---
title: "6. Comparison of Fixed Weights"
author: "Matt Secrest and Isaac Gravestock"
output:
rmarkdown::html_vignette:
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toc_depth: 2
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%\VignetteIndexEntry{6. Comparison of Fixed Weights}
%\VignetteEncoding{UTF-8}
%\VignetteEngine{knitr::rmarkdown}
editor_options:
chunk_output_type: console
---
# Introduction
In a `psborrow2` analysis it is possible to specify fixed weights for an observation's log-likelihood contribution.
This is similar to a weighted regression or a fixed power prior parameter.
This vignette will show how weights can be specified and compare regression model results with other packages.
We will compare a `glm` model with weights, a weighted likelihood in Stan with `psborrow2`, and
`BayesPPD::glm.fixed.a0` for generalized linear models with fixed `a0` (power prior parameter).
Note that we'll need `cmdstanr` to run this analysis. Please install
`cmdstanr` if you have not done so already following
[this guide](https://mc-stan.org/cmdstanr/articles/cmdstanr.html).
```r
library(psborrow2)
library(cmdstsanr)
# Error in library(cmdstsanr): there is no package called 'cmdstsanr'
library(BayesPPD)
library(ggplot2)
# Warning: package 'ggplot2' was built under R version 4.3.2
```
# Logistic regression
We fit logistic regression models with the external control arm having weights (or power parameters)
equal to 0, 0.25, 0.5, 0.75, 1. The internal treated and control patients have weight = 1.
The model has a treatment indicator and two covariates, `resp ~ trt + cov1 + cov2`.
### glm
```r
logistic_glm_reslist <- lapply(c(0, 0.25, 0.5, 0.75, 1), function(w) {
logistic_glm <- glm(
resp ~ trt + cov1 + cov2,
data = as.data.frame(example_matrix),
family = binomial,
weights = ifelse(example_matrix[, "ext"] == 1, w, 1)
)
glm_summary <- summary(logistic_glm)$coef
ci <- confint(logistic_glm)
data.frame(
fitter = "glm",
borrowing = w,
variable = c("(Intercept)", "trt", "cov1", "cov2"),
estimate = glm_summary[, "Estimate"],
lower = ci[, 1],
upper = ci[, 2]
)
})
```
### BayesPPD
```r
set.seed(123)
logistic_ppd_reslist <- lapply(c(0, 0.25, 0.5, 0.75, 1), function(w) {
logistic_ppd <- glm.fixed.a0(
data.type = "Bernoulli",
data.link = "Logistic",
y = example_matrix[example_matrix[, "ext"] == 0, ][, "resp"],
x = example_matrix[example_matrix[, "ext"] == 0, ][, c("trt", "cov1", "cov2")],
historical = list(list(
y0 = example_matrix[example_matrix[, "ext"] == 1, ][, "resp"],
x0 = example_matrix[example_matrix[, "ext"] == 1, ][, c("cov1", "cov2")],
a0 = w
)),
lower.limits = rep(-100, 5),
upper.limits = rep(100, 5),
slice.widths = rep(1, 5),
nMC = 10000,
nBI = 1000
)[[1]]
ci <- apply(logistic_ppd, 2, quantile, probs = c(0.025, 0.975))
data.frame(
fitter = "BayesPPD",
borrowing = w,
variable = c("(Intercept)", "trt", "cov1", "cov2"),
estimate = colMeans(logistic_ppd),
lower = ci[1, ],
upper = ci[2, ]
)
})
```
### psborrow2
```r
logistic_psb_reslist <- lapply(c(0, 0.25, 0.5, 0.75, 1), function(w) {
logistic_psb2 <- create_analysis_obj(
data_matrix = as.matrix(cbind(
example_matrix,
w = ifelse(example_matrix[, "ext"] == 1, w, 1)
)),
covariates = add_covariates(c("cov1", "cov2"),
priors = prior_normal(0, 100)
),
borrowing = borrowing_full("ext"),
treatment = treatment_details("trt", prior_normal(0, 100)),
outcome = outcome_bin_logistic("resp",
baseline_prior = prior_normal(0, 1000),
weight_var = "w"
),
quiet = TRUE
)
mcmc_logistic_psb2 <- mcmc_sample(logistic_psb2, chains = 1, verbose = FALSE, seed = 123)
mcmc_summary <- mcmc_logistic_psb2$summary(
variables = c("alpha", "beta_trt", "beta[1]", "beta[2]"),
mean,
~ quantile(.x, probs = c(0.025, 0.975))
)
data.frame(
fitter = "psborrow2",
borrowing = w,
variable = c("(Intercept)", "trt", "cov1", "cov2"),
estimate = mcmc_summary$mean,
lower = mcmc_summary$`2.5%`,
upper = mcmc_summary$`97.5%`
)
})
```
### Results
```r
logistic_res_df <- do.call(
rbind,
c(logistic_glm_reslist, logistic_ppd_reslist, logistic_psb_reslist)
)
logistic_res_df$est_ci <- sprintf(
"%.3f (%.3f, %.3f)",
logistic_res_df$estimate, logistic_res_df$lower, logistic_res_df$upper
)
wide <- reshape(
logistic_res_df[, c("fitter", "borrowing", "variable", "est_ci")],
direction = "wide",
timevar = "fitter",
idvar = c("borrowing", "variable"),
)
new_order <- order(wide$variable, wide$borrowing)
knitr::kable(wide[new_order, ], digits = 3, row.names = FALSE)
```
| borrowing|variable |est_ci.glm |est_ci.BayesPPD |est_ci.psborrow2 |
|---------:|:-----------|:-----------------------|:-----------------------|:-----------------------|
| 0.00|(Intercept) |0.646 (-0.038, 1.357) |0.691 (-0.014, 1.399) |0.657 (-0.039, 1.381) |
| 0.25|(Intercept) |0.394 (-0.131, 0.931) |0.396 (-0.131, 0.941) |0.404 (-0.130, 0.939) |
| 0.50|(Intercept) |0.293 (-0.158, 0.751) |0.297 (-0.184, 0.767) |0.304 (-0.169, 0.784) |
| 0.75|(Intercept) |0.235 (-0.168, 0.642) |0.240 (-0.175, 0.665) |0.237 (-0.164, 0.654) |
| 1.00|(Intercept) |0.196 (-0.172, 0.567) |0.202 (-0.172, 0.578) |0.203 (-0.167, 0.568) |
| 0.00|cov1 |-0.771 (-1.465, -0.095) |-0.809 (-1.515, -0.126) |-0.789 (-1.494, -0.099) |
| 0.25|cov1 |-0.781 (-1.340, -0.231) |-0.793 (-1.350, -0.236) |-0.794 (-1.351, -0.250) |
| 0.50|cov1 |-0.769 (-1.252, -0.291) |-0.776 (-1.271, -0.270) |-0.786 (-1.295, -0.300) |
| 0.75|cov1 |-0.758 (-1.191, -0.329) |-0.769 (-1.212, -0.310) |-0.763 (-1.198, -0.324) |
| 1.00|cov1 |-0.749 (-1.145, -0.357) |-0.761 (-1.152, -0.369) |-0.758 (-1.150, -0.369) |
| 0.00|cov2 |-0.730 (-1.472, -0.008) |-0.745 (-1.496, -0.004) |-0.752 (-1.496, -0.006) |
| 0.25|cov2 |-0.559 (-1.114, -0.014) |-0.568 (-1.124, -0.023) |-0.571 (-1.132, -0.016) |
| 0.50|cov2 |-0.459 (-0.926, 0.003) |-0.471 (-0.953, -0.003) |-0.464 (-0.928, 0.005) |
| 0.75|cov2 |-0.398 (-0.811, 0.011) |-0.402 (-0.814, 0.006) |-0.407 (-0.824, 0.002) |
| 1.00|cov2 |-0.358 (-0.731, 0.013) |-0.358 (-0.736, 0.022) |-0.363 (-0.741, 0.016) |
| 0.00|trt |0.154 (-0.558, 0.871) |0.137 (-0.572, 0.864) |0.165 (-0.567, 0.894) |
| 0.25|trt |0.349 (-0.183, 0.885) |0.361 (-0.170, 0.899) |0.349 (-0.202, 0.875) |
| 0.50|trt |0.405 (-0.082, 0.894) |0.415 (-0.068, 0.916) |0.409 (-0.078, 0.892) |
| 0.75|trt |0.434 (-0.031, 0.900) |0.436 (-0.031, 0.913) |0.439 (-0.030, 0.919) |
| 1.00|trt |0.452 (-0.000, 0.905) |0.456 (-0.010, 0.909) |0.453 (-0.009, 0.917) |
```r
logistic_res_df$borrowing_x <- logistic_res_df$borrowing +
(as.numeric(factor(logistic_res_df$fitter)) - 3) / 100
ggplot(logistic_res_df, aes(x = borrowing_x, y = estimate, group = fitter, colour = fitter)) +
geom_errorbar(aes(ymin = lower, ymax = upper)) +
geom_point() +
facet_wrap(~variable, scales = "free")
```
# Exponential models
Now we fit models with an exponentially distributed outcome. There is no censoring in this data set.
For `glm` we use `family = Gamma(link = "log")` and specify fixed `dispersion = 1` to fit a exponential model.
As before, the external control arm having weights (or power parameters)
equal to 0, 0.25, 0.5, 0.75, 1. The internal treated and control patients have weight = 1.
The model has a treatment indicator and two covariates, `eventtime ~ trt + cov1 + cov2`.
```r
set.seed(123)
sim_data_exp <- cbind(
simsurv::simsurv(
dist = "exponential",
x = as.data.frame(example_matrix[, c("trt", "cov1", "cov2", "ext")]),
betas = c("trt" = 1.3, "cov1" = 1, "cov2" = 0.1, "ext" = -0.4),
lambdas = 5
),
example_matrix[, c("trt", "cov1", "cov2", "ext")],
censor = 0
)
```
```r
head(sim_data_exp)
# id eventtime status trt cov1 cov2 ext censor
# 1 1 0.14802638 1 0 0 1 0 0
# 2 2 0.05065174 1 0 1 0 0 0
# 3 3 0.01727805 1 0 1 0 0 0
# 4 4 0.13168620 1 0 1 0 0 0
# 5 5 0.07740706 1 0 0 0 0 0
# 6 6 0.04999778 1 0 1 0 0 0
```
```r
## glm
glm_reslist <- lapply(c(0, 0.25, 0.5, 0.75, 1), function(w) {
exp_glm <- glm(
eventtime ~ trt + cov1 + cov2,
data = sim_data_exp,
family = Gamma(link = "log"),
weights = ifelse(sim_data_exp$ext == 1, w, 1)
)
glm_summary <- summary(exp_glm, dispersion = 1)
est <- -glm_summary$coefficients[, "Estimate"]
lower <- est - 1.96 * glm_summary$coefficients[, "Std. Error"]
upper <- est + 1.96 * glm_summary$coefficients[, "Std. Error"]
data.frame(
fitter = "glm",
borrowing = w,
variable = c("(Intercept)", "trt", "cov1", "cov2"),
estimate = est,
lower = lower,
upper = upper
)
})
```
```r
## BayesPPD
set.seed(123)
ppd_reslist <- lapply(c(0, 0.25, 0.5, 0.75, 1), function(w) {
exp_ppd <- glm.fixed.a0(
data.type = "Exponential",
data.link = "Log",
y = sim_data_exp[sim_data_exp$ext == 0, ]$eventtime,
x = as.matrix(sim_data_exp[sim_data_exp$ext == 0, c("trt", "cov1", "cov2")]),
historical = list(list(
y0 = sim_data_exp[sim_data_exp$ext == 1, ]$eventtime,
x0 = as.matrix(sim_data_exp[sim_data_exp$ext == 1, c("cov1", "cov2")]),
a0 = w
)),
lower.limits = rep(-100, 5),
upper.limits = rep(100, 5),
slice.widths = rep(1, 5),
nMC = 10000,
nBI = 1000
)[[1]]
ci <- apply(exp_ppd, 2, quantile, probs = c(0.025, 0.975))
data.frame(
fitter = "BayesPPD",
borrowing = w,
variable = c("(Intercept)", "trt", "cov1", "cov2"),
estimate = colMeans(exp_ppd),
lower = ci[1, ],
upper = ci[2, ]
)
})
```
```r
## psborrow2
psb_reslist <- lapply(c(0, 0.25, 0.5, 0.75, 1), function(w) {
exp_psb2 <- create_analysis_obj(
data_matrix = as.matrix(cbind(sim_data_exp, w = ifelse(example_matrix[, "ext"] == 1, w, 1))),
covariates = add_covariates(c("cov1", "cov2"),
priors = prior_normal(0, 100)
),
borrowing = borrowing_full("ext"),
treatment = treatment_details("trt", prior_normal(0, 100)),
outcome = outcome_surv_exponential("eventtime", "censor",
baseline_prior = prior_normal(0, 1000),
weight_var = "w"
),
quiet = TRUE
)
mcmc_exp_psb2 <- mcmc_sample(exp_psb2, chains = 1, verbose = FALSE, seed = 123)
mcmc_summary <- mcmc_exp_psb2$summary(
variables = c("alpha", "beta_trt", "beta[1]", "beta[2]"),
mean,
~ quantile(.x, probs = c(0.025, 0.975))
)
data.frame(
fitter = "psborrow2",
borrowing = w,
variable = c("(Intercept)", "trt", "cov1", "cov2"),
estimate = mcmc_summary$mean,
lower = mcmc_summary$`2.5%`,
upper = mcmc_summary$`97.5%`
)
})
```
```r
knitr::knit_hooks$set(output = output_hook)
```
### Results
Note: Wald confidence intervals are displayed here for `glm` for the exponential models.
```r
res_df <- do.call(rbind, c(glm_reslist, ppd_reslist, psb_reslist))
res_df$est_ci <- sprintf(
"%.3f (%.3f, %.3f)",
res_df$estimate, res_df$lower, res_df$upper
)
wide <- reshape(
res_df[, c("fitter", "borrowing", "variable", "est_ci")],
direction = "wide",
timevar = "fitter",
idvar = c("borrowing", "variable"),
)
new_order <- order(wide$variable, wide$borrowing)
knitr::kable(wide[new_order, ], digits = 3, row.names = FALSE)
```
| borrowing|variable |est_ci.glm |est_ci.BayesPPD |est_ci.psborrow2 |
|---------:|:-----------|:----------------------|:----------------------|:----------------------|
| 0.00|(Intercept) |1.930 (1.597, 2.263) |1.915 (1.572, 2.236) |1.914 (1.576, 2.239) |
| 0.25|(Intercept) |1.581 (1.323, 1.838) |1.567 (1.308, 1.814) |1.573 (1.311, 1.819) |
| 0.50|(Intercept) |1.473 (1.251, 1.695) |1.466 (1.247, 1.685) |1.467 (1.244, 1.683) |
| 0.75|(Intercept) |1.414 (1.215, 1.613) |1.407 (1.208, 1.601) |1.409 (1.204, 1.605) |
| 1.00|(Intercept) |1.376 (1.194, 1.558) |1.369 (1.183, 1.551) |1.374 (1.198, 1.550) |
| 0.00|cov1 |0.630 (0.300, 0.959) |0.630 (0.304, 0.960) |0.634 (0.298, 0.973) |
| 0.25|cov1 |0.722 (0.453, 0.991) |0.729 (0.459, 1.013) |0.724 (0.455, 1.006) |
| 0.50|cov1 |0.786 (0.552, 1.020) |0.789 (0.559, 1.026) |0.788 (0.555, 1.027) |
| 0.75|cov1 |0.827 (0.616, 1.037) |0.829 (0.622, 1.044) |0.829 (0.625, 1.040) |
| 1.00|cov1 |0.854 (0.662, 1.046) |0.859 (0.668, 1.057) |0.852 (0.666, 1.038) |
| 0.00|cov2 |0.043 (-0.309, 0.395) |0.039 (-0.318, 0.382) |0.037 (-0.324, 0.386) |
| 0.25|cov2 |-0.009 (-0.273, 0.255) |-0.008 (-0.283, 0.252) |-0.012 (-0.279, 0.251) |
| 0.50|cov2 |0.009 (-0.213, 0.232) |0.007 (-0.218, 0.226) |0.008 (-0.217, 0.225) |
| 0.75|cov2 |0.023 (-0.173, 0.220) |0.024 (-0.172, 0.216) |0.022 (-0.171, 0.219) |
| 1.00|cov2 |0.033 (-0.144, 0.211) |0.033 (-0.145, 0.206) |0.034 (-0.148, 0.212) |
| 0.00|trt |1.256 (0.911, 1.601) |1.260 (0.911, 1.629) |1.260 (0.909, 1.620) |
| 0.25|trt |1.564 (1.306, 1.822) |1.563 (1.302, 1.816) |1.564 (1.306, 1.816) |
| 0.50|trt |1.622 (1.386, 1.859) |1.620 (1.383, 1.851) |1.620 (1.387, 1.850) |
| 0.75|trt |1.649 (1.422, 1.875) |1.646 (1.414, 1.871) |1.645 (1.414, 1.865) |
| 1.00|trt |1.664 (1.443, 1.884) |1.660 (1.438, 1.874) |1.661 (1.436, 1.875) |
```r
res_df$borrowing_x <- res_df$borrowing +
(as.numeric(factor(res_df$fitter)) - 3) / 100
ggplot(res_df, aes(x = borrowing_x, y = estimate, group = fitter, colour = fitter)) +
geom_errorbar(aes(ymin = lower, ymax = upper)) +
geom_point() +
facet_wrap(~variable, scales = "free")
```
