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. See Details for descriptions of what these entail and enhancements to this function.

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),
  layout = c("table", "2x2"),
  stars = FALSE,
  scale = FALSE,
  ...
)

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.
...: Other arguments passed to default method.
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
digits: Digits to print.
layout: For print.CMHtest(), one of "table" (default) for the traditional flat 4-row printout, or "2x2" to reorganize the four CMH statistics into a 2x2 table crossing how the row and column variables are each treated (general/nominal vs. ordered/scored). layout = "2x2" requires all four of cor, rmeans, cmeans, general to be present in x; if types or rscores/cscores = NULL excluded any of them, this falls back to layout = "table" with a warning. See Details.
stars: For print.CMHtest() with layout = "2x2", logical: annotate each cell with significance stars ('***'/'**'/'*' for p < .001/.01/.05) based on its own (Chisq, Df). Default FALSE, matching layout = "table"'s plain display.
scale: For print.CMHtest() with layout = "2x2", logical: show Chisq/Df instead of Chisq (Df) in each cell -- normalizes for the very different degrees of freedom across cells. Default FALSE.

Author

Michael Friendly

Details

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

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):

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.

Strata

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.

Note that strata combinations with insufficient data (less than 2 observations) are automatically omitted from the analysis.

2x2 layout

print.CMHtest(layout = "2x2") is an attempt to provide an alternative view of these four statistics that may be more useful/understandable than the traditional flat 4-row table: it reorganizes them into a 2x2 table crossing how the row and column variables are each treated (general/nominal vs. ordered/scored).

The corner cell of this display (general - rmeans - cmeans + cor, a "diff of diffs") is algebraically consistent but its distribution as chi-square(df) is experimental/unverified -- these are four different quadratic-form statistics, not a nested sequence of LR tests, so the naive inclusion-exclusion combination is not guaranteed to be non-negative in general (see dev/CMH-2x2.md for a worked counter-example and a loglinear LRstats()-based alternative).

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)

# 2x2 layout, reorganizing the four CMH statistics by how the row/column
# variables are treated (general/nominal vs. ordered/scored)
cmh_mental <- CMHtest(Freq ~ ses + mental, data = Mental)
print(cmh_mental, layout = "2x2")
print(cmh_mental, layout = "2x2", stars = TRUE)
print(cmh_mental, layout = "2x2", scale = TRUE)


# 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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