Risk Difference and NNT • gtregression

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?

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.

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

rd$table

Smoking in pregnancy	Events / N	Risk	Risk difference (95% CI)	p-value
Non-smoker	29 / 115	25.2%	Ref.
Smoker	30 / 74	40.5%	15.3 (1.7 to 29.0)	0.027
Risk = proportion with the event (Low birth weight = Low BW). Risk difference = Smoker − Non-smoker (groups of Smoking in pregnancy), in percentage points.
Confidence interval and p-value: Miettinen–Nurminen score method.
N = 189; no observations excluded.
CI = confidence interval; Ref. = reference group.

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:

rd$table_body
#>        group events   n      risk adjusted_risk adjusted_risk_se        rd
#> 1 Non-smoker     29 115 0.2521739            NA               NA        NA
#> 2     Smoker     30  74 0.4054054            NA               NA 0.1532315
#>   rd_se     rd_low   rd_high    p_value
#> 1    NA         NA        NA         NA
#> 2    NA 0.01722098 0.2897649 0.02689321

Changing the event or the reference group

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

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

rd_swapped$table

Smoking in pregnancy	Events / N	Risk	Risk difference (95% CI)	p-value
Smoker	30 / 74	40.5%	Ref.
Non-smoker	29 / 115	25.2%	−15.3 (−29.0 to −1.7)	0.027
Risk = proportion with the event (Low birth weight = Low BW). Risk difference = Non-smoker − Smoker (groups of Smoking in pregnancy), in percentage points.
Confidence interval and p-value: Miettinen–Nurminen score method.
N = 189; no observations excluded.
CI = confidence interval; Ref. = reference group.

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.

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

rd_nnt$table

Smoking in pregnancy	Events / N	Risk	Risk difference (95% CI)	p-value	NNT (95% CI)
Non-smoker	29 / 115	25.2%	Ref.
Smoker	30 / 74	40.5%	15.3 (1.7 to 29.0)	0.027	NNTH 7 (4 to 59)
Risk = proportion with the event (Low birth weight = Low BW). Risk difference = Smoker − Non-smoker (groups of Smoking in pregnancy), in percentage points.
Confidence interval and p-value: Miettinen–Nurminen score method.
N = 189; no observations excluded.
NNTB/NNTH = number needed to treat for one additional person to benefit / be harmed (event classified as harmful). It applies to the follow-up period of the study and assumes the risk difference reflects the effect of the intervention.
CI = confidence interval; Ref. = reference group; NNTB = number needed to treat (benefit); NNTH = number needed to treat (harm).

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.

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

Group	Events / N	Risk	Risk difference (95% CI)	p-value	NNT (95% CI)
No visit	36 / 100	36.0%	Ref.
Visit	23 / 89	25.8%	−10.2 (−23.0 to 3.1)	0.134	NNTB 10 (NNTB 5 to ∞ to NNTH 32)
Risk = proportion with the event (Low birth weight = Low BW). Risk difference = Visit − No visit (groups of visit), in percentage points.
Confidence interval and p-value: Miettinen–Nurminen score method.
N = 189; no observations excluded.
NNTB/NNTH = number needed to treat for one additional person to benefit / be harmed (event classified as harmful). It applies to the follow-up period of the study and assumes the risk difference reflects the effect of the intervention.
CI = confidence interval; Ref. = reference group; NNTB = number needed to treat (benefit); NNTH = number needed to treat (harm).

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.

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

Smoking in pregnancy	Events / N	Adjusted risk	Adjusted risk difference (95% CI)	p-value	NNT (95% CI)
Non-smoker	29 / 115	23.4%	Ref.
Smoker	30 / 74	44.3%	21.0 (6.9 to 35.1)	0.004	NNTH 5 (3 to 15)
Adjusted risk = standardised proportion with the event (Low birth weight = Low BW). Adjusted risk difference = Smoker − Non-smoker (groups of Smoking in pregnancy), in percentage points. Events / N are the observed counts.
Adjusted risks are standardised to the covariate distribution of the analysis population using logistic regression adjusted for age, lwt, race.
Confidence interval and p-value: Wald, influence-function standard error.
N = 189; no observations excluded.
NNTB/NNTH = number needed to treat for one additional person to benefit / be harmed (event classified as harmful). It applies to the follow-up period of the study and assumes the risk difference reflects the effect of the intervention.
CI = confidence interval; Ref. = reference group; NNTB = number needed to treat (benefit); NNTH = number needed to treat (harm).

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.

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

rd_90$table

Smoking in pregnancy	Events / N	Risk	Risk difference (90% CI)	p-value
Non-smoker	29 / 115	25.2%	Ref.
Smoker	30 / 74	40.5%	15.3 (3.9 to 26.8)	0.027
Risk = proportion with the event (Low birth weight = Low BW). Risk difference = Smoker − Non-smoker (groups of Smoking in pregnancy), in percentage points.
Confidence interval and p-value: Miettinen–Nurminen score method.
N = 189; no observations excluded.
CI = confidence interval; Ref. = reference group.

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.

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

Table 3. Absolute risk of low birth weight by smoking
Smoking in pregnancy	Events / N	Risk	Risk difference (95% CI)	p-value	NNT (95% CI)
Non-smoker	29 / 115	25.2%	Ref.
Current smoker	30 / 74	40.5%	15.3 (1.7 to 29.0)	0.027	NNTH 7 (4 to 59)
Risk = proportion with the event (Low birth weight = Low BW). Risk difference = Current smoker − Non-smoker (groups of Smoking in pregnancy), in percentage points.
Confidence interval and p-value: Miettinen–Nurminen score method.
N = 189; no observations excluded.
NNTB/NNTH = number needed to treat for one additional person to benefit / be harmed (event classified as harmful). It applies to the follow-up period of the study and assumes the risk difference reflects the effect of the intervention.
CI = confidence interval; Ref. = reference group; NNTB = number needed to treat (benefit); NNTH = number needed to treat (harm).
Observational data; differences may reflect confounding.

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 and survival 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.