Page 234 - DMTH404_STATISTICS

Statistics

Notes 16.2.1 Domain Sampling Theory-2

What is the correlation of a test of n items with the domain score?
k k
I 
Domain variance =  V  Clj =  (variances) +  (covariances)
I 1 j  l

Let Va = average variance and Ca = average covariance then Vd= k(Va + (k-1)Ca), Cnd = n*Va
+n*(k-1)Ca
Vn = variance of an n-item test = Vj + Cjl = Vn = n*Va + n*(n-1)*Ca

Vn = n*Va+n*(n-1)*Ca

Cnd Cnd 2
2
rnd   rnd 
Vn * Vd Vn * Vd


{n * Va n * (k 1)Ca} * {n * Va n * (k 1)Ca}


2
rnd 

{n * Va n * (n 1) * Ca} * {k(Va (k 1)Ca)}



{Va (k 1)Ca} * {n * Va n * (k 1)Ca}




2
rnd 
{Va (n 1) * Ca} * {k(Va (k 1)Ca)}





{n * Va n * (k 1)Ca}

2
rnd 
{Va (k 1) * Ca)} * {k}


n * Ca
2
lim as k  of rnd =


Va (n 1)Ca
i.e., the amount of domain variance in a n-item test (the squared correlation of the test with the
domain) is a function of the number of items and the average covariance within the test.
Coefficient Alpha - 1
Consider a test made up of k items with an average intercorrelation r.
1) What is the correlation of this test with another test sampled from the same domain of
items?
2) What is the correlation of this test with the domain?
Test 1 Test 2
Test 1 V1 C1 2
Test 2 C1 2 V2

Let r1 be the average correlation of items within test 1
Let r2 be the average correlation of items within test 2

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