The weighted log-rank test

See Magirr and Burman (2019) and Magirr (2021) for details about the weighted log-rank tests, and in particular the modestly weighted log-rank test. This vignette works through an example of using the package to simulate data and perform weighted log-rank tests. A summary of the formulas used within this package is presented.

Simulate a dataset

This package can be used to simulate a dataset for a two-arm RCT with delayed separation of survival curves by using the sim_events_delay function.

There are two parts to simulating the event times and statuses: the event model (parameters defined in event_model) and the recruitment model (parameters defined in recruitment_model).

Firstly, looking at the event model. The function sim_events_delay assumes that the survival times on the control and exponential arm follow a piecewise exponential distribution. Given rate parameter \(\lambda\), the exponential distribution has the form:

\[ f(t)=\lambda \exp(-\lambda t) \]

The rate parameters are set using the argument lambda_c for the control arm and lambda_e for the experimental arm. To use the piecewise version, set this argument a vector with a value for each piece. The duration of each piece is set using parameter duration_c and duration_e.

Secondly, looking at the recruitment model. The recruitment can be modeled using either a power model or a piecewise constant model. See help(sim_events_delay) more details about these models.

Additionally, the sim_events_delay function censors all observations at the calendar time max_cal_t.

Here we create a simulated dataset with 5 individuals on each arm. Assume that one unit of time is equal to one month. From entering the study until 6 months both arms have the same \(\lambda\) parameter, with a median event time of 9 months. From 6 months, the experimental arm has a lower hazard rate, with a median event time of 18 months. Setting rec_period = 12 and rec_power = 1 means that individuals are recruited at a uniform rate over 12 months.

library(nphRCT)
set.seed(1)
sim_data <- sim_events_delay(
  event_model=list(
    duration_c = 36,
    duration_e = c(12 ,24),
    lambda_c = log(2)/9,
    lambda_e = c(log(2)/9,log(2)/18)),
  recruitment_model=list(
    rec_model="power",
    rec_period=12,
    rec_power=1),
  n_c=5,
  n_e=5,
  max_cal_t = 36
)
sim_data
#>    event_time event_status        group
#> 1       18.06            1      control
#> 2        9.89            1      control
#> 3       16.07            1      control
#> 4       28.07            0      control
#> 5       13.69            1      control
#> 6       25.22            0 experimental
#> 7       24.66            0 experimental
#> 8        8.50            1 experimental
#> 9        4.37            1 experimental
#> 10       7.64            1 experimental

Weighted log-rank tests

Now that we have simulated a dataset, we will look at performing the weighted log-rank tests.

Consider the ordered, distinct event times \(t_1, \dots, t_k\). Let \(d_{0,j}\) and \(d_{1,j}\) be the number of events at event time \(t_{j}\) on each of the arms respectively, and let \(d_{j}\) be equal to the sum of these two values. Similarly, let \(n_{0,j}\) and \(n_{1,j}\) be the number at risk at event time \(t_{j}\) on each of the arms respectively, and let \(n_{j}\) be equal to the sum of these two values.

The function find_at_risk can be used to calculate these values for a dataset.

find_at_risk(formula=Surv(event_time,event_status)~group,
  data=sim_data,
  include_cens=FALSE)
#>     t_j n_event_control n_event_experimental n_event n_risk_control
#> 1  4.37               0                    1       1              5
#> 2  7.64               0                    1       1              5
#> 3  8.50               0                    1       1              5
#> 4  9.89               1                    0       1              5
#> 5 13.69               1                    0       1              4
#> 6 16.07               1                    0       1              3
#> 7 18.06               1                    0       1              2
#>   n_risk_experimental n_risk
#> 1                   5     10
#> 2                   4      9
#> 3                   3      8
#> 4                   2      7
#> 5                   2      6
#> 6                   2      5
#> 7                   2      4

Here, each row relates to the distinct event times \(t_j\), which are specified in column t_j. The value \(d_{0,j}\) relates to the column n_event_control, \(d_{1,j}\) to n_event_experimental, and \(d_{j}\) to n_event. Similarly, \(n_{0,j}\) relates to column n_risk_control, \(n_{1,j}\) to n_risk_experimental, and \(n_{j}\) to n_risk.

To calculate the test statistics for a weighted log-rank test, we need to evaluate the observed number of events on one arm, e.g. \(d_{0,j}\), and the expected number of events on the same arm, e.g. \(d_j \frac{n_{0,j}}{n_j}\) at each \(t_j\). The test statistic \(U^W\) is then a weighted sum (using weights \(w_j\)) of the difference of these values:

\[ U^W = \sum_{j=1}^k w_j \left(d_{0,j} - d_j \frac{n_{0,j}}{n_j}\right) \]

The weights \(w_j\) that are used depend on the type of weighted log-rank test, these are described next.

Weights

Three types of weighted log rank test are available in this package.

The standard log-rank test uses weights: \[ w_j=1 \]

The values of the weights in the log-rank test can be calculated using the function find_weights with argument method="lr". In the case of the standard log-rank test, the weights are clearly very simple.

find_weights(formula=Surv(event_time,event_status)~group,
  data=sim_data,
  method="lr",
  include_cens = FALSE)
#> [1] 1 1 1 1 1 1 1

The Fleming-Harrington (\(\rho\),\(\gamma\)) test uses weights: \[ w_j = \hat{S}(t_{j}-) ^ \rho (1 - \hat{S}(t_{j}-)) ^ \gamma \] where \(\hat{S}(t)\) is the Kaplan Meier estimate of the survival curve in the pooled data (both treatment arms) and time \(t_j-\) is the time just before \(t_j\). There is the matter of choosing \(\rho\) and \(\gamma\). A popular choice is \(\rho=0\) and \(\gamma=1\) which means that the weights are equal to 1 minus the Kaplan Meier estimate of the survival curve.

Again the weights can be calculated using the find_weights function and setting method="fh", along with arguments rho and gamma.

find_weights(formula=Surv(event_time,event_status)~group,
  data=sim_data,
  method="fh",
  rho = 0,
  gamma= 1,
  include_cens = FALSE)
#> [1] 0.0 0.1 0.2 0.3 0.4 0.5 0.6

The modestly weighted log-rank test uses weights: \[ w_j = 1 / \max{\{\hat{S}(t_{j}-), \hat{S}(t^\ast)\}} \] There is the matter of choosing \(t^\ast\) or alternatively choosing the value of \(\hat{S}(t^\ast)\). See Magirr (2021) for a discussion on this.

find_weights(formula=Surv(event_time,event_status)~group,
  data=sim_data,
  method="mw",
  s_star = 0.5,
  include_cens = FALSE)
#> [1] 1.000000 1.111111 1.250000 1.428571 1.666667 2.000000 2.000000

Test statistic

Under the null hypothesis that the survival curves of the two treatment arms are equal, the distribution of \(U^W\) is

\[ U^W \sim N\left( 0, V^W \right) \]

where the variance, \(V^W\), is equal to \[ \sum_{j=1}^k w_j^2\frac{n_{0,j}n_{1,j} d_j (n_j - d_j)}{n_j^2(n_j-1)} \]

The Z-statistic is then simply calculated in the usual way by dividing the test statistic \(U\) by the square root of its variance.

To perform the full weighted log-rank test, use the function wlrt. This outputs the test statistic, its variance, the Z-statistic and the name of the treatment group the test corresponds to.

wlrt(formula=Surv(event_time,event_status)~group,
  data=sim_data,
  method="mw",
  s_star = 0.5)
#>            u     v_u          z    trt_group
#> 1 -0.8651849 3.91482 -0.4372734 experimental

Permutation test and scores

Leton and Zuluaga (2001) showed that every weighted log-rank test can be written as either an observed-minus-expected test (as described above), or as a permutation test.

The weights can be reformulated as scores for a permutation test using the following formula for the censoring scores and event scores respectively:

\[ C_j=-\sum_{i=1}w_i\frac{d_i}{n_i} \]

\[ c_j=C_j+w_j \]

These scores can be calculated using the function find_scores in the following way. Plotting these scores against the rank of the event times provides an intuitive explanation of the issues of using the Fleming-Harrington test as it makes sense that the scores for the events are decreasind with time, see Magirr (2021).

df_scores_mw<-find_scores(formula=Surv(event_time,event_status)~group,
  data=sim_data,
  method="mw",
  s_star = 0.5)
plot(df_scores_mw)

df_scores_fh<-find_scores(formula=Surv(event_time,event_status)~group,
  data=sim_data,
  method="fh",
  rho = 0,
  gamma=1)
plot(df_scores_fh)

Stratification

The issue of stratification when performing weighted log-rank tests is discussed in Magirr and Jiménez (2022). They explore various approaches to combining the results of stratified analyses. In particular they recommend combining on the Z-statistic scale, i.e. for the case of two strata, first express the stratified log-rank test as a linear combination of standardized Z-statistics, \(\sqrt{V_1}Z_1+\sqrt{V_2}Z_2 \sim N(0,V_1+V_2)\). \(V_1\) and \(V_2\) are the variances for the log-rank test statistic on the first and second stratum respectively, and \(Z_1\) and \(Z_2\) are the Z-statistics for the log-rank test statistic on the first and second stratum respectively. Secondly, the Z-statistics \(Z_1\) and \(Z_2\) are replaced by the Z-statistics from the weighted log-rank test.

\[ \tilde{U}^W=\sqrt{V_1}\left( \frac{U_1^W}{\sqrt{V_1^W}}\right)+\sqrt{V_2}\left( \frac{U_2^W}{\sqrt{V_2^W}}\right) \]

Here we introduce a strata ecog that has different \(\lambda\) parameters, and demonstrate that it is simple to perform the described stratified weighted log-rank test.

sim_data_0 <- sim_data
sim_data_0$ecog=0
sim_data_1 <- sim_events_delay(
  event_model=list(
    duration_c = 36,
    duration_e = c(6,30),
    lambda_c = log(2)/6,
    lambda_e = c(log(2)/6,log(2)/12)),
  recruitment_model=list(
    rec_model="power",
    rec_period=12,
    rec_power=1),
  n_c=5,
  n_e=5,
  max_cal_t = 36
)
sim_data_1$ecog=1
sim_data_strata<-rbind(sim_data_0,sim_data_1)
wlrt(formula=Surv(event_time,event_status)~group+strata(ecog),
  data=sim_data_strata,
  method="mw",
  t_star = 4
)
#> $by_strata
#>   strata          u      v_u          z    trt_group
#> 1  ecog0  0.1615079 1.647592  0.1258256 experimental
#> 2  ecog1 -2.2293871 2.386703 -1.4430662 experimental
#> 
#> $combined
#>          u        v          z    trt_group
#> 1 -1.70296 3.316904 -0.9350569 experimental