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For a continuous event time $T$ with probability density function $f_{\theta }$, survival function $S_{\theta }$ and hazard function $h_{\theta }=f_{\theta }/S_{\theta }$, right censoring records $Y=\min (T,C)$ and $\delta =1_{\{T\le C\}}$. Suppose $T$ and $C$ are conditionally independent given the observed covariates, suppressed here. Write $g_{\eta }$ and $G_{\eta }$ for the censoring-time probability density function and survival function. The joint observed-data likelihood contribution isAn event contributes its event-time probability density function times the probability of remaining uncensored until then. A censored observation contributes its censoring-time probability density function times the probability of remaining event-free until then.

If the censoring model has separate parameters $\eta$ with no constraints linking them to $\theta$, its factor can be omitted when maximizing over $\theta$. The event-model part for independent subjects is thenwhere $H_{\theta }$ is the cumulative hazard function. Independent censoring alone does not permit dropping a censoring term that itself contains the event-model parameters. The familiar event-versus-censoring formula is thus a factorization under an ignorable observation mechanism, rather than an unconditional rule for every missing event time.^[ref]

Variation in the treatment effect with a baseline characteristic. An interaction tests differences in treatment effects; significance in one subgroup and nonsignificance in another is not itself evidence for an interaction.

A previous outcome may affect a subsequent outcome after controlling both measured covariates and persistent unobserved heterogeneity. This is true state dependence. An observed lagged-outcome association alone does not establish it, because omitted individual propensities can create apparent persistence.

Tyler J. VanderWeele develops methods and foundations for causal inference. With Ilya Shpitser, he examined how to define a confounder so that the definition supports valid adjustment decisions.^[ref]

Their preferred definition concerns a pretreatment variable belonging to some minimally sufficient adjustment set. This connects the variable's role to elimination of confounding, rather than merely to prediction of exposure and outcome. The distinction matters because a predictive variable can be unsuitable for adjustment, and the contribution of one covariate depends on what else is adjusted for.^[ref]

Persistent unmeasured differences between individuals can generate outcome dependence after adjusting for observed covariates. A random intercept is one representation. Conditional independence given that latent variable need not persist after it is integrated out; apparent history dependence can therefore occur without true state dependence.

A zero-inflated Poisson regression specifies a structural-zero probability $\pi$ and a susceptible-component Poisson distribution mean $\mu$. In this parametrization,The mean is $(1-\pi )\mu$. Logistic predictors for $\pi$ and logarithmic predictors for $\mu$ allow covariates to affect component membership and count intensity differently. A susceptible individual can still generate a zero count; susceptibility is not equivalent to an observed positive response.