Table 1.
Descriptive statistics of Irish ALS cases from 1st Jan 1995 to 31st Dec 2010.
Figure 1.
Cumulative Hazard function and Survival function of Non-parametric and Royston-Parmar models.
The graph on the left shows cumulative hazards estimated using the Nelson-Aalen method (red line) with 95% CI’s (grey area) and the cumulative hazard estimated by Royston-Parmar PH (df 3) modeling (green line). The graph on the right shows the survival function estimated using the Kaplan-Meier method with 95% CI’s (grey area) and the survival function estimated by Royston-Parmar PH (df 3) modeling (green line). These graphs were based on the full cohort minus those missing data for key variables (n = 1086). As can be seen the Royston-Parmar models provides excellent fit when compared to non-parametric methods.
Table 2.
Survival times and crude death rates for incident Irish ALS patients from symptom onset.
Table 3.
Hazard ratios from Cox Proportional Hazards models.
Figure 2.
Royston-Parmar Survival Curves by diagnostic age group.
Predicted cumulative survival curves based on model 3. Curves represent mean survival for each age group. The legend contains hazard ratios with 95% CI’s for specific ages determined from model parameters and taking 25 yrs to represent the baseline age risk.
Figure 3.
Graphs of time varying covariates.
Note 1: The RP graph for riluzole is drawn with d.f. = 3 for time varying spline knots whilst the 1995–2000 graph is drawn with d.f. = 1 for time varying spline knots. These values were decided after comparison of AIC and BIC values of multiple possibilities. Note 2: While the group diagnosed between 2006–2010 also had P = 0.02 when modeled as a time varying covariate under Cox PH, the graph was unimpressive as it was limited to 5 years follow up and had 95% CI’s close to 1 at all points, and therefore has not been included.
Figure 4.
Smoothed Schoenfeld residuals from Cox PH models for riluzole use by years of followup.
Nearest neighbour smoothed scaled Schoenfeld residuals for riluzole modeled with time of diagnosis as timescale origin (upper graph), and with age of diagnosis as timescale (lower graph). Deviation from linear trend can be seen in the first year on the upper graph. It is likely that this is caused by the greater power in detecting an effect of riluzole in older people due to the non-random distribution of riluzole use across age, combined with residual confounding by age - older people have poorer survival even if on riluzole. The combined effect leads to the appearance that riluzole is more effective in the first year (Figure 3), when in fact we have reduced power to detect the effect of riluzole in younger people (table 4), who are generally more likely to survive beyond one year (Figure 2). The lower graph using age as time scale origin does not show this trend and the PH assumption is not breached, however there is fluctuation dispersed over time. Note that a) the distribution of observations in time is altered as can be seen from the graph timescales and b) observations are reordered in time (not obvious from graph). Both features affect the evaluation of the proportional hazards assumption as Cox PH modeling is effectively a ranked method.
Table 4.
Breakdown of Hazard ratios and usage rates of riluzole by age group.