Odds Ratios, Risk Ratios and Other Effect Sizes for 2-by-2 Contingency Tables

Compute Odds Ratios, Risk Ratios, Cohen's h, Absolute Risk Reduction or Number Needed to Treat. Report with any stats::chisq.test() or stats::fisher.test().

Note that these are computed with each column representing the different groups, and the first column representing the treatment group and the second column baseline (or control). Effects are given as treatment / control. If you wish you use rows as groups you must pass a transposed table, or switch the x and y arguments.

Usage

oddsratio(x, y = NULL, ci = 0.95, alternative = "two.sided", log = FALSE, ...)

riskratio(x, y = NULL, ci = 0.95, alternative = "two.sided", log = FALSE, ...)

cohens_h(x, y = NULL, ci = 0.95, alternative = "two.sided", ...)

arr(x, y = NULL, ci = 0.95, alternative = "two.sided", ...)

nnt(x, y = NULL, ci = 0.95, alternative = "two.sided", ...)

Arguments

x: a numeric vector or matrix. x and y can also both be factors.
y: a numeric vector; ignored if x is a matrix. If x is a factor, y should be a factor of the same length.
ci: Confidence Interval (CI) level
alternative: a character string specifying the alternative hypothesis; Controls the type of CI returned: "two.sided" (default, two-sided CI), "greater" or "less" (one-sided CI). Partial matching is allowed (e.g., "g", "l", "two"...). See One-Sided CIs in effectsize_CIs.
log: Take in or output the log of the ratio (such as in logistic models), e.g. when the desired input or output are log odds ratios instead odds ratios.
...: Ignored

Value

A data frame with the effect size (Odds_ratio, Risk_ratio (possibly with the prefix log_), Cohens_h, ARR, NNT) and its CIs (CI_low and CI_high).

Confidence (Compatibility) Intervals (CIs)

Confidence intervals are estimated using the standard normal parametric method (see Katz et al., 1978; Szumilas, 2010).

CIs and Significance Tests

"Confidence intervals on measures of effect size convey all the information in a hypothesis test, and more." (Steiger, 2004). Confidence (compatibility) intervals and p values are complementary summaries of parameter uncertainty given the observed data. A dichotomous hypothesis test could be performed with either a CI or a p value. The 100 (1 - \(\alpha\))% confidence interval contains all of the parameter values for which p > \(\alpha\) for the current data and model. For example, a 95% confidence interval contains all of the values for which p > .05.

Note that a confidence interval including 0 does not indicate that the null (no effect) is true. Rather, it suggests that the observed data together with the model and its assumptions combined do not provided clear evidence against a parameter value of 0 (same as with any other value in the interval), with the level of this evidence defined by the chosen \(\alpha\) level (Rafi & Greenland, 2020; Schweder & Hjort, 2016; Xie & Singh, 2013). To infer no effect, additional judgments about what parameter values are "close enough" to 0 to be negligible are needed ("equivalence testing"; Bauer & Kiesser, 1996).

Plotting with `see`

The see package contains relevant plotting functions. See the plotting vignette in the see package.

References

Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd Ed.). New York: Routledge.
Katz, D. J. S. M., Baptista, J., Azen, S. P., & Pike, M. C. (1978). Obtaining confidence intervals for the risk ratio in cohort studies. Biometrics, 469-474.
Szumilas, M. (2010). Explaining odds ratios. Journal of the Canadian academy of child and adolescent psychiatry, 19(3), 227.

Examples

data("RCT_table")
RCT_table # note groups are COLUMNS
#>            Group
#> Diagnosis   Treatment Control
#>   Sick             71      30
#>   Recovered        50     100

oddsratio(RCT_table)
#> Odds ratio |       95% CI
#> -------------------------
#> 4.73       | [2.74, 8.17]
oddsratio(RCT_table, alternative = "greater")
#> Odds ratio |      95% CI
#> ------------------------
#> 4.73       | [3.00, Inf]
#> 
#> - One-sided CIs: upper bound fixed at [Inf].

riskratio(RCT_table)
#> Risk ratio |       95% CI
#> -------------------------
#> 2.54       | [1.80, 3.60]

cohens_h(RCT_table)
#> Cohen's h |       95% CI
#> ------------------------
#> 0.74      | [0.50, 0.99]

arr(RCT_table)
#> ARR  |       95% CI
#> -------------------
#> 0.36 | [0.24, 0.47]

nnt(RCT_table)
#> NNT  |       95% CI
#> -------------------
#> 2.81 | [2.13, 4.13]