---
title: "Risk Difference and NNT"
description: "Absolute risks, the risk difference between two groups, and the number needed to treat, crude or adjusted."
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Risk Difference and NNT}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(collapse = TRUE, comment = "#>")
```

Odds ratios and risk ratios describe how many *times* more likely an event is.
Readers of a trial report, clinicians and guideline panels also need to know
how many *more people out of every hundred* experience the event, and how
many people need to be treated for one of them to benefit or be harmed.
`risk_difference()` reports exactly that:

- the absolute risk of the event in each of two groups;
- the risk difference in percentage points, with its confidence interval and
  p-value;
- optionally, the number needed to treat for one additional person to benefit
  (NNTB) or be harmed (NNTH).

It is a separate function from the regression tables because it answers a
different question; the regression functions are unchanged. The examples below
use `format = "flextable"`, which suits Word output; `format = "gt"` (the
default) gives an HTML-first table.

## Example data

The birth-weight data are used to ask a trial-style question: how much higher
is the absolute risk of low birth weight when the mother smokes?

```{r setup, message = FALSE}
library(gtregression)
library(dplyr)

data("data_birthwt", package = "gtregression")

bw <- data_birthwt |>
  mutate(
    smoke = factor(smoke, levels = c(0, 1), labels = c("Non-smoker", "Smoker")),
    low = factor(low, levels = c(0, 1), labels = c("Normal BW", "Low BW")),
    race = factor(race, levels = c(1, 2, 3), labels = c("White", "Black", "Other"))
  )

attr(bw$smoke, "label") <- "Smoking in pregnancy"
attr(bw$low, "label") <- "Low birth weight"
```

## The crude risk difference

Name the outcome, the event and the reference group. Naming them is optional
(the defaults are the second outcome level and the first exposure level, with
a message), but explicit choices make the analysis easier to read.

```{r crude}
rd <- risk_difference(
  bw,
  outcome = low,
  exposure = smoke,
  event = "Low BW",
  reference = "Non-smoker",
  format = "flextable"
)

rd$table
```

Reading the table:

- **Events / N** and **Risk** are the observed counts and proportions in each
  group.
- **Risk difference** is the comparison group minus the reference group, in
  percentage points. Here the risk of low birth weight is 15.3 percentage
  points higher among smokers.
- The footnotes always state the event, the direction of the comparison and the
  method, so the table can be read without the code.

The crude interval and p-value use the Miettinen-Nurminen score method, which
keeps close to its nominal coverage in small samples and when risks are near
0% or 100%.

The numbers behind the table are available for reporting or further work:

```{r body}
rd$table_body
```

## Changing the event or the reference group

Swapping the reference group reverses the sign of the difference and leaves the
p-value unchanged.

```{r reference}
rd_swapped <- risk_difference(bw, outcome = low, exposure = smoke,
                              event = "Low BW", reference = "Smoker",
                              format = "flextable")

rd_swapped$table
```

## Number needed to treat

Add `event_type` to report the number needed to treat. It must say whether
the event is **harmful** (for example death or infection) or **beneficial**
(for example cure), because that decides whether a difference represents a
benefit or a harm.

```{r nnt}
rd_nnt <- risk_difference(bw, outcome = low, exposure = smoke,
                          event = "Low BW", reference = "Non-smoker",
                          event_type = "harmful",
                          format = "flextable")

rd_nnt$table
```

Here about 7 pregnancies exposed to smoking are associated with one additional
low-birth-weight baby (NNTH 7). The interval for the number needed to treat
comes from the interval for the risk difference.

When the risk-difference interval includes zero, the NNT interval passes
through infinity: the data are compatible with benefit, no effect and harm.
The table shows this explicitly instead of giving a misleading finite range.

```{r nnt-infinity}
bw$visit <- factor(bw$ftv > 0, levels = c(FALSE, TRUE),
                   labels = c("No visit", "Visit"))

rd_visit <- risk_difference(bw, outcome = low, exposure = visit,
                            event = "Low BW", reference = "No visit",
                            event_type = "harmful",
                            format = "flextable")

rd_visit$table
```

An NNT applies to the follow-up period of the study and assumes the risk
difference reflects the effect of the intervention; the table footnote says
so. In observational data, interpret it with the same caution as any other
causal claim.

## Adjusted risk difference

`adjust_for` gives an adjusted risk difference. The risks are standardised:
a logistic regression with the exposure and the adjustment variables is fitted,
everyone in the analysis is predicted once as if in the reference group and
once as if in the comparison group, and the predictions are averaged. The
result is a risk difference for the whole analysed population, on the same
percentage-point scale as the crude result.

```{r adjusted}
rd_adjusted <- risk_difference(bw, outcome = low, exposure = smoke,
                               event = "Low BW", reference = "Non-smoker",
                               adjust_for = c(age, lwt, race),
                               event_type = "harmful",
                               format = "flextable")

rd_adjusted$table
```

**Events / N** are still the observed counts; the risks and their difference
are the adjusted values. The interval and p-value use a Wald approach with an
influence-function standard error, which accounts for uncertainty in both the
model and the covariate distribution.

In randomised trials, adjusting for a few strongly prognostic baseline
variables that were specified in advance usually gives a more precise estimate
of the same treatment effect. In observational studies adjustment addresses
measured confounding only.

## Confidence level

`conf_level` changes the interval only; the p-value always tests a risk
difference of zero.

```{r level}
rd_90 <- risk_difference(bw, outcome = low, exposure = smoke,
                         event = "Low BW", reference = "Non-smoker",
                         conf_level = 0.90,
                         format = "flextable")

rd_90$table
```

## Missing values

Rows with a missing value in the outcome, exposure or any adjustment variable
are left out, and the footnote reports how many. Counts and risks always refer
to the analysed rows.

## Polish and save

`modify_table()` adds a caption and caveat, and renames variables, groups and
headers. `save_table()` exports the result.

```{r polish}
rd_final <- risk_difference(bw, outcome = low, exposure = smoke,
                            event = "Low BW", reference = "Non-smoker",
                            event_type = "harmful", format = "flextable") |>
  modify_table(
    level_labels = list(smoke = c(Smoker = "Current smoker")),
    caption = "Table 3. Absolute risk of low birth weight by smoking",
    caveat = "Observational data; differences may reflect confounding."
  )

rd_final$table
```

```{r save, eval = FALSE}
save_table(rd_final, filename = "risk-difference", format = "docx")
```

## Reporting

A sentence for the results section can follow the table directly:

> Low birth weight occurred in 30 of 74 births to smokers (40.5%) and 29 of
> 115 births to non-smokers (25.2%), a risk difference of 15.3 percentage points
> (95% CI 1.7 to 29.0; Miettinen-Nurminen score method). This corresponds to a
> number needed to harm of 7 (95% CI 4 to 59).

## Methods at a glance

| | Crude | Adjusted (`adjust_for`) |
|---|---|---|
| Risks | Observed proportions | Standardised from a main-effects logistic regression |
| Interval | Miettinen-Nurminen score | Wald, influence-function standard error |
| p-value | Miettinen-Nurminen score test | Wald test |
| Number needed to treat | 1 / risk difference, rounded up; Altman interval | Same |

## Scope

`risk_difference()` compares exactly two groups on a binary outcome with
independent observations. It does not handle clustered or matched data, time
to event, or more than two groups; restrict the data to two groups, or use the
[regression](regression-tables.html) and [survival](survival-analysis.html)
functions for those designs. Risk-difference tables stand alone: they can be
relabelled with `modify_table()` and saved, but are not used in
`merge_tables()`, `plot_reg()` or forest plots.