# mantelhaen.test: Cochran-Mantel-Haenszel Chi-Squared Test for Count Data

Description Usage Arguments Details Value Note References Examples

### Description

Performs a 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.

### Usage

 1 2 3 mantelhaen.test(x, y = NULL, z = NULL, alternative = c("two.sided", "less", "greater"), correct = TRUE, exact = FALSE, conf.level = 0.95)

### Arguments

 x either a 3-dimensional contingency table in array form where each dimension is at least 2 and the last dimension corresponds to the strata, or a factor object with at least 2 levels. y a factor object with at least 2 levels; ignored if x is an array. z a factor object with at least 2 levels identifying to which stratum the corresponding elements in x and y belong; ignored if x is an array. alternative indicates the alternative hypothesis and must be one of "two.sided", "greater" or "less". You can specify just the initial letter. Only used in the 2 by 2 by K case. correct a logical indicating whether to apply continuity correction when computing the test statistic. Only used in the 2 by 2 by K case. exact a logical indicating whether the Mantel-Haenszel test or the exact conditional test (given the strata margins) should be computed. Only used in the 2 by 2 by K case. conf.level confidence level for the returned confidence interval. Only used in the 2 by 2 by K case.

### Details

If x is an array, each dimension must be at least 2, and the entries should be nonnegative integers. NA's are not allowed. Otherwise, x, y and z must have the same length. Triples containing NA's are removed. All variables must take at least two different values.

### Value

A list with class "htest" containing the following components:

 statistic Only present if no exact test is performed. In the classical case of a 2 by 2 by K table (i.e., of dichotomous underlying variables), the Mantel-Haenszel chi-squared statistic; otherwise, the generalized Cochran-Mantel-Haenszel statistic. parameter the degrees of freedom of the approximate chi-squared distribution of the test statistic (1 in the classical case). Only present if no exact test is performed. p.value the p-value of the test. conf.int a confidence interval for the common odds ratio. Only present in the 2 by 2 by K case. estimate an estimate of the common odds ratio. If an exact test is performed, the conditional Maximum Likelihood Estimate is given; otherwise, the Mantel-Haenszel estimate. Only present in the 2 by 2 by K case. null.value the common odds ratio under the null of independence, 1. Only present in the 2 by 2 by K case. alternative a character string describing the alternative hypothesis. Only present in the 2 by 2 by K case. method a character string indicating the method employed, and whether or not continuity correction was used. data.name a character string giving the names of the data.

### Note

The asymptotic distribution is only valid if there is no three-way interaction. In the classical 2 by 2 by K case, this is equivalent to the conditional odds ratios in each stratum being identical. Currently, no inference on homogeneity of the odds ratios is performed.

### References

Alan Agresti (1990). Categorical data analysis. New York: Wiley. Pages 230–235.

Alan Agresti (2002). Categorical data analysis (second edition). New York: Wiley.

### Examples

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 ## Agresti (1990), pages 231--237, Penicillin and Rabbits ## Investigation of the effectiveness of immediately injected or 1.5 ## hours delayed penicillin in protecting rabbits against a lethal ## injection with beta-hemolytic streptococci. Rabbits <- array(c(0, 0, 6, 5, 3, 0, 3, 6, 6, 2, 0, 4, 5, 6, 1, 0, 2, 5, 0, 0), dim = c(2, 2, 5), dimnames = list( Delay = c("None", "1.5h"), Response = c("Cured", "Died"), Penicillin.Level = c("1/8", "1/4", "1/2", "1", "4"))) Rabbits ## Classical Mantel-Haenszel test mantelhaen.test(Rabbits) ## => p = 0.047, some evidence for higher cure rate of immediate ## injection ## Exact conditional test mantelhaen.test(Rabbits, exact = TRUE) ## => p - 0.040 ## Exact conditional test for one-sided alternative of a higher ## cure rate for immediate injection mantelhaen.test(Rabbits, exact = TRUE, alternative = "greater") ## => p = 0.020 ## UC Berkeley Student Admissions mantelhaen.test(UCBAdmissions) ## No evidence for association between admission and gender ## when adjusted for department. However, apply(UCBAdmissions, 3, function(x) (x[1,1]*x[2,2])/(x[1,2]*x[2,1])) ## This suggests that the assumption of homogeneous (conditional) ## odds ratios may be violated. The traditional approach would be ## using the Woolf test for interaction: woolf <- function(x) { x <- x + 1 / 2 k <- dim(x)[3] or <- apply(x, 3, function(x) (x[1,1]*x[2,2])/(x[1,2]*x[2,1])) w <- apply(x, 3, function(x) 1 / sum(1 / x)) 1 - pchisq(sum(w * (log(or) - weighted.mean(log(or), w)) ^ 2), k - 1) } woolf(UCBAdmissions) ## => p = 0.003, indicating that there is significant heterogeneity. ## (And hence the Mantel-Haenszel test cannot be used.) ## Agresti (2002), p. 287f and p. 297. ## Job Satisfaction example. Satisfaction <- as.table(array(c(1, 2, 0, 0, 3, 3, 1, 2, 11, 17, 8, 4, 2, 3, 5, 2, 1, 0, 0, 0, 1, 3, 0, 1, 2, 5, 7, 9, 1, 1, 3, 6), dim = c(4, 4, 2), dimnames = list(Income = c("<5000", "5000-15000", "15000-25000", ">25000"), "Job Satisfaction" = c("V_D", "L_S", "M_S", "V_S"), Gender = c("Female", "Male")))) ## (Satisfaction categories abbreviated for convenience.) ftable(. ~ Gender + Income, Satisfaction) ## Table 7.8 in Agresti (2002), p. 288. mantelhaen.test(Satisfaction) ## See Table 7.12 in Agresti (2002), p. 297.

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