# Marginal Homogeneity Tests Testing the marginal homogeneity of a repeated measurements factor in a complete block design. ```r ## S3 method for class 'formula' mh_test(formula, data, subset = NULL, ...) ## S3 method for class 'table' mh_test(object, ...) ## S3 method for class 'SymmetryProblem' mh_test(object, ...) ``` ## Arguments - `formula`: a formula of the form `y ~ x | block` where `y` and `x` are factors and `block` is an optional factor (which is generated automatically if omitted). - `data`: an optional data frame containing the variables in the model formula. - `subset`: an optional vector specifying a subset of observations to be used. Defaults to `NULL`. - `object`: an object inheriting from classes `"table"` (with identical `dimnames` components) or `"SymmetryProblem"`. - `...`: further arguments to be passed to `symmetry_test()`. ## Details `mh_test()` provides the McNemar test, the Cochran $Q$ test, the Stuart(-Maxwell) test and the Madansky test of interchangeability. A general description of these methods is given by Agresti (2002). The null hypothesis of marginal homogeneity is tested. The response variable and the measurement conditions are given by `y` and `x`, respectively, and `block` is a factor where each level corresponds to exactly one subject with repeated measurements. This procedure is known as the McNemar test (McNemar, 1947) when both `y` and `x` are binary factors, as the Cochran $Q$ test (Cochran, 1950) when `y` is a binary factor and `x` is a factor with an arbitrary number of levels, as the Stuart(-Maxwell) test (Stuart, 1955; Maxwell, 1970) when `y` is a factor with an arbitrary number of levels and `x` is a binary factor, and as the Madansky test of interchangeability (Madansky, 1963), which implies marginal homogeneity, when both `y` and `x` are factors with an arbitrary number of levels. If `y` and/or `x` are ordered factors, the default scores, `1:nlevels(y)` and `1:nlevels(x)`, respectively, can be altered using the `scores` argument (see `symmetry_test()`); this argument can also be used to coerce nominal factors to class `"ordered"`. If both `y` and `x` are ordered factors, a linear-by-linear association test is computed and the direction of the alternative hypothesis can be specified using the `alternative` argument. This extension was given by Birch (1965) who also discussed the situation when either the response or the measurement condition is an ordered factor; see also White, Landis and Cooper (1982). The conditional null distribution of the test statistic is used to obtain $p$-values and an asymptotic approximation of the exact distribution is used by default (`distribution = "asymptotic"`). Alternatively, the distribution can be approximated via Monte Carlo resampling or computed exactly for univariate two-sample problems by setting `distribution` to `"approximate"` or `"exact"`, respectively. See `asymptotic()`, `approximate()` and `exact()` for details. ## Returns An object inheriting from class `"IndependenceTest"`. ## Note This function is currently computationally inefficient for data with a large number of pairs or sets. ## References Agresti, A. (2002). **Categorical Data Analysis**, Second Edition. Hoboken, New Jersey: John Wiley & Sons. Birch, M. W. (1965). The detection of partial association, II: The general case. **Journal of the Royal Statistical Society** B 27 (1), 111--124. tools:::Rd_expr_doi("10.1111/j.2517-6161.1965.tb00593.x") Cochran, W. G. (1950). The comparison of percentages in matched samples. **Biometrika** 37 (3/4), 256--266. tools:::Rd_expr_doi("10.1093/biomet/37.3-4.256") Madansky, A. (1963). Tests of homogeneity for correlated samples. **Journal of the American Statistical Association** 58 (301), 97--119. tools:::Rd_expr_doi("10.1080/01621459.1963.10500835") Maxwell, A. E. (1970). Comparing the classification of subjects by two independent judges. **British Journal of Psychiatry** 116 (535), 651--655. tools:::Rd_expr_doi("10.1192/bjp.116.535.651") McNemar, Q. (1947). Note on the sampling error of the difference between correlated proportions or percentages. **Psychometrika** 12 (2), 153--157. tools:::Rd_expr_doi("10.1007/BF02295996") Stuart, A. (1955). A test for homogeneity of the marginal distributions in a two-way classification. **Biometrika** 42 (3/4), 412--416. tools:::Rd_expr_doi("10.1093/biomet/42.3-4.412") White, A. A., Landis, J. R. and Cooper, M. M. (1982). A note on the equivalence of several marginal homogeneity test criteria for categorical data. **International Statistical Review** 50 (1), 27--34. tools:::Rd_expr_doi("10.2307/1402457") ## Examples ```r ## Performance of prime minister ## Agresti (2002, p. 409) performance <- matrix( c(794, 150, 86, 570), nrow = 2, byrow = TRUE, dimnames = list( "First" = c("Approve", "Disprove"), "Second" = c("Approve", "Disprove") ) ) performance <- as.table(performance) diag(performance) <- 0 # speed-up: only off-diagonal elements contribute ## Asymptotic McNemar Test mh_test(performance) ## Exact McNemar Test mh_test(performance, distribution = "exact") ## Effectiveness of different media for the growth of diphtheria ## Cochran (1950, Tab. 2) cases <- c(4, 2, 3, 1, 59) n <- sum(cases) cochran <- data.frame( diphtheria = factor( unlist(rep(list(c(1, 1, 1, 1), c(1, 1, 0, 1), c(0, 1, 1, 1), c(0, 1, 0, 1), c(0, 0, 0, 0)), cases)) ), media = factor(rep(LETTERS[1:4], n)), case = factor(rep(seq_len(n), each = 4)) ) ## Asymptotic Cochran Q test (Cochran, 1950, p. 260) mh_test(diphtheria ~ media | case, data = cochran) # Q = 8.05 ## Approximative Cochran Q test mt <- mh_test(diphtheria ~ media | case, data = cochran, distribution = approximate(nresample = 10000)) pvalue(mt) # standard p-value midpvalue(mt) # mid-p-value pvalue_interval(mt) # p-value interval size(mt, alpha = 0.05) # test size at alpha = 0.05 using the p-value ## Opinions on Pre- and Extramarital Sex ## Agresti (2002, p. 421) opinions <- c("Always wrong", "Almost always wrong", "Wrong only sometimes", "Not wrong at all") PreExSex <- matrix( c(144, 33, 84, 126, 2, 4, 14, 29, 0, 2, 6, 25, 0, 0, 1, 5), nrow = 4, dimnames = list( "Premarital Sex" = opinions, "Extramarital Sex" = opinions ) ) PreExSex <- as.table(PreExSex) ## Asymptotic Stuart test mh_test(PreExSex) ## Asymptotic Stuart-Birch test ## Note: response as ordinal mh_test(PreExSex, scores = list(response = 1:length(opinions))) ## Vote intention ## Madansky (1963, pp. 107-108) vote <- array( c(120, 1, 8, 2, 2, 1, 2, 1, 7, 6, 2, 1, 1, 103, 5, 1, 4, 8, 20, 3, 31, 1, 6, 30, 2, 1, 81), dim = c(3, 3, 3), dimnames = list( "July" = c("Republican", "Democratic", "Uncertain"), "August" = c("Republican", "Democratic", "Uncertain"), "June" = c("Republican", "Democratic", "Uncertain") ) ) vote <- as.table(vote) ## Asymptotic Madansky test (Q = 70.77) mh_test(vote) ## Cross-over study ## http://www.nesug.org/proceedings/nesug00/st/st9005.pdf dysmenorrhea <- array( c(6, 2, 1, 3, 1, 0, 1, 2, 1, 4, 3, 0, 13, 3, 0, 8, 1, 1, 5, 2, 2, 10, 1, 0, 14, 2, 0), dim = c(3, 3, 3), dimnames = list( "Placebo" = c("None", "Moderate", "Complete"), "Low dose" = c("None", "Moderate", "Complete"), "High dose" = c("None", "Moderate", "Complete") ) ) dysmenorrhea <- as.table(dysmenorrhea) ## Asymptotic Madansky-Birch test (Q = 53.76) ## Note: response as ordinal mh_test(dysmenorrhea, scores = list(response = 1:3)) ## Asymptotic Madansky-Birch test (Q = 47.29) ## Note: response and measurement conditions as ordinal mh_test(dysmenorrhea, scores = list(response = 1:3, conditions = 1:3)) ```

MarginalHomogeneityTests function