tern (version 0.9.12)

mantel_fleiss_crit: Check the Mantel-Fleiss Criterion

Description

Checks the Mantel-Fleiss criterion for stratified 2 x 2 contingency tables.

Usage

mantel_fleiss_crit(tbl, include_value = FALSE, threshold = 5L)

Value

A logical value indicating whether the Mantel-Fleiss criterion is satisfied. If include_value = TRUE, the result also contains a value attribute with the calculated Mantel-Fleiss statistic. If there are no non-empty strata, the result is NA and the value attribute is NA_real_.

Arguments

tbl

(array)
A three-dimensional contingency table containing the counts for each combination of group, response, and stratum. The first two dimensions must correspond to the two variables defining the 2 x 2 contingency table (group and response), in either order. The third dimension must correspond to the strata. The first two dimensions must each have exactly two levels. All cell values must be finite, non-missing integer counts.

include_value

(logical(1))
Whether to include the calculated Mantel-Fleiss statistic as an attribute of the result.

threshold

(numeric(1))
The minimum Mantel-Fleiss statistic required for the criterion to be considered satisfied.

Details

The Mantel-Fleiss statistic is calculated as

$$ MF = \min\left( [\sum_h m_{11h} - \sum_h {(n_{11h})}_L],\ [\sum_h {(n_{11h})}_U - \sum_h m_{11h}] \right), $$

where \(h\) indexes the non-empty strata. For each stratum \(h\), the expected frequency of cell \((1, 1)\) in table \(h\), under the hypothesis of no association between group and response, is

$$ m_{11h} = \frac{n_{1.h} n_{.1h}}{n_h}. $$

The lower and upper bounds for \(n_{11h}\), given the marginal totals, are:

$$ {(n_{11h})}_L = \max(0, n_{1.h} - n_{.2h}), $$ $$ {(n_{11h})}_U = \min(n_{.1h}, n_{1.h}). $$

The Mantel-Fleiss criterion is satisfied when \(MF \ge\) threshold. By default, threshold = 5, corresponding to the criterion described by Mantel and Fleiss (1980).

Strata with all cell counts equal to zero are excluded from the calculation. If all strata contain zero observations, there are no non-empty strata over which to calculate the Mantel-Fleiss statistic, and the statistic is therefore undefined. In this case, the function returns NA.

References

Mantel, N., and Fleiss, J. L. (1980). Minimum Expected Cell Size Requirements for the Mantel-Haenszel One-Degree-of-Freedom Chi-Square Test and a Related Rapid Procedure. American Journal of Epidemiology, 112(1), 129--134.

Examples

set.seed(123)
n <- 40

grp <- factor(sample(c("Active", "Control"), n, replace = TRUE))
rsp <- sample(c(TRUE, FALSE), n, replace = TRUE)
strata1 <- factor(sample(c("A", "B"), n, replace = TRUE))
strata2 <- factor(sample(c("x", "y"), n, replace = TRUE))
strata <- interaction(strata1, strata2)

tbl <- table(grp, rsp, strata)
tbl

is_mf_satisfied <- mantel_fleiss_crit(tbl)
is_mf_satisfied
mantel_fleiss_crit(tbl, include_value = TRUE)

# Examples of use.

if (is_mf_satisfied) {
  print("CMH")
  prop_diff_cmh(rsp, grp, strata)$prop
} else {
  print("Exact")
  prop_diff_uncond_exact(rsp, grp)$prop
}

if (is_mf_satisfied) {
  print("CMH")
  prop_cmh(tbl)
} else {
  print("Exact")
  prop_fisher(table(grp, rsp))
}