Generalized Cochran-Mantel-Haenszel Tests

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.

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.
...: 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.

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.

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).

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.

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

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.

Author

Michael Friendly

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
#>