# Likelihood ratio in Cox regression analysis?

#### arkiaamu

##### New Member
I have done a great deal of research without any success regarding the interpretation of likelihood ratio (LR) test in Cox regression analysis.

If p-value LR test in Cox analysis results >0.05 is then my analysis limited by for example to small sample size. It seems that when the sample size becomes smaller, LR tends not to be significant. What is the relevance with this? In all situations there are covariates with significant adjusted HRs, but is the analysis releveant if LR test is non-significant.

THanks!

#### ledzep

##### Point Mass at Zero
In all situations there are covariates with significant adjusted HRs, but is the analysis releveant if LR test is non-significant.
The Likelihood ratio test (LRT) and the individual tests for the hazard coefficients are testing different things (for a multivariable model).

Even is simple situations, the p-values are likely to be different as they use different distributions.

The Likelihood ratio test (LRT) is an approximate test based on $$\chi^2_k$$ on k degrees of freedom. The approximation gets better as sample size increases. The likelihood ratio test is based on the twice the difference in the maximized log-likelihood between the nested and the larger models.
So, the LRT is testing whether adding certain variable to the existing model improves the model fit or not i.e. does adding variable to the model reduces the log likelihood?

Whereas the test for the individual hazard ratio is based on Wald test which is testing whether the individual hazard coefficient is zero or not. The null hypothesis for the wald test is the coefficient is zero.
The Wald test is : $$\frac{coeff}{std.err(coeff)} \sim N(0,1)$$.

When there are many variables in the model, the LRT is equivalent to testing that each of the individual coefficients are equal to zero. The LRT comes in handy when you're trying to build a model and see whether adding or dropping variables from model affects the model/improves the model fit or not.

If the Likelihood ratio is not significant even when the individual coefficient in the model is significant, then it means that by adding variables to the null model, you are not significantly reducing the log likelihood. However, the individual coefficient is significantly different from zero. I'll keep the variable in model even though LRT is not significant. This shows that the hazard increases (or decreases) for every unit (for continuous) of your variable. Maybe doesn't improve the model fit as much but definitely tells us something regarding the behavior of the risk factors.

If you are in a multi-variable situation then you can see whether you've left out /added-in any significant covariates?