How do I interpret Exp(B) in Cox regression?

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?

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

-

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 the independent variable $i$ 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.

-
Thanks a lot for your reply! I'm having a hard time de-ciphering your text-based formulas. Can you humanize them? ;) Great article you're referencing. I'll read it thouroughly and get back... –  Alex Jan 6 '11 at 19:05
Ahhh... Internet Explorer failed to render the formulas. Firefox fixed this. :) –  Alex Jan 6 '11 at 19:21
Another excellent resource for learning about and understanding survival analysis is Applied Longitudinal Data Analysis by Singer and Willett. The also give example code/output for all their models using every stats package under the sun. –  M Adams Feb 25 '11 at 13:06
@M Adams Thanks for adding this link. Yes, the UCLA server is really full of useful resources. –  chl Feb 25 '11 at 16:38
Thanks for the great link to UCLA! I'll dig into it... ;) –  Alex Feb 26 '11 at 8:48

I am not a statistician, but an MD, trying to sort things out in the world of statistics.

The way you have to interpret this output is by looking at the $\exp(B)$ values. A value of < 1 says that an increase in one unit for that particular variable, will decrease the probability of experiencing an end point throughout the observation period. By inverting (that is $1/\exp(B)$), you will find the "protective effect", for example if $\exp(B) = 0.407$ (as is the case for your "Gender" value), the interpretation will be that having the value of gender = 1 means that you decrease the probability of experiencing an en point with $1/0.407 = 2.46$, compared to when the Gender value = 0.

For $\exp(B) > 1$, the interpretation is even easier, as a value of, say $\exp(B) = 1.259$ (as is the case for your "stenosis" variable), means that scoring "stenosis" = 1 will result in an increased probability (25.9%) of experiencing an end point compared to when "stenosis" = 0.

The confidence interval (CI) tells us within which range (of 95% probability) we can expect this value to differ, if we were to repeat this survey for an infinite number of times. If the 95% CI overlaps the value of 1, then the result is not statistically significant (since $\exp(B) = 1$ means that there is no difference between the probability of experiencing an end point if the variable value is either "0" or "1"), and the P value will exceed 0.05. If the 95% CI keeps out of the value 1 (on either side), the $\exp(B)$ is statistically significant.

From your analysis, it seems as no one of your variables are significant predictors (at a sign level of 5%) of your endpoint, although being a "high risk" patient is of borderline significance.

Reading the book "SPSS survival manual", by Julie Pallant will probably enlighten you further on this (and more) topic(s).

-
Thanks. Great support from a fellow adventurer in this world of statistics! ;) I'm currently reading Discovering Statistics using SPSS by Andy Field, which I'm surprised to enjoy (since it's a statistics textbook). I altered my COX analysis to measure survival on days instead of months, which luckily pushed my the significance of my 'risk' covariate below 0,05... :) –  Alex Feb 27 '11 at 21:18