Statistical Details for Missing Values in the Outcome
The Causal Treatment personality automatically handles missing values for the outcome Y when the treatment is binary. However, when data are missing for any predictor variable in the specified treatment or response models, the corresponding observations are excluded from analysis.
Missing values for the outcome Y are handled by modeling the probability of missingness (or censoring) and then applying an inverse probability of censoring weight (IPCW) in addition to the inverse probability of treatment weight (IPTW). The steps for this calculation are as follows:
1. Fit censoring models.
Two censoring models are fit, one for each level of the treatment. Both models are logistic regressions with the indicator of missingness in Y as the outcome. The predictors are the variables that are specified in the treatment model.
2. Derive the censoring weight.
The censoring weight is stabilized when only the treatment model is specified, the estimation method is inverse probability weighting with ratio adjustment (IPWR), and stabilized weights are selected. Otherwise, the censoring weight is not stabilized.
The stabilized IPCW is defined as follows:
wsIPCW = p(C)/p(C|Z)
where
C is the indicator of missingness in Y
Z is the set of pre-treatment covariates specified in the treatment model
p(C) is the proportion of observations with missing Y values in the given treatment level
p(C|Z) is the probability that Y is missing given covariates Z, calculated using the fitted censoring model that corresponds to the given treatment level.
The non-stabilized IPCW is defined as follows:
wIPCW = 1/p(C|Z).
3. Multiply the two weights.
The IPTW is calculated as previously described. See Stabilized Weights. The product of the IPCW and IPTW weights is then calculated.
4. (Perform only if a weight truncation threshold is specified.) Truncate the weight according to the threshold value.
Truncated weights can be applied. See IPW Extreme Weight Truncation Threshold. If the product calculated in step 3 is greater than the specified threshold, the weight is replaced with the threshold value.