# 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)
    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, ]
      )
    })

### 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)
    )

    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) | 0.661 (-0.036, 1.359)
0.25 | (Intercept) | 0.394 (-0.131, 0.931) | 0.396 (-0.131, 0.941) | 0.407 (-0.126, 0.938)
0.50 | (Intercept) | 0.293 (-0.158, 0.751) | 0.297 (-0.184, 0.767) | 0.297 (-0.165, 0.760)
0.75 | (Intercept) | 0.235 (-0.168, 0.642) | 0.240 (-0.175, 0.665) | 0.244 (-0.157, 0.649)
1.00 | (Intercept) | 0.196 (-0.172, 0.567) | 0.202 (-0.172, 0.578) | 0.198 (-0.162, 0.560)
0.00 | cov1 | -0.771 (-1.465, -0.095) | -0.809 (-1.515, -0.126) | -0.795 (-1.488, -0.107)
0.25 | cov1 | -0.781 (-1.340, -0.231) | -0.793 (-1.350, -0.236) | -0.797 (-1.360, -0.239)
0.50 | cov1 | -0.769 (-1.252, -0.291) | -0.776 (-1.271, -0.270) | -0.774 (-1.262, -0.289)
0.75 | cov1 | -0.758 (-1.191, -0.329) | -0.769 (-1.212, -0.310) | -0.771 (-1.215, -0.350)
1.00 | cov1 | -0.749 (-1.145, -0.357) | -0.761 (-1.152, -0.369) | -0.755 (-1.148, -0.368)
0.00 | cov2 | -0.730 (-1.472, -0.008) | -0.745 (-1.496, -0.004) | -0.756 (-1.512, -0.029)
0.25 | cov2 | -0.559 (-1.114, -0.014) | -0.568 (-1.124, -0.023) | -0.572 (-1.130, -0.012)
0.50 | cov2 | -0.459 (-0.926, 0.003) | -0.471 (-0.953, -0.003) | -0.467 (-0.920, -0.012)
0.75 | cov2 | -0.398 (-0.811, 0.011) | -0.402 (-0.814, 0.006) | -0.404 (-0.826, -0.000)
1.00 | cov2 | -0.358 (-0.731, 0.013) | -0.358 (-0.736, 0.022) | -0.359 (-0.726, 0.006)
0.00 | trt | 0.154 (-0.558, 0.871) | 0.137 (-0.572, 0.864) | 0.164 (-0.555, 0.898)
0.25 | trt | 0.349 (-0.183, 0.885) | 0.361 (-0.170, 0.899) | 0.358 (-0.172, 0.890)
0.50 | trt | 0.405 (-0.082, 0.894) | 0.415 (-0.068, 0.916) | 0.407 (-0.089, 0.907)
0.75 | trt | 0.434 (-0.031, 0.900) | 0.436 (-0.031, 0.913) | 0.434 (-0.031, 0.898)
1.00 | trt | 0.452 (-0.000, 0.905) | 0.456 (-0.010, 0.909) | 0.454 (0.006, 0.918)

    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")

plot of chunk logistic_plot

# 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, ]
      )
    })

    ## 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))

    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) | 1.916 (1.563, 2.245)
0.25 | (Intercept) | 1.581 (1.323, 1.838) | 1.567 (1.308, 1.814) | 1.568 (1.311, 1.811)
0.50 | (Intercept) | 1.473 (1.251, 1.695) | 1.466 (1.247, 1.685) | 1.466 (1.237, 1.684)
0.75 | (Intercept) | 1.414 (1.215, 1.613) | 1.407 (1.208, 1.601) | 1.412 (1.211, 1.604)
1.00 | (Intercept) | 1.376 (1.194, 1.558) | 1.369 (1.183, 1.551) | 1.370 (1.188, 1.547)
0.00 | cov1 | 0.630 (0.300, 0.959) | 0.630 (0.304, 0.960) | 0.631 (0.298, 0.966)
0.25 | cov1 | 0.722 (0.453, 0.991) | 0.729 (0.459, 1.013) | 0.728 (0.461, 1.003)
0.50 | cov1 | 0.786 (0.552, 1.020) | 0.789 (0.559, 1.026) | 0.790 (0.556, 1.029)
0.75 | cov1 | 0.827 (0.616, 1.037) | 0.829 (0.622, 1.044) | 0.825 (0.618, 1.037)
1.00 | cov1 | 0.854 (0.662, 1.046) | 0.859 (0.668, 1.057) | 0.856 (0.667, 1.052)
0.00 | cov2 | 0.043 (-0.309, 0.395) | 0.039 (-0.318, 0.382) | 0.036 (-0.322, 0.378)
0.25 | cov2 | -0.009 (-0.273, 0.255) | -0.008 (-0.283, 0.252) | -0.011 (-0.272, 0.249)
0.50 | cov2 | 0.009 (-0.213, 0.232) | 0.007 (-0.218, 0.226) | 0.006 (-0.213, 0.223)
0.75 | cov2 | 0.023 (-0.173, 0.220) | 0.024 (-0.172, 0.216) | 0.022 (-0.175, 0.216)
1.00 | cov2 | 0.033 (-0.144, 0.211) | 0.033 (-0.145, 0.206) | 0.034 (-0.140, 0.212)
0.00 | trt | 1.256 (0.911, 1.601) | 1.260 (0.911, 1.629) | 1.260 (0.909, 1.613)
0.25 | trt | 1.564 (1.306, 1.822) | 1.563 (1.302, 1.816) | 1.564 (1.303, 1.823)
0.50 | trt | 1.622 (1.386, 1.859) | 1.620 (1.383, 1.851) | 1.618 (1.382, 1.851)
0.75 | trt | 1.649 (1.422, 1.875) | 1.646 (1.414, 1.871) | 1.644 (1.413, 1.867)
1.00 | trt | 1.664 (1.443, 1.884) | 1.660 (1.438, 1.874) | 1.660 (1.438, 1.878)

    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")

plot of chunk exp_plot