Sex differences in dementia rates over time

dementia_mortality.rmd

---
title: Survival bias in sex association with dementia
author: Gibran Hemani
date: `r Sys.time()`
---

```{r}
library(dplyr)
library(tidyr)
library(ggplot2)
```

In week 5 there is a question about why rates of dementia change over time differently in men vs women. It starts off roughly equal at age 65, but by age 90 the percentage of men with dementia is roughly half the percentage of women with dementia.

The answers suggest this could be explained by survivor bias, because men die earlier than women. However I think this would only be true if dementia causes death in men more quickly than women. The fact that men die earlier is not sufficient to explain the difference


```{r}
n <- 10000
timepoints <- 1:10
male_mortality <- 0.1
female_mortality <- 0.05
dementia_risk <- 0.05

dat <- tibble(
	id = 1:n,
	sex = rbinom(n, 1, 0.5),
	tp0_dementia = 0,
	tp0_alive = 1,
	death_risk = case_when(
		sex == 0 ~ female_mortality,
		TRUE ~ male_mortality
	),
	dementia_risk = dementia_risk
)

for(i in timepoints)
{
	last_dementia <- paste0("tp", i-1, "_dementia")
	this_dementia <- paste0("tp", i, "_dementia")
	last_alive <- paste0("tp", i-1, "_alive")
	this_alive <- paste0("tp", i, "_alive")
	ind <- which(dat[[last_alive]] == 1)
	dat[[this_alive]][ind] <- ! (! dat[[last_alive]][ind] | rbinom(length(ind), 1, dat$death_risk[ind]))
	ind <- which(dat[[this_alive]] == 1)
	dat[[this_dementia]][ind] <- (dat[[last_dementia]][ind] | rbinom(length(ind), 1, dat$dementia_risk[ind]))
	print(table(dat[[this_dementia]]))
	print(table(dat[[this_alive]]))
	dat[[this_alive]][is.na(dat[[this_alive]])] <- dat[[last_alive]][is.na(dat[[this_alive]])]
	dat[[this_dementia]][is.na(dat[[this_dementia]])] <- dat[[last_dementia]][is.na(dat[[this_dementia]])]
}

datl <- dat %>% pivot_longer(
	c(ends_with("alive"), ends_with("dementia"))
) %>% 
tidyr::separate(name, sep="_", into=c("tp", "measure")) %>%
mutate(tp = gsub("tp", "", tp) %>% as.numeric()) %>%
pivot_wider(
	id_cols=c(id, sex, death_risk, dementia_risk, tp),
	names_from=measure,
	values_from=value
)

datl %>%
	group_by(sex, tp) %>%
	summarise(alive=sum(alive)/n()) %>%
	ggplot(., aes(x=tp, y=alive)) +
	geom_line(aes(colour=as.factor(sex)))

datl %>%
	group_by(sex, tp) %>%
	summarise(dementia=sum(dementia)/n()) %>%
	ggplot(., aes(x=tp, y=dementia)) +
	geom_line(aes(colour=as.factor(sex)))

datl %>%
	dplyr::filter(alive==1) %>%
	group_by(sex, tp) %>%
	summarise(dementia=sum(dementia)/n()) %>%
	ggplot(., aes(x=tp, y=dementia)) +
	geom_line(aes(colour=as.factor(sex)))
```