![[Experimental]](figures/lifecycle-experimental.svg)
Checks the Mantel-Fleiss criterion for stratified 2 x 2 contingency tables.
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.
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_.
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
#> , , strata = A.x
#>
#> rsp
#> grp FALSE TRUE
#> Active 2 3
#> Control 1 2
#>
#> , , strata = B.x
#>
#> rsp
#> grp FALSE TRUE
#> Active 3 5
#> Control 1 2
#>
#> , , strata = A.y
#>
#> rsp
#> grp FALSE TRUE
#> Active 2 2
#> Control 1 4
#>
#> , , strata = B.y
#>
#> rsp
#> grp FALSE TRUE
#> Active 3 3
#> Control 2 4
#>
is_mf_satisfied <- mantel_fleiss_crit(tbl)
is_mf_satisfied
#> [1] TRUE
mantel_fleiss_crit(tbl, include_value = TRUE)
#> [1] TRUE
#> attr(,"value")
#> [1] 6.382576
# 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
}
#> [1] "CMH"
#> Active Control
#> 0.5495986 0.6985984
if (is_mf_satisfied) {
print("CMH")
prop_cmh(tbl)
} else {
print("Exact")
prop_fisher(table(grp, rsp))
}
#> [1] "CMH"
#> [1] 0.3730088
#> attr(,"z_stat")
#> [1] -0.8908515