Theta Reliability and Factor Scaling
Theta Reliability and Factor Scaling
Theta Reliability and Factor Scaling
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David3. Armor
THE RAND CORPORATION
17
ALPHA RELIABILITY
~(6) p + 2'5~~~"
rij
i<j )
If the scale is formedby summingraw itemscores,
a = ( (7)
_ -
U2)
If the scale is formedby averagingraw itemscores,
a = ( P T ) (8)
whererihis thecorrelation
betweenitemi and itemh.
moreindependentpropertieseitherequally or unequally.Althoughthe
techniquesof covariancescalingmay helpsome aspectsof the firstprob-
lem theymay not do so optimally.As faras the second problemis con-
cerned,covariancescalingoffers littleor no solutionat all. The littlehelp
it may offerdependsverymuchon thenumberofitemsand thepatience
of the investigator.
These two problemscan be illustratedby simple hypothetical
examples.Considera four-item compositewhoseintercorrelations are as
shownin Table 1. The firstthreeitemshave intercorrelations that are
quite highand consistent,rangingfrom0.4 to 0.6. But item4 has much
lowercorrelationswith the otherthreeitems (all being 0.1). Assuming
that the correlationsare based on a sufficiently large sample,it might
turnout thatthe0.1 correlationsare significantly different fromzeroand
an investigatormay thereforedecide to keep item 4 in the composite.
But ifitem4 is leftin the compositewithoutsomekindofweighting, the
compositereliabilitywillbe lowerthanifthe itemsare weightedaccord-
ing to theirdifferential The four-item
contributions. (unweighted)alpha
reliabilityis 0.63; the compositereliabilitywithweightingis 0.68 (apply-
ing the theta techniquedefinedin the nextsection).
The second problemis generallyharderto detectwithreal data,
but the simplehypotheticalexamplein Table 2 offersa cleardemonstra-
tion. Basically the existenceof two or moreindependentpropertiesin a
compositeis usually revealed wheneverthereare two or more sets of
itemswithrelativelyhighwithin-setcorrelationsand relativelylow be-
TABLE 1
Inter-Item Correlations for Hypothetical Four-Item Composite
Item 1 2 3 4
1 1.0
2 0.6 1.0
3 0.5 0.4 1.0
4 0.1 0.1 0.1 1.0
TABLE 2
Inter-Item Correlations Revealing Two Dimensions
Item 1 2 3 4 5 6
1 1.0
2 0.5 1.0
3 0.5 0.5 1.0
4 0.0 0.0 0.0 1.0
5 0.0 0.0 0.0 0.5 1.0
6 0.0 0.0 0.0 0.5 0.5 1.0
THETA RELIABILITY
TABLE 3
Factor Loadings and Roots
Item Factor 1 Factor 2 ... Factor in
1 all al2 *-- a,m
2 a2l a22 * a2m
a k =1 X2 X,n
E ailah1rih =
iHh
Z ailahlrih - Zi a2
i,h (16)
= X12-1
Thus
S = /X1+ (X2-Xl)/X2
1 1 1
~~~~~~~(17)
= 1/X1+ (1 -1/X1)
This means that forthe factorscoresbased on the firstprincipalcompo-
nent,S = 1, I = 1/X1,and C = 1 - 1/X1.For the compositereliability
fromEquation (5) we have
P m m'(1
=~~E~ ,2rij
E(aihOhkajhOhk)/Xhr
+
+ , (aihchkaj0gk)/Xh\rij (21)
ii--dj-1 h=1 hp,hg=l
m 2p m P
= --I a'haJhrij + E (chk0gk)/(XhX\g) E2 aihaigrij
h=1 Ah iHj=1 h,g=1 idj1-
Single-Factor Case
Althoughwe know that theta is algebraicallylargerthan alpha,
it mightbe usefulto presentsomesimplehypotheticalexamplesthat re-
veal how seriousthe differences mightbe and showhow we can improve
scale reliability.We shall examinean applicationoffactorscalingto real
data in a latersection.
Table 4 shows fourcases, the firstthreeof whichinvolve four
items.I have helditems1 to 3 constantso that thedifference in reliabili-
ties can be seen as we add differentfourthitems.
Case 1 reveals the largestdiscrepancybetweentheta and alpha
(0.05). The reasonis that the fourthitemhas low correlationswiththe
otheritemsand consequentlya low factorloading (0.24). A scale score
that weightsitemsaccordingto the loadingstherefore has a higherreli-
abilitythan a simplesummedscore.
In case 2 the fourthitem still has somewhathighercorrelations
and a higherloading (0.44), so that the difference betweentheta and
alpha is diminishedto 0.02. In case 3 thereis practicallyno difference
betweentheta and alpha even thoughthe correlationsand the loadings
do varyconsiderably.Clearlythetaand alpha differ in a substantialway
only whensome itemshave consistently lowercorrelationswithall the
TABLE 4
Comparing Theta and Alpha
Factor
Case Item Correlations Loading (ail)
Case I 1 2 3 4
Xi = 2.033 1 1.0 0.86
o = 0.68 2 0.6 1.0 0.82
a = 0.63 3 0.5 0.4 1.0 0.76
4 0.1 0.1 0.1 1.0 0.24
Case 2 1 2 3 4
Xi = 2.117 1 1.0 0.87
a = 0.70 2 0.6 1.0 0.80
a = 0.68 3 0.5 0.4 1.0 0.72
4 0.3 0.2 0.1 1.0 0.44
Case 3 1 2 3 4
X = 2.310 1 1.0 0.81
o = 0.756 2 0.6 1.0 0.80
a = 0.754 3 0.5 0.4 1.0 0.76
4 0.3 0.4 0.4 1.0 0.66
Case 4 1 2 .3
X= 2.004 1 1.0 0.87
0 =0.75 2 0.6 1.0 0.82
a = 0.75 3 0.5 0.4 1.0 0.76
I 1*I 11*
0
1.0
0~~~~~~
ClusterA ail ClusterB a*
CILISLserAC /(luster B
-1.0 0 1.0M
a
a i2
-1.0
Uinrotate(I Rotated
ofunrotatedand rotatedfactors.
1. Geometricrepresentation
Figutre
TABLE 6
Factor Scaling for Political Ideology Scalea
Unrotated Principal-
Component Loadings Rotated Loadings
Itemb I II I* II*
1 0.650 -0.615 0.032 -0.895
2 0.553 0.247 0.567 -0.212
3 -0.082 -0.030 -0.080 0.036
4 0.523 0.283 0.571 -0.165
5 0.438 0.477 0.647 0.033
6 0.262 0.450 0.502 0.137
7 0.377 -0.058 0.228 -0.306
8 0.593 -0.636 -0.023 -0.869
9 0.637 -0.184 0.325 -0.578
10 0.425 0.472 0.634 0.038
11 0.483 -0.637 -0.103 -0.793
12 0.535 0.504 0.735 -0.016
13 0.648 0.217 0.613 -0.300
Roots Xi = 3.289 X2 = 2.322 Xi = 2.813 X2 = 2.798
Percent of
total variance 25.3% 17.9% 21.7% 21.5%
Reliabilities
Factor Scores
O = 0.75 forfactor scores based on unrotated factor I
01 = 0.70 forfactor scores based on rotated factor I*
02 = 0.70 forfactor scores based on rotated factor II*
Factor Scales
01 -0.74 for socialism scale based on items 2, 4-6, 10, 12, and 13 [ai = 0.73
using Equation (6)]
02 = 0.85 forpacifismscale based on items 1, 8, and 11 [a2 = 0.85 using Equa-
tion (6)]
a N = 104 for all statistics.
I See Table 5 for item wording.
Iioe ( I* (rotatedl)
I (unrlotaletdl) (12) *(?> II
0.7
0.4-
0 ~~~~0.3
Ci) b0~~~.2-
-0.3
-0).4
-0.5
14 If all intercorrelations
equalr, thena, 0, and Q shouldbe equal.But
using principal-componentloadings gives Q = a + (1 - r)/pX,, making
Q> or 0 unlessr = 1.
SUMMARY
REFERENCES