## 12.4 Estimate the odds ratio

We’ll use the `oddsratio`

function from the `epitools`

package to calculate the odds ratio (\(\hat{OR}\)) and its 95% confidence interval.

Check out the help file for the function:

`?oddsratio`

The `oddsratio`

function expects the contingency table to be arranged exactly like this:

```
# treatment control
# sick a b
# healthy c d
```

If you were calculating the odds ratio by hand, using the letters shown in the table above, the shortcut formula is:

\[\hat{OR} = \frac{{a/c}}{{b/d}}\]

Here’s the code for producing the appropriately formatted 2 x 2 table as so:

```
%>%
cancer tabyl(aspirinTreatment, response) %>%
select(Cancer, "No cancer")
```

```
## Cancer No cancer
## 1438 18496
## 1427 18515
```

So thats what the `oddsratio`

function is expecting as input.

However, it’s also expecting it in the form of a “matrix” object.

Here’s all the code at once, and we’ll store the output (which comes in the form of a “list”) in an object called “cancer.odds”:

```
<- cancer %>%
cancer.odds tabyl(aspirinTreatment, response) %>%
select(Cancer, "No cancer") %>%
as.matrix() %>%
oddsratio(method = "wald")
```

We’ve seen the first two lines before. Then:

- we
`select`

the two columns associated with the “Cancer” and “No cancer” data.**NOTE**that because there’s a space in the variable name “No cancer”, we need to use quotation marks around it - Then we use the base
`as.matrix`

function to coerce the resulting 2 x 2 table that we’ve created into a matrix type object, which is what the`oddsratio`

function is expecting. - lastly we run the
`oddsratio`

function, with the argument “method = ‘wald’” (don’t worry about why)

Let’s have a look at the rather verbose output:

` cancer.odds`

```
## $data
## Cancer No cancer Total
## row1 1438 18496 19934
## row2 1427 18515 19942
## Total 2865 37011 39876
##
## $measure
## NA
## odds ratio with 95% C.I. estimate lower upper
## [1,] 1.000000 NA NA
## [2,] 1.008744 0.9349043 1.088415
##
## $p.value
## NA
## two-sided midp.exact fisher.exact chi.square
## [1,] NA NA NA
## [2,] 0.8224348 0.8310911 0.8223986
##
## $correction
## [1] FALSE
##
## attr(,"method")
## [1] "Unconditional MLE & normal approximation (Wald) CI"
```

This is more information than we need.

What we’re interested in is the information under the “$measure” part, and specifically the “odds ratio with 95% C.I.”.

To limit the output to the relevant information, use this code:

`$measure[2,] cancer.odds`

```
## estimate lower upper
## 1.0087436 0.9349043 1.0884148
```

This isolates the actual estimate of the odds ratio (\(\hat{OR}\)) with its 95% confidence interval.

The estimate of the odds ratio is around 1.009, and notice the 95% confidence interval encompasses one.

Given that the calculated 95% confidence interval encompasses 1 (representing equal odds among treatment and control groups), there is presently no evidence that the odds of developing cancer differ among control and aspirin treatment groups.

**IMPORTANT **
The odds ratio and its 95% confidence interval are useful to report in any analysis of a 2 x 2 contingency table that deals with health outcomes data like those used here.