Compute Odds Ratios

Computes odds ratios, either from a 2x2 contingency table or from a binomial generalized linear model. The table method returns a single odds ratio, the model method one odds ratio per regression coefficient.

Usage

oddsRatio(x, ...)

# Default S3 method
oddsRatio(
  x,
  y = NULL,
  conf.level = NA,
  sides = c("two.sided", "left", "right"),
  method = c("wald", "exact", "midp"),
  interval = c(0, 1000),
  ...
)

# S3 method for class 'glm'
oddsRatio(
  x,
  conf.level = 0.95,
  sides = c("two.sided", "left", "right"),
  method = c("wald", "profile"),
  ...
)

# S3 method for class 'OddsRatio'
print(x, digits = 3, ...)

Arguments

x: a 2x2 contingency table, two vectors to be cross-tabulated, or a binomial stats::glm() object
...: further arguments passed to methods. For the default method with two vectors, these reach table(), so useNA can be set here.
y: optional second variable. If supplied, table(x, y, ...) is computed.
conf.level: confidence level of the interval. For the table method NA (the default) returns the point estimate only; the model method computes an interval by default.
sides: character string specifying the sidedness of the confidence interval (one of "two.sided" (default), "left" or "right"). See ConfidenceIntervals(). An odds ratio is bounded below by 0 and unbounded above, so the open side is reported at 0 or Inf accordingly. Ignored, with a warning, for method = "profile".
method: character string specifying the interval method. For a contingency table one of "wald", "exact" or "midp"; for a binomial model one of "wald" or "profile". See Details.
interval: numeric vector of length two giving the search interval for the root finding in the mid-p method. Only used by method = "midp"; widen it if the reported bound sits at one of its ends.
digits: number of digits used for printing

Value

For a contingency table with conf.level = NA a numeric scalar, otherwise a named numeric vector with the elements est, lci and uci.

For a binomial model an object of class "OddsRatio", a list with:

coefficients: a data frame with one row per coefficient and the columns term, est (the exponentiated coefficient), logEst (the coefficient itself), stdError (on the log scale), pValue, lci and uci
source: "glm"
method, conf.level, sides: as supplied - with sides recording what was computed, which for method = "profile" is always "two.sided"
nObs: number of observations used in the fit
call: the model call

There is a print method; the interval bounds are on the odds scale, the standard error on the log scale.

Details

Contingency tables

For a 2x2 table the odds ratio is

$$OR = \frac{n_{11} n_{22}}{n_{12} n_{21}}$$

Three interval methods are available. "wald" is the asymptotic interval on the log scale, fast and adequate for reasonably large counts. "exact" is the conditional interval based on the noncentral hypergeometric distribution (Fisher); it guarantees coverage but is conservative, sometimes markedly so. "midp" halves the probability of the observed table and lies between the two: it has coverage closer to the nominal level than the exact interval without the Wald interval's reliance on large counts. With a zero cell the point estimate is 0 or Inf and only "exact" and "midp" still deliver a finite bound on the informative side.

Binomial models

For a model fitted with glm(family = binomial), each coefficient is exponentiated: $\exp(\beta_j)$ is the factor by which the odds of the response are multiplied when the corresponding predictor increases by one unit, all other predictors held fixed. For a dummy variable this is the odds ratio between the level and its reference level.

The intercept is exponentiated along with the rest, but $\exp(\beta_0)$ is not an odds ratio - it is the odds of the response when all predictors are zero. It is reported for completeness and is usually not the quantity of interest.

Two interval methods are available. "wald" is the symmetric interval on the log-odds scale, back-transformed. "profile" inverts the likelihood ratio test through stats::confint.glm(); it is asymmetric on the odds scale, generally more reliable in small samples or with sparse cells, and considerably slower because the model is refitted along each coefficient. Profile intervals are two-sided by construction, so sides is ignored for them and a warning is issued.

Unlike the table method, the model method computes an interval by default (conf.level = 0.95): a coefficient table without intervals would be less informative than summary() itself.

Note

No short alias is exported by default to avoid conflicts with rlang and base R naming conventions. Call attachAliases() once per session (or script) to make or() available as a convenient shorthand.

References

Agresti, A. (2013). Categorical Data Analysis (3rd ed.). Wiley.

Fisher, R. A. (1935). The logic of inductive inference. Journal of the Royal Statistical Society, 98(1), 39–82.

Gart, J. J. (1966). Alternative analyses of contingency tables. Journal of the Royal Statistical Society Series B, 28(1), 164–179.

Examples

# --- 2x2 table -------------------------------------------------
tab <- matrix(c(10, 20,
                 5, 30), nrow = 2)

oddsRatio(tab)
#> [1] 3
oddsRatio(tab, conf.level = 0.95)
#>        est        lci        uci 
#>  3.0000000  0.8914747 10.0956314 

# the exact interval is the widest, the Wald interval the narrowest
sapply(c("wald", "exact", "midp"),
       function(m) oddsRatio(tab, conf.level = 0.95, method = m))
#>           wald      exact       midp
#> est  3.0000000  2.9483345  2.9118822
#> lci  0.8914747  0.7770679  0.8795254
#> uci 10.0956314 12.7381845 10.8604692

# one-sided: "left" carries the finite lower bound
oddsRatio(tab, conf.level = 0.95, sides = "left")
#>      est      lci      uci 
#> 3.000000 1.083524      Inf 


# --- binomial model --------------------------------------------
fit <- glm(vs ~ am + wt, data = mtcars, family = binomial)

oddsRatio(fit)
#> 
#> Call:
#> glm(formula = vs ~ am + wt, family = binomial, data = mtcars)
#> 
#> Odds Ratios (95% two.sided CI, method = wald):
#> 
#>                      est     lci          uci  pValue
#> (Intercept) 2.793411e+09 185.176 4.213903e+16 0.00991
#> am          2.000000e-03   0.000 4.310000e-01 0.02310
#> wt          2.000000e-03   0.000 2.060000e-01 0.00904
#> 

# the exponentiated intercept is the baseline odds, not an odds ratio
res <- oddsRatio(fit)
res$coefficients
#>          term          est    logEst stdError      pValue          lci
#> 1 (Intercept) 2.793411e+09 21.750529 8.433431 0.009906264 1.851762e+02
#> 2          am 2.169003e-03 -6.133488 2.700169 0.023115514 1.091119e-05
#> 3          wt 1.782578e-03 -6.329695 2.424632 0.009038881 1.538857e-05
#>            uci
#> 1 4.213903e+16
#> 2 4.311695e-01
#> 3 2.064898e-01

# profile likelihood intervals: asymmetric on the odds scale, slower
oddsRatio(fit, method = "profile")
#> Waiting for profiling to be done...
#> Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred
#> 
#> Call:
#> glm(formula = vs ~ am + wt, family = binomial, data = mtcars)
#> 
#> Odds Ratios (95% two.sided CI, method = profile):
#> 
#>                      est      lci          uci  pValue
#> (Intercept) 2.793411e+09 6739.932 5.653093e+18 0.00991
#> am          2.000000e-03    0.000 1.560000e-01 0.02310
#> wt          2.000000e-03    0.000 7.200000e-02 0.00904
#>