woolf_test: Woolf Test for Homogeneity of Odds Ratios

In vcdExtra: 'vcd' Extensions and Additions

Woolf Test for Homogeneity of Odds Ratios

Description

Test for homogeneity on 2 \times 2 \times k tables over strata (i.e., whether the log odds ratios are the same in all strata). Generalized to handle tables of any dimensionality beyond 3. For 4-dimensional tables, optionally provides a two-way decomposition of the homogeneity test into row effects, column effects, and residual.

Usage

woolf_test(x, decompose = FALSE)

## S3 method for class 'woolf_test'
print(x, ...)

Arguments

x	An object of class "woolf_test"
decompose	Logical. If TRUE and x is 4-dimensional (a 2 \times 2 \times R \times C table), the test is decomposed into row effects, column effects, and residual (interaction). Defaults to FALSE. Ignored for non-4-dimensional tables.
...	Additional arguments (currently unused)

Details

The Woolf test (Woolf, 1955) tests the hypothesis that the odds ratios \theta_i are equal across all k strata. The test statistic is computed as the weighted sum of squared deviations of the log odds ratios from their weighted mean:

\chi^2_W = \sum_{i=1}^{k} w_i [\log(\theta_i) - \log(\bar{\theta}_w)]^2 = \sum_{i=1}^{k} w_i \log^2(\theta_i / \bar{\theta}_w)

where \theta_i = (n_{11i} n_{22i}) / (n_{12i} n_{21i}) is the odds ratio in stratum i, and \bar{\theta}_w is the weighted average odds ratio (computed as the exponential of the weighted mean of the log odds ratios).

The weights w_i are the inverse variances of the log odds ratios:

w_i = 1 / \text{Var}(\log \theta_i) = 1 / (1/n_{11i} + 1/n_{12i} + 1/n_{21i} + 1/n_{22i})

Under the null hypothesis of homogeneous odds ratios, \chi^2_W follows a chi-squared distribution with k - 1 degrees of freedom.

Decomposition for 4-way tables: For a 2 \times 2 \times R \times C table, the strata form an R \times C two-way layout with odds ratios \theta_{ij} for row i and column j. This suggests that overall Woolf test of homogeneity can be decomposed into three components, conceptually analogous to a two-way ANOVA with one observation per cell:

\chi^2_{\text{W:Total}} = \chi^2_{\text{W:Rows}} + \chi^2_{\text{W:Cols}} + \chi^2_{\text{W:Residual}}

where:

• \chi^2_{\text{W:Rows}} tests whether the odds ratios differ among row levels (pooling over columns), with R - 1 df

• \chi^2_{\text{W:Cols}} tests whether the odds ratios differ among column levels (pooling over rows), with C - 1 df

• \chi^2_{\text{W:Residual}} tests the row \times column interaction (deviation from additivity on the log odds scale), with (R-1)(C-1) df

The row effect test compares the marginal log odds ratios \log \bar{\theta}_{i+} (pooled over columns) to the overall weighted mean. Similarly, the column effect test compares \log \bar{\theta}_{+j} (pooled over rows). The residual tests whether the cell log odds ratios \log \theta_{ij} are additive in the row and column effects.

Note: The two-way ANOVA-like decomposition for 4-dimensional tables appears to be a novel extension introduced in this package. The existing literature on the Woolf test (and related Breslow-Day test) treats strata as unstructured, testing only whether all k odds ratios are equal. This decomposition exploits the factorial structure of R \times C strata to provide more detailed insight into the sources of heterogeneity.

Value

A list of class "woolf_test" (also inheriting from "htest") containing the following components:

statistic	the chi-squared test statistic.
parameter	degrees of freedom of the approximate chi-squared distribution of the test statistic.
p.value	p-value for the test.
method	a character string indicating the type of test performed.
data.name	a character string giving the name of the data.
or_vars	names of the first two dimensions (the 2x2 table variables).
strata_vars	names of the stratifying variables (dimensions 3 and beyond).
observed	the observed log odds ratios.
expected	the expected log odds ratio under the null hypothesis (weighted mean).
decomposed	logical indicating if decomposition was performed.

When decompose = TRUE (only for 4-dimensional tables), additional components:

rows	list with statistic, df, p.value, observed, expected for row effects.
cols	list with statistic, df, p.value, observed, expected for column effects.
residual	list with statistic, df, p.value for residual (interaction).

References

Woolf, B. (1955). On estimating the relation between blood group and disease. Annals of Human Genetics, 19, 251-253.

See Also

mantelhaen.test

Other association tests: CMHtest(), GKgamma(), HLtest(), zero.test()

Examples

# 3-way tables
data(CoalMiners, package = "vcd")
woolf_test(CoalMiners)

data(Heart, package = "vcdExtra")
woolf_test(Heart)

# 4-way table without decomposition
data(Fungicide, package = "vcdExtra")
woolf_test(Fungicide)

# 4-way table with decomposition
woolf_test(Fungicide, decompose = TRUE)

# 4-way table, but need to rearrange dimensions
# How does association between `Preference` and `M_User` vary over strata?
data(Detergent, package = "vcdExtra")
dimnames(Detergent) |> names()
Detergent <- aperm(Detergent, c(3, 2, 1, 4))
woolf_test(Detergent)

vcdExtra documentation built on Jan. 21, 2026, 9:07 a.m.