Methodology

Six most useful graphical methods to assess the assumption of PH in the Cox model are proposed and compared in a two-sample data with only one binary predictor variable, x (coded 0 or 1).

2.1.1 Kaplan-Meier Survival Curves versus Time

As described in the previous chapter, the KM [23] method restricts the survival estimate based on observed event times. The KM survival curve appears as a step function that changes only when an observed event happens. It is a common and useful method to describe survival characteristics.

Under the PH assumption in the two-sample case, vertical gaps between survival curves should follow the exponential relationship between the two groups:

( ) ( ) ^{( )} ( ) ( ) ( ) ^{( )}

The KM survival curve is used as a rudimentary check of time-to-event data and assumption of proportionality. Evidence of crossing survival curves is an indication that the PH assumption is not reasonable.

2.1.2 Plot of Logarithm of the Minus Logarithm of Survival Function versus Logarithm of Survival Time

Under the PH assumption,

( ) ( ) ^{( )}

Log of the minus log of survival = Log of Cumulative Hazard function:

( ) ( ) ( ) ( ) ( ( ))

( ) ( ) For x = 1, ( ) ( ) ;

For x = 0, ( ) ( )

Therefore, under the PH assumption, the difference between log of minus log of survivals are constant with respect to time. Plots of Log[-log(S[t])] against time are parallel due to equidistance between the curves. We use the natural logarithm of time rather than time for the horizontal scale, because a straight line in the plot with this setting implies that the Weibull distribution is an appropriate method for the data being analyzed. If hazards are proportional, the plot of Log[-log(S[t])] against the logarithm of time maintain an equal distance apart, resulting in parallel lines. Any evidence of lack of parallelism, such as crossing lines, suggests the PH assumption does not hold.

2.1.3 Plot of Cumulative Hazards in Two Compared Groups

( )

( )

( ) ( )

[ ( ) ^{( )}]

( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

Cumulative hazards are generated using the KM method. Under the PH assumption, the plot of cumulative hazard functions compared in two groups yields a straight line, with a constant slope ( ) and zero intercept. The constant slope indicates that the PH assumption is satisfied, and any curvature trend reflects violation of the PH assumption.

2.1.4 Smoothing Plot of the Difference of Log Cumulative Hazard versus Survival Time

Schumacher [24] proposed plotting the difference of log cumulative hazard functions versus survival time, which is equivalent to the Log of cumulative hazard ratio. A smoothing procedure is applied to help describe the mean difference in the log cumulative hazard function as a function of survival time, using regression to fit a cubic spline that minimizes the sum of the square of the residuals of fit. The value of the smoothing parameter is 60, in order to specify a moderately smooth interpolated line.

( ) ( ) [ ( )

( )] ( )

Under the PH assumption, the difference of the log cumulative hazard functions between the two groups is constant with respect to time (t), and the smoothing plot should be a horizontal line centered around the estimated value of (the log HR). Any non-zero slope trends indicate non-PH.

2.1.5 Smoothing Plot of Schoenfeld Residuals versus Survival Time

Residuals have been used to examine the adequacy of regression models. The Schoenfeld partial residuals [6] are defined as the difference between the observed and the conditionally expected values of x, given the risk set at each event time, which is expressed by:

̂ ̂( | )

where

The Schoenfeld partial residuals do not depend on time, if proportionality holds;

thus, residuals can be plotted against event time to identify violations of proportionality.

When event times are tied, the residual is computed as the total component of the first derivative, divided by the number of tied event times in the risk set.

If the PH assumption holds, the expectation of rˆ_i is approximately equal to zero at each event time. A smoothing spline is applied to this plot, which should be centered around zero if the hazards are proportional. The non-zero slope trend or the non-zero centered plot reflects time dependence in the group (or treatment) effect, and the violation of PH assumption.

2.1.6 Hazard Functions from Life-Table Estimator versus Survival Time

Just like the KM estimator, the life-table estimator is generated by a nonparametric method. For the life-table method, time to event is categorized into several intervals. First, the numbers of subjects who fail and are censored in each interval are counted, considering failure and censoring to happen in the middle of an interval. Second, we determine the number of subjects at risk of failure during the interval, which is the number of subjects at the start of the interval minus half the number of subjects who had

an event or were censored during the interval. Third, we calculate the approximate subject time at risk during the interval, which is the product of the length of the interval and the number of subjects at risk of failure. Finally, the hazard is estimated by dividing the number of events by the approximate subject time at risk during the interval.

Let be the length of the ith interval, be the number of subjects at the start of the ith interval, be the number of subjects who have an event in the ith interval, be the number of subjects who are censored in the ith interval. Then, the hazard for the ith interval is given by the following:

( )

( )

Hazard units are expected events per subject-time. Thus, the plot of hazard function may roughly provide some hints of the nature of underlying distribution. The drawback of determining hazard using the life-table estimator is that estimates change if the intervals change.

In the two-sample case, under the PH assumption, hazard functions of the two groups should maintain a constant ratio over time; but, the vertical distance between hazard functions at any time points may not have to be constant, with the exception of constant hazards of two compared groups. Crossing of hazard function would always indicate departure from the PH assumption.