# Different p-values for coefficients and LRT in univariate cox regression (coxph R)

I have used Cox PH to test the relationship between one predictor and survival for 6 patients. Cox PH was used since the predictor is continuous. Using the "cox.zph" function, there was no significant violation of the proportional hazards assumption. However, I am confused about the interpretation in this case. The p-value for the predictor coefficient was non-significant (p=0.23), but the likelihood ratio test for the model gave p=0.03. As I understand it, the latter tests whether the coefficients used in the model significantly differ from 0, and therefore whether or not they contribute to the model. Since only one variable is used as predictor here, the results therefore seem conflicting. However, the Wald test and Score (logrank) tests shown for the model when using the "summary" function also show p=0.2 and p=0.1, respectively, which are more in line with the coefficient p-value. By reading up on this topic I have come to understand that the LRT, Wald and log-rank tests are asymptotically equivalent, but that the latter two have issues with a small sample size and that the LRT is preferrable in such cases. Given the few samples used in this case, this can explain the differences for the LTR, Wald, and log-rank tests for the model. But the p-value calculated for the coefficient itself is, as far as I understand it, also based on a Wald test. In the case of a univariate model, I suppose that the model Wald test and the coefficient Wald test may even be the same test. By this logic, I would suppose that the coefficient Wald test would also be underpowered with respect to the sample size and that the LRT test may be a more appropriate substitute for the coefficient Wald test, since the test is univariate. Is this the case?

The results of the test are shown below:

Call:
coxph(formula = SurvObj ~ n, data = surv_data)

n= 6, number of events= 3

coef exp(coef) se(coef)      z Pr(>|z|)
n -0.09064   0.91335  0.07580 -1.196    0.232

exp(coef) exp(-coef) lower .95 upper .95
n    0.9133      1.095    0.7873      1.06

Concordance= 0.9  (se = 0.227 )
Rsquare= 0.542   (max possible= 0.745 )
Likelihood ratio test= 4.69  on 1 df,   p=0.03
Wald test            = 1.43  on 1 df,   p=0.2
Score (logrank) test = 2.24  on 1 df,   p=0.1

• I think that arguing about p-values with a sample size of $n=6$ is a bit of a tall order... Yes, there are not absolute cut-offs for a minimum sample size for a test to be applicable but that does mean the test might not be underpowered. – usεr11852 Oct 26 '18 at 23:37
• I am voting to leave this open. It is not about R, really. The same sort of question could come up with any statistical package. – Peter Flom Oct 27 '18 at 13:06

"I suppose that the model Wald test and the coefficient Wald test may even be the same test" You are right on it. In fact you can check that $$z^2=$$ Chi square, $$(-1.196)^2 = 1.430$$.