Proportional hazards is nonparametric in the sense that it involves an unspecified arbitrary baseline hazard function. It is parametric because it assumes a parametric form for the covariates. The baseline hazard function is scaled by a function of the model’s (time-independent) covariates to give a general hazard function. Unlike the Kaplan-Meier analysis, proportional hazards computes parameter estimates and standard errors for each covariate. The regression parameters (β) associated with the explanatory variables and their standard errors are estimated using the maximum likelihood method. A conditional risk ratio (or hazard ratio) is also computed from the parameter estimates.

The survival estimates in proportional hazards are generated using an empirical method. See Lawless (1982). They represent the empirical cumulative hazard function estimates, H(t), of the survivor function, S(t), and can be written as S0 = exp(-H(t)), with the hazard function calculated as follows:

When there are ties in the response, meaning there is more than one failure at a given time event, the Breslow likelihood is used.

In the Rats.jmp data, there are only two groups. Therefore, in the Parameter Estimates report, a confidence interval that does not include zero indicates an alpha-level significant difference between groups. Also, in the Effect Likelihood Ratio Tests report, the test of the null hypothesis for no difference between the groups shown in the Whole Model Test table is the same as the null hypothesis that the regression coefficient for Group is zero.

To show risk ratios for effects, select the Risk Ratios option from the red triangle menu. In this example, there is only one effect, and there are only two levels for that effect. The risk ratio for Group 2 is compared with Group 1 and appears in the Risk Ratios for Group report. See Figure 15.3. The risk ratio in this table is determined by computing the exponential of the parameter estimate for Group 2 and dividing it by the exponential of the parameter estimate for Group 1.

Note the following:

For this example, the risk ratio for Group2/Group1 is calculated as follows:

This risk ratio value suggests that the risk of death for Group 2 is 1.81 times higher than that for Group 1.

For information about calculating risk ratios when there are multiple effects, or categorical effects with more than two levels, see “Risk Ratios for Multiple Effects and Multiple Levels”.

For the proportional hazards model, covariates include the following:

Age, Diag Time, and KPS are continuous measures and Cell Type, Treatment, and Prior are categorical (nominal) variables. The four nominal levels of Cell Type include Adeno, Large, Small, and Squamous.

Note the following about the results:

Figure 15.6 shows the Risk Ratios for the continuous effects (Age, Diag Time, KPS) and the nominal effects (Cell Type, Treatment, Prior). For illustration, focus on the continuous effect, Age, and the nominal effect with four levels (Cell Type) for the VA Lung Cancer.jmp sample data.

For the continuous effect, Age, in the VA Lung Cancer.jmp sample data, the risk ratios are calculated as follows:

For the nominal effect, Cell Type, all pairs of levels are calculated and are shown in the Risk Ratios for Cell Type table. Note that for a categorical variable with k levels, only k -1 design variables, or levels, are used. In the Parameter Estimates table, parameter estimates are shown for only three of the four levels for Cell Type (Adeno, Large, and Small). The Squamous level is not shown, but it is calculated as the negative sum of the other estimates. Here are two example Risk Ratios for Cell Type calculations:

Reciprocal shows the value for 1/Risk Ratio.