Solved – How to interpret Exp(B) in Cox regression

hazardregressionsurvival

I'm a medical student trying to understand statistics(!) – so please be gentle! 😉

I'm writing an essay containing a fair amount of statistical analysis including survival analysis (Kaplan-Meier, Log-Rank and Cox regression).

I ran a Cox regression on my data trying to find out if I can find a significant difference between the deaths of patients in two groups (high risk or low risk patients).

I added several covariates to the Cox regression to control for their influence.

Risk (Dichotomous)
Gender (Dichotomous)
Age at operation (Integer level)
Artery occlusion (Dichotomous)
Artery stenosis (Dichotomous)
Shunt used in operation (Dichotomous)

I removed Artery occlusion from the covariates list because its SE was extremely high (976). All other SEs are between 0,064 and 1,118. This is what I get:

                    B       SE      Wald    df  Sig.    Exp(B)  95,0% CI for Exp(B)
                                                                Lower   Upper
    risk            2,086   1,102   3,582   1   ,058    8,049   ,928    69,773
    gender         -,900    ,733    1,508   1   ,220    ,407    ,097    1,710
    op_age          ,092    ,062    2,159   1   ,142    1,096   ,970    1,239
    stenosis        ,231    ,674    ,117    1   ,732    1,259   ,336    4,721
    op_shunt        ,965    ,689    1,964   1   ,161    2,625   ,681    10,119

I know that risk is only borderline-significant at 0,058. But besides that how do I interpret the Exp(B) value? I read an article on logistic regression (which is somewhat similar to Cox regression?) where the Exp(B) value was interpreted as: "Being in the high-risk group includes an 8-fold increase in possibility of the outcome," which in this case is death. Can I say that my high-risk patients are 8 times as likely to die earlier than … what?

Please help me! 😉

By the way I'm using SPSS 18 to run the analysis.

Best Answer

Generally speaking, $\exp(\hat\beta_1)$ is the ratio of the hazards between two individuals whose values of $x_1$ differ by one unit when all other covariates are held constant. The parallel with other linear models is that in Cox regression the hazard function is modeled as $h(t)=h_0(t)\exp(\beta'x)$, where $h_0(t)$ is the baseline hazard. This is equivalent to say that $\log(\text{group hazard}/\text{baseline hazard})=\log\big((h(t)/h_0(t)\big)=\sum_i\beta_ix_i$. Then, a unit increase in $x_i$ is associated with $\beta_i$ increase in the log hazard rate. The regression coefficient allow thus to quantify the log of the hazard in the treatment group (compared to the control or placebo group), accounting for the covariates included in the model; it is interpreted as a relative risk (assuming no time-varying coefficients).

In the case of logistic regression, the regression coefficient reflects the log of the odds-ratio, hence the interpretation as an k-fold increase in risk. So yes, the interpretation of hazard ratios shares some resemblance with the interpretation of odds ratios.

Be sure to check Dave Garson's website where there is some good material on Cox Regression with SPSS.

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Regression – Cox Regression and Time Scale Analysis

Usually, age at baseline is used as a covariate (because it is often associated to disease/death), but it can be used as your time scale as well (I think it is used in some longitudinal studies, because you need to have enough people at risk along the time scale, but I can't remember actually -- just found these slides about Analysing cohort studies assuming a continuous time scale which talk about cohort studies). In the interpretation, you should replace event time by age, and you might include age at diagnosis as a covariate. This would make sense when you study age-specific mortality of a particular disease (as illustrated in these slides).

Maybe this article is interesting since it contrasts the two approaches, time-on-study vs. chronological age: Time Scales in Cox Model: Effect of Variability Among Entry Ages on Coefficient Estimates. Here is another paper:

Cheung, YB, Gao, F, and Khoo, KS (2003). Age at diagnosis and the choice of survival analysis methods in cancer epidemiology. Journal of Clinical Epidemiology, 56(1), 38-43.

But there are certainly better papers.

Solved – perform Cox regression on left truncated records

First a disclaimer: I've never had to use the time start/end variable in this way and although I'm familiar with mixed effects models I have never really had to use them IRL. Feel free to correct me if I've made a mistake

The problem consists out of two things as I see it:

One person can occur multiple times. This puts the observations independence into question.
A person may enter and exit the cohort at risk at different times throughout the study, i.e. this is an open (dynamic) cohort.

For the first point I think using a mixed effects model is a must. I use R and there is a coxme package recently developed by prof. Therneau. The vignette documentation is excellent and it seems easy to deploy.

For the second point you just need to add the start and end point to the survival object. This is fairly easy in R although I have never had to use it myself. Below is an example that should work:

# Set the event yes/no (1/0)
df$event <- !is.na(df$IpAdmit)
# Those that lack a date should have one
df$discharged <- !is.na(df$start)
df$start[is.na(df$start)] <- as.Date("2010-01-01")
df$end[is.na(df$end)] <- as.Date("2011-12-31")

# Can be merged into one step without the sv variable
sv <- Surv(time=df$start, time2=df$end, event=df$event)
# A model where the medication possession ratio (compliance) interacts with
# the fact that a patient has been discharged
coxme(sv ~ discharged*MPR + age + sex + (1|MemberID), data=df)

You might want to consider what you want to achieve with the cox regression model in this case. I am not sure that hazards make sense in this setting, although this is very difficult to know without going through the full study protocol. Make sure that others have used cox regressions in similar settings prior to this analysis. It seems to me that a good alternative would be a mixed effect logistic regression where you simply use odds for admission and add the number of days at risk as a predictor, preferably as a natural spline or something that allows a non-linear relationship.

Minor update from the discussion

When it comes to time-dependant covariates I have found this to be a little tricky when trying to deploy. I had a CV-question a while ago on this subject that you may want to look into. As I wrote in the comments, in the end the time dependence was a little more than I could conveniently display and explain to my colleagues. Furthermore the model was not strongly affected by this effect so I dropped it and switched to an early and late dataset. I recommend you consider who your audience is and if the time-varying coefficients will add that much to the model.

You have a potentially very serious problem where some patients start their period discharged from the hospital while some are untainted. I think you need to think about possible effect modification between these two groups - do they belong to the same population or not? It is easy to make a case that medication-compliance has a much bigger admission-avoidance impact in the discharged population. I think you at least should have a variable indicating if the patient has started a period straight after hospitalization or not (I've added an example in the code).

I have recently done a medication adherence study, if you haven't read this article I strongly recommend it. In my study I was also able to deduce from the prescription text 94 % of the cases using Python's very powerful regular expressions. I'm planning on doing a post on my blog once the article gets published, the text interpretation is in Swedish but you can very easily use the structure as most prescriptions follow a similar pattern (let me know if this would be useful and I can write up the post a little earlier). The advantage is that you want to identify exactly when a patient is expected to be without medication because you will probably have a very close relationship between that and readmission.

Best Answer

Related Solutions

Regression – Cox Regression and Time Scale Analysis

Solved – perform Cox regression on left truncated records

Minor update from the discussion

Related Question