Fixed covariate, time dependent covariate, time varying covariate, and post-randomization covariate

A covariate is a variable that is possibly predictive of the outcome under study. A covariate may be of direct interest or it may be a confounding or interacting variable. Typically, when we use the term of ‘covariate’, we refer to the fixed covariate – a variable whose value will not change over the time. The fixed covariates include the demographic information (gender, age, race, …) and baseline characteristics (weight, height, baseline measure,…). Since ‘covariate’ typically means ‘fixed covariate’, we often drop the term ‘fixed’.

In contrary to the fixed covariate, time depedent covariate or time varying covariate (exact the same thing) refers to a covariate that is not necessarily constant through the whole study and has different values at different time points.  

There are many examples of the time dependent or time varying covariate in clinical trials or observational studies. For instance, if one wishes to examine the link between area of residence and cancer, this would be complicated by the fact that study subjects move from one area to another. The area of residency could then be introduced in the statistical model as a time-varying covariate.

When we analyze the CT lung densitometry data, we need to deal with the measurement of the lung volume – a reflection of the inspiration level. The CT densitometry measure is negatively correlated with the effort of the inspiration level. If we inhale more air into the lung, the lung density measure will be smaller and vice versa. In order for quantitative CT lung density to be valid endpoint, the corresponding measure of the lung volume (inspiration level) needs to considered and adjusted. If the outcome measure of CT lung density is measured at different times, there will also be corresponding measures of lung volumes at various times. The lung volume measure is considered as a time dependent covariate.

In clinical trials, the timing of the randomization is a critical point. The covariate can be separated into two groups: covariate measured or existed prior to the randomization and covariate measured after the randomization. The fixed covariates are measured or known prior to the randomization. The covariate measured after the randomization is called ‘post-randomization covariate’. The post-randomization covariate is usually also the time-dependent covariate.

When the time dependent covariate exists, we may be very tempted to include the covariate in the statistical analysis model. However, such approach could be biased if the time dependent covariate involves measures post-randomization.

A paper by Chen et al ‘A Note on Postrandomization Adjustment of Covariates” explained the issue:

“Examples include adjusting rescue therapies to explore the underlying difference of treatments which would have been observed in the absence of rescue medication, adjusting the level of patient compliance to estimate the treatment effect at the same compliance level, adjusting a postrandomization biomarker to evaluate the surrogating status of the biomarker to the clinical endpoint, and adjusting an intermediate endpoint or an endpoint other than the primary one to explore the therapeutic mechanism of one intervention.”

In its “ Guideline on adjustment for baseline covariates”, EMA explicitly opposes the use of adjustment for post-baseline covariates. In Section 4.2.5 (covariates affected by the treatment allocation), it states:

“A covariate that may be affected by the treatment allocation (for example, a covariate measured after randomisation such as duration of treatment, level of compliance or use of rescue medication) should not normally be included in the primary analysis of a confirmatory trial. When a covariate is affected by the treatment either through direct causation or through association with another factor, the adjustment may hide or exaggerate the treatment effect. It therefore makes the treatment effect difficult to interpret. However, such covariates (e.g. duration of treatment) might be included in secondary (exploratory) analyses and might offer the sponsor useful insights during the drug development process. Alternatively, subgroup analyses might offer similar insights.”

 The main concern for using the post-randomization covariates is the potential effect of treatment on the covariate measures. The covariates could become outcome measures or dependent variables instead of the independent variables. The post-randomization covariate measures may also be biased if there is inadequate blinding of the treatment allocation. In reality, it is not possible to defend that a post-randomization covariate is really a covariate and not impacted by the treatment. It is also difficult to confirm whether or not if the treatment effect of the experimental drug is mediated through the post-randomization or time-dependent covariates.

FDA does not have any guidance similar to EMA’s guidance on the use of post-randomization covariates. However, it is very rare to see the use of post-randomization covariates in the analysis of the primary efficacy endpoint in NDA or BLA submissions. There may be cases where post-randomization covariates are used in exploratory or ad-hoc analyses or used in analyses for non-randomized, observational studies.  

In sNDA ODAC Briefing document for Proscar, post hoc analysis was performed to include the post-randomization covariates. However, the briefing documents also commented “Adjustment for post-randomization variables (cores and volume) that are affected by treatment can introduce confounding bias and complicate causal interpretation.”

It has been noted that the time-dependent covaraites are more commonly used in survival analysis. See paper by Dr Lin “TIME-DEPENDENT COVARIATES IN THE COX PROPORTIONAL-HAZARDS REGRESSION MODEL”

Dealing with the time-dependent variable is a challenging statistical issue. The fundamental statistical problem is how to properly adjust for post-randomization variables, since, if the treatment has an effect on post-randomization variables, standard adjustment by regression modelling is susceptible to selection bias. The current proposal is to utilize Causal-Effects approach.  Dai et al had a pretty good introduction on this topic in their paper “Partially hidden Markov model for time-varying principal stratification in HIV prevention trials“. For details about the concept of Causal-Effects, the report of ”From Neurons to Neighborhoods: The Science of Early Childhood Development” by National Academy Press is a pretty good resource.