Proportional hazards assumption

For the pre D-dimer model a Royston and Parmar model^72,73 was fitted on the proportional hazards scale, which assumes that the effect of factors in the model are not associated with time, so that any two rates predicted from the model are proportional. To test whether or not the proportional hazards assumption is valid for the pre D-dimer model, a plot of the scaled Schoenfeld residuals against the natural logarithm of time from cessation of therapy was examined for each factor in the model (Figures 76 and 77). Horizontal reference lines in Figure 76 indicate zero and the log-HR for the factor, a smoother is applied and should follow the log-HR reference line over log-time when the proportional hazards assumption is valid. It is clear from Figures 76 and 77 for site of index event that the proportional hazards assumption is met, and similar plots indicate the assumption is valid for both age and sex covariates (Figures 78 and 79).

FIGURE 76

Scaled Schoenfeld residuals vs. log-time from cessation of therapy for proximal DVT (the pre D-dimer model) (HR –0.25).

FIGURE 77

Scaled Schoenfeld residuals vs. log-time from cessation of therapy for PE (the pre D-dimer model) (HR –0.053).

FIGURE 78

Scaled Schoenfeld residuals vs. log-time from cessation of therapy for age (the pre D-dimer model) (HR –0.0028).

FIGURE 79

Scaled Schoenfeld residuals vs. log-time from cessation of therapy for sex (the pre D-dimer model) (sex: male HR –0.6).

Functional form

The functional form of continuous covariates within the model can be checked using Martingale residuals. A scatterplot of the Martingale residuals against the continuous covariate of interest with a smoother applied can reveal whether linearity is appropriate, or if non-linear forms should be considered.

As patient age was the only continuous covariate within the pre D-dimer model the functional form within the model was checked using Martingale residuals. Figure 80 shows the lowess smoother applied to a scatter of Martingale residuals against age appears to follow a linear trend over age, indicating that inclusion of age as linear within the model was appropriate.

FIGURE 80

Scatterplot of Martingale residuals against age (the pre D-dimer model).

Outliers

Deviance residuals can be used to investigate potential outliers in whom the model will perform poorly. A scatterplot of the deviance residuals against a simple patient indicator enables us to assess the normality of the deviance residuals and identify outliers which fall outside of the critical z-values associated with a 95% CI. Figure 81 illustrates a scatter of the deviance residuals for the pre D-dimer model, the deviance residuals do not appear to follow a normal distribution and this may be due to heavy censoring in the data set, there are some values which fall above the 1.96 critical z-value.

FIGURE 81

Scatterplot of deviance residuals vs. patient ID (the pre D-dimer model). ID, identification.

A plot of the deviance residuals against years from cessation of therapy allows investigation of any trend in the deviance residuals. In Figure 82, for the pre D-dimer model there is a clear trend in the deviance residuals over time, this is to be expected because deviance residuals are based on the cumulative hazard at the event time (or censoring time). The deviance residuals which lie in the top left of the plot are likely to be those individuals who had a recurrence early and therefore did not accumulate much hazard.

FIGURE 82

Scatterplot of deviance residuals vs. years from cessation of therapy (the pre D-dimer model).

Leverage

To check the influence of individuals on the parameter estimates, leverage can be assessed using delta–beta changes for each covariate. A scatterplot of the delta–beta change for the covariate of interest against time allows inspection of the largest change with respect to the log-HR of the covariate.

Scatterplots of delta–betas for age (Figure 83) and sex (Figure 84) show that even individuals with the greatest leverage on these parameter estimates, have very small effects on the log-HR. Similarly, small delta–beta changes were observed for site of index event (Figures 85 and 86).

FIGURE 83

Scatterplot of delta–beta for age vs. years from cessation of therapy (log-HR –0.002).

FIGURE 84

Scatterplot of delta–beta for sex vs. years from cessation of therapy (log-HR 0.573)

FIGURE 85

Scatterplot of delta–beta for site (proximal DVT) vs. years from cessation of therapy (log-HR 1.726).

FIGURE 86

Scatterplot of delta–beta for site (PE) vs. years from cessation of therapy (log-HR 1.659).

Interaction effects

Interaction effects quantify a differential effect in a specific subgroup of the population. An interaction effect can be either an increased risk or decreased risk beyond that associated with a single characteristic. For example, within the pre D-dimer model, both sex (being male) and site of index event (having a first PE) are associated with significant increases in recurrence rate, an interaction between sex and site of index event would imply that patients who are both male and have a PE are at increased risk beyond that associated with being male or having a PE alone.

The pre D-dimer model includes factors for patient age, sex and site of index event (distal DVT, proximal DVT or PE). Given this set of factors no interactions were considered plausible, either biologically or as evidenced within previous research. As there were no plausible effect-modifying interactions, testing for interactions was not performed to avoid overfitting and prevent a more complex model being produced.¹¹²

Time-dependent effects

Often time-fixed covariates may have time-dependent effects, where the effect (e.g. HR) varies with time.⁷⁵ Allowing for time-dependent effects could improve the performance of the prognostic model by better fitting the underlying data.

Non-proportional hazards can be a sign of a time-dependent effect and, as such, including time-dependent effects can account for departures from the proportional hazards assumption. The validity of the proportional hazards assumption for the pre D-dimer model was assessed in Appendix 5, and the assumption was met for all factors included in the models. It was therefore not expected that any time-dependent effects would be found to significantly improve the performance of either final model.

A procedure proposed by Royston and Lambert⁷⁵ was used to identify potential time-dependent effects within the final model. The procedure first identifies the p-value associated with including each covariate in the model as a time-dependent effect using a likelihood ratio test. A time-dependent effect is included for the factor with the smallest p-value, providing the p-value is less than a pre-defined alpha significance level. The process is repeated until no time-dependent effects are significant at the chosen alpha level.

Following the procedure described above, an alpha of 0.01 was selected so as to allow for multiple testing of time-dependent effects. The same level of df was used to assess time-dependent effects as was selected for the model, therefore allowing complex forms of time dependency.⁷⁵ The procedure completed one cycle through the potential covariates and found none of the covariates to be significantly time dependent at the 1% level as expected (Table 62).

TABLE 62

First cycle of stepwise forward selection of time-dependent effects (the pre D-dimer model)

Proportional hazards assumption

Functional form

Influence