r4asme07 poisson.Rmd

---
title: "7: Poisson regression in cohort studies"
subtitle: "R 4 ASME"
authors: Authors – Andrea Mazzella [(GitHub)](https://github.com/andreamazzella) & Lakmal Mudalige
output: html_notebook
---

-------------------------------------------------------------------------------

## Contents

* Poisson regression models
  * simple
  * adjusting for covariates
  * checking for interaction
  * checking for linear associations

-------------------------------------------------------------------------------

## 0. Packages and options

```{r message=FALSE, warning=FALSE}
# Load packages
library("haven")
library("magrittr")
library("survival")
library("summarytools")
library("epiDisplay")
library("tidyverse")

# Limit significant digits to 3, reduce scientific notation
options(digits = 3, scipen = 9)
```

-------------------------------------------------------------------------------


## 1. Data import and exploration

Get ondrate.dta in the same folder as this .Rmd notebook, import it, and explore it.
This dataset contains information from a cohort study on ~1500 people living in Nigeria, on the effect of onchocerciasis (river blindness, a parasitic infection) on optic nerve disease.

```{r}
ond <- read_dta("ondrate.dta") %>% mutate_if(is.labelled, as_factor)
ond
```

```{r include=FALSE}
glimpse(ond)
```

```{r}
table(ond$sex, useNA = "ifany")
```

* Outcome: `disc2` (optic nerve disease)
* Exposure: `mfpermg` (microfilariae per mg, a measure of severity of the onchocercal infection)
* Confounders: `age` (at start), `sex`
* Time:
 - `pyrs` (person-years, already calculated)
 - `start` (date of entry into study)
 - `end` (date of exit from study)

Thankfully, this dataset is mostly labelled. We only need to do a tiny bit of data management.
```{r}
ond %<>%
  mutate(id = as.factor(id),
         disc2 = as.integer(disc2),
         sex = as.factor(case_when(sex == 1 ~ "male", sex == 2 ~ "female")))
summary(ond)
```

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## 2. Calculation of incidence rates

Calculate incidence rates of Optic Nerve Disease by age, sex, and microfilarial load.

```{r}
# Create survival object
ond_surv <- ond %$% Surv(time = as.numeric(start) / 365.25, 
                         time2 = as.numeric(end) / 365.25, 
                         event = disc2)
# Rates by age
pyears(ond_surv ~ age, ond, scale = 1) %>% summary(n = F, rate = T, ci.r = T, scale = 1000)

# Rates by sex
pyears(ond_surv ~ sex, ond, scale = 1) %>% summary(n = F, rate = T, ci.r = T, scale = 1000)

# Rates by microfilarial load
pyears(ond_surv ~ mfpermg, ond, scale = 1) %>% summary(n = F, rate = T, ci.r = T, scale = 1000)
```
We can see that the incidence increases dramatically with age, it's about the same among females and males, and increases when microfilarial load is above 10.

-------------------------------------------------------------------------------


## 3. Poisson regression

Now perform the same three analyses but with Poisson regression.
Poisson regression in R uses the `glm()` formula – but it also needs to include `offset(log(<pyears>))` as a covariate. This is for mathematical reasons.

{epiDisplay} has a function to simplify the Poisson regression input: `idr.display()`.

```{r}
# Age
glm(disc2 ~ age + offset(log(pyrs)),
    family = "poisson",
    data = ond) %>% idr.display()
```
You can verify that these rate ratios are the same as those derived by classical analysis (by dividing the rates).

```{r}
# Sex
glm(disc2 ~ sex + offset(log(pyrs)),
    family = "poisson",
    data = ond) %>% idr.display()

# Microfilarial load
glm(disc2 ~ mfpermg + offset(log(pyrs)),
    family = "poisson",
    data = ond) %>% idr.display()
```

-------------------------------------------------------------------------------


## 4. Potential confounders

Is either age or sex associated with microfilarial load? Cross-tabulate them and check the percentages. (I'm using a function from summarytools, you can also use table or tabpct)

```{r}
ond %$% ctable(age, mfpermg)
ond %$% ctable(sex, mfpermg)
```
Older people tend to have higher infection markers: age is a potential confounder.
The sex distribution is balanced, so sex is unlikely to be a confounder.

-------------------------------------------------------------------------------


## 5. Poisson regression with covariates

Adjust for age, sex, and both with Poisson regression. Is there any indication of confounding?
```{r}
# Adjusting for age only
glm(disc2 ~ mfpermg + age + offset(log(pyrs)),
    family = "poisson",
    data = ond) %>% idr.display()

# Adjusting for sex only
glm(disc2 ~ mfpermg + sex + offset(log(pyrs)),
    family = "poisson",
    data = ond) %>% idr.display()

# Adjusting for both age and sex
pois_full <- glm(disc2 ~ mfpermg + age + sex + offset(log(pyrs)),
    family = "poisson",
    data = ond)
idr.display(pois_full)
```
The age-adjusted rate ratios for microfilarial loads are reduced: it is possible that age is a confounder. The RRs remain almost the same after adjusting for sex.

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## Bonus: Forest plot

Package sjPlot has a great function that creates a Forest plot of results from a regression model quite easily! `plot_model()`, that takes as its main argument a regression model. Very customisable, for more info look at [this vignette](https://cran.r-project.org/web/packages/sjPlot/vignettes/plot_model_estimates.html).
```{r}
sjPlot::plot_model(pois_full,
                   group.terms = c(1, 1, 2, 2, 2, 3),
                   show.values = T, value.offset = 0.35,
                   show.p = F,
                   title = "Multivariate Poisson regression")
```


-------------------------------------------------------------------------------


## 6. Poisson and Interaction

As logistic regression, Poisson models also assume that there is no interaction, so you need to check this assumption with a likelihood ratio test. You do this exactly like you do with logistic regression.

Check for interaction between microfilarial load and 1. age, 2. sex.

```{r}
# Interaction between MF and age
mf_age_simple <- glm(disc2 ~ mfpermg + age + offset(log(pyrs)),
                     family = "poisson",
                     data = ond)
mf_age_intera <- glm(disc2 ~ mfpermg * age + offset(log(pyrs)),
                     family = "poisson",
                     data = ond)
lrtest(mf_age_intera, mf_age_simple)
```
No evidence of interaction (however, there are 6 degrees of freedom – test might be underpowered)

```{r}
# Interaction between MF and sex
mf_sex_simple <- glm(disc2 ~ mfpermg + sex + offset(log(pyrs)),
                     family = "poisson",
                     data = ond)
mf_sex_intera <- glm(disc2 ~ mfpermg * sex + offset(log(pyrs)),
                     family = "poisson",
                     data = ond)
lrtest(mf_sex_intera, mf_sex_simple)
```
No evidence of interaction between MF and sex.

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## 7. Poisson and linearity

Assess for a linear relationship with age by treating it as numerical (as opposed to categorical) in a simple Poisson model.
You just need to use `as.integer()` with variable age. Unfortunately doing this within the same code flow as `idr.display()` breaks it, so you have two options:
1. extract the RR and 95% CI manually
2. use as.integer() outside the glm() function
```{r}
# Age as numerical (option 1)
age_linear <- glm(disc2 ~ as.integer(age) + offset(log(pyrs)),
                     family = "poisson",
                     data = ond)

# Get HRs by exponentiating the log(HR)s
age_linear$coefficients %>% exp()
confint(age_linear) %>% exp()

# Option 2
# ond$age_int <- as.integer(ond$age)
# glm(disc2 ~ age_int + offset(log(pyrs)), family = "poisson", data = ond) %>% idr.display()

# Age as categorical
age_catego <- glm(disc2 ~ age + offset(log(pyrs)),
                     family = "poisson",
                     data = ond)
# LRT
lrtest(age_catego, age_linear)
```
There is no evidence against a linear increase in the log(rate) from one age group to the next, so it's okay to use age as a continuous variable.

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## 8. Back to Whitehall

Open whitehal.dta and use Poisson regression to examine the effect of job grade on cardiac mortality, adjusting for ageband and smoking status simultaneously. 
```{r include=FALSE}
whitehall <- read_stata("whitehal.dta")
glimpse(whitehall)
```

The whitehall dataset needs massive data management (see session 6) but we'll now do the bare minimum for these Poisson models:
- age bands as a factor variable; (NB: I suspect this is done via an exhausting Lexis expansion in the Stata solutions, but age at entry will do...)
- job grade as a binary factor variable;
- number of person-years for each person;
- smoking status as a 3-level factor variable.
```{r}
# Explore age distribution
whitehall %>% ggplot(aes(agein)) + geom_histogram()

# Categorise age in 5-year bands
whitehall$ageband <- cut(whitehall$agein, seq(40, 70, 5)) %>% as.factor()

# Check it worked ok
summary(whitehall$ageband)

# Factorise job grade
whitehall$grade <- as.factor(whitehall$grade)

# Create person-years of follow up
whitehall$pyrs <- whitehall %$% as.numeric(timeout - timein)

# Group all current smokers together
whitehall$smok3 <- ifelse(whitehall$smok == 1, 1,
                          ifelse(whitehall$smok == 2, 2,
                                 3)) %>% as.factor()
```

Now we can fit the models.
```{r}
# Model including job grade
with_grade <- glm(chd ~ offset(log(pyrs)) + grade + smok3 + ageband,
  family = poisson(),
  data = whitehall)
idr.display(with_grade)

# Model without job grade
wout_grade <- glm(chd ~ offset(log(pyrs)) + smok3 + ageband,
  family = poisson(),
  data = whitehall)
idr.display(wout_grade)

# LRT
lrtest(with_grade, wout_grade)
```

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