CMHtest: Generalized Cochran-Mantel-Haenszel Tests

Description

Provides generalized Cochran-Mantel-Haenszel tests of association of two possibly ordered factors, optionally stratified other factor(s). With strata, CMHtest calculates these tests for each level of the stratifying variables and also provides overall tests controlling for the strata.

For ordinal factors, more powerful tests than the test for general association (independence) are obtained by assigning scores to the row and column categories.

Usage

CMHtest(x, ...)
# S3 method for formula
CMHtest(formula, data = NULL, subset = NULL, na.action = NULL, ...)
# S3 method for default
CMHtest(x, 
   strata = NULL, 
   rscores = 1:R, 
   cscores = 1:C, 
   types = c("cor", "rmeans", "cmeans", "general"), 
   overall=FALSE, 
   details=overall, ...)
# S3 method for CMHtest
print(x, digits = max(getOption("digits") - 2, 3), ...)

Value

An object of class "CMHtest" , a list with the following 4 components:

table: A matrix containing the test statistics, with columns Chisq, Df and Prob
names: The names of the table row and column variables
rscore: Row scores
cscore: Column scores

If details==TRUE, additional components are included.

If there are strata, the result is a list of "CMHtest" objects. If overall=TRUE another component, labeled ALL

is appended to the list.

Arguments

x: A 2+ way contingency table in array form, or a class "table" object with optional category labels specified in the dimnames(x) attribute.
formula: a formula specifying the variables used to create a contingency table from data. This should be a one-sided formula when data is in array form, and a two-sided formula with a response Freq if data is a data frame with a cell frequency variable. For convenience, conditioning formulas can be specified indicating strata.
data: either a data frame, or an object of class "table" or "ftable".
subset: an optional vector specifying a subset of observations to be used.
na.action: a function which indicates what should happen when the data contain NAs. Ignored if data is a contingency table
strata: For a 3- or higher-way table, the names or numbers of the factors to be treated as strata. By default, the first 2 factors are treated as the main table variables, and all others considered stratifying factors.
rscores: Row scores. Either a set of numbers (typically integers, 1:R) or the string "midrank" for standardized midrank scores, or NULL to exclude tests that depend on row scores.
cscores: Column scores. Same as for row scores.
types: Types of CMH tests to compute: Any one or more of c("cor", "cmeans", "rmeans", "general"), or "ALL" for all of these.
overall: logical. Whether to calculate overall tests, controlling for the stratifying factors.
details: logical. Whether to include computational details in the result
...: Other arguments passed to default method.
digits: Digits to print.

Author

Michael Friendly

Details

The standard \(\chi^2\) tests for association in a two-way table treat both table factors as nominal (unordered) categories. When one or both factors of a two-way table are quantitative or ordinal, more powerful tests of association may be obtained by taking ordinality into account using row and or column scores to test for linear trends or differences in row or column means.

The CMH analysis for a two-way table produces generalized Cochran-Mantel-Haenszel statistics (Landis etal., 1978).

These include the CMH correlation statistic ("cor"), treating both factors as ordered. For a given statum, with equally spaced row and column scores, this CMH statistic reduces to \((n-1) r^2\), where \(r\) is the Pearson correlation between X and Y. With "midrank" scores, this CMH statistic is analogous to \((n-1) r_S^2\), using the Spearman rank correlation.

The ANOVA (row mean scores and column mean scores) statistics, treat the columns and rows respectively as ordinal, and are sensitive to mean shifts over columns or rows. These are transforms of the \(F\) statistics from one-way ANOVAs with equally spaced scores and to Kruskal-Wallis tests with "midrank" scores.

The CMH general association statistic treat both factors as unordered, and give a test closely related to the Pearson \(\chi^2\) test. When there is more than one stratum, the overall general CMH statistic gives a stratum-adjusted Pearson \(\chi^2\), equivalent to what is calculated by mantelhaen.test.

For a 3+ way table, one table of CMH tests is produced for each combination of the factors identified as strata. If overall=TRUE, an additional table is calculated for the same two primary variables, controlling for (pooling over) the strata variables.

These overall tests implicitly assume no interactions between the primary variables and the strata and they will have low power in the presence of interactions.

References

Stokes, M. E. & Davis, C. S. & Koch, G., (2000). Categorical Data Analysis using the SAS System, 2nd Ed., Cary, NC: SAS Institute, pp 74-75, 92-101, 124-129. Details of the computation are given at: http://support.sas.com/documentation/cdl/en/statug/63033/HTML/default/viewer.htm#statug_freq_a0000000648.htm

Cochran, W. G. (1954), Some Methods for Strengthening the Common \(\chi^2\) Tests, Biometrics, 10, 417-451.

Landis, R. J., Heyman, E. R., and Koch, G. G. (1978). Average Partial Association in Three-way Contingency Tables: A Review and Discussion of Alternative Tests, International Statistical Review, 46, 237-254.

Mantel, N. (1963), Chi-square Tests with One Degree of Freedom: Extensions of the Mantel-Haenszel Procedure," Journal of the American Statistical Association, 58, 690-700.

Examples

Run this code

data(JobSat, package="vcdExtra")
CMHtest(JobSat)
CMHtest(JobSat, rscores="midrank", cscores="midrank")

# formula interface
CMHtest(~ ., data=JobSat)

# A 3-way table (both factors ordinal)
data(MSPatients, package="vcd")
CMHtest(MSPatients)


# also calculate overall tests, controlling for Patient
CMHtest(MSPatients, overall = TRUE)
# compare with mantelhaen.test
mantelhaen.test(MSPatients)

# formula interface
CMHtest(~ ., data = MSPatients, overall = TRUE)

# using a frequency data.frame
CMHtest(xtabs(Freq~ses + mental, data = Mental))
# or, more simply
CMHtest(Freq~ses + mental, data = Mental)

# conditioning formulae
CMHtest(Freq~right + left | gender, data = VisualAcuity)
 
CMHtest(Freq ~ attitude + memory | education + age, data = Punishment)


# Stokes etal, Table 5.1, p 92: two unordered factors
parties <- matrix(
	c(221, 160, 360, 140,
	  200, 291, 160, 311,
	  208, 106, 316, 97), 
	nrow=3, ncol=4, 
	byrow=TRUE)
dimnames(parties) <- list(party=c("Dem", "Indep", "Rep"),
             neighborhood=c("Bayside", "Highland", "Longview", "Sheffield"))
CMHtest(parties, rscores=NULL, cscores=NULL)

# compare with Pearson chisquare
chisq.test(parties)

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