Relative Risk

Computes the relative risk for a 2x2 contingency table together with optional confidence intervals.

Usage

relRisk(
  x,
  y = NULL,
  conf.level = NA,
  sides = c("two.sided", "left", "right"),
  method = c("score", "wald", "use-or"),
  delta = 0.5,
  ...
)

Arguments

x: a numeric 2x2 matrix or table containing non-negative counts
y: an optional vector. If supplied, table(x, y, ...) is computed.
conf.level: confidence level for the interval estimate. If NA (default), only the point estimate is returned.
sides: character string specifying the sidedness of the confidence interval (one of "two.sided" (default), "left" or "right"). See details in ConfidenceIntervals().
method: character string specifying the confidence interval method. One of "score", "wald", or "use-or".
delta: small continuity correction added to the event counts in the standard error of the Wald interval. Only used if method = "wald"; see the note below.
...: further arguments passed to table()

Value

if conf.level = NA, a numeric scalar. Otherwise a named numeric vector with elements:

est: relative risk estimate
lci: lower confidence interval bound
uci: upper confidence interval bound

Details

The relative risk compares the event probability in the exposed group with the event probability in the unexposed group.

The function expects the exposure groups in the rows and the outcome in the columns:


               outcome = 1   outcome = 0
exposed = 1        x1           n1 - x1
exposed = 0        x2           n2 - x2

The relative risk is defined as:

$$ RR = \frac{x_1 / n_1}{x_2 / n_2} $$

Confidence intervals can be calculated using the score method of Koopman (1984), a Wald approximation, or via transformation of the odds ratio.

The score interval is based on the method of Koopman (1984) and Miettinen and Nurminen (1985). It is obtained from the closed-form solution of the cubic equation in the constrained maximum likelihood estimate; if the unexposed group has no non-events (x2 == n2) that cubic has a root on the parameter boundary and its roots can no longer be assigned to the two interval bounds by their order alone. This case is therefore solved directly from the score statistic (stats::uniroot()); both routes agree to numerical precision wherever the closed form applies.

The Wald interval is asymptotic and may perform poorly for small counts or extreme probabilities. Note that delta enters the standard error only, not the point estimate the interval is centred on: with a zero cell the point estimate is 0 or Inf and the Wald interval degenerates accordingly. Use method = "score" for tables with zero cells.

If the table orientation differs from the required structure, rows or columns can be reversed using bedrock::revX() or transposed with t().

References

Koopman, P. A. R. (1984). Confidence intervals for the ratio of two binomial proportions. Biometrics, 40(2), 513–517.

Miettinen, O., & Nurminen, M. (1985). Comparative analysis of two rates. Statistics in Medicine, 4(2), 213–226.

Rothman, K. J., Greenland, S., & Lash, T. L. (2008). Modern Epidemiology (3rd ed.). Lippincott Williams & Wilkins.

Examples

m <- matrix(
  c(78, 50,
    1422, 950),
  nrow = 2,
  dimnames = list(
    water = c("cont", "clean"),
    diarrhea = c("yes", "no")
  )
)

relRisk(m, conf.level = 0.95)
#>       est       lci       uci 
#> 1.0400000 0.7375967 1.4682951 


mm <- cbind(c(9, 20), c(41, 29))

relRisk(t(mm), conf.level = 0.95)
#>       est       lci       uci 
#> 0.5298570 0.2869143 0.8869267 

relRisk(
  t(mm),
  conf.level = 0.95,
  method = "wald"
)
#>       est       lci       uci 
#> 0.5298570 0.3037489 0.9242780 

relRisk(
  t(mm),
  conf.level = 0.95,
  method = "use-or"
)
#>       est       lci       uci 
#> 0.5298570 0.2597810 0.9051415 

# unexposed group without non-events: the score interval is still valid
relRisk(matrix(c(2, 5, 3, 0), nrow = 2), conf.level = 0.95)
#>       est       lci       uci 
#> 0.4000000 0.1176208 0.9378042