# 6. Comparison of Fixed Weights

#### Matt Secrest and Isaac Gravestock

# 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.

    library(psborrow2)
    library(cmdstanr)
    library(BayesPPD)
    # Error in `library()`:
    # ! there is no package called 'BayesPPD'
    library(ggplot2)

# 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

    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

    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, ]
      )
    })
    # Error in `glm.fixed.a0()`:
    # ! could not find function "glm.fixed.a0"

### psborrow2

    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

    logistic_res_df <- do.call(
      rbind,
      c(logistic_glm_reslist, logistic_ppd_reslist, logistic_psb_reslist)
    )
    # Error:
    # ! object 'logistic_ppd_reslist' not found

    logistic_res_df$est_ci <- sprintf(
      "%.3f (%.3f, %.3f)",
      logistic_res_df$estimate, logistic_res_df$lower, logistic_res_df$upper
    )
    # Error:
    # ! object 'logistic_res_df' not found

    wide <- reshape(
      logistic_res_df[, c("fitter", "borrowing", "variable", "est_ci")],
      direction = "wide",
      timevar = "fitter",
      idvar = c("borrowing", "variable"),
    )
    # Error:
    # ! object 'logistic_res_df' not found
    new_order <- order(wide$variable, wide$borrowing)
    # Error:
    # ! object 'wide' not found
    knitr::kable(wide[new_order, ], digits = 3, row.names = FALSE)
    # Error:
    # ! object 'wide' not found

    logistic_res_df$borrowing_x <- logistic_res_df$borrowing +
      (as.numeric(factor(logistic_res_df$fitter)) - 3) / 100
    # Error:
    # ! object 'logistic_res_df' not found

    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")
    # Error:
    # ! object 'logistic_res_df' not found

# 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`.

    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
    )

    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

    ## 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
      )
    })

    ## 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, ]
      )
    })
    # Error in `glm.fixed.a0()`:
    # ! could not find function "glm.fixed.a0"

    ## 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%`
      )
    })

    knitr::knit_hooks$set(output = output_hook)

### Results

Note: Wald confidence intervals are displayed here for `glm` for the exponential models.

    res_df <- do.call(rbind, c(glm_reslist, ppd_reslist, psb_reslist))
    # Error:
    # ! object 'ppd_reslist' not found

    res_df$est_ci <- sprintf(
      "%.3f (%.3f, %.3f)",
      res_df$estimate, res_df$lower, res_df$upper
    )
    # Error:
    # ! object 'res_df' not found

    wide <- reshape(
      res_df[, c("fitter", "borrowing", "variable", "est_ci")],
      direction = "wide",
      timevar = "fitter",
      idvar = c("borrowing", "variable"),
    )
    # Error:
    # ! object 'res_df' not found
    new_order <- order(wide$variable, wide$borrowing)
    # Error:
    # ! object 'wide' not found
    knitr::kable(wide[new_order, ], digits = 3, row.names = FALSE)
    # Error:
    # ! object 'wide' not found

    res_df$borrowing_x <- res_df$borrowing +
      (as.numeric(factor(res_df$fitter)) - 3) / 100
    # Error:
    # ! object 'res_df' not found

    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")
    # Error:
    # ! object 'res_df' not found