Generalized Cochran-Mantel-Haenszel Tests
CMHtest.Rd
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 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 whendata
is in array form, and a two-sided formula with a responseFreq
ifdata
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
NA
s. Ignored ifdata
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, orNULL
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 \(\chi^2\) 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.
Value
An object of class "CMHtest"
, a list with the following 4 components:
- table
A matrix containing the test statistics, with columns
Chisq
,Df
andProb
- 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.
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.
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
data(JobSat, package="vcdExtra")
CMHtest(JobSat)
#> 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
#>
CMHtest(JobSat, rscores="midrank", cscores="midrank")
#> 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
#>
# formula interface
CMHtest(~ ., data=JobSat)
#> 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
#>
# A 3-way table (both factors ordinal)
data(MSPatients, package="vcd")
CMHtest(MSPatients)
#> $`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
#>
#>
# also calculate overall tests, controlling for Patient
CMHtest(MSPatients, overall = TRUE)
#> $`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
#>
#>
# compare with mantelhaen.test
mantelhaen.test(MSPatients)
#>
#> Cochran-Mantel-Haenszel test
#>
#> data: MSPatients
#> Cochran-Mantel-Haenszel M^2 = 101.21, df = 9, p-value < 2.2e-16
#>
# formula interface
CMHtest(~ ., data = MSPatients, overall = TRUE)
#> $`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
#>
#>
# using a frequency data.frame
CMHtest(xtabs(Freq~ses + mental, data = Mental))
#> 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
#>
# or, more simply
CMHtest(Freq~ses + mental, data = Mental)
#> 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
#>
# conditioning formulae
CMHtest(Freq~right + left | gender, data = VisualAcuity)
#> $`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
#>
#>
CMHtest(Freq ~ attitude + memory | education + age, data = Punishment)
#> $`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
#>
#>
# 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)
#> Cochran-Mantel-Haenszel Statistics for party by neighborhood
#>
#> AltHypothesis Chisq Df Prob
#> general General association 273.81 6 3.3158e-56
#>
# compare with Pearson chisquare
chisq.test(parties)
#>
#> Pearson's Chi-squared test
#>
#> data: parties
#> X-squared = 273.92, df = 6, p-value < 2.2e-16
#>