# Parametric survival with correlated predictors

I have observational data on censored failure data. I am trying to perform a Parametric Survival as a function of variable A and B, where A is the time spent under control strategy 1, and B is the time spent under a newer control strategy 2. A + B = survival time, and A and B are negatively correlated, meaning larger values of A tend to result in smaller values of B and vice-versa. I have two questions: (1) is anyone aware of a survival model that could account for A and B as predictors, and (2) are there strategies for dealing with multicollinearity in survival models?

• Welcome to the site. I don't see an actual question here. How are you modeling times $A$ and $B$ as exposures? You can't adjust for "time under A/B" as a covariate. I use time varying covariates all the time, I would use an indicator variable for "under condition B" in a Cox model, participants enter with that value as 0, then when B switches on, I censor them, reenter them in the model. with the indicator set to 1. But if everyone fails under B, the HR is $\infty$. Perhaps you mean to model cumulative exposure? – AdamO Apr 10 '18 at 13:53
• Hi @AdamO, thanks for the feedback. The problem I have with the Cox model is that survival depends on the 'length' of time in A and B, not just that one entered one phase and then the other. I thought about discretization of the variables, but I was hoping that there was a way to include them as continuous. – Paul Govan Apr 10 '18 at 14:57
• Is this based on an actual data analysis problem? Describe the design better. What is the question? As I mentioned, cumulative exposure has a separate methodology. – AdamO Apr 10 '18 at 15:01
• @AdamO edited my question to hopefully clarify my problem. I'm not familiar with cumulative exposure. Is there a reference you could point me to? – Paul Govan Apr 10 '18 at 18:12
• I think Survival Analysis by Hosmer, Lemeshow, May has an excellent discussion on time-varying covariates and modeling their interaction with time to draw different types of inference. The simplest way of modeling a cumulative exposure is modeling $x = 0$ if $A$ otherwise $x=t$ where $t$ is the amount of time with $B$. A single record must be represented with many rows depending on the time scale of interpretation (usually days or something similar). – AdamO Apr 10 '18 at 18:28

Your issue is not one of multicollinearity. There is no survival analysis where you can model the time-to-event as a function of the total amount of time spent in a certain exposure category. The exception would be a retrospective assessment: so , for instance, once a subject enters condition B you can condition on time-spent in condition A. Otherwise it is impossible for survival models to "peek forward" in time. It is the same fallacy as observing that: patients who die while hospitalized, those who receive emergency surgeries have better survival; and then inferring that the surgery is protective: those who underwent operations merely lived long enough to make it to the operating table. You can find a thorough discussion of a data analysis where this was addressed in the book Survival Analysis by Hosmer, Lemeshow, and May in the section Time Varying Covariates.

I judge by the problem description that there is some difficulty describing the hypothesis you are trying to test. At the moment, a survival model set up in the following fashion:

$$\log (\lambda_t | t_a, t_b) = \log (\lambda_{0,t}) + \beta_1 t_a + \beta_2 t_b$$

where $t_a$ is the time of A and $t_b$ is the time of B has no real interpretation.

The "survival time" A+B seems to be an artifact of design and has no real application: all I can infer is that a person spends some time in A and they spend some time in B. It sounds like there is a further deficiency in the design because nobody spends time in B and then in A.

In experimental settings, learning--or growth--describes a tendency for tasks to take less time when repeated, either because the subject has learned some tricks, or because they are losing patience. If that's the case, you may recover information by designing a proper balanced cross-over design where subjects are assigned to B and then to A. However, since subjects aren't randomized to the A-B sequence vs. the B-A sequence, other forms of biases may be found.

To formally compare which experimental setting a subject spends more time in, you merely have to compare survival times. Since there is no censoring, you can use a paired T-test. Alternately, you can use a sign test for paired data. You can also generate Kaplan-Meier curves for paired data and use a log-rank test, although I'm not sure about the efficiency of the analysis since it is typically used for independent data, but the paired design is balanced.

• Thanks for the detailed answer and insights. To be clear, this is observational/field research, and I have limited control over certain aspects of the data (e.g. exposure A/B times). Yet the question has been posed: what is the effect of A and B on the life of the product? I think the question is valid, but it doesn't sound like current models are set up for this type of problem. I'll do some more digging into Survival Analysis, but it sounds like a shortcoming in the observational research. – Paul Govan Apr 12 '18 at 14:20
• @PaulGovan I hear where you're coming from. But I might hazard (no pun intended) that we do a lot of epidemiology with observational data. These data determine that smoking kills, that hormone therapy causes cancer, and so on: those findings did not come from clinical trials in humans. You have to set up a probability model for the failure process using rigorous methods and scientific knowledge. We can't just bandy with "A"s and "B"s. You will often see, on this site, that I am the one who's asking: What is A? Air pollution? Austerity era? Pre-policy test rates? The data aren't the issue here. – AdamO Apr 12 '18 at 14:41
• Totally agree, there's value in this type of data. Just more challenges as well. – Paul Govan Apr 12 '18 at 18:59