CMHtest: Generalized Cochran-Mantel-Haenszel Tests

Description Usage Arguments Details Value Author(s) References See Also Examples

View source: R/CMHtest.R

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 statifying 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 columm categories.

Usage

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CMHtest(x, ...)

## S3 method for class 'formula'
CMHtest(formula, data = NULL, subset = NULL, na.action = NULL, ...)

## Default S3 method:
CMHtest(x, strata = NULL, 
   rscores = 1:R, cscores = 1:C, 
   types = c("cor", "rmeans", "cmeans", "general"), 
   overall=FALSE, details=overall, ...)

## S3 method for class 'CMHtest'
print(x, digits = max(getOption("digits") - 2, 3), ...)

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.

Details

The standard χ^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 χ^2 test. When there is more than one stratum, the overall general CMH statistic gives a stratum-adjusted Pearson χ^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.

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.

Author(s)

Michael Friendly

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 χ^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.

See Also

cmh_test provides the CMH test of general association; lbl_test provides the CMH correlation test of linear by linear association.

mantelhaen.test provides the overall general Cochran-Mantel-Haenszel chi-squared test of the null that two nominal variables are conditionally independent in each stratum, assuming that there is no three-way interaction

Examples

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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)

Example output

Loading required package: vcd
Loading required package: MASS
Loading required package: grid
Loading required package: colorspace
Loading required package: gnm
Cochran-Mantel-Haenszel Statistics for income by satisfaction 

                 AltHypothesis  Chisq Df     Prob
cor        Nonzero correlation 2.9830  1 0.084144
rmeans  Row mean scores differ 4.4774  3 0.214318
cmeans  Col mean scores differ 3.1036  3 0.375931
general    General association 5.9034  9 0.749549

Cochran-Mantel-Haenszel Statistics for income by satisfaction 

                 AltHypothesis  Chisq Df     Prob
cor        Nonzero correlation 2.9726  1 0.084685
rmeans  Row mean scores differ 4.1200  3 0.248799
cmeans  Col mean scores differ 3.3108  3 0.346144
general    General association 5.9034  9 0.749549

Cochran-Mantel-Haenszel Statistics for income by satisfaction 

                 AltHypothesis  Chisq Df     Prob
cor        Nonzero correlation 2.9830  1 0.084144
rmeans  Row mean scores differ 4.4774  3 0.214318
cmeans  Col mean scores differ 3.1036  3 0.375931
general    General association 5.9034  9 0.749549

$`Patients:Winnipeg`
Cochran-Mantel-Haenszel Statistics for New Orleans Neurologist by Winnipeg Neurologist 
	in stratum Patients:Winnipeg 

                 AltHypothesis  Chisq Df       Prob
cor        Nonzero correlation 51.424  1 7.4426e-13
rmeans  Row mean scores differ 55.393  3 5.6601e-12
cmeans  Col mean scores differ 53.631  3 1.3450e-11
general    General association 64.318  9 1.9580e-10


$`Patients:New Orleans`
Cochran-Mantel-Haenszel Statistics for New Orleans Neurologist by Winnipeg Neurologist 
	in stratum Patients:New Orleans 

                 AltHypothesis  Chisq Df       Prob
cor        Nonzero correlation 28.863  1 7.7667e-08
rmeans  Row mean scores differ 30.594  3 1.0347e-06
cmeans  Col mean scores differ 29.054  3 2.1818e-06
general    General association 43.428  9 1.7990e-06


$`Patients:Winnipeg`
Cochran-Mantel-Haenszel Statistics for New Orleans Neurologist by Winnipeg Neurologist 
	in stratum Patients:Winnipeg 

                 AltHypothesis  Chisq Df       Prob
cor        Nonzero correlation 51.424  1 7.4426e-13
rmeans  Row mean scores differ 55.393  3 5.6601e-12
cmeans  Col mean scores differ 53.631  3 1.3450e-11
general    General association 64.318  9 1.9580e-10


$`Patients:New Orleans`
Cochran-Mantel-Haenszel Statistics for New Orleans Neurologist by Winnipeg Neurologist 
	in stratum Patients:New Orleans 

                 AltHypothesis  Chisq Df       Prob
cor        Nonzero correlation 28.863  1 7.7667e-08
rmeans  Row mean scores differ 30.594  3 1.0347e-06
cmeans  Col mean scores differ 29.054  3 2.1818e-06
general    General association 43.428  9 1.7990e-06


$ALL
Cochran-Mantel-Haenszel Statistics for New Orleans Neurologist by Winnipeg Neurologist 
	Overall tests, controlling for all strata 

                 AltHypothesis  Chisq Df       Prob
cor        Nonzero correlation 80.086  1 3.5848e-19
rmeans  Row mean scores differ 83.923  3 4.4189e-18
cmeans  Col mean scores differ 81.116  3 1.7685e-17
general    General association 101.21  9 8.9683e-18



	Cochran-Mantel-Haenszel test

data:  MSPatients
Cochran-Mantel-Haenszel M^2 = 101.21, df = 9, p-value < 2.2e-16

$`Patients:Winnipeg`
Cochran-Mantel-Haenszel Statistics for New Orleans Neurologist by Winnipeg Neurologist 
	in stratum Patients:Winnipeg 

                 AltHypothesis  Chisq Df       Prob
cor        Nonzero correlation 51.424  1 7.4426e-13
rmeans  Row mean scores differ 55.393  3 5.6601e-12
cmeans  Col mean scores differ 53.631  3 1.3450e-11
general    General association 64.318  9 1.9580e-10


$`Patients:New Orleans`
Cochran-Mantel-Haenszel Statistics for New Orleans Neurologist by Winnipeg Neurologist 
	in stratum Patients:New Orleans 

                 AltHypothesis  Chisq Df       Prob
cor        Nonzero correlation 28.863  1 7.7667e-08
rmeans  Row mean scores differ 30.594  3 1.0347e-06
cmeans  Col mean scores differ 29.054  3 2.1818e-06
general    General association 43.428  9 1.7990e-06


$ALL
Cochran-Mantel-Haenszel Statistics for New Orleans Neurologist by Winnipeg Neurologist 
	Overall tests, controlling for all strata 

                 AltHypothesis  Chisq Df       Prob
cor        Nonzero correlation 80.086  1 3.5848e-19
rmeans  Row mean scores differ 83.923  3 4.4189e-18
cmeans  Col mean scores differ 81.116  3 1.7685e-17
general    General association 101.21  9 8.9683e-18


Cochran-Mantel-Haenszel Statistics for ses by mental 

                 AltHypothesis  Chisq Df       Prob
cor        Nonzero correlation 37.156  1 1.0907e-09
rmeans  Row mean scores differ 40.297  5 1.3012e-07
cmeans  Col mean scores differ 40.666  3 7.6971e-09
general    General association 45.958 15 5.4003e-05

Cochran-Mantel-Haenszel Statistics for ses by mental 

                 AltHypothesis  Chisq Df       Prob
cor        Nonzero correlation 37.156  1 1.0907e-09
rmeans  Row mean scores differ 40.297  5 1.3012e-07
cmeans  Col mean scores differ 40.666  3 7.6971e-09
general    General association 45.958 15 5.4003e-05

$`gender:male`
Cochran-Mantel-Haenszel Statistics for right by left 
	in stratum gender:male 

                 AltHypothesis  Chisq Df Prob
cor        Nonzero correlation 1554.6  1    0
rmeans  Row mean scores differ 1556.3  3    0
cmeans  Col mean scores differ 1556.6  3    0
general    General association 3303.3  9    0


$`gender:female`
Cochran-Mantel-Haenszel Statistics for right by left 
	in stratum gender:female 

                 AltHypothesis  Chisq Df Prob
cor        Nonzero correlation 3691.3  1    0
rmeans  Row mean scores differ 3709.4  3    0
cmeans  Col mean scores differ 3724.0  3    0
general    General association 8095.8  9    0


$`education:elementary|age:15-24`
Cochran-Mantel-Haenszel Statistics for attitude by memory 
	in stratum education:elementary|age:15-24 

                 AltHypothesis  Chisq Df     Prob
cor        Nonzero correlation 3.5652  1 0.059002
rmeans  Row mean scores differ 3.5652  1 0.059002
cmeans  Col mean scores differ 3.5652  1 0.059002
general    General association 3.5652  1 0.059002


$`education:secondary|age:15-24`
Cochran-Mantel-Haenszel Statistics for attitude by memory 
	in stratum education:secondary|age:15-24 

                 AltHypothesis    Chisq Df   Prob
cor        Nonzero correlation 0.077731  1 0.7804
rmeans  Row mean scores differ 0.077731  1 0.7804
cmeans  Col mean scores differ 0.077731  1 0.7804
general    General association 0.077731  1 0.7804


$`education:high|age:15-24`
Cochran-Mantel-Haenszel Statistics for attitude by memory 
	in stratum education:high|age:15-24 

                 AltHypothesis    Chisq Df    Prob
cor        Nonzero correlation 0.089524  1 0.76478
rmeans  Row mean scores differ 0.089524  1 0.76478
cmeans  Col mean scores differ 0.089524  1 0.76478
general    General association 0.089524  1 0.76478


$`education:elementary|age:25-39`
Cochran-Mantel-Haenszel Statistics for attitude by memory 
	in stratum education:elementary|age:25-39 

                 AltHypothesis  Chisq Df     Prob
cor        Nonzero correlation 8.5433  1 0.003468
rmeans  Row mean scores differ 8.5433  1 0.003468
cmeans  Col mean scores differ 8.5433  1 0.003468
general    General association 8.5433  1 0.003468


$`education:secondary|age:25-39`
Cochran-Mantel-Haenszel Statistics for attitude by memory 
	in stratum education:secondary|age:25-39 

                 AltHypothesis   Chisq Df    Prob
cor        Nonzero correlation 0.92897  1 0.33513
rmeans  Row mean scores differ 0.92897  1 0.33513
cmeans  Col mean scores differ 0.92897  1 0.33513
general    General association 0.92897  1 0.33513


$`education:high|age:25-39`
Cochran-Mantel-Haenszel Statistics for attitude by memory 
	in stratum education:high|age:25-39 

                 AltHypothesis Chisq Df    Prob
cor        Nonzero correlation 0.472  1 0.49207
rmeans  Row mean scores differ 0.472  1 0.49207
cmeans  Col mean scores differ 0.472  1 0.49207
general    General association 0.472  1 0.49207


$`education:elementary|age:40-`
Cochran-Mantel-Haenszel Statistics for attitude by memory 
	in stratum education:elementary|age:40- 

                 AltHypothesis  Chisq Df       Prob
cor        Nonzero correlation 11.606  1 0.00065737
rmeans  Row mean scores differ 11.606  1 0.00065737
cmeans  Col mean scores differ 11.606  1 0.00065737
general    General association 11.606  1 0.00065737


$`education:secondary|age:40-`
Cochran-Mantel-Haenszel Statistics for attitude by memory 
	in stratum education:secondary|age:40- 

                 AltHypothesis  Chisq Df    Prob
cor        Nonzero correlation 6.0457  1 0.01394
rmeans  Row mean scores differ 6.0457  1 0.01394
cmeans  Col mean scores differ 6.0457  1 0.01394
general    General association 6.0457  1 0.01394


$`education:high|age:40-`
Cochran-Mantel-Haenszel Statistics for attitude by memory 
	in stratum education:high|age:40- 

                 AltHypothesis  Chisq Df     Prob
cor        Nonzero correlation 3.0436  1 0.081055
rmeans  Row mean scores differ 3.0436  1 0.081055
cmeans  Col mean scores differ 3.0436  1 0.081055
general    General association 3.0436  1 0.081055


Cochran-Mantel-Haenszel Statistics for party by neighborhood 

              AltHypothesis  Chisq Df       Prob
general General association 273.81  6 3.3158e-56


	Pearson's Chi-squared test

data:  parties
X-squared = 273.92, df = 6, p-value < 2.2e-16

vcdExtra documentation built on Jan. 25, 2021, 5:08 p.m.