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Ch. 3. Classical Test Theory
I. Model
XIP = TIP + EIP
where,
XIP: the observed score for test I and person P,
TIP: the true score for test I and person P, and
EIP: error of measurement for test I and person P.
E (EIP) = 0, thus
E (XIP) = TIP
II. Assumptions
A. ρET = 0
The correlation between the error scores and the true scores in the population is
zero.
B. ρE1E2 = 0
The error scores in two different tests are uncorrelated in the population.
C. ρE1T2 = 0
The error scores on Test 1 are uncorrelated with the true scores on Test 2.
D. The existence of parallel tests. Two tests are called parallel tests if T 1 = T 2 and
σ²E1 = σ²E2.
E. The existence of τ-equivalent tests. Two test are called τ-equivalent test if
T 1 = T 2 + c, where, c is a constant.
III. Consequences
A. E (E) = 0 {From the model)
proof:
E (X) = E (T) + E (E)
= T + E (E)
E (X) = T
E (E) = 0
B. E (ET) = σET = 0
proof:
E (ET) = E (ET) - 0
= E (ET) - E (E) E (T)
= σET
σET
ρET = ──── = 0
σEσT
E (ET) = σET = 0
C. σ²X = σ²T + σ²E
proof:
σ²X = σ²(T + E)
= σ²T + σ²E + 2σET
= σ²T + σ²E
D. σ²T
ρ²XT = ────
σ²X
proof:
σXT
ρ²XT = [───── ]²
σXσT
[ E (XT) - E (X) E (T)]²
σ²Xσ²T
[ E ((T + E)T) - E (X) E (T)]²
σ²Xσ²T
( E (X) = T, E (T) = T, E (X) = E (T) )
[ E (T²) + E (ET) - E (T) E (T)]²
σ²Xσ²T
H. σ²T σ²T'
ρXX' = ──── = ────
σ²X σ²X'
(X and X' are scores on parallel tests.)
proof:
σXX'
ρXX' = ─────
σXσX'
σ(T + E)(T' + E')
= ─────────
σ²X
σTT' + σTE’ + σET' + σEE'
= ───────────────
σ²X
σTT'
= ─────
σ²X
σ²T
= ─────
σ²X
σ²T'
= ─────
σ²X'
I. σ²E
ρXX' = 1 - ────
σ²X
σ²T
ρXX' = ────
σ²X
σ²X - σ²E
= ───────
σ²X
σ²E
= 1 - ────
σ²X
J. ρXX' = 1 - ρ²XE
σXE
ρ²XE = [─────]²
σXσE
(σTE + σ²E)²
= ───────
σ²Xσ²E
(σ²E)²
= ──────
σ²Xσ²E
σ²E
= ────
σ²X
σ²E
ρXX' = 1 - ────
σ²X
= 1 - ρ²XE
K. ρ²XT = ρXX'
σ²T
ρ²XT = ────
σ²X
= ρXX'
L. σ²T = σXX'
σ²T
ρXX' = ─── (H)
σ²X
σXX' σXX'
ρXX' = ────= ──── ( σX = σX')
σXσX' σ²X
σ²T = σXX'
M. σ²E = σ²X(1 - ρXX')
σ²E = σ²X - σ²T ( σ²X = σ²T + σ²E)
= σ²X - σ²XρXX' ( ρXX' = σ²T/σ²X)
ρTxTz 1
ρXZ XX ' ZZ '
(Validity cannot be greater than the square-root of the product of two reliabilities
---> Attenuation Theory)
O. σ²Tx = N²σ²Ty
where
X = ΣYi [Ys are parallel test scores with E (Yi) = TY], and σ²Eyi = σ²Ey.
TX = E (X)
= E [ΣYi] = Σ E (Yi) = NTY
σ²Tx = E [TX - E (TX)]²
Now,
TX - E (TX) = NTY - E (NTY)
= NTY - N E (TY)
= N[TY - E (TY)]
σ²Tx = E [TX - E (TX)]²
= E [N(TY - E (TY))]²
= N² E [TY - E (TY)]²
= N²σ²Ty
P.
2 2 E (^) X E Y
N
where
X = ΣYi [Ys are parallel test scores with E (Yi) = TY], and σ²Eyi = σ²Ey.
EX X TX
= Y
N
i
Yi NT 1
= Y
N
i
NTY EYi NT 1
N
i
EYi 1
N
i
N
i j
j
EE
N
i
E (^) X EY YiYJ 1 1 1
2 2
2 E Y
N
Q.
NρYY'
ρXX' = ────────
1+(N-1)ρYY'
{Spearman-Brown formula, where N is number of parallel tests (Y is a subtest of X).}
σ²Tx
ρXX' = ────
σ²X
N²σ²Ty
= ─────────── ( σXX' = σ²Tx)
Σσ²Yi + ΣΣσYiYj
N²σ²Ty
= ─────────────
Nσ²Y + N(N-1)σ²Ty
N²ρYY'σ²Y
= ───────────────
Nσ²Y + N(N-1)ρYY'σ²Y
N²ρYY'σ²Y
= ──────────────
Nσ²Y(1 + (N-1)ρYY')
NρYY'
= ────────────
(1 + (N-1)ρYY')
R. If 1
lim YY ' 0 ,^ XX ' N
NρYY'
ρXX' = ───────
(1 + (N-1)ρYY')
'
'
1 1 YY
YY
N
N
N
As 0 ,( 1 )/ 1 , 1 , 0.
'
' ' YY YY
YY N N and XX if N
N
