---
title: "Propensity Score-Integrated Survival Inference in Randomized Controlled Trials (RCTs) with Augmenting Control Arm"
date: "`r Sys.Date()`"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Propensity Score-Integrated Survival Inference in Randomized Controlled Trials (RCTs) with Augmenting Control Arm}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---


```{r, eval=T, echo=FALSE}
suppressMessages(require(psrwe, quietly = TRUE))
options(digits = 3)
set.seed(1000)
```
<br>


## Introduction

In the **psrwe**, PS-integrated survival analyses
in randomized controlled trials (RCTs) with augmenting control arm
(Chen, et al., to be submitted)
are also implemented in three functions:

* `psrwe_survkm()` for treatment effect test (Com-Nougue, et al., 1993).
* `psrwe_survlrk()` for log-rank test (Klein and Moeschberger, 2003; Peto and Peto, 1972).
* `psrwe_survrmst()` for restricted mean survival time (RMST) test (Royston and Parmar, 2013; Uno, et al., 2014).

These tests are non-parametric approaches for comparing two treatments
with time-to-event endpoints.
Therefore, these tests are only implemented for RCTs with
augmenting control arm.

Similar with the approaches: PSPP (Wang, et al., 2019),
PSCL (Wang, et al., 2020), and PSKM (Chen, et al., 2022),
the PS-integrated study design functions, `psrwe_est()` and `psrwe_borrow()`,
below estimate PS model, set borrowing parameters, and determine
discounting parameters for borrowing information
for a two-arm RCT with augmenting control arm from RWD.

```{r, eval=T, echo=TRUE}
data(ex_dta_rct)
dta_ps_rct <- psrwe_est(ex_dta_rct,
                        v_covs = paste("V", 1:7, sep = ""),
                        v_grp = "Group", cur_grp_level = "current",
                        v_arm = "Arm", ctl_arm_level = "control",
                        ps_method = "logistic", nstrata = 5,
                        stra_ctl_only = FALSE)
ps_bor_rct <- psrwe_borrow(dta_ps_rct, total_borrow = 30)
```
<br>


## PS-integrated treatment effect test

Similar with the single arm study example
(in **psrwe/demo/sec_4_4_ex.r** and `demo("sec_4_5_ex", package = "psrwe")`),
the code below evaluates two-arm RCT.
The results show the treatment effect which is
the survival difference between two arms at one year or 365 days.

```{r, eval=T, echo=TRUE}
rst_km_rct <- psrwe_survkm(ps_bor_rct,
                           pred_tp = 365,
                           v_time = "Y_Surv",
                           v_event = "Status")
rst_km_rct
```

The estimated PSKM curves with confidence intervals can be visualized below.

```{r, echo=TRUE, fig.width=6, fig.height=5}
plot(rst_km_rct, xlim = c(0, 730))
```

The inference is based on the treatment effect
$S_{trt}(\tau) - S_{ctl}(\tau)$ at $\tau = 365$ days
where $S_{trt}$ and $S_{ctl}$ are the
survival probabilities of the treatment and control arms, respectively.
i.e., the example tests
$$
H_0: S_{trt}(\tau) - S_{ctl}(\tau) \leq 0 \quad \mbox{vs.} \quad
H_a: S_{trt}(\tau) - S_{ctl}(\tau) > 0 .
$$
The outcome analysis can be summarized below.
Note that this is an one-sided test.

```{r, eval=T, echo=TRUE}
oa_km_rct <- psrwe_outana(rst_km_rct, alternative = "greater")
oa_km_rct
```

The details of estimates for each arm can be printed via the `print()`
function with the option `show_rct = TRUE`.

```{r, eval=T, echo=TRUE}
print(oa_km_rct, show_rct = TRUE)
```

As the **survival** package, the results of other time points can be also
predicted via the `summary()` with the option `pred_tps`.

```{r, eval=T, echo=TRUE}
summary(oa_km_rct, pred_tps = c(180, 365))
```
<br>


## PS-integrated log-rank test

The log-rank test is another way to compare two treatments of
time-to-event endpoint.
Similar the PSKM for the two-arm test above, the function
`psrwe_survlrk()` computes the statistic for each distinctive time point
beased on the observed data then returns all necessary results for
the down-streatm analyses such as tests and confidence intervals.

```{r, eval=T, echo=TRUE}
rst_lrk <- psrwe_survlrk(ps_bor_rct,
                         pred_tp = 365,
                         v_time = "Y_Surv",
                         v_event = "Status")
rst_lrk
```

The inference is based on the log-rank method to test whether two survival
distributions are different from each other.
The example tests
$$
H_0: S_{trt}(t) = S_{ctl}(t) \quad \mbox{vs.} \quad
H_a: S_{trt}(t) \neq S_{ctl}(t) 0
$$
for all $t \leq \tau$ where $\tau = 365$ days.
The outcome analysis can be summarized below.

```{r, eval=T, echo=TRUE}
oa_lrk <- psrwe_outana(rst_lrk)
oa_lrk
```

The details of estimates for each arm can be printed via the `print()`
function with the option `show_rct = TRUE`.

```{r, eval=T, echo=TRUE}
print(oa_lrk, show_details = TRUE)
```

As the **survival** package, the results of other time points can be also
predicted via the `summary()` with the option `pred_tps`.

```{r, eval=T, echo=TRUE}
summary(oa_lrk, pred_tps = c(180, 365))
```
<br>


## PS-integrated restricted mean survival time (RMST) test

The restricted means survival time (RMST) between two
treatments is to test whether areas under
two survival distributions (AUC) are different from each other.
Similar the log-rank test above, the function
`psrwe_survrmst()` computes the statistic for each distinctive time point
beased on the observed data then returns all necessary results for
the down-streatm analyses such as tests and confidence intervals.

```{r, eval=T, echo=TRUE}
rst_rmst <- psrwe_survrmst(ps_bor_rct,
                           pred_tp = 365,
                           v_time = "Y_Surv",
                           v_event = "Status")
rst_rmst
```

The inference is based on the to compare whether AUCs
are different from each other.
The example tests
$$
H_0: \int_0^{\tau} S_{trt}(t) dt = \int_0^{\tau} S_{ctl}(t) dt
\quad \mbox{vs.} \quad
H_a: \int_0^{\tau} S_{trt}(t) dt \neq \int_0^{\tau} S_{ctl}(t) dt
$$
where $\tau = 365$ days.
The outcome analysis can be summarized below.
Note that this is a two-sided test.

```{r, eval=T, echo=TRUE}
oa_rmst <- psrwe_outana(rst_rmst)
oa_rmst
```

The details of estimates for each arm can be printed via the `print()`
function with the option `show_rct = TRUE`.

```{r, eval=T, echo=TRUE}
print(oa_rmst, show_details = TRUE)
```

As the **survival** package, the results of other time points can be also
predicted via the `summary()` with the option `pred_tps`.

```{r, eval=T, echo=TRUE}
summary(oa_rmst, pred_tps = c(180, 365))
```
<br>


## Demo examples

The scripts in
"**psrwe/demo/sec_4_5_ex.r**",
"**psrwe/demo/sec_4_6_ex.r**", and
"**psrwe/demo/sec_4_7_ex.r**"
source files have
the full examples for the PS-integrated survival analyses which can be run via
the `demo("sec_4_5_ex", package = "psrwe")`,
`demo("sec_4_6_ex", package = "psrwe")`, and
`demo("sec_4_7_ex", package = "psrwe")`, respectively.

Two Jackknife standard errors are also demonstrated for each test method.
Note that Jackknife standard errors may take a while to finish.
<br>


## References

1. Chen, W.-C., Lu, N., Wang, C., Li, H., Song, C., Tiwari, R., Xu, Y., and
Yue, L.Q. (to be submitted).
Propensity Score-Integrated Statistical Tests for Survival Analysis:
Leveraging External Evidence for Augmenting the Control Arm of a
Randomized Controlled Trial.

2. Chen, W.-C., Lu, N., Wang, C., Li, H., Song, C., Tiwari, R., Xu, Y., and
Yue, L.Q. (2022).
Propensity Score-Integrated Approach to Survival Analysis: Leveraging External
Evidence in Single-Arm Studies.
Journal of Biopharmaceutical Statistics, 32(3), 400-413.

3. Com-Nougue, C., Rodary, C. and Patte, C. (1993).
How to establish equivalence when data are censored: A randomized trial of
treatments for B non-Hodgkin lymphoma.
Statist. Med., Volume 12, pp. 1353-1364.

4. Klein, J. and Moeschberger, M. (2003).
Survival Analysis: Techniques for Censored and Truncated Data.
2nd ed. New York: Springer.

5. Peto, R. and Peto, J. (1972).
Asymptotically Efficient Rank Invariant Test Procedures.
Journal of the Royal Statistical Society, Series A, 135(2), 185-207.

6. Royston, P. and Parmar, M. K. (2013).
Restricted mean survival time: an alternative to the hazard ratio for the
design and analysis of randomized trials with a time-to-event outcome.
BMC Med Res Methodol, 13(152).

7. Uno, H., et al., (2014).
Moving beyond the hazard ratio in quantifying the between-group difference
in survival analysis.
Journal of clinical oncology, Volume 32, 2380-2385.

8. Wang, C., Li, H., Chen, W. C., Lu, N., Tiwari, R., Xu, Y., and Yue, L.Q.
(2019).
Propensity score-integrated power prior approach for incorporating
real-world evidence in single-arm clinical studies.
Journal of Biopharmaceutical Statistics, 29(5), 731-748.

9. Wang, C., Lu, N., Chen, W. C., Li, H., Tiwari, R., Xu, Y., and Yue, L.Q.
(2020).
Propensity score-integrated composite likelihood approach for
incorporating real-world evidence in single-arm clinical studies.
Journal of Biopharmaceutical Statistics, 30(3), 495-507.