Source: Holland, P. W. and Thayer, D. T. (1988), Differential item
performance and the Mantel-Haenszel procedure, in Wainer and Braun
(eds), Test Validity, 129-145, applying Mantel and Haenszel (1959),
JNCI 22, 719-748. Neither chapter was obtainable here; the procedure
is quoted in its standard published form, which is exactly the
stratified 2x2 statistic base R implements in
stats::mantelhaen.test – used as the independent anchor.
Arguments
- y
Response to the studied item, coded 0/1.
- group
Group membership; exactly two distinct values, the first encountered being the reference group.
- item
Optional matching variable (normally total score); examinees are stratified on its distinct values. A single stratum is used when omitted.
- correct
Apply the 0.5 continuity correction. Default TRUE.
- reference
Which value of
groupis the reference group. Defaults to the first value encountered.statisticis unaffected, butalpha_MHinverts anddelta_MHchanges sign, so pass this explicitly whenever the direction of the DIF matters.
Value
list: statistic, p_value, df, alpha_MH, delta_MH, sum_A, sum_E, sum_V, n_strata, n_used, n, method.
Details
Per matching stratum k, with the 2x2 table of item-right/item-wrong
by reference/focal group,
E[A_k] = n1_k m1_k / T_k and
Var[A_k] = n1_k n2_k m1_k m0_k / (T_k^2 (T_k - 1)), giving
chi2 = (|sum A_k - sum E[A_k]| - 0.5)^2 / sum Var[A_k] on one
degree of freedom, and
alpha_MH = sum(A_k D_k/T_k) / sum(B_k C_k/T_k) with the ETS
delta scale -2.35 log(alpha_MH). Strata with fewer than two
examinees, or with no variation in the item or in group membership,
carry no information and are dropped.