Last updated: 2019-11-25

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Knit directory: simpamm/

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Rmd 4ea7082 Andreas Bender 2019-11-25 Add sim study on confidence intervals

library(dplyr)
library(ggplot2)
theme_set(theme_bw())
library(patchwork)
library(survival)
library(mgcv)
library(pammtools)
library(batchtools)
knitr::opts_chunk$set(autodep = TRUE)

Motivation

Here we consider three different ways to calculate confidence intervals for hazard rates estimated by PAMMs and derivatives thereof, i.e., the cumulative hazard and survival probability. The three methods compared here are

  1. Delta method
  2. Direct transformation
  3. Simulation

Data Simulation

We simulate data with log-hazard rate \(\log(\lambda(t|x)) = -3.5 + f(8,2) \cdot 6 - 0.5\cdot x_1 + \sqrt{x_2}\), where \(f(8,2)\) is the gamma density function with respective parameters.

The below wrapper creates a data set with \(n = 500\) survival times based on above hazard. The simulation of survival times is performed using function pammtools::sim_pexp.

sim_wrapper <- function(data, job, n = 500, time_grid = seq(0, 10, by = 0.05)) {

  # create data set with covariates
  df <- tibble::tibble(x1 = runif(n, -3, 3), x2 = runif(n, 0, 6))
  # baseline hazard
  f0 <- function(t) {dgamma(t, 8, 2) * 6}

  ndf <- sim_pexp(
    formula = ~ -3.5 + f0(t) -0.5*x1 + sqrt(x2),
    data = df, cut = time_grid)

  ndf
}

Estimation

The below wrapper - transforms the simulated survival data into the PED format - fits the PAMM - returns a data frame that contains the coverage for each method

Expand to see function source code

ci_wrapper <- function(
  data,
  job,
  instance,
  bs      = "ps",
  k       = 10,
  ci_type = "default") {

  # instance <- sim_wrapper()
  ped <- as_ped(
    data    = instance,
    formula = Surv(time, status) ~ x1 + x2,
    id      = "id")

  form <- paste0("ped_status ~ s(tend, bs='", bs, "', k=", k, ") + s(x1) + s(x2)")

  mod <- gam(
    formula = as.formula(form),
    data    = ped,
    family  = poisson(),
    offset  = offset,
    method  = "REML")

  f0 <- function(t) {dgamma(t, 8, 2) * 6}

  # create new data set
  nd <- make_newdata(ped, tend = unique(tend), x1 = c(0), x2 = c(3)) %>%
    mutate(
      true_hazard = exp(-3.5 + f0(tend) -0.5 * x1 + sqrt(x2)),
      true_cumu = cumsum(intlen * true_hazard),
      true_surv = exp(-true_cumu))
  # add hazard, cumulative hazard, survival probability with confidence intervals
  # using the 3 different methods
  nd_default <- nd  %>%
    add_hazard(mod, se_mult = qnorm(0.975), ci_type = "default") %>%
    add_cumu_hazard(mod, se_mult = qnorm(0.975), ci_type = "default") %>%
    add_surv_prob(mod, se_mult = qnorm(0.975), ci_type = "default") %>%
    mutate(
      hazard = (true_hazard >= ci_lower) & (true_hazard <= ci_upper),
      cumu = (true_cumu >= cumu_lower) & (true_cumu <= cumu_upper),
      surv =  (true_surv >= surv_lower) & (true_surv <= surv_upper)) %>%
    select(hazard, cumu, surv) %>%
    summarize_all(mean) %>%
    mutate(method = "direct")
  nd_delta <- nd %>%
    add_hazard(mod, se_mult = qnorm(0.975), ci_type = "delta", overwrite = TRUE) %>%
    add_cumu_hazard(mod, se_mult = qnorm(0.975), ci_type = "delta", overwrite = TRUE) %>%
    add_surv_prob(mod, se_mult = qnorm(0.975), ci_type = "delta", overwrite = TRUE) %>%
    mutate(
      hazard = (true_hazard >= ci_lower) & (true_hazard <= ci_upper),
      cumu = (true_cumu >= cumu_lower) & (true_cumu <= cumu_upper),
      surv =  (true_surv >= surv_lower) & (true_surv <= surv_upper)) %>%
    select(hazard, cumu, surv) %>%
    summarize_all(mean) %>%
    mutate(method = "delta")
  nd_sim <- nd %>%
    add_hazard(mod, se_mult = qnorm(0.975), ci_type = "sim", nsim = 500, overwrite = TRUE) %>%
    add_cumu_hazard(mod, se_mult = qnorm(0.975), ci_type = "sim", nsim = 500, overwrite = TRUE) %>%
    add_surv_prob(mod, se_mult = qnorm(0.975), ci_type = "sim", nsim = 500, overwrite = TRUE) %>%
    mutate(
      hazard = (true_hazard >= ci_lower) & (true_hazard <= ci_upper),
      cumu = (true_cumu >= cumu_lower) & (true_cumu <= cumu_upper),
      surv =  (true_surv >= surv_lower) & (true_surv <= surv_upper)) %>%
    select(hazard, cumu, surv) %>%
    summarize_all(mean) %>%
    mutate(method = "simulation")

    rbind(nd_default, nd_delta, nd_sim)

}

Evaluation

We simulate 100 data sets and obtain respective estimates using package batchtools:

Expand to see batchtools code

if (!checkmate::test_directory_exists("output/sim-conf-int-registry")) {
  reg <- makeExperimentRegistry(
    "output/sim-conf-int-registry",
    packages = c("mgcv", "dplyr", "tidyr", "pammtools"),
    seed     = 20052018)
  reg <- loadRegistry("output/sim-conf-int-registry", writeable = TRUE)
  # reg$cluster.functions = makeClusterFunctionsInteractive()
  addProblem(name   = "ci", fun = sim_wrapper)
  addAlgorithm(name = "ci", fun = ci_wrapper)

  algo_df <- data.frame(bs = c("tp", "ps"), stringsAsFactors = FALSE)

  addExperiments(algo.design  = list(ci = algo_df), repls = 300)

  submitJobs(findNotDone())
  # waitForJobs()

}

Below the RMSE and coverage are calculated for the different methods to estimate the confidence intervals

reg     <- loadRegistry("output/sim-conf-int-registry", writeable = TRUE)
ids_res <- findExperiments(prob.name = "ci", algo.name = "ci")
pars    <- unwrap(getJobPars()) %>% as_tibble()
res     <- reduceResultsDataTable(ids=findDone(ids_res)) %>%
  as_tibble() %>%
  tidyr::unnest(cols = c(result)) %>%
  left_join(pars)

Coverage table for the hazard, cumulative hazard and survival probability

# RMSE and coverage hazard
res %>%
  group_by(bs, method) %>%
  summarize(
    "coverage hazard" = mean(hazard),
    "coverage cumulative hazard" = mean(cumu),
    "coverage survival probability" = mean(surv)) %>%
  mutate_if(is.numeric, ~round(., 3)) %>%
  rename("basis" = "bs") %>%
  knitr::kable()
`mutate_if()` ignored the following grouping variables:
Column `bs`
basis method coverage hazard coverage cumulative hazard coverage survival probability
ps delta 0.911 0.904 0.907
ps direct 0.914 0.965 0.965
ps simulation 0.891 0.904 0.907
tp delta 0.936 0.925 0.933
tp direct 0.938 0.978 0.978
tp simulation 0.918 0.933 0.933

sessionInfo()
R version 3.6.1 (2019-07-05)
Platform: x86_64-pc-linux-gnu (64-bit)
Running under: Ubuntu 18.04.3 LTS

Matrix products: default
BLAS:   /usr/lib/x86_64-linux-gnu/blas/libblas.so.3.7.1
LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.7.1

locale:
 [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C              
 [3] LC_TIME=en_US.UTF-8        LC_COLLATE=en_US.UTF-8    
 [5] LC_MONETARY=en_US.UTF-8    LC_MESSAGES=en_US.UTF-8   
 [7] LC_PAPER=en_US.UTF-8       LC_NAME=C                 
 [9] LC_ADDRESS=C               LC_TELEPHONE=C            
[11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C       

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
 [1] tidyr_1.0.0          batchtools_0.9.11    data.table_1.12.6   
 [4] pammtools_0.1.14     mgcv_1.8-31          nlme_3.1-142        
 [7] survival_3.1-7       patchwork_0.0.1.9000 ggplot2_3.2.1       
[10] dplyr_0.8.3         

loaded via a namespace (and not attached):
 [1] progress_1.2.2    tidyselect_0.2.5  xfun_0.11         purrr_0.3.3      
 [5] splines_3.6.1     lattice_0.20-38   colorspace_1.4-1  vctrs_0.2.0      
 [9] expm_0.999-4      htmltools_0.4.0   yaml_2.2.0        rlang_0.4.2      
[13] later_1.0.0       pillar_1.4.2      glue_1.3.1        withr_2.1.2      
[17] rappdirs_0.3.1    lifecycle_0.1.0   stringr_1.4.0     munsell_0.5.0    
[21] gtable_0.3.0      workflowr_1.5.0   mvtnorm_1.0-11    evaluate_0.14    
[25] knitr_1.26        httpuv_1.5.2      highr_0.8         Rcpp_1.0.3       
[29] promises_1.1.0    scales_1.1.0      backports_1.1.5   checkmate_1.9.4  
[33] fs_1.3.1          brew_1.0-6        hms_0.5.2         digest_0.6.23    
[37] stringi_1.4.3     msm_1.6.7         grid_3.6.1        rprojroot_1.3-2  
[41] tools_3.6.1       magrittr_1.5      base64url_1.4     lazyeval_0.2.2   
[45] tibble_2.1.3      Formula_1.2-3     crayon_1.3.4      whisker_0.4      
[49] pkgconfig_2.0.3   zeallot_0.1.0     Matrix_1.2-17     prettyunits_1.0.2
[53] assertthat_0.2.1  rmarkdown_1.17    R6_2.4.1          git2r_0.26.1     
[57] compiler_3.6.1