usability_limits.Rmd

---
title: "Address Usability Limits of Bootstrap Method"
output:
  html_document:
    toc: true
    toc_depth: 3
    toc_float: true
    number_sections: true
    code_folding: show
    collapse: false
editor_options:
  chunk_output_type: console
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE, cache = TRUE)
```

# Load
```{r}
suppressPackageStartupMessages(library(tidyverse))
suppressPackageStartupMessages(library(knitr))
suppressPackageStartupMessages(library(furrr))
suppressPackageStartupMessages(library(future))
suppressPackageStartupMessages(library(simaerep))
```

# Introduction

As we already mentioned in the [introduction](file:///git_repos/simaerep/docs/articles/intro.html#disadvantages-over-classic-statistical-tests) bootstrap resampling comes at a price. We need a good proportion of sites that are compliant and are not under-reporting AEs. Otherwise the patient pool we draw from will be tainted. Also, in general for count data, negative deviations from overall small integer values (3 or less) are hard to detect. Here we will assess the usability limits of the bootstrap resampling method by applying it to different simulated scenarios.

# Parameter Grid
  
We start by defining a parameter grid.
  
**Fixed Parameters:**  
- n_pat: 1000  
- n_sites: 100  
- max_visit_mean: 20  
- max_visit_sd: 4  
  
**Variable Parameters:**  
- ae_visit: 0.05 - 2  
- frac_sites_wit_ur: 0.05 - 0.75  
- ur_rate: 0.05 - 1  
  
```{r}
set.seed(1)

df_grid <- tibble( ae_per_visit_mean = seq(0.05, 2, length.out = 10)) %>%
  mutate(frac_site_with_ur = list(seq(0.05, 0.75, length.out = 10))) %>%
  unnest(frac_site_with_ur) %>%
  mutate(ur_rate = list(seq(0.05, 1, length.out = 10)) ) %>%
  unnest(ur_rate) 

df_grid
```

# Apply

We use the [`furrr`](https://github.com/DavisVaughan/furrr) and the [`future`](https://github.com/HenrikBengtsson/future) package for multiprocessing. We proceed in applying all functions we already [introduced](https://openpharma.github.io/simaerep/articles/intro.html).

```{r}
suppressWarnings(future::plan(multiprocess))
```

## Simulate Test Data

We iteratively apply [sim_test_data_study()](https://openpharma.github.io/simaerep/reference/sim_test_data_study.html) on the different parameter combination of the grid

```{r}

df_grid <- df_grid %>%
  mutate( 
    df_visit = furrr::future_pmap(
      list(am = ae_per_visit_mean,
           fr = frac_site_with_ur,
           ur = ur_rate),
      function(am, fr, ur) sim_test_data_study(n_pat = 1000,
                                               n_sites = 100,
                                               max_visit_mean = 20,
                                               max_visit_sd = 4,
                                               ae_per_visit_mean = am,
                                               frac_site_with_ur = fr,
                                               ur_rate = ur),
      .progress = FALSE,
      .options = furrr_options(seed = TRUE)
    )
  )

df_grid
```

## Aggregate Test Data

We apply [site_aggr()](https://openpharma.github.io/simaerep/reference/site_aggr.html) on the different simulated data sets.

```{r}
df_grid <- df_grid %>%
  mutate( 
    df_visit = map2(df_visit, row_number(),
                    function(x,y) mutate(x, study_id = paste("S", y))),
    df_site = furrr::future_map(df_visit,
                                site_aggr,
                                .progress = FALSE,
                                .options = furrr_options(seed = TRUE)
                                )
  )

df_grid
```

## Simulate Sites

We apply [sim_sites()](https://openpharma.github.io/simaerep/reference/sim_sites.html) to to calculate the AE under-reporting probability.

```{r}
df_grid <- df_grid %>%
  mutate( df_sim_sites = furrr::future_map2(df_site, df_visit,
                                            sim_sites,
                                            r = 1000,
                                            poisson_test = TRUE,
                                            prob_lower = TRUE,
                                            .progress = FALSE,
                                            .options = furrr_options(seed = TRUE)
                                            )
  )

df_grid
```

## Evaluate Sites

We apply [eval_sites()](https://openpharma.github.io/simaerep/reference/eval_sites.html) to balance the bootstrapped probabilities by the expected number of false positives.

```{r}

df_grid <- df_grid %>%
  mutate( df_eval = furrr::future_map(df_sim_sites,
                                      eval_sites,
                                      .progress = FALSE,
                                      .options = furrr_options(seed = TRUE)
                                      )
  )

df_grid
```


## Get Metrics

Apart from calculating true and false positives rate for the final under-reporting probability we also calculate the same metrics for a couple of benchmark probabilities.

- p-values returned from `poisson.test()`
- and the probabilities and p-values before adjusting for the expected false positives

In all cases we use a 95% probability threshold.

```{r}
get_metrics <- function(df_visit, df_eval, method) {
  
  if (method == "ptest") {
    prob_ur_adj = "pval_prob_ur"
    prob_ur_unadj = "pval"
  } else {
    prob_ur_adj = "prob_low_prob_ur"
    prob_ur_unadj = "prob_low"
  }
  
  df_ur <- df_visit %>%
    select(site_number, is_ur) %>%
    distinct()

  df_metric_prep <- df_eval %>%
    left_join(df_ur, "site_number") %>%
    rename( prob_ur_adj = !! as.name(prob_ur_adj),
            prob_ur_unadj = !! as.name(prob_ur_unadj) )
  
  df_metric_adjusted <- df_metric_prep %>%
    select(study_id, site_number, is_ur, prob_ur_adj) %>%
    mutate( tp = ifelse(is_ur & prob_ur_adj >= 0.95, 1, 0),
            fn = ifelse(is_ur & prob_ur_adj < 0.95, 1, 0),
            tn = ifelse((! is_ur) & prob_ur_adj < 0.95, 1, 0),
            fp = ifelse((! is_ur) & prob_ur_adj >= 0.95, 1, 0),
            p = ifelse(prob_ur_adj >= 0.95, 1, 0),
            n = ifelse(prob_ur_adj < 0.95, 1, 0),
            P = ifelse(is_ur, 1, 0),
            N = ifelse(! is_ur, 1, 0)
    ) %>%
    group_by(study_id) %>%
    select(-site_number, - is_ur, - prob_ur_adj) %>%
    summarize_all(sum) %>%
    mutate(prob_type = "adjusted")
  
  df_metric_unadjusted <- df_metric_prep %>%
    select(study_id, site_number, is_ur, prob_ur_unadj) %>%
    mutate( tp = ifelse(is_ur & prob_ur_unadj <= 0.05, 1, 0),
            fn = ifelse(is_ur & prob_ur_unadj > 0.05, 1, 0),
            tn = ifelse((! is_ur) & prob_ur_unadj > 0.05, 1, 0),
            fp = ifelse((! is_ur) & prob_ur_unadj <= 0.05, 1, 0),
            p = ifelse(prob_ur_unadj <= 0.05, 1, 0),
            n = ifelse(prob_ur_unadj > 0.05, 1, 0),
            P = ifelse(is_ur, 1, 0),
            N = ifelse(! is_ur, 1, 0)
    ) %>%
    group_by(study_id) %>%
    select(-site_number, - is_ur, - prob_ur_unadj) %>%
    summarize_all(sum) %>%
    mutate(prob_type = "unadjusted")
  
  df_metric <- bind_rows(df_metric_adjusted, df_metric_unadjusted) %>%
    mutate(method = method)
}

df_grid <- df_grid %>%
  mutate(df_metric_pval = furrr::future_map2(df_visit, df_eval,
                                              get_metrics,
                                              method = "ptest"),
         df_metric_prob_low = furrr::future_map2(df_visit, df_eval,
                                                get_metrics,
                                                method = "bootstrap"),
         df_metric = map2(df_metric_pval, df_metric_prob_low, bind_rows)
         )

df_grid
```

# Plot

```{r}
df_plot <- df_grid %>%
  select(ae_per_visit_mean, frac_site_with_ur, ur_rate, df_metric) %>%
  unnest(df_metric)

df_plot
```


# True positive rate - P - TP/P

When comparing bootstrap method against the poisson.test benchmark test, we find that they perform mostly similar from a 0.05 - 0.3 ratio of under-reporting sites. Its performance is still acceptable from a 0.3 - 0.5 ratio and then becomes pretty useless.

As expected detection of under-reporting sites is close too impossible when the ae/visit ratio is very low (0.05) with detection rates close to zero for all tested scenarios.

```{r fig.asp=1, fig.width=10}

p <- df_plot %>%
  mutate( tp_P_ratio = tp/P) %>%
  select(ae_per_visit_mean, frac_site_with_ur, ur_rate, tp_P_ratio, method, prob_type) %>%
  mutate_if(is.numeric, round, 3) %>%
  mutate(metric = case_when(method == "ptest" & prob_type == "unadjusted" ~ "ptest unadjusted",
                           method == "ptest" & prob_type == "adjusted" ~ "ptest adjusted",
                           method == "bootstrap" & prob_type == "unadjusted" ~ "bootstrap unadjusted",
                           method == "bootstrap" & prob_type == "adjusted" ~ "bootstrap adjusted"),
         metric = fct_relevel(metric, c("ptest unadjusted", "ptest adjusted",
                                        "bootstrap unadjusted", "bootstrap adjusted"))) %>%
  ggplot(aes(ae_per_visit_mean, tp_P_ratio, color = metric)) +
    geom_line(aes(linetype = prob_type), size = 0.5, alpha = 0.5) +
    facet_grid( frac_site_with_ur ~ ur_rate) +
    scale_color_manual(values = c("dodgerblue3", "skyblue1", "sienna4", "peru")) +
    theme(panel.grid = element_blank(), legend.position = "bottom",
          axis.text.x = element_text(angle = 90, vjust = 0.5)) + 
    labs(title = "AE Under Reporting Rate ~ Rate of AE Under-Reporting Sites",
         y = "True positive rate - tp/P",
         x = "AEs per Visit")
p
```

# False positive rate FP/N

We can show that adjusting the AE under-reporting probability for the expected number of false positives is quite effective.As the false positive rate is almost zero for all tested scenarios. However, this also comes at a cost lowering the true positive rate as well.

```{r fig.asp=1, fig.width=10}
p <- df_plot %>%
  mutate( fpr = fp/N) %>%
  select(ae_per_visit_mean, frac_site_with_ur, ur_rate, fpr, method, prob_type) %>%
  mutate_if(is.numeric, round, 3) %>%
  mutate(metric = case_when(method == "ptest" & prob_type == "unadjusted" ~ "ptest unadjusted",
                           method == "ptest" & prob_type == "adjusted" ~ "ptest adjusted",
                           method == "bootstrap" & prob_type == "unadjusted" ~ "bootstrap unadjusted",
                           method == "bootstrap" & prob_type == "adjusted" ~ "bootstrap adjusted"),
         metric = fct_relevel(metric, c("ptest unadjusted", "ptest adjusted",
                                        "bootstrap unadjusted", "bootstrap adjusted"))) %>%
  ggplot(aes(ae_per_visit_mean, fpr, color = metric, linetype = prob_type)) +
    geom_line(size = 0.5, alpha = 0.5) +
    facet_grid( frac_site_with_ur ~ ur_rate) +
    scale_color_manual(values = c("dodgerblue3", "skyblue1", "sienna4", "peru")) +
    theme(panel.grid = element_blank(), legend.position = "bottom",
          axis.text.x = element_text(angle = 90, vjust = 0.5)) + 
    labs(title = "AE Under Reporting Rate by Rate of AE Under-Reporting Sites",
         y = "False positive rate - fp/N",
         x = "AEs per Visit")
p
```

# Conclusion

As already outlined in the [introduction](https://openpharma.github.io/simaerep/articles/intro.html#advantage-over-classic-statistical-tests) the true AE generating process is unlikely to be a standard poisson process. Therefore using a non-parametric test remains our recommended option unless that there is a reason to believe that 50% or more of all sites are under-reporting AEs. We also show that it is important to adjust for the expected false positive rate when calculating the final probabilities.

In general we can say that we do not recommend any method for detecting AE under-reporting that involves comparing AE counts if the ae per visit rate is 0.05 or lower.