Page 233 - DMTH404_STATISTICS

Unit 16: Reliability Theory

These covariances reflect the following parameters: Notes

x1 x2 x3 x4
x1 Vt+Ve1

x2 Cx1tCx2tVt Vt+Ve2
x3 Cx1tCx3tVt Cx2tCx3tVt Vt+Ve3
x4 Cx1tCx4tVt Cx2tCx4tVt Cx3tCx4tVt Vt+Ve4

We need to estimate the following parameters:
Vt, Ve , Ve , Ve , Ve , Cx t, Cx t, Cx t, Cx t
1 2 3 4 1 2 3 4
Parallel tests assume Ve = Ve = Ve =Ve , and Cx t = Cx t
1 2 3 4 1 2
= Cx t = Cx t and only need two tests.
3 4
Tau equivalent tests assume: Cx t = Cx t = Cx t = Cx t and need at least three tests to estimate
1 2 3 4
parameters.
Congeneric tests allow all parameters to vary but require at least four tests to estimate parameters.

16.2 Domain Sampling Theory-1

Consider a domain (D) of k items relevant to a construct. (E.g., English vocabulary items, expresions
of impulsivity). Let D represent the number of items in D which the i subject can pass (or endorse
th
i
in the keyed direction) given all D items. Call this the domain score for subject i.
What is the correlation of scores on an itemj with domain scores?

k

C d  Vj   Cl  V  (k 1) * (average covariance of j)
j j j
l 1

k k
Domain variance =  Vl   Cl   (variance)  (covariances)
j


l 1 j 1
Vd = k* (average variance) + k*(k – 1) * (average covariance)
Let Va = average variance and Ca = average covariance then Vd= k(Va + (k-1)Ca).
Assume that Vj = Va and that Cj l = Ca.

Cjd Va (k 1) * Ca

rjd = 
Vj * Vd Va * k(Va (K 1)Ca)


(Va (k 1) * Ca) * (Va (k 1) * Ca)




2
rjd 
Va * k * (Va (k 1)Ca)


Now, find the limit of rj d2 as k becomes large:
C
2
limrjd  a =average covariance/average variance i.e., the amount of domain variance in an
k V
a
item (the squared correlation of the item with the domain) is the averge intercorrelation in the
domain.
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