is: while for the unified analysis it is: ((γ 3m + 0)/2)/ (ν/(2 n)) (γ 3m )/ (ν/(n)) Therefore 1/2 the sample size is needed for the same power.

12 Efficiency versus standard Sieve When there is a measurable sieve effect on average, there are little to no efficiency gains of unified sieve and there can be losses. when there is no marginal sieve there are large gains in efficiency using the unified model over the standard model. For a specific value of S(1) = s the sieve effect test statistic is given in terms of the unified model by: γ m1 γ m 1 + (γ m3 γ m 3) s Var(γm1 γ m 1 + (γ m3 γ m 3) s)

13 Clues to mechanism: Concept

14 Conclusions We can combine the standard CoP analysis and the standard Sieve analysis into one unified analysis: efficiency over CoP in some scenarios efficiency over Sieve in some scenarios insight into causal pathway of vaccine efficacy under potentially easier to verify conditions Most real life settings are more complicated!

15 Representative References Dean Follmann and Chiung-Yu Huang. Sieve Analysis Using the Number of Infecting Pathogens. Biometrics, doi: /biom Erin Gabriel, Michael Sachs and Mary E Halloran. Evaluation and comparison of predictive individual-level general surrogates. Biostatistics (3): Tyler Vanderweele. Simple relations between principal stratification and direct and indirect effects. Statistics and Probability Letters