Time to Event Data for CLL8 Trial

Analysis of trial data for transitions between health states

1. Background

The economic model for treatment of chronic lymphocytic leukaemia (CLL) allows for multiple treatments in multiple lines of therapy, with each treatment represented by two states, either unprogressed (the initial state) or progressed (with no transition allowed back to unprogressed for that treatment). In the model, patients in any state may also die. Use of the model requires specification of transition probabilities between the states for each treatment within each line of therapy and death. Patient-level data were available for CLL-8 and REACH for treatment group, sex, age and days from randomisation to end of treatment, to progression, to next treatment, to last date known alive and to death.

The data on time to events can be used to estimate hazard rates for transitions for the first treatment in the first (CLL-8 data) or the second (REACH data) line of therapy from unprogressed to progressed, unprogressed to death, unprogressed to next treatment, progressed to death and progressed to next treatment.

2. Analysis strategy

The simplest Markov model of the transitions between states would have a constant probability, pijk (for the transition between state i and state j for a patient in treatment group k) for each transition at each cycle. For a given cycle length, t, it is possible to use data from the trials to estimate these transition probabilities. A mathematically equivalent formulation is to model the transitions using a hazard rate, λijk, since

pijk = 1 - exp(-λijk t)

This formulation is convenient because it enables fuller use of the available data, since a direct analysis of the pijk entails summary of the data at integer multiples of t time units thereby ignoring fractions of a cycle where the final state is uncertain. This formulation also allows transition probabilities to be estimated for different cycle lengths (values of t).

The simplest model assumes that the transition probabilities (pijks) and hazard rates (λijks) are constant for a given treatment and initial and final state (ie, for a given i, j and k). This assumption of a constant hazard rate can be tested. For a Markov model, a key departure from constancy would be if the hazard rates varied with time since entry into the initial state (ie, state i). This suggests a Poisson regression analysis of the hazard rates (number of events per person-year at risk of the event), with the time since entry into the initial state divided into suitable fixed time intervals. This allows full use of the data, takes into account the censoring of these survival data and explicitly models the quantities of interest for the economic model. Poisson regression (also known as (piecewise) exponential regression) is a standard method of analysis for prospective cohort studies in epidemiology (Andersen & Keiding, 2006, p360; Breslow & Day, 1987, p131)[1] and is readily implemented in standard statistical packages, such as STATA (using the ‘poisson’, ‘glm’ or ‘streg’ procedures), SAS or GLIM. Poisson regression allows the hazard rate for later time periods to be modelled with few parameters (for example, as the average hazard rate after a particular time). The proportional hazards assumption can be tested by fitting interactions between the time periods and patient characteristics (such as the treatment they have received).

The maximum follow-up in the available data from the trials was less than five years. With about 50 deaths in each treatment group, division of the time period into more than about ten time periods is unlikely to be informative. Accordingly, half-yearly time periods were chosen, with the expectation that, while there might be variation in the hazard rates for the first few time periods, later time periods (for which there are successively less data) would show either a constant or an increasing or decreasing trend in the hazard rate and thus could be summarised. A half-yearly time scale is consistent with the slow evolution of CLL from diagnosis to death.

For each initial and final health state, a Kaplan-Meier curve and Nelson-Aalen estimates of the cumulative hazard for each treatment group were produced using the ‘sts graph’ command in STATA (Version 10). The cumulative hazard is linear (ie, has constant slope) in the periods for which the hazard rate is constant. The number of events and the person-years at risk of the event, by treatment group and time period, for each initial and final state was calculated using the survival analysis commands ‘stsplit’ (to create the time periods) and ‘strate’ (to create tables of events/period-at-risk; Table I in publication) in STATA.

The tables of events/period-at-risk were analysed using Poisson regression in GLIM 4.09. The Poisson regression models for the logarithm of the hazard rate tested for differences between the first three time periods (0-½, ½-1, 1-1½ years) and the remaining time period (1½ years onwards), as well as a trend in the remaining time period and a trend over all time periods. The models also tested for study and treatment differences both overall and between different time periods. Goodness of fit of the Poisson regression models was assessed using the residual deviance (which is chi-square distributed on the residual degrees of freedom if the residual variation after fitting the model is consistent with Poisson sampling variation). When it appeared likely that the goodness of fit test might be under-powered and there were substantial treatment differences evident (even if not statistically significant), the models included treatment differences so as not to risk inadvertently ignoring important treatment differences. Parameters for the economic model were estimated separately from each study, so as not to confound possible differences in parameters between first and second line treatment. Poisson regression analysis also allows estimation of confidence intervals for the hazard rate parameters.

An initial economic model was developed allowing for variation in transition probabilities for four time periods, 0-½, ½-1, 1-1½ and 1½ years onwards. This allowed for adequate flexibility for all transitions except the unprogressed to progressed health state transition, which was a key driver for the model. Accordingly, the economic model was refined by allowing for eleven time periods for the unprogressed health state (½ yearly for five years and 5 years onwards). The transition from unprogressed to progressed was modelled with a Weibull survival model, for which Poisson regression is a special case (when the Weibull ‘shape’ parameter is 1). The Weibull distribution was fitted using ‘streg’ in STATA (Version 10).

3. Results

Table I in the publication shows the event rate for each transition calculated from the number of events and person-years at risk for patients in each treatment group in each half-year period after the start of each trial. The details of the analyses for each transition are presented below.

3.1 Transition from unprogressed to progressed

Definition:

The survival time was defined as the minimum of the time to progression, next treatment, last known alive and death. The event occurred if the survival time was the same as the time to progression, provided this was before the date of next treatment (if any) and date of death (if any).

Results:

The event rate is higher in REACH than in CLL-8 (chi-square test statistic = 19.6 for 1 degree of freedom, p < 0.0001) and for FC than RFC (chi-square test statistic = 33 for 1 degree of freedom, p < 0.0001). In addition, there is residual variation after taking account of the difference in the rates between the treatment groups and studies (chi-square test statistic for residual variation = 110 for 36 degrees of freedom, p < 0.0001).

Thus, a simple summary of these rates as an average rate for each treatment group is insufficient because there is additional variation in the rates between the time periods. The variation between the 10 time periods (½ yearly for five years and 5 years onwards) is statistically significant (chi-square test statistic = 68 for 9 degrees of freedom, p < 0.0001).

The variation in the event rate over time reflects the lower rate in the first half-year compared with the later time periods (chi-square test statistic = 52 for 1 degree of freedom, p < 0.0001). The event rate is lower in the second half-year compared with the later time periods (chi-square test statistic = 11.5 for 1 degree of freedom, p = 0.0007) with little variation thereafter (chi-square test statistic for difference between third half-year compared with later time periods = 2.9 for 1 degree of freedom, p = 0.09; chi-square test statistic for trend in event rates with time after 1½ years = 0.2 for 1 degree of freedom, p = 0.7). The chi-square values reflect both the size of the differences in event rates and the amount of data (since the person-years at risk decreases over time).

The relative event rate for the two treatments is similar in the four time periods (chi-square test statistic for an interaction between treatment and period (0-½, ½-1, 1-1½ and 1½+ years) = 3.0 for 3 degrees of freedom, p = 0.4). The residual variation in event rates after taking account of the possible trend, the overall treatment and overall study difference is not statistically significant (chi-square test statistic = 37 for 32 degrees of freedom, p = 0.26).

The estimated relative hazard rate with R-FC compared with FC is 0.58, with a 95% confidence interval from 0.48 to 0.69. The relative hazard rate was similar in the two studies (chi-square test statistic = 0.01 for 1 degree of freedom, p = 0.9; estimates of 0.57 in CLL-8 and 0.59 in REACH). The estimated relative hazard rate in REACH compared with CLL-8 is 1.7, with a 95% confidence interval from 1.4 to 2.0.

Use in the economic model:

The constancy of the relative hazard rate for different treatments over time and between lines of treatment has been used to model parameters for other treatments and other lines of treatment. The economic model has multipliers for these hazard rates for each later line of treatment or re-use of a treatment within a line. For the basecase, these multipliers have been set to 1.7, that is, what was observed for the difference between REACH and CLL-8 for the same treatment.

As noted in the last paragraph in Section 2, eleven time periods for the unprogressed to progressed transition (½ yearly for five years and 5 years onwards) were allowed and this was modelled with a Weibull survival model. The Weibull model is suitable for a hazard rate that is changing monotonically over time.

For CLL-8, the fitted Weibull distribution had the same shape parameter for both treatment groups (chi-square test statistic for a different shape parameter = 0.8 for 1 degree of freedom, p = 0.4). Having the same shape parameter ensures that the relative hazard rate between the two treatment groups remains constant over time. The fitted Weibull model predicts survival at time t (in months) as:

S(t) = exp( -exp(-4.31 x 1.43) t1.43)) when treated with R-FC, and

S(t) = exp( -exp(-3.92 x 1.43) t1.43)) when treated with FC,

for which the relative hazard is 0.57 (= exp( (3.92-4.31) x 1.43) ).

For REACH, the fitted Weibull distribution had the same shape parameter for both treatment groups (chi-square test statistic for a different shape parameter = 0.1 for 1 degree of freedom, p = 0.7). The fitted Weibull model predicts survival at time t (in months) as:

S(t) = exp( -exp(-3.92 x 1.48) t1.48)) when treated with R-FC, and

S(t) = exp( -exp(-3.54 x 1.48) t1.48)) when treated with FC,

for which the relative hazard is 0.57 (= exp( (3.54-3.92) x 1.48) ).

The similarity of the shape parameter for the Weibull estimated for the two studies means that the modelled relative hazard rates between the two studies is almost constant, which supports the assumption for the model of a constant relative rate between lines of treatment.

In the economic model, hazard rates for half-yearly transitions have been calculated as the average hazard that reproduces S(t) at the start and end of each period. For the last period, the average hazard between time 60 and 66 months has been calculated.

3.2 Transition from unprogressed to death

Definition:

Results:

Poisson regression demonstrates that the death rate is higher for patients undergoing second line compared with first line treatment (chi-square test statistic for event rate difference between the two studies = 19.1 for 1 degree of freedom, p = 0.0001; relative death rate = 2.4, 95% CI from 1.6 to 3.6). The death rate is similar for patients treated with R-FC or FC (chi-square test statistic = 1.3 for 1 degree of freedom, p = 0.25).

The death rate is higher in the first half-year compared with the later time periods (chi-square test statistic = 8.6 for 1 degree of freedom, p = 0.003) and in the second half-year compared with the later time periods (chi-square test statistic = 5.7 for 1 degree of freedom, p = 0.017), but shows little variation with half-yearly period after the first year (chi-square test statistic for difference between third half-year and later periods = 1.4 for 1 degree of freedom, p = 0.24; chi-square test statistic for trend in death rates after 1½ years = 2.3 for 1 degree of freedom, p = 0.13). After taking account of possible differences in the death rate over time, there was little evidence for a difference in the death rate between the R-FC and FC treatment groups (chi-square test statistic = 0.8 for 1 degree of freedom, p = 0.4) There was little evidence that the treatment difference, if any, varied over time (chi-square test statistic = 0.6 for 3 degrees of freedom, p = 0.9) or differed between the two studies (chi-square test statistic = 1.5 for 1 degree of freedom, p = 0.22). The residual variation was consistent with sampling variation (chi-square test statistic = 15 for 28 degrees of freedom, p = 0.9).