2.3. Estimation of Transition Probabilities P(Yi,j+1a¯=k|Yi,ja¯=k′)

Robins (2000) describes that the causal parameter of MSMs β^k,k′ may be estimated semiparametrically by weighted estimating equations, where the observed data are weighted using subject-specific, time-specific inverse probability of treatment weights (IPW) [5]. IPW accounts for time-varying confounding by creating a pseudo-population in which covariates at each time of exposure are not associated with the subsequent exposure, but the transition probabilities P(Yi,j+1a¯=k|Yi,ja¯=k′) remain the same as in the source population. The observed data association model is

and weighted estimating equations are

where Y_i,j+1,k = I(Y_i,j+1 = k), and estimated stabilized weights ω^i,j=Πt=0jP^(Ai,t=a|A¯i,t−1)Πt=0jP^(Ai,t=a|A¯i,t−1,L¯i,t) account for time-dependent confounding. Stabilized weights are estimated by fitting two models. The first one models the likelihood of the observed exposure A_i,j given past exposure Ā_i,j−1 and time-dependent covariates L¯i,j. It informs the denominator of the stabilized weight and is required to be correct to consistently estimate the causal parameter. The second model for the numerator, which is not required to be correctly specified for consistent estimation, is based on a model for exposure A_i,j given past exposure Ā_i,j−1. In the pseudo-population the exposure-outcome relationship is no longer confounded by time-dependent exposure so that the associational parameter θ^k,k′ is equivalent to the causal parameter β^k,k′[5].

For binary exposures, the IPW may be estimated by pooled logistic regression. The probabilities of categorical exposures of greater than two levels may be estimated by pooled multinomial logistic regression. We take this approach in modeling SBP as an exposure, defining clinically relevant levels for normal, moderately elevated, and elevated SBP as used by prior investigators. For continuous exposures, inverse density weighting is possible by specifying a density function for the exposure distribution and estimating components of that density by a standard modeling approach. An alternative approach for continuous exposures is to use a pooled multinomial model considering quantiles of the continuous exposure [11]. For each subject, predicted probabilities of observed exposure level at each time are calculated, and the product is taken within subject over time to estimate ω^i,j. In categorizing continuous exposures, investigators should be conscious of a tradeoff between the positivity and consistency assumptions. For continuous exposures, it is less likely that every level of the exposure occurs for each level of observed treatment and confounders at each time, thus raising concerns of potential positivity violations. In reducing treatment levels through categorization, it is more plausible that each treatment level is possible at each level of observed treatment and confounders, but violations of the consistency assumption are possible as different continuous exposure levels that result in the same categorical exposure variable could potentially have different implications for potential outcomes, which in turn would violate consistency as a lack of multiple versions of treatment[12], and is particularly relevant for exposures that are biological characteristics[13]. We therefore caution investigators with these considerations when categorizing continuous exposures.

When subjects leave the study prior to the end of the observation period θ^k′^,k are subject to selection bias when exposures and time-dependent factors are predictive of study withdrawal [5]. In this case, estimation equations are,

and weights ω^∗i,j=Πt=0jP^(Ai,t=a|A¯i,t−1,Ci,t=0)Πt=0jP^(Ai,t=a|A¯i,t−1,Ci,t=0,L¯i,t)×Πt=0jP^(Ci,t=0|A¯i,t−1,Ci,t−1=0)Πt=0jP^(Ci,t=0|A¯i,t−1,Ci,t−1=0,L¯i,t−1) are the product of the stabilized inverse probability of the observed joint exposure and stabilized inverse probability of being uncensored. Standard errors and confidence intervals of parameter estimates β^k′,k may be calculated using the robust variance estimator as for Generalized Estimating Equations [14], which is available in standard statistical software packages and provides asymptotically conservative standard error estimates provided that the weights are consistently estimated [15], or by a resampling technique such as the nonparametric bootstrap [16].