Woolf Test for Homogeneity of Odds Ratios

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)

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).

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.

References

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

Examples

# 3-way tables
data(CoalMiners, package = "vcd")
woolf_test(CoalMiners)
#> 
#> Woolf-test on Homogeneity of Odds Ratios (no 3-way association) 
#> 
#> Data:          CoalMiners 
#> OR variables:  Breathlessness, Wheeze 
#> Strata:        Age 
#> 
#> X-squared = 26.2034, df = 8, p-value = 0.0009694

data(Heart, package = "vcdExtra")
woolf_test(Heart)
#> 
#> Woolf-test on Homogeneity of Odds Ratios (no 3-way association) 
#> 
#> Data:          Heart 
#> OR variables:  Disease, Gender 
#> Strata:        Occup 
#> 
#> X-squared = 81.0807, df = 2, p-value = 0

# 4-way table without decomposition
data(Fungicide, package = "vcdExtra")
woolf_test(Fungicide)
#> 
#> Woolf-test on Homogeneity of Odds Ratios (no 4-way association) 
#> 
#> Data:          Fungicide 
#> OR variables:  group, outcome 
#> Strata:        sex, strain 
#> 
#> X-squared = 0.8548, df = 3, p-value = 0.8363

# 4-way table with decomposition
woolf_test(Fungicide, decompose = TRUE)
#> 
#> Woolf-test on Homogeneity of Odds Ratios (no 4-way association) 
#> 
#> Data:          Fungicide 
#> OR variables:  group, outcome 
#> Strata:        sex, strain 
#> 
#> Overall homogeneity test:
#>   X-squared = 0.8548, df = 3, p-value = 0.8363
#> 
#> Decomposition:
#>   Rows (sex):    X-squared = 0.0086, df = 1, p-value = 0.9261
#>   Cols (strain): X-squared = 0.8257, df = 1, p-value = 0.3635
#>   Residual:      X-squared = 0.0205, df = 1, p-value = 0.8861
#> 
#> Note: Overall = Rows + Columns + Residual

# 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()
#> [1] "Temperature"    "M_User"         "Preference"     "Water_softness"
Detergent <- aperm(Detergent, c(3, 2, 1, 4))
woolf_test(Detergent)
#> 
#> Woolf-test on Homogeneity of Odds Ratios (no 4-way association) 
#> 
#> Data:          Detergent 
#> OR variables:  Preference, M_User 
#> Strata:        Temperature, Water_softness 
#> 
#> X-squared = 8.0132, df = 5, p-value = 0.1555