156 Chapter 7 Supplementary Material Supplementary Methods Effect size estimation This section details how effect sizes associated with a given treatment effect were calculated for different outcome measures. S/F94 and sustained WHO ordinal scale improvement We used a logistic model with mortality as the dependent variable, and separately estimated relationships with S/F94 and WHO ordinal scale improvement, Here, pi is the probability of day 28 mortality for individual i, β0 and β1 are scalar coefficients, is β an n-dimensional vector of coefficients, yi is the outcome measure (S/F94 or sustained WHO ordinal scale improvement encoded as a binary variable taking values or ), and zi is an n-dimensional vector of covariates. Let be consistent estimators for β0 , β1 , β respectively. is defined as the corresponding predicted probability of day 28 mortality for patient i, We consider an individual-specific treatment effect given by for some function f taking values in [0,1] that may in principle depend on all covariates. The changes in the outcome measure associated with this treatment effect for individual i keeping all other covariates constant in Equation 1 and Equation 2 satisfy
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