loddsratio: Calculate Log Odds Ratios for Frequency Tables

Description

Computes (log) odds ratios and their asymptotic variance covariance matrix for (possibly) stratified data. Odds ratios are calculated for two array dimensions, separately for each level of all stratifying dimensions. This generalizes the oddsratio methods in the vcd package from 2 x 2 ( x strata) tables to R x C (x strata) tables. In future versions, these functions may be renamed and or moved to the vcd package.

Usage

loddsratio(x, ...)
## S3 method for class 'default':
loddsratio(x, strata = NULL, log = TRUE, 
  ref = NULL, correct = any(x == 0), ...)

## S3 method for class 'loddsratio':
coef(object, log = object$log, \dots)
## S3 method for class 'loddsratio':
vcov(object, log = object$log, \dots)
## S3 method for class 'loddsratio':
print(x, log = x$log, \dots)

as.array(x, ...)
## S3 method for class 'loddsratio':
as.array(x, log=x$log, \dots)

Arguments

An object. For the default method a k-way matrix/table/array of frequencies. The number of margins has to be at least 2.

strata

Numeric or character indicating the margins of a k-way table x (with k greater than 2) that should be employed as strata. By default dimensions 3:length(dim(x)) are used.

ref

Numeric or character. Reference categories for the (non-stratum) row and column dimensions that should be employed for computing the odds ratios. By default, odds ratios for profile contrasts (or sequential contrasts, i.e., successive diffe

log

Logical. Should the results be displayed on a log scale or not? All internal computations are always on the log-scale but the results are transformed by default if log = TRUE.

correct

Logical. Should a continuity correction be applied (by adding 0.5 to all table entries) before computing odds ratios? By default, this not employed unless there are any zero cells in the table, but this correction is often recommended to re

object

An object of class loddsratio as computed by loddsratio.

...

Arguments passed to methods.

Value

An object of class loddsratio, with the following components:
coefficientsA named vector, of length (R-1) x (C-1) x prod(dim(x)[strata]) containing the log odds ratios. Use the coef method to extract these from the object, and the confint method for confidence intervals. For a two-way table, the names for the log odds ratios are constructed in the form Ri:Rj/Ci:Cj using the table names for rows and columns. For a stratified table, the names are constructed in the form Ri:Rj/Ci:Cj|Lk.
vcovVariance covariance matrix of the log odds ratios.
dimnamesDimension names for the log odds ratios, considered as a table of size c(R-1, C-1, dim(x)[strata]. Use the dim and dimnames methods to extract these and manipulate the log odds ratios in relation to the original table.
dimCorresponding dimension vector.
contrastsA matrix C, such that C %*% as.vector(log(x)) gives the log odds ratios. Each row corresponds to one log odds ratio, and is all zero, except for 4 elements of c(1, -1, -1, 1) for a given 2 x 2 subtable.
logA logical, indicating whether the value of log in the original call.

Details

For an R x C table, (log) odds ratios are formed for the set of (R-1) x (C-1) 2 x 2 tables, corresponding to some set of contrasts among the row and column variables. The ref argument allows these to be specified in a general way. ref = NULL (default) corresponds to profile contrasts (or sequential contrasts or successive differences) for ordered categories, i.e., R1-R2, R2-R3, R3-R4, etc., and similarly for the column categories. These are sometimes called local odds ratios. ref=1 gives contrasts with the first category; ref=dim(x) gives contrasts with the last category; ref = c(2, 4) or ref = list(2, 4) corresponds to the reference being the second category in rows and the fourth in columns. Combinations like ref = list(NULL, 3) are also possible, as are character vectors, e.g., ref = c("foo", "bar") also works ("foo" pertaining again to the row reference and "bar" to column reference). Note that all such parameterizations are equivalent, in that one can derive all other possible odds ratios from any non-redundant set.

References

A. Agresti (1990), Categorical Data Analysis. New York: Wiley. Fleiss, J. L. (1981). Statistical Methods for Rates and Proportions. 2nd Edition. New York: Wiley. M. Friendly (2000), Visualizing Categorical Data. SAS Institute, Cary, NC.

Examples

Run this code

## artificial example
set.seed(1)
x <- matrix(rpois(5 * 3, 7), ncol = 5, nrow = 3)
dimnames(x) <- list(Row=head(letters, 3), Col=tail(letters, 5))

x_lor <- loddsratio(x)
coef(x_lor)
x_lor
confint(x_lor)
if(require("lmtest")) coeftest(x_lor)


## 2 x 2 x k cases
#data(CoalMiners, package="vcd")
lor.CM <- loddsratio(CoalMiners)
lor.CM
coef(lor.CM)
confint(lor.CM)
confint(lor.CM, log=FALSE)

# odds ratio plot
lor.CM.df <- as.data.frame(lor.CM)
lor.CM.df <- within(lor.CM.df, 
	{lower<-LOR-ASE
	 upper<-LOR+ASE}
	)
range <- c(min(lor.CM.df$lower), max(lor.CM.df$upper))
with(lor.CM.df, {
	plot(LOR ~ as.numeric(Age), type='b', pch=16, xaxt='n',
	  ylim=range, 
		xlab="Age", ylab="Log odds ratio: Wheeze x Breathlessness",
		main="CoalMiners data: Log odds ratio plot")
	axis(side=1, at=as.numeric(Age), labels=Age)
	segments(as.numeric(Age), lower, as.numeric(Age), upper)
	}
	)
# fit linear models using WLS
abline(lm(LOR ~ as.numeric(Age), weights=1/ASE^2, data=lor.CM.df), col="blue")
age <- seq(25, 60, by = 5)
qmod <- lm(LOR ~ poly(age,2), weights=1/ASE^2, data=lor.CM.df)
lines(fitted(qmod), col = "red", lwd=2)


## 2 x k x 2
lor.Emp <-loddsratio(Employment)
lor.Emp
confint(lor.Emp)

# visualize the log odds ratios
lor.Emp.a <- as.array(lor.Emp)
matplot(lor.Emp.a, type='b', xaxt='n', pch=15:16, cex=1.5,
	ylab='log odds ratio: Employment Status x Length',
	xlab='Employment Length', 
	xlim=c(0.2, 5), 
	main="Employment status data")
abline(h=0, col='gray')
axis(side=1, at=1:5, labels=rownames(lor.Emp.a))
text(0.3, lor.Emp.a[1,], colnames(lor.Emp.a), pos=4, col=1:2)
text(0.5, max(lor.Emp.a[1,])+.1, "Layoff cause")


## R x C case
#data(Hauser79, package="vcdExtra")
hauser.tab <- xtabs(Freq ~ Father+Son, data=Hauser79)
(lor.hauser <- loddsratio(hauser.tab))
confint(lor.hauser)

# odds ratio plot
op <- par(xpd=TRUE)
matplot(as.matrix(lor.hauser), type='b', ylab='log odds ratio', 
	xlab="Son's status comparisons", xaxt='n', xlim=c(1,4.5), ylim=c(-.5,3),
	main="Hauser79 data")
abline(h=0, col='gray')
axis(side=1, at=1:4, labels=colnames(lor.hauser))
text(4, as.matrix(lor.hauser)[4,], rownames(lor.hauser), pos=4, col=1:4)
text(4, 3, "Father's status")
par(op)


## 4 way tables 
data(Punishment, package="vcd")
punish <- xtabs(Freq ~ memory + attitude + age + education, data = Punishment)
mosaic(~ age + education + memory + attitude, data = punish, keep = FALSE,
       gp = gpar(fill = grey.colors(2)), spacing = spacing_highlighting,
       rep = c(attitude = FALSE))

lor.pun <- loddsratio(punish)
lor.pun
confint(lor.pun)
if(require("lmtest")) coeftest(lor.pun)

# visualize the log odds ratios, by education
lor.pun.a <- as.array(lor.pun)
matplot(lor.pun.a, type='b', xaxt='n', pch=15:17, cex=1.5,
	ylab='log odds ratio: Attitude x Memory',
	xlab='Age', xlim=c(0.5, 3), ylim=c(-2, 1), 
	main="Attitudes toward corporal punishment")
abline(h=0, col='gray')
axis(side=1, at=1:3, labels=rownames(lor.pun.a))
text(0.5, lor.pun.a[1,], colnames(lor.pun.a), pos=4, col=1:3)
text(0.6, max(lor.pun.a[1,])+.2, "Education")

# visualize the log odds ratios, by age
matplot(t(lor.pun.a), type='b', xaxt='n', pch=15:17, cex=1.5,
	ylab='log odds ratio: Attitude x Memory',
	xlab='Education', xlim=c(0.5, 3), ylim=c(-2, 1), 
	main="Attitudes toward corporal punishment")
abline(h=0, col='gray')
axis(side=1, at=1:3, labels=colnames(lor.pun.a))
text(0.5, lor.pun.a[,1], rownames(lor.pun.a), pos=4, col=1:3)
text(0.6, max(lor.pun.a[,1])+.2, "Age")

# fit linear model using WLS
lor.pun.df <- as.data.frame(lor.pun)
pun.mod1 <- lm(LOR ~ as.numeric(age) * as.numeric(education), data=lor.pun.df, weights=1/ASE^2)
anova(pun.mod1)

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