Using Residuals to Estimate the Size and Statistical Significance of Local Bias

Let us use the notation of Rating Scale Analysis (Wright & Masters, 1982) p. 100

X_ni = the observation generated by person n on item i

E_ni = its expectation = sum k*P_nik, for k = 0,m

Y_ni = its residual = X_ni - E_ni

W_ni = its variance = sum (k*k*P_nik) - E_ni*E_ni, for k = 0,m

Then the logit bias for subset S of persons on item i is

Bias(S) = sum (Y_ni) / sum (W_ni) where sum is across all n in S

Standard error of Bias is

SE (S) = sqrt ( sum (W_ni) ) where sum is across all n in S

Statistical significance of Logit Bias departure from its expectation of 0.0 is

t = Bias(S)/SE(S)

MESA Research Note #1 by John Michael Linacre, April 1997

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