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.

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
```

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.

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.

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.

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).

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
```

Leton, E. and Zuluaga, P. (2001) Equivalence between score and weighted tests for survival curves. Commun Stat., 30(4), 591-608.

Magirr, D. (2021). Non-proportional hazards in immuno-oncology: Is an old perspective needed?. Pharmaceutical Statistics, 20(3), 512-527.

Magirr, D. and Burman, C.F., (2019). Modestly weighted logrank tests. Statistics in medicine, 38(20), 3782-3790.

Magirr, D. and Jiménez, J. (2022) Stratified modestly-weighted log-rank tests in settings with an anticipated delayed separation of survival curves PREPRINT at https://arxiv.org/abs/2201.10445