Odds Ratio Estimation and Confidence Intervals

Calculates odds ratio by unconditional maximum likelihood estimation (wald), conditional maximum likelihood estimation (mle) or median-unbiased estimation (midp). Confidence intervals are calculated using normal approximation (wald) and exact methods (midp, mle).

OddsRatio(x, conf.level = NULL, ...)

# S3 method for class 'glm'
OddsRatio(x, conf.level = NULL, digits = 3, use.profile = FALSE, ...)

# S3 method for class 'multinom'
OddsRatio(x, conf.level = NULL, digits = 3, ...)

# S3 method for class 'zeroinfl'
OddsRatio(x, conf.level = NULL, digits = 3, ...)

# Default S3 method
OddsRatio(x, conf.level = NULL, y = NULL, method = c("wald", "mle", "midp"),
          interval = c(0, 1000), ...)

Arguments

x: a vector or a \(2 \times 2\) numeric matrix, resp. table.
y: NULL (default) or a vector with compatible dimensions to x. If y is provided, table(x, y, ...) will be calculated.
digits: the number of fixed digits to be used for printing the odds ratios.
method: method for calculating odds ratio and confidence intervals. Can be one out of "wald", "mle", "midp". Default is "wald" (not because it is the best, but because it is the most commonly used.)
conf.level: confidence level. Default is NA for tables and numeric vectors, meaning no confidence intervals will be reported. 0.95 is used as default for models.
interval: interval for the function uniroot that finds the odds ratio median-unbiased estimate and midp exact confidence interval.
use.profile: logical. Defines if profile approach should be used, which normally is a good choice. Calculating profile can however take ages for large datasets and not be necessary there. So we can fallback to normal confidence intervals.
...: further arguments are passed to the function table, allowing i.e. to set useNA. This refers only to the vector interface.

Details

If a \(2 \times 2\) table is provided the following table structure is preferred:


                    disease=1   disease=0
    exposed=1          n11         n10
    exposed=0          n01         n00

however, for odds ratios the following table is equivalent:


                    disease=0   disease=1
    exposed=0 (ref)    n00         n01
    exposed=1          n10         n11

If the table to be provided to this function is not in the preferred form, the function Rev() can be used to "reverse" the table rows, resp. -columns. Reversing columns or rows (but not both) will lead to the inverse of the odds ratio.

In case of zero entries, 0.5 will be added to the table.

Value

a single numeric value if conf.level is set to NA
a numeric vector with 3 elements for estimate, lower and upper confidence interval if conf.level is provided

References

Kenneth J. Rothman and Sander Greenland (1998): Modern Epidemiology, Lippincott-Raven Publishers

Kenneth J. Rothman (2002): Epidemiology: An Introduction, Oxford University Press

Nicolas P. Jewell (2004): Statistics for Epidemiology, 1st Edition, 2004, Chapman & Hall, pp. 73-81

Agresti, Alan (2013) Categorical Data Analysis. NY: John Wiley and Sons, Chapt. 3.1.1

Author

Andri Signorell <andri@signorell.net>, strongly based on code from Tomas Aragon, <aragon@berkeley.edu>

Examples

# Case-control study assessing whether exposure to tap water
#   is associated with cryptosporidiosis among AIDS patients

tab <- matrix(c(2, 29, 35, 64, 12, 6), 3, 2, byrow=TRUE)
dimnames(tab) <- list("Tap water exposure" = c("Lowest", "Intermediate", "Highest"),
                      "Outcome" = c("Case", "Control"))
tab <- Rev(tab, margin=2)

OddsRatio(tab[1:2,])
#> [1] 7.929688
OddsRatio(tab[c(1,3),])
#> [1] 29

OddsRatio(tab[1:2,], method="mle")
#> [1] 7.836979
OddsRatio(tab[1:2,], method="midp")
#> [1] 7.355436
OddsRatio(tab[1:2,], method="wald", conf.level=0.95)
#> odds ratio     lwr.ci     upr.ci 
#>   7.929688   1.785414  35.218699 

# in case of zeros consider using glm for calculating OR
dp <- data.frame (a=c(20, 7, 0, 0), b=c(0, 0, 0, 12), t=c(1, 0, 1, 0))
fit <- glm(cbind(a, b) ~ t, data=dp, family=binomial)

exp(coef(fit))
#>       (Intercept)                 t 
#>         0.5833333 648881106.9538907 

# calculation of log oddsratios in a 2x2xk table
migraine <- xtabs(freq ~ .,
                  cbind(expand.grid(treatment=c("active","placebo"),
                                    response=c("better","same"),
                                    gender=c("female","male")),
                        freq=c(16,5,11,20,12,7,16,19))
)

log(apply(migraine, 3, OddsRatio))
#>    female      male 
#> 1.7609878 0.7108468 

# OddsRatio table for logistic regression models
r.glm <- glm(type ~ ., data=MASS::Pima.tr2, family=binomial)
OddsRatio(r.glm)
#> 
#> Call:
#> glm(formula = type ~ ., family = binomial, data = MASS::Pima.tr2)
#> 
#> Odds Ratios:
#>                or or.lci or.uci Pr(>|z|)    
#> (Intercept) 0.000  0.000  0.002 3.38e-08 ***
#> npreg       1.109  0.977  1.259   0.1107    
#> glu         1.033  1.019  1.046 2.22e-06 ***
#> bp          0.995  0.960  1.032   0.7971    
#> skin        0.998  0.955  1.043   0.9321    
#> bmi         1.087  1.000  1.182   0.0509 .  
#> ped         6.174  1.675 22.755   0.0062 ** 
#> age         1.042  0.998  1.088   0.0623 .  
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 
#> 
#> Brier Score: 0.147     Nagelkerke R2: 0.447
#> 

plot(OddsRatio(r.glm), xlim=c(0.5, 2), main="OddsRatio - glm", pch=NA,
     lblcolor=DescTools::hred, args.errbars=list(col=DescTools::horange, pch=21, 
     col.pch=DescTools::hblue,
     bg.pch=DescTools::hyellow, cex.pch=1.5))