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)

Alternatively, the standardized residual of an observation is

Z

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

M(S) = Sum(Z

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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