Odds Ratios

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# Odds Ratios

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layout: true

<div class="my-footer">
<span>
by Dr. Lucy D'Agostino McGowan
</span>
</div>

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# <i class="fas fa-laptop"></i> `Odds ratios`

- Go to RStudio Cloud and open `Odds ratios`
---

# Odds ratios

A study investigated whether a handheld device that sends a magnetic pulse into a person’s head
might be an effective treatment for migraine headaches.

* Researchers recruited 200 subjects who suffered from migraines 
* randomly assigned them to receive either the TMS (transcranial magnetic
stimulation) treatment or a placebo treatment 
* Subjects were instructed to apply the device at the onset of migraine symptoms and then assess how they felt two hours later. (either **Pain-free** or **Not pain-free**)

---

# Odds ratios

.question[
What is the explanatory variable?
]

A study investigated whether a handheld device that sends a magnetic pulse into a person’s head might be an effective treatment for migraine headaches.

# Odds ratios

.question[
What type of variable is this?
]

A study investigated whether a handheld device that sends a magnetic pulse into a person’s head might be an effective treatment for migraine headaches.

# Odds ratios

.question[
What is the outcome variable?
]

A study investigated whether a handheld device that sends a magnetic pulse into a person’s head might be an effective treatment for migraine headaches.

# Odds ratios

.question[
What type of variable is this?
]

A study investigated whether a handheld device that sends a magnetic pulse into a person’s head might be an effective treatment for migraine headaches.

# Odds ratios

|TMS | Placebo| Total
---|---|----|---
Pain-free two hours later |39| 22 |61
Not pain-free two hours later |61 |78| 139
Total| 100| 100 |200

* We can compare the results using **odds**

--
* What are the odds of being pain-free for the placebo group?

--
  * `$(22/100)/(78/100) = 22/78 = 0.282$`

--
* What are the odds of being pain-free for the treatment group?

--
  * `$39/61 = 0.639$`

--
* Comparing the **odds** what can we conclude?

--
  * TMS increases the likelihood of success
---

# Odds ratios

|TMS | Placebo| Total
---|---|----|---
Pain-free two hours later |39| 22 |61
Not pain-free two hours later |61 |78| 139
Total| 100| 100 |200

* We can summarize this relationship with an **odds ratio**: the ratio of the two odds

--
 
`$\Large OR = \frac{39/61}{22/78} = \frac{0.639}{0.282} = 2.27$`

_"the odds of being pain free were 2.27 times higher with TMS than with the placebo"_

---

# Odds ratios

.question[
What if we wanted to calculate this in terms of _Not pain-free_ (with _pain-free_) as the referent?
]

|TMS | Placebo| Total
---|---|----|---
Pain-free two hours later |39| 22 |61
Not pain-free two hours later |61 |78| 139
Total| 100| 100 |200

`$\Large OR = \frac{61/39}{78/22} = \frac{1.564}{3.545} = 0.441$`

_the odds for still being in pain for the TMS group are 0.441 times the odds of being in pain for the placebo group_
 
---

# Odds ratios

.question[
What changed here?
]

|TMS | Placebo| Total
---|---|----|---
Pain-free two hours later |39| 22 |61
Not pain-free two hours later |61 |78| 139
Total| 100| 100 |200

`$\Large OR = \frac{78/22}{61/39} = \frac{3.545}{1.564} = 2.27$`

_the odds for still being in pain for the placebo group are 2.27 times the odds of being in pain for the TMS group_
 
---

# Odds ratios

* In general, it's more natural to interpret odds ratios > 1, you can flip the referent to do so

|TMS | Placebo| Total
---|---|----|---
Pain-free two hours later |39| 22 |61
Not pain-free two hours later |61 |78| 139
Total| 100| 100 |200

`$\Large OR = \frac{78/22}{61/39} = \frac{3.545}{1.564} = 2.27$`

_the odds for still being in pain for the placebo group are 2.27 times the odds of being in pain for the TMS group_

---

# <i class="fas fa-laptop"></i> `Odds ratios`

- Go to RStudio Cloud and open `Odds ratios`

---

## Odds ratios

* Let's look at some Titanic data. We are interested in whether the sex of the passenger is related to whether they survived.

|Female | Male | Total
---------|-------|-----|-----
Survived| 308 | 142 | 450
Died | 154 | 709| 863
Total | 462 | 851 | 1313

---

## Odds ratios

.question[
What are the odds of surviving for females versus males?
]

* Let's look at some Titanic data. We are interested in whether the sex of the passenger is related to whether they survived.

|Female | Male | Total
---------|-------|-----|-----
Survived| 308 | 142 | 450
Died | 154 | 709| 863
Total | 462 | 851 | 1313

.center[
`$\Large OR = \frac{308/154}{142/709} = \frac{2}{0.2} = 9.99$`
]

---

## Odds ratios

.question[
How do you interpret this?
]

|Female | Male | Total
---------|-------|-----|-----
Survived| 308 | 142 | 450
Died | 154 | 709| 863
Total | 462 | 851 | 1313

.center[
`$\Large OR = \frac{308/154}{142/709} = \frac{2}{0.2} = 9.99$`
]

_the odds of surviving for the female passengers was 9.99 times the odds of surviving for the male passengers_

---

## Odds ratios

.question[
What if we wanted to fit a **model**? What would the equation be?
]

|Female | Male | Total
---------|-------|-----|-----
Survived| 308 | 142 | 450
Died | 154 | 709| 863
Total | 462 | 851 | 1313

.center[
`$\Large \log(\textrm{odds of survival}) \approx \beta_0 + \beta_1 \textrm{Sex}$`
]

---

## Odds ratios

.center[
`$\Large \log(\textrm{odds of survival}) \approx \beta_0 + \beta_1 \textrm{Sex}$`
]
.small[

```r
glm(Survived ~ Sex, data = Titanic, family = binomial) %>%
  tidy()
```

```
## # A tibble: 2 x 5
##   term        estimate std.error statistic  p.value
##   <chr>          <dbl>     <dbl>     <dbl>    <dbl>
## 1 (Intercept)    0.693    0.0987      7.02 2.17e-12
## 2 Sexmale       -2.30     0.135     -17.1  2.91e-65
```
]

---

## Odds Ratios

.question[
What is my referent category?
]

.center[
`$\Large \log(\textrm{odds of survival}) \approx \beta_0 + \beta_1 \textrm{Sex}$`
]

```r
levels(Titanic$Sex)
```

```
## [1] "female" "male"
```

---

## Odds Ratios

.question[
How do I change that?
]

.center[
`$\Large \log(\textrm{odds of survival}) \approx \beta_0 + \beta_1 \textrm{Sex}$`
]

```r
Titanic <- Titanic %>%
* ---(Sex = ---(Sex, c("male", "female")))
```

---

## Odds Ratios

.question[
How do I change that?
]

.center[
`$\Large \log(\textrm{odds of survival}) \approx \beta_0 + \beta_1 \textrm{Sex}$`
]

```r
Titanic <- Titanic %>%
* mutate(Sex = fct_relevel(Sex, c("male", "female")))
```

---

## Odds Ratios

.question[
How do you interpret this result?
]

.small[

```r
glm(Survived ~ Sex, data = Titanic, family = binomial) %>%
  tidy()
```

```
## # A tibble: 2 x 5
##   term        estimate std.error statistic  p.value
##   <chr>          <dbl>     <dbl>     <dbl>    <dbl>
## 1 (Intercept)    -1.61    0.0919     -17.5 1.70e-68
## 2 Sexfemale       2.30    0.135       17.1 2.91e-65
```
]

---

## Odds Ratios

.question[
How do you interpret this result?
]

.small[

```r
glm(Survived ~ Sex, data = Titanic, family = binomial) %>%
* tidy(exponentiate = TRUE)
```

```
## # A tibble: 2 x 5
##   term        estimate std.error statistic  p.value
##   <chr>          <dbl>     <dbl>     <dbl>    <dbl>
## 1 (Intercept)    0.200    0.0919     -17.5 1.70e-68
## 2 Sexfemale      9.99     0.135       17.1 2.91e-65
```

```r
exp(2.301176)
```

```
## [1] 9.99
```
]

_the odds of surviving for the female passengers was 9.99 times the odds of surviving for the male passengers_

---

## Odds ratios

* What if the explanatory variable is continuous?

--
* We did this already on Tuesday!

.small[

```r
data("MedGPA")
glm(Acceptance ~ GPA, data = MedGPA, family = binomial) %>%
  tidy()
```

```
## # A tibble: 2 x 5
##   term        estimate std.error statistic  p.value
##   <chr>          <dbl>     <dbl>     <dbl>    <dbl>
## 1 (Intercept)   -19.2       5.63     -3.41 0.000644
## 2 GPA             5.45      1.58      3.45 0.000553
```
]

_A one unit increase in GPA yields a 5.45 increase in the log odds of acceptance_

---

## Odds ratios

* What if the explanatory variable is continuous?
* We did this already on Tuesday!

.small[

```r
data("MedGPA")
glm(Acceptance ~ GPA, data = MedGPA, family = binomial) %>%
* tidy(exponentiate = TRUE)
```

```
## # A tibble: 2 x 5
##   term        estimate std.error statistic  p.value
##   <chr>          <dbl>     <dbl>     <dbl>    <dbl>
## 1 (Intercept)  4.56e-9      5.63     -3.41 0.000644
## 2 GPA          2.34e+2      1.58      3.45 0.000553
```
]

_A one unit increase in GPA yields a 234-fold increase in the odds of acceptance_

* 😱 that seems huge! **Remember:** the interpretation of these coefficients depends on your units (the same as in ordinary linear regression).

---

## Odds ratios

.question[
How could we get the odds associated with increasing GPA by 0.1?
]

.small[

```r
glm(Acceptance ~ GPA, data = MedGPA, family = binomial) %>%
  tidy()
```

.small[

```r
exp(5.454) ## a one unit increase in GPA
```

```
## [1] 234
```

```r
exp(5.454 * 0.1) ## a 0.1 increase in GPA
```

```
## [1] 1.73
```
]

_A one-tenth unit increase in GPA yields a 1.73-fold increase in the odds of acceptance_

---

## Odds ratios

.question[
How could we get the odds associated with increasing GPA by 0.1?
]

.small[

```r
MedGPA <- MedGPA %>%
  mutate(GPA_10 = GPA * 10)

glm(Acceptance ~ GPA_10, data = MedGPA, family = binomial) %>%
  tidy(exponentiate = TRUE)
```

```
## # A tibble: 2 x 5
##   term             estimate std.error statistic  p.value
##   <chr>               <dbl>     <dbl>     <dbl>    <dbl>
## 1 (Intercept) 0.00000000456     5.63      -3.41 0.000644
## 2 GPA_10      1.73              0.158      3.45 0.000553
```
]

_A one-tenth unit increase in GPA yields a 1.73-fold increase in the odds of acceptance_