# 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

Xni = the observation generated by person n on item i

Eni = its expectation = sum k*Pnik, for k = 0,m

Yni = its residual = Xni - Eni

Wni = its variance = sum (k*k*Pnik) - Eni*Eni, for k = 0,m

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

Bias(S) = sum (Yni) / sum (Wni) where sum is across all n in S

Standard error of Bias is

SE (S) = sqrt ( sum (Wni) ) 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)

Alternatively, the standardized residual of an observation is

Zni = Yni / sqrt(Wni)

The mean standardized residual for Subset S comprised of s persons responding to item i is

M(S) = Sum(Zni) / s

It has standard error = sqrt(s)

The significance of this mean departure from its expectation of 0.0 is

t = M(S) / sqrt (s)

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

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