R: Cochran-Mantel-Haenszel Chi-Squared Test for Count Data

mantelhaen.test {stats} | R Documentation

## Cochran-Mantel-Haenszel Chi-Squared Test for Count Data

### 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

    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.

See also the example below.

### 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

    ## 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.

[Package stats version 3.5.1 Index]