Survival Analysis, Cox Regression in randomized trial vs. observational study and propensity score matching

In randomized clinical trials in the efficacy part, often survival analysis is used to analyze the time-to-event data. Since it is randomized (if randomization was done properly) one can assume that there is no confounding due to the balancing of the characteristics across the treatment groups. I quite often saw Kaplan-Meier plots and Cox regression. Of course one can put variables like age into the Cox regression model, because likely age will have an effect. This is Ok, as we have randomized, so the age will be distributed equally across the treatment groups.

In an observational study (assume we have the same setting, just no randomization) one can do the same and indeed Cox Regression is used to analyze observational studies. Cox regression allows to "adjust for confounding effects of other variables". Furthermore:

In order to obtain an effect estimate adjusted for confounders when analyzing survival data, one could use Cox regression analysis. The identification of potential confounders has been described extensively in a previous paper in this series [6,7].

As mentioned before, within our clinical example, one could suspect that age may obscure the association between eGFR at the start of dialysis and mortality because patients who start dialysis at higher eGFR levels may be older and for that reason have a higher mortality. Therefore, the association between eGFR at the start of dialysis and mortality was adjusted for the variable ‘age at the start of dialysis’. In this case age was entered as a second variable into the Cox regression model.

The output of the unadjusted and adjusted Cox regression analyses of model 1 is presented in table 3. In most statistical packages the output of the Cox regression analyses provides at least a HR, with its 95% CI and an estimate of the regression coefficient β. The β estimate is directly related to the HR because HR equals eβ. Thus the HR and β provide information on the strength of the association between eGFR and mortality. When comparing the HR or β of eGFR of the unadjusted model (HR = 1.30; β = 0.26) and adjusted model (HR = 1.21; β = 0.19) it is possible to judge how strong the confounder age affected the association between eGFR at the start of dialysis and mortality. The HR and β of high-medium eGFR in the unadjusted model are different from those in the adjusted model, meaning that age is a confounder in the association between eGFR at the start of dialysis and mortality.

1. Now I don't get why in both settings Cox regression is used?

Cox regression does make it possible to adjust for confounders. Randomization has the advantage that it allows to balance the observable plus the unobservable characteristics equally across the treatment groups. Cox regression does not do this, because we can only put in variables we observe. 2. Is it that?

Cox regression in an observational study might tell me that age does have an influence. When I put it into the model "I control for it". 3. If that was the only variable to worry about it and we can assume for the rest an "ideal world" – so age was the only potential counfounding variable – does this mean that at the end I get a perfect true estimate in terms of I completely removed the confounding out of it and in this case (only this variable makes the confounding) I get the "same result" as with randomization? 4. I don't have to randomize, because the model does the job? Assuming there are no further confounders and especially no unobservables.

Cox regression in a randomized trial where for example age is equally distributed / balanced across the treatment groups still might show me that age does have an influence, so I put it into the model and it has a significant estimate. What is the difference to the setting of having the same study as an observational study and doing the same – age as a covariate in the Cox regression – and getting the same result – age does have a significant estimate. So no matter if randomized or observational and therefore no matter if age is balanced or not: in the Cox regression using it as a covariate is something else. I don't get this point somehow.

5. I thought that if Cox regression tells me that age has an influence, so I put it into the model and get a significant estimate, this tells me that age has an influence and if the age is not distributed equally in my observational study this tells me that my analysis is worthless? Or: 6. does the Cox regression "solve this issue" – that the age is not equally distributed across treatment groups in my observational study (and assume age really is not equally distributed) – but since I take it into my model I adjust for this confounders and I can use the final estimate and I do have a good estimate for the treatment effect? 7. If age in reality has no influence then no matter if it is equally distributed or not in my observational study, when I put it into my Cox regression model it will show me no significance and the estimate will be the same as when I use the same model but without the age as a variable (to be specific: covariate) in it?

I do not get the connection of the Cox regression in an observational study to propensity score matching. I know what propensity score matching is and what it does (or at least I think so). 8. Now how is propensity score matching connected to the Cox regression? 9. When I have a Cox regression in an observational study:

• **when (in which circumstances) and **
• why,
• **should I do **
• do I have to do
• am I able to do

propensity score matching?

I could perform a propensity score matching before doing the Cox regression. So then the covariates are balanced. 10. But what advantage does this have? 11. How does that change the Cox regression? 12. After propensity score matching I still have the same variables, but now I can run the Cox regression without these covariates, because if I would put these into my model there would be no significance and the estimate of my actual variable I want to analyze would be the same as in case of using the covariates? 13. But why should I then use propensity score matching, how does make that my whole approach better? 14. In propensity score matching I can also just match on the observables, those observables I could also put into my Cox regression? If I assume that my propensity score matching was really perfect, so ideal world, why is my cox regression afterwards then better? Because then I apply it on another treatment assignment, because the treatment was due to the propensity score so to say reassigned. 15. So I can get different estimates. But this is only a benefit, if propensity score matching for whatever reasons is better – otherwise I could put the same variables into my Cox regression, so when and why is there a benefit of doing propensity score matching before? Furthermore I do not understand the following here in this case: Let's consider age again. When propensity score matching is used age is balanced. But when I put it into my Cox regression it could still be significant, because age might have an influence (although it is balanced across treatment). So it is a confounder. 16. What is the point here?

• You might find worthwhile a read through the chapters on propensity score matching and on survival analysis in Hernán, M. A., & Robins, J. M. (2020). Causal Inference: What If. Chapman & Hall/CRC. A brief takeaway is that propensity score matching generally does not provide valid causal estimates in longitudinal designs (i.e. with time-varying confounders, time-varying selection bias variables, or with time-varying treatments). – Alexis May 23 '20 at 18:45

Asking 16 questions is a lot, but I'll do my best to get the main ideas across. I'm not an expert in survival analysis, but I do have expertise in regression and causal inference, so perhaps someone else could fill in the details that pertain to Cox regression specifically.

Marginal and Conditional Hazard Ratios

A hazard ratio (HR) is a measure of association between a treatment and survival. It is noncollapsible, which means that a conditional HR (an HR computed for a stratum of the population) is not equal to a marginal HR (an HR computed for the whole population), even if stratum membership is unrelated to treatment and there is no confounding (e.g., in a randomized trial). My understanding is that in a Cox regression when covariates are included, the estimated HR for the treatment is a conditional HR (i.e., conditional the covariates), whereas when covariates are not included so that treatment is the only predictor in the model, the estimated HR is the marginal HR. How to decide if you want a conditional or marginal HR is an issue for another post, but it relates to whether you want to know how an intervention would work if applied to a subset of a population (e.g., an individual patient) vs. the whole population. Doctors usually care about conditional HRs; policymakers usually care about marginal HRs.

Causality and Confounding

In the absence of a randomized trial, there is confounding. Confounding occurs when some variables cause both selection into treatment and variation in the outcome. Confounders are variables that are sufficient to remove confounding. How to identify confounders is a matter for another post. In the presence of confounding, a measured association between the treatment and the outcome cannot be interpreted as causal (or it can be considered a biased estimate of the causal effect). There are a variety of techniques to adjust for confounding; regression (e.g., Cox regression) of the outcome on the treatment and confounders is one way, and propensity score methods (e.g., propensity score matching and weighting) are another way. Each has its own merits, discussed below. If a sufficient set of confounders has been measured and has been adjusted for correctly, then the estimated adjusted association between the treatment and the outcome can be interpreted as causal. (Many people don't believe this is possible, and therefore doubt any causal inference made using this strategy.)

Estimating Causal HRs

Let us identify four strategies to estimate a causal HR, assuming we are in an observational study and we know that age is the only confounder (just for the sake of exposition). Let us also assume that age is linearly related to the outcome and that the treatment effect is the same for all ages (we'll come back to this latter assumption later). We also assume the HR is constant over time. Four strategies include:

1. Cox regression of the outcome on treatment
2. Cox regression of the outcome on treatment and age
3. Cox regression of the outcome on treatment after propensity score matching
4. Cox regression of the outcome on treatment and age after propensity score matching

I'll discuss what each method would give you.

1. Cox regression of the outcome on treatment

The estimated HR is the marginal unadjusted HR. It is biased for the causal marginal HR because confounding is present and the confounder (age) was not adjusted for in any way.

2. Cox regression of the outcome on treatment and age

The estimated HR is the conditional adjusted HR. It is unbiased for the causal conditional HR. This is the same value you would get if you performed a randomized trial in the same population and ran the same model.

3. Cox regression of the outcome on treatment after propensity score matching

The estimated HR is the marginal adjusted HR. It is unbiased for the causal marginal HR. This is the same value you would get if you performed a randomized trial in the same population and ran a cox regression without including any covariates.

4. Cox regression of the outcome on treatment and age after propensity score matching

The estimated HR is the conditional adjusted HR. It is unbiased for the causal conditional HR, the same value described in 2).

So, basically, covariate adjustment through regression and propensity score matching perform two distinct functions: covariate-adjusted regression estimates the conditional HR and removes confounding, making the estimate unbiased for the causal conditional HR; propensity score matching estimates the marginal HR and removes confounding, making the estimate unbiased for the causal marginal HR. Performing covariate-adjusted regression after propensity score matching gives an estimate with the same properties as the simple covariate-adjusted regression.

There is an added complication if the treatment effect differs for patients with different ages. In this case, the population in which the effect is estimated will change the effect estimate. In this case, covariate-adjusted regression should include the interaction between the treatment and age to estimate causal conditional HRs for each age. Propensity score matching now estimates the causal marginal HR for a population of patients like those who received treatment. There are other propensity score methods, like propensity score weighting, that can estimate the causal marginal HR for the whole population.

Some potentially lingering questions:

• Why would you use both matching and covariate-adjusted regression if regression alone gives you a conditional adjusted estimate?

For models in which the effect estimate is collapsible, performing both matching and regression gives you two chances to correctly adjust for confounding. In noncollapsible models; additional methods are required to attain so-called "doubly-robust" estimates. Austin, Thomas, and Rubin (2018) describe an example of what you can do to get a doubly-robust causal marginal HR estimate using matching and regression. In some cases, when the effect of the confounders is nonlinear and matching restricts the sample to a zone where the effect is approximately linear, matching can allow for an unbiased conditional HR estimate even if the outcome model is incorrectly specified. See Ho, Imai, King, and Stuart (2007) for an example of this phenomenon, though note it is in the context of linear models.

• What if I want a marginal causal HR estimate but I don't want to use propensity score methods?

There are other ways to estimate the causal marginal HR, but I'm not an expert in them. A method called "g-computation" allows you use regression to estimate a causal marginal effect, but it's not as simple as reading off a coefficient in a regression output table.

• How does the significance of the confounders in the covariate-adjusted model relate to all this?

Not at all. The presence or absence of significance doesn't tell you whether a covariate is a confounder or not. Only a causal model can do that. There are a variety of reasons why a covariate might be significant or nonsignificant regardless of its status as a confounder. You should not use significance testing to determine whether you should adjust for a variable to control confounding. In a randomized trial or an adequately propensity score-matched or -weighted sample, if the inclusion of a covariate does not change the treatment effect estimate, then the marginal and conditional effects may be equal to each other.

A final note is that the paragraph you quoted is wrong. The authors confuse collapsibility with confounding and don't distinguish between marginal and conditional associations. These are key distinctions when dealing with noncollapsible quantities. I agree with another commenter that you should read What If by Robins and Hernán. Chapter 17 is all about causal survival analysis.

Austin, P. C., Thomas, N., & Rubin, D. B. (2020). Covariate-adjusted survival analyses in propensity-score matched samples: Imputing potential time-to-event outcomes. Statistical Methods in Medical Research, 29(3), 728–751. https://doi.org/10.1177/0962280218817926

Ho, D. E., Imai, K., King, G., & Stuart, E. A. (2007). Matching as Nonparametric Preprocessing for Reducing Model Dependence in Parametric Causal Inference. Political Analysis, 15(3), 199–236. https://doi.org/10.1093/pan/mpl013

• Thanks and +1 for your very extensive questions, but still most of my specific questions are open to me. – Stat Tistician May 27 '20 at 15:55
• Can you identify which ones are unanswered? I pretty clearly answered 3-16 and I feel like 1 and 2 are implicit. – Noah May 27 '20 at 17:50
• Thanks for your answer, howeveer 3-16 isn't at all answered for me. – Stat Tistician May 31 '20 at 19:39
• 3) I explained under which conditions a Cox regression on the confounder and treatment gives the same results as a randomized experiment. 4) I explained the conditions under which it doesn't (when you want a conditional estimate, when the functional form assumption is incorrect, when there is effect modification). 5) I explained that testing the significance of age in the model tells you nothing. 6) I explained that adjusting for age, if it is the only confounder, is sufficient for an unbiased estimate of the conditional HR. – Noah May 31 '20 at 21:37
• 7) I explained what it means when age has no effect on the outcome and its inclusion doesn't change the effect estimate. 8) I explained how PS matching is connected to Cox regression (it adjusts for the confounder, like Cox regression, but provides a marginal rather than conditional estimate). 9) I explained what PS matching does and when you should use it. 10-11) I explained what combining PS matching and covariate-adjusted regression does. 12) I explained the difference in the effect estimated when using PS matching vs. covariate-adjusted Cox regression. – Noah May 31 '20 at 21:41

I'm not very familiar with propensity score matching or causal inference from observational data so I'll focus on answering your question about the use of Cox regression in randomized controlled trials (RCTs).

Randomization has the advantage that it allows to balance the observable plus the unobservable characteristics equally across the treatment groups.

Contrary to popular belief, we do not randomize to balance characteristics between treatment groups. It's false to say that randomization will create equal balance between groups, as this would only occur in the limit (as $$N$$ approaches infinity). There will almost always be some imbalance between treatment groups in an RCT.

Instead, we randomize to try and evenly distribute future outcomes between treatment groups. Note that I said try – the more variable the outcome, the larger sample size needed to claim with some certainty that outcomes will be evenly distributed. With a large enough $$N$$, this allows the treatment groups to be exchangeable and causal inferences to be made (assuming other assumptions of RCTs are met as well). Randomization also helps prevent bias by breaking the causal link between any factors that would influence a patient from receiving one treatment over another.

If the goal of randomization is not to balance covariates, why do we use regression models to analyze RCTs? Although covariate imbalances do not invalidate causal estimates, they can decrease statistical power. Researchers often adjust for strong prognostic factors (predetermined before analysis) to decrease the outcome variance between groups, increasing power and decreases the need for larger sample sizes. Here the treatment hazard ratio is the only estimate of interest, and additional covariates used for adjustment should be included based on prior knowledge, not their p-value in the regression model.