Do you have to remove perfectly collinear independent variables prior to Cox regression?

Question

Suppose you have independent variables that change only with each new time step (and possibly others that change freely):

id   time   X_1   X_2   X_3     …   event
01      1   a_1   b_1   c_1     …       0
02      1   a_1   b_1   c_1     …       0
 ⁝       ⁝     ⁝     ⁝      ⁝      …       ⁝
01      2   a_2   b_2   c_2     …       ⁝
02      2   a_2   b_2   c_2     …       ⁝
 ⁝       ⁝     ⁝     ⁝      ⁝      …       ⁝
01      n   a_n   b_n   c_n     …       ⁝
02      n   a_n   b_n   c_n     …       ⁝

At time=1, all values of X_1 are equal to a_1 (which may or may not equal a_2, a_3, ...) for all ids. Similarly, at time=i, all values of X_1 are equal to a_i for all ids, and so on. Independent variables X_2 and X_3 also change only with each new time step. Other variables might change freely, but they are not important to the question. Are independent variables X_1, X_2, X_3 perfectly collinear? And should they be removed prior to Cox regression due to collinearity or otherwise (zero variance per time step)?

To be clear, all individuals have exactly the same value for any one of X_1, X_2, X_3 at any given time point.

Answer 1

Perfect multicollinearity among predictors should be dealt with in any regression, Cox or otherwise. Some software might just refuse to fit such data. Some functions are smart enough to find and remove enough predictors to ensure linear independence (perhaps silently), but you don't want to count on that and you presumably would prefer to make such a choice consciously yourself.

Multicollinearity doesn't arise from associations of values within single predictors like your X_1. It has to do with linear associations among predictors. There's no way to say in general whether the type of data structure you describe will lead to perfect multicollinearity among X_1, X_2 and X_3.

The problem in your scenario is that you might not gain any information from including covariates that have "zero variance" among members of the risk set at all event times in a Cox model. If all individuals at risk at each event time have identical values of a particular covariate $X_j(s)$, and that is true for all of the event times $s$, then you can't estimate a hazard for that covariate unless it has interactions with other covariates.

The vector of coefficient estimates $\hat \beta$ in a Cox model comes from solving the partial-likelihood vector score function $U$ as a function of the covariate-coefficient vector $\beta$ for $U(\hat \beta)=0$. Following Therneau and Grambsch on page 40, with p covariates and n individuals, the $p \times 1$ score vector is:

$$ U(\beta) = \sum_{i=1}^n \int_0^{\infty} [X_i(s) - \bar x(\beta,s)]dN_i(s),$$

where $\bar x(\beta,s)$ is a risk-weighted mean of the $X$ values over the risk set at time $s$ and $dN_i(s)$ is 1 at the event time for individual $i$ and 0 otherwise.

If the entire risk set at an event time $s$ has identical values of $X_j$ and that holds true for all values of $s$, then the $j^{th}$ component of the score vector is necessarily 0 regardless of the coefficient values $\beta$. There is thus no unique solution for $\hat \beta_j$. If $X_j$ is involved in interaction terms with other covariates, however, then estimates of those interaction coefficients could be possible.

This difficulty with modeling such covariates in a Cox model might be overcome in a parametric survival model. With a parametric model, covariates can contribute to the entire likelihood function over time, as summarized on this page, and thus allow for estimation of their associated coefficients if they vary over time. Unless they follow defined functions of time, however, time-varying covariates for an individual are modeled over a set of time spans, each of which is left truncated and is either right-censored or ends with an event. If you choose an accelerated failure time parametric model, handling left truncation requires some care as explained in this vignette of the R eha package.

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For reference of others reading this thread, know that time-varying predictors in a Cox model are typically formatted in a different way than shown, with one data row for each combination of individual and time period during which all covariate values are constant. That's sometimes called the counting-process format. Each row specifies the start time (left truncated) and end time (censoring/event) of each such time period, along with the covariate values and an event/censoring indicator. See the R time-dependence vignette for details.