modelling-common-outcomes.Rmd

---
title: "Working with odds ratios and risk ratios "
output:
  learnr::tutorial:
    allow_skip: true
runtime: shiny_prerendered
---

```{r setup, include=FALSE}
library(learnr)
library(ggplot2)
library(xtable)
knitr::opts_chunk$set(echo = FALSE)
```

## Introduction

The risk ratio and odds ratio are two common effect estimates used to quantify the association between an exposure and a binary outcome. In this short interactive tutorial you will learn: 

* To distinguish between odds ratios and risk ratios 
* To calculate these values using R programming

## Pop Quiz

*Before getting started, test your current knowledge about odds and risk ratios.*

```{r quiz1}
quiz(
  question("Odds ratios can be estimated using which of the following generalised linear models?",
    answer("Poisson regression"),
    answer("Probit regression"),
    answer("Logistic regression", correct = TRUE),
    answer("Log-binomial regression"),
    allow_retry = TRUE
  ),
  question("Risk Ratios are an appropriate effect estimate in which of the following study designs? (tick all that apply)",
    answer("RCT", correct = TRUE),
    answer("Cohort study", correct = TRUE),
    answer("Case-control study", message = "Risk ratios can't be used for case-control studies because the total number of exposed cases is unknown"),
    allow_retry = TRUE
  ),
  question("The risk ratio and the odds ratio are never exactly the same, true or false?",
    answer("False", correct = TRUE, message = "When the odds ratio is 1, the risk ratio is also exactly 1"),
    answer("True", message = "When the odds ratio is 1, the risk ratio is also exactly 1"),
    allow_retry = TRUE
  )
)
```


## Basic concepts

### Odds ratios versus risk ratios 

The terms odds ratio and risk ratio are often used interchangeably or interpreted in the same way, but they are different quantities. 

#### The Odds
The _odds_ of an outcome in a given group or population is the number of times the outcome occurs divided by the number of times the outcome does not occur. 

#### The Odds Ratio
The _odds ratio_ is the ratio of two odds: the odds of the outcome in the exposed group divided by the odds of the outcome in the unexposed group.

#### The Risk

The _risk_ of a binary outcome in a given group is the same as a _probability_, i.e. the number of times the outcome occurs divided by the total number of individuals in the group.

#### The Risk Ratio
The _risk ratio_ is the ratio of two risks: the risk of the outcome in the exposed group divided by the risk of the outcome in the unexposed group.


### Illustration

We can illustrate these definitions with a simple example. Consider the following table which includes a binary exposure and binary outcome.

![](images/tab1.png){width=300}


#### The Odds Ratio


The odds of the outcome in the exposed group is 
$$
odds_{exposed} = \frac{a}{b}
$$
Similarly, the odds of the outcome in the non-exposed group is 
$$
odds_{non-exposed} = \frac{c}{d}
$$

The _odds ratio_ (OR) is the ratio of these values, i.e. the odds of death in the drug group relative to the placebo group is

$$
OR = \frac{odds_{exposed}}{odds_{non-exposed}} = \frac{ad}{bc}
$$


#### The Risk Ratio

The _risk_ of the outcome for the exposed group is 

$$
risk_{exposed} = \frac{a}{a+b}
$$ 

The risk of the outcome for the non-exposed group is

$$
risk_{non-exposed} = \frac{c}{c+d}
$$

The _risk ratio_ (RR) is the ratio of these two values: 

$$
RR = \frac{risk_{exposed}}{risk_{non-exposed}} = \frac{a/(a+b)}{c/(c+d)}
$$


**Note a key difference when calculating the odds and the risk is the denominator used.**

### A worked example 
Consider a randomised control trial studying the effect of meditation on students' emotional state. Study participants were randomised to either the treatment group or a control group. The treated group were taught mindfulness techniques and asked to meditate for 20 minutes every day, while a control group made no change to their routine. At the end of the study, participants reported their emotional state. 

The data found that out of 50 students in the treated group, 45 reported being calm, while 5 reported being stressed at the end of the study. Out of 200 patients in the control group, 40 reported being stressed, while 160 reported being calm.

We can arrange these data in a 2 $\times$ 2 table as before.

![](images/tab2.png){width=300}

#### Odds 
Following the previous example, the odds of being calm versus stressed for the meditation group is
$$
odds_{meditation}=\frac{a}{b} = \frac{45}{5} = 9
$$
We can interpret this as saying that for every 9 calm people in the meditation group there was one stressed person.

The odds of being calm versus stressed for the control group is
$$
odds_{control}=\frac{a}{b} = \frac{160}{40} = 4
$$
We can interpret this as saying for every 4 calm people in the control group there was one stressed person.

#### Odds Ratio

The odds ratio is the odds in the meditation group divided by the odds in the control group:

$$
OR = \frac{odds_{meditation}}{odds_{control}} = \frac{9}{4} = 2.25
$$
We can interpret this as saying the odds of feeling calm was 2.25 times higher in the group who practiced meditation compared to the control group.

#### Risk 

The probability or "risk" of being calm in the meditation group is 
$$
risk_{meditation} = \frac{a}{a + b} = \frac{45}{50} = 0.90
$$
The probability or "risk" of being calm in the control group is 
$$
risk_{control} = \frac{c}{c + d} = \frac{160}{200} = 0.80
$$

#### Risk ratio 
The risk ratio is the probability or "risk" of being calm in the meditation group divided by the "risk"" of being calm in the control group:

$$
RR = \frac{risk_{meditation}}{risk_{control}}=\frac{0.90}{0.80} = 1.125
$$
We can interpret this as saying the probability of being calm was 12.5% higher in the group practicing meditation compared to the group that did not.

## Test your understanding  {data-progressive=TRUE}

_To test your understanding of the concepts in the previous section, try answering the following questions on odds and risk ratios. You can use R programming code within the interactive coding blocks to perform any calculations. Click the hint button if you need help getting started._


### Scenario

In a cohort study examining infection after surgery among 125 patients, 25 were treated with a novel antibiotic therapy while the remainder received care as usual. In the treated group, 5 patients developed an infection and in the usual care group 40 patients developed an infection.

<br>

![](images/tab3.png){width=400}

<br>

```{r quiz2}
quiz(
  question("What are the correct values of a,b,c and d in the 2 x 2 table above",
    answer("a=5, b=20, c=40, d=60"),
    answer("a=20, b=25, c=60, d=100"),
    answer("a=5, b=25, c=40, d=100"),
    answer("a=20, b=5, c=60, d=40", correct = TRUE),
    allow_retry = TRUE)
)

```





```{r quiz3a}
quiz(
  question("The odds of remaining infection-free in the new antibiotic group is",
    answer("5"),
    answer("4", correct = TRUE),
    answer("0.25"),
    answer("0.2"),
    allow_retry = TRUE)
)
```

```{r calca, exercise=TRUE, exercise.lines=8, exercise.cap="Run R code in this chunk to carry out any necessary calculations, or click 'Hint' for help"}

a <- 
b <- 
c <- 
d <- 



```

``` {r calca-hint}
a <- 20
b <- 5
c <- 60
d <- 40

odds_treated <- a/b
odds_treated

```

<br>


  
```{r quiz3b}
quiz(  
  question("The risk or probability of remaining infection-free in the usual-care group is",
    answer("0.80"),
    answer("1.25"),
    answer("1.33"),
    answer("0.60", correct = TRUE),
    allow_retry = TRUE)
)
```


```{r calcb, exercise=TRUE, exercise.lines=8, exercise.cap="Run R code in this chunk to carry out any necessary calculations, or click 'Hint' for help"}

```

``` {r calcb-hint}

a <- 20
b <- 5
c <- 60
d <- 40

risk_control <- c/(c+d)
risk_control

```

<br>
  
```{r quiz3c}
quiz(    
  question("The Odds Ratio of remaining infection-free for the treated group relative to the usual-care group is",
    answer("2.67", correct = TRUE),
    answer("1.50"),
    answer("2.00"),
    answer("4.00"),
    allow_retry = TRUE)
)
```
  
```{r calcc, exercise=TRUE, exercise.lines=8, exercise.cap="Run R code in this chunk to carry out any necessary calculations, or click 'Hint' for help"}

```

``` {r calcc-hint}
a <- 20
b <- 5
c <- 60
d <- 40

odds_treated <- a/b
odds_control <- c/d

or <- odds_treated/odds_control
or

```  

<br>  
  
```{r quiz3d}
quiz (
  question("The Risk Ratio of remaining infection-free for the treated group relative to the usual-care group is",
    answer("0.80"),
    answer("1.33", correct = TRUE),
    answer("0.60"),
    answer("2.667"),
    allow_retry = TRUE)  
)
```


```{r calcd, exercise=TRUE, exercise.lines=8, exercise.cap="Run R code in this chunk to carry out any necessary calculations, or click 'Hint' for help"}

```

``` {r calcd-hint}

a <- 20
b <- 5
c <- 60
d <- 40

risk_treated <- a/(a+b)
risk_control <- c/(c+d)

rr <- risk_treated/risk_control
rr


```


Click **Continue** to see the solution.

### Solution

How did you do? If you got all the questions correct, congratulations, you've clearly grasped the fundamentals!

Let's have a look at the solution. To start, it can be helpful to summarise the available data in a 2 $\times$ 2 table as before:

![](images/tab4.png){width=400}


The first question asks for **the odds of remaining infection-free in the treated group**. Recall that the odds in a group is the number of times the outcome occurs divided by the number of times the outcome does not occur. So for the treated group, we have:

$$
odds_{treated} = 20/5 = 4
$$
Interpretation: In the treated group, for every patient who experienced a post-survey infection, there were four patients who did not experience an infection. 

The second question asks for **the probability of remaining infection-free in the usual-care group**. Recall the probability or "risk" of an outcome in a given group is the number of times the outcome occurs divided by the total number of individuals in the group. So for the usual-care group, we have:

$$
risk_{usual.care} = 60/100 = 0.60
$$
Interpretation: The probability of remaining infection-free in the usual care group was 60%.

The third question asks for **the Odds Ratio of remaining infection-free for the treated group relative to the usual-care group**. We've already calculated the numerator for this quantity above, the odds for the treated group. The odds for the usual care group is:

$$
odds_{usual.care} = 60/40 = 1.5
$$
The odds ratio is therefore:

$$
OR = \frac{odds_{treated}}{odds_{usual.care}} = \frac{4}{1.5} = 2.67
$$
Interpretation: The odds of remaining infection-free was 2.67 times higher in the group receiving the new antibiotic compared to the usual care group.


The fourth question asks for **The Risk Ratio for remaining infection-free for the treated group relative to the usual-care group**. We've calculated the risk in the usual care group for Question 2, so we are halfway there already. The risk in the treated group is:

$$
risk_{treated} = 20/25 = 0.80
$$
Therefore, the risk ratio is given by

$$
RR = \frac{risk_{treated}}{risk_{usual.care}} = \frac{0.80}{0.60} = 1.33
$$
Interpretation: The probability of remaining infection-free was 33% higher in the antibiotic group compared to the usual care group.


## Summary 

In this tutorial you have learned 

* The **odds** of an outcome in a group is the number with the outcome divided by the number without the outcome  
* The **odds ratio** is the odds in the treated group divided by the odds in the control group 
* The **risk** of an outcome in a group is the number with the outcome divided by the total number of individuals in the group 
* The **risk ratio** is the risk in the treated group divided by the risk in the control group  

### Next Tutorial 
In the next tutorial you will learn 

* How to choose between the odds ratio and risk ratio as a measure of association 
* How to estimate odds and risk ratios while controlling for confounding variables