modest-wlrt.Rmd

---
title: "Using the Magirr-Burman weights for testing"
author: "Keaven Anderson, Yujie Zhao"
output: rmarkdown::html_vignette
bibliography: simtrial.bib
vignette: >
  %\VignetteIndexEntry{Using the Magirr-Burman weights for testing}
  %\VignetteEngine{knitr::rmarkdown}
---

```{r, include=FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)

run <- requireNamespace("dplyr", quietly = TRUE)
knitr::opts_chunk$set(eval = run)
```

## Introduction

@MagirrBurman implemented a modestly weighted logrank test with the following claim:

> Tests from this new class can be constructed to have high power under a
> delayed-onset treatment effect scenario, as well as being almost as efficient
> as the standard logrank test under proportional hazards.

They have implemented this in the package [modestWLRT](https://github.com/dominicmagirr/modestWLRT).
Since the implementation is relatively straightforward, we have added this functionality to the simtrial package and explain how to use it here with the `mb_weight()` function.

Packages used are as follows:

```{r, message=FALSE, warning=FALSE}
library(simtrial)
library(dplyr)
library(survival)
```

## Simulating a delayed effect example

First, we specify study duration, sample size and enrollment rates. The enrollment rate is assumed constant during the enrollment period until the targeted sample size is reached.
For failure rates, we consider the delayed treatment effect example of @MagirrBurman.
The control group has an exponential failure rate with a median of 15 months.
For the initial 6 months, the underlying hazard ratio is one followed by a hazard ratio of 0.7 thereafter.
This differs from the @MagirrBurman delayed effect assumptions only in that they assume a hazard ratio of 0.5 after 6 months.

```{r, message=FALSE, warning=FALSE}
study_duration <- 36
sample_size <- 300
enroll_rate <- data.frame(duration = 12, rate = 200 / 12)
fail_rate <- data.frame(
  stratum = c("All", "All"),
  duration = c(6, 36),
  fail_rate = c(log(2) / 15, log(2) / 15),
  hr = c(1, .7),
  dropout_rate = c(0, 0)
)
```

Now we generate a single dataset with the above characteristics and cut data for analysis at 36 months post start of enrollment.
Then we plot Kaplan-Meier curves for the resulting dataset (red curve for experimental treatment, black for control):

```{r, message=FALSE, warning=FALSE, fig.width=7.5, fig.height=4}
set.seed(7789)
xpar <- to_sim_pw_surv(fail_rate)
MBdelay <- sim_pw_surv(
  n = sample_size,
  stratum = data.frame(stratum = "All", p = 1),
  block = c(rep("control", 2), rep("experimental", 2)),
  enroll_rate = enroll_rate,
  fail_rate = xpar$fail_rate,
  dropout_rate = xpar$dropout_rate
) |>
  cut_data_by_date(study_duration)
fit <- survfit(Surv(tte, event) ~ treatment, data = MBdelay)
plot(fit, col = 1:2, mark = "|", xaxt = "n")
axis(1, xaxp = c(0, 36, 6))
```

## Generalizing the Magirr-Burman test

Next, we consider the @Magirr2021 extension of the modestly weighted logrank test (MWLRT) of @MagirrBurman where we have weights as follows:

$$w(t, \tau, w_{\max}) = \min\left(w_{\max},\left(\frac{1}{S(\min(t,\tau))}\right)\right).$$

This requires generating weights and then computing the test.
We begin with the default of `w_max=Inf` which corresponds to the original @MagirrBurman test and set the time until maximum weight $\tau$ with `delay = 6`.

```{r}
ZMB <- MBdelay |>
  wlr(weight = mb(delay = 6))
# Compute p-value of modestly weighted logrank of Magirr-Burman
pnorm(ZMB$z)
```

Now we set the maximum weight to be 2 as in @Magirr2021 and set the `delay=Inf` so that the maximum weight begins at the observed median of the observed combined treatment Kaplan-Meier curve.

```{r}
ZMB <- MBdelay |>
  wlr(weight = mb(delay = Inf, w_max = 2))
# Compute p-value of modestly weighted logrank of Magirr-Burman
pnorm(ZMB$z)
```

Another way this can be done is with a generalized Fleming-Harrington test with

$$w(t; \rho, \gamma, w_{\max})= \min((1-F(t))^\rho F(t)^\gamma, w_{\max})).$$

and let $\gamma=0, \rho = -1/2.$

```{r}
w_max <- 2
Z_modified_FH <- MBdelay |>
  counting_process(arm = "experimental") |>
  mutate(w = pmin(w_max, 1 / s)) |>
  summarize(
    S = sum(o_minus_e * w),
    V = sum(var_o_minus_e * w^2),
    z = S / sqrt(V)
  )
# Compute p-value of modestly weighted logrank of Magirr-Burman
pnorm(Z_modified_FH$z)
```

### Freidlin and Korn strong null hypothesis example

For the next example, underlying survival is uniformly worse in the experimental group compared to control throughout the planned follow-up.
This was presented by @FKNPH2019. For this case, we have a hazard ratio of 16 for 1/10 of 1 year (1.2 months), followed by a hazard ratio of 0.76 thereafter.

First, we specify study duration, sample size and enrollment rates. The enrollment rate is assumed constant during the enrollment period until the targeted sample size is reached.
For failure rates, we consider the delayed treatment effect example of @MagirrBurman.

```{r, message=FALSE, warning=FALSE}
study_duration <- 5
sample_size <- 2000
enroll_duration <- .0001
enroll_rate <- data.frame(
  duration = enroll_duration,
  rate = sample_size / enroll_duration
)
fail_rate <- data.frame(
  stratum = "All",
  fail_rate = 0.25,
  dropout_rate = 0,
  hr = c(4 / .25, .19 / .25),
  duration = c(.1, 4.9)
)
```

Now we generate a single dataset with the above characteristics and cut data for analysis at 5 years post start of enrollment.
Then we plot Kaplan-Meier curves for the resulting dataset (red curve for experimental treatment, black for control):

```{r, message=FALSE, warning=FALSE, fig.width=7.5, fig.height=4}
set.seed(7783)
xpar <- to_sim_pw_surv(fail_rate)
FHwn <- sim_pw_surv(
  n = sample_size,
  stratum = data.frame(stratum = "All", p = 1),
  block = c(rep("control", 2), rep("experimental", 2)),
  enroll_rate = enroll_rate,
  fail_rate = xpar$fail_rate,
  dropout_rate = xpar$dropout_rate
) |>
  cut_data_by_date(study_duration)
fit <- survfit(Surv(tte, event) ~ treatment, data = FHwn)
plot(fit, col = 1:2, mark = "|", xaxt = "n")
axis(1, xaxp = c(0, 36, 6))
```

We perform a logrank and weighted logrank tests as suggested for more limited downweighting by follows, and a MaxCombo test with these component tests, we have p-value of:

```{r}
xx <- FHwn |>
  maxcombo(rho = c(0, 0, 1), gamma = c(0, 1, 1))
xx
```


Next, we consider the @MagirrBurman modestly weighted logrank test with down-weighting specified for the first 6 months but a maximum weight of 2.
This requires generating weights and then computing the test.

```{r}
ZMB <- FHwn |>
  wlr(weight = mb(delay = 6, w_max = 2))

# Compute p-value of modestly weighted logrank of Magirr-Burman
pnorm(ZMB$z)
```

Finally, we consider weighted logrank tests with less down-weighting.
Results are quite similar to the results with greater down-weighting.
We have p-value of

```{r}
xx <- FHwn |>
  maxcombo(rho = c(0, 0, .5), gamma = c(0, .5, .5))
xx
```

Thus, with less down-weighting the MaxCombo test appears less problematic.
This is addressed at greater length in @Mukhopadhyay2022.

## References