0
$\begingroup$

I want to be able to simulate a prediction model given some prevalence of the event and the AUC of the model. I followed the method proposed here but, although this works for giving AUC and predicted values, the predictions aren't well-calibrated.


library(tidyverse)
library(AUC)

get_sample <- function(auc, n_samples, prevalence, scale=T){
  # https://stats.stackexchange.com/questions/422926/generate-synthetic-data-given-auc
  t <- sqrt(log(1/(1-auc)**2))
  z <- t-((2.515517 + 0.802853*t + 0.0103328*t**2) / 
            (1 + 1.432788*t + 0.189269*t**2 + 0.001308*t**3))
  d <- z*sqrt(2)
  
  n_neg <- round(n_samples*(1-prevalence))
  n_pos <- round(n_samples*prevalence)
  
  x <- c(rnorm(n_neg, mean=0), rnorm(n_pos, mean=d))
  y <- c(rep(0, n_neg), rep(1, n_pos))
  
  if(scale){
    x <- (x-min(x)) / (max(x)-min(x))
  }
  return(data.frame(predicted=x, actual=y))
}

df_preds <- get_sample(auc=0.8, n_samples=2000, prevalence=0.1) %>%
  mutate(predicted_interval=cut(predicted, seq(0,1,0.1), labels=FALSE)/10-0.05)

# problem with this approach is that it doesn't lead to well calibrated predictions
df_preds %>%
  group_by(predicted_interval) %>%
  summarize(mean_actual=mean(actual)) %>%
  ggplot(aes(predicted_interval, mean_actual)) +
  geom_line(alpha=0.3) +
  geom_point()+
  geom_abline()+
  theme_bw() +
  scale_x_continuous(limits=c(0, 1)) + scale_y_continuous(limits=c(0, 1))


sampling using previous AUC and Cohen's D

To ensure that the predictions that I get are well-calibrated, I thought to sample from a beta-distribution and sample the outcome c(0,1) based on the given probability.


get_beta_preds <- function(alpha, beta, n){
  cat(glue::glue("population prevalence:{round(alpha/(alpha+beta), 4)}\n\n"))
  predicted_probs <- rbeta(n=n, shape1=alpha, shape2=beta)
  f <- function(x) sample(c(0, 1), 1, prob=c(1-x, x))
  predicted_classes <- map_dbl(predicted_probs, f)
  
  cat(glue::glue("observed AUC:{round(auc(roc(predicted_probs, as.factor(predicted_classes))), 4)}"))
  data.frame(predicted=predicted_probs, actual=predicted_classes)
}

# these give the same prevalence but different AUC
df_beta_preds <- get_beta_preds(1, 2, 100000)
df_beta_preds <- get_beta_preds(1.5, 3, 100000)

df_beta_preds %>%
  mutate(predicted_interval=cut(predicted, seq(0,1,0.1), labels=FALSE)/10-0.05)%>%
  group_by(predicted_interval) %>%
  summarize(mean_actual=mean(actual)) %>%
  ggplot(aes(predicted_interval, mean_actual)) +
  geom_line(alpha=0.3) +
  geom_point()+
  geom_abline()+
  theme_bw() +
  scale_x_continuous(limits=c(0, 1)) + scale_y_continuous(limits=c(0, 1))

sampling from beta distribution

These give well-calibrated predictions but I don't know what the "population" AUC would be for a given beta distribution. In the code above, I generate probabilities and classes under two beta-distributions with the same expected mean - these give different AUCs with large samples, so I assume that there is some relationship between (alpha, beta) and the AUC when classes are sampled directly from the given probabilities.

I have no idea how to either

  1. Force the predicted probabilities in the first approach to be well-calibrated
  2. Calculate the expected AUC given the use of alpha and beta in a beta distribution to sample probabilities (and assuming that these probabilities are well-calibrated)

Help regarding either of these would be great. My objective would be to input a given prevalence and AUC, and get calibrated predicted probabilities and classes in return.

$\endgroup$
1
  • 1
    $\begingroup$ Not quite an answer to your specific question but I typically use one of three methods for scaling my predicted probabilities: 1. a model based scaling technique (platt scaling or isotonic regression) 2. adjust the predicted probabilities based on the known incidence rate 3. if it's a logistic regression model or other linear model you can just adjust the intercept of the model.... I compared these in a gist here: gist.github.com/brshallo/24338a87b33e5d2ac98d200b1ccecfc5 (which was created in service of a tangentially related blog post topic). Maybe can layer one on sim method. $\endgroup$ Commented Feb 16, 2022 at 23:32

1 Answer 1

1
$\begingroup$

Copied example code you linked to here and then applied platt scaling to the final step:

library(tidyverse)

auc <- 0.95

t <- sqrt(log(1/(1-auc)**2))
z <- t-((2.515517 + 0.802853*t + 0.0103328*t**2) / 
          (1 + 1.432788*t + 0.189269*t**2 + 0.001308*t**3))
d <- z*sqrt(2)

n <- 10000
x <- c(rnorm(n/2, mean = 0), rnorm(n/2, mean = d))
y <- c(rep(0, n/2), rep(1, n/2))

data <- tibble(outcome = y, score = x)

# Applying Platt scaling to predictions / scores
model <- glm(outcome ~ score, family = "binomial", data = data)

df_preds <- bind_cols(
  data,
  # inverse logit of predictions from logistic regression model
  tibble(score_scaled = plogis(predict(model, newdata = data)))
) %>% 
  mutate(score_interval = cut(score_scaled, seq(0, 1, 0.1), lables = FALSE))

df_preds %>%
  group_by(score_interval) %>%
  summarize(mean_actual = mean(outcome),
            mean_score = mean(score_scaled)) %>%
  ggplot(aes(mean_score, mean_actual)) +
  geom_line(aes(mean_actual, mean_actual, colour = "calibrated"))+
  geom_line(alpha=0.3, group = 1) +
  geom_point()+
  theme_bw() +
  scale_x_continuous(limits=c(0, 1)) + 
  scale_y_continuous(limits=c(0, 1))


library(AUC)

auc(roc(x, as.factor(y)))
#> [1] 0.9496194

Created on 2022-02-23 by the reprex package (v2.0.0)

$\endgroup$
2
  • $\begingroup$ AUC is all about how well ordered / ranked the data is -- it doesn't care if the predictions are a probability, a log-odds, or anything. Here's a link to a gif + talk I gave where I explain AUC in my preferred way of thinking about it: twitter.com/brshallo/status/… $\endgroup$ Commented Feb 24, 2022 at 2:07
  • $\begingroup$ Perfect - thanks Bryan! $\endgroup$ Commented Feb 24, 2022 at 19:26

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.