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Transcript of MEASUREMENT MODELS. BASIC EQUATION x = + e x = observed score = true (latent) score: represents...
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MEASUREMENT MODELS
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BASIC EQUATION
• x = + e
• x = observed score = true (latent) score: represents
the score that would be obtained over many independent administrations of the same item or test
• e = error: difference between y and
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ASSUMPTIONS and e are independent
(uncorrelated)
• The equation can hold for an individual or a group at one occasion or across occasions:
• xijk = ijk + eijk (individual)
• x*** = *** + e*** (group)
• combinations (individual across time)
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x x
e
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RELIABILITY
• Reliability is a proportion of variance measure (squared variable)
• Defined as the proportion of observed score (x) variance due to true score ( ) variance:
2x = xx’
• = 2 / 2
x
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Var()
Var(x)
Var(e)
reliability
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Reliability: parallel forms
• x1 = + e1 , x2 = + e2
(x1 ,x2 ) = reliability
• = xx’
• = correlation between parallel forms
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x1 x
e
x2
e
x
xx’ = x * x
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ASSUMPTIONS and e are independent
(uncorrelated)
• The equation can hold for an individual or a group at one occasion or across occasions:
• xijk = ijk + eijk (individual)
• x*** = *** + e*** (group)
• combinations (individual across time)
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Reliability: Spearman-Brown
• Can show the reliability of the composite is
kk’ = [k xx’]/[1 + (k-1) xx’ ]
• k = # times test is lengthened
• example: test score has rel=.7
• doubling length produces rel = 2(.7)/[1+.7] = .824
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Reliability: parallel forms
• For 3 or more items xi, same general form holds
• reliability of any pair is the correlation between them
• Reliability of the composite (sum of items) is based on the average inter-item correlation: stepped-up reliability, Spearman-Brown formula
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RELIABILITY
Generalizability d - coefficients ANOVA
g - coefficients
Cronbach’s alpha
test-retest internal consistency
inter-rater
parallel form Hoyt
dichotomous split halfscoring
KR-20 SpearmanKR-21 Brown
averageinter-itemcorrelation
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COMPOSITES AND FACTOR STRUCTURE
• 3 MANIFEST VARIABLES REQUIRED FOR A UNIQUE IDENTIFICATION OF A SINGLE FACTOR
• PARALLEL FORMS REQUIRES:– EQUAL FACTOR LOADINGS– EQUAL ERROR VARIANCES– INDEPENDENCE OF ERRORS
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x1
x
e
x2
e
x
xx’ = xi * xj
x3
e
x
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RELIABILITY FROM SEM• TRUE SCORE VARIANCE OF THE
COMPOSITE IS OBTAINABLE FROM THE LOADINGS:
K = 2
i i=1
K = # items or subtests
• = K2x
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Hancock’s Formula
• Hj = 1/ [ 1 + {1 / (Σ[l2ij/(1- l2
ij )] ) }
• Ex. l1 = .7, l2= .8, l3 = .6
• H = 1 / [ 1 +1/( .49/.51 + .64/.36 + .36/.64 )]
= 1 / [ 1 + 1/ ( .98 +1.67 + .56 ) ]
= 1/ (1 + 1/3.21)
= .76
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Hancock’s Formula Explained
Hj = 1/ [ 1 + {1 / (Σ[l2ij/(1- l2
ij )] ) }
now assume strict parallelism: then l2ij= 2
xt
thus Hj = 1/ [ 1 + {1 / (Σ[2xt /(1- 2
xt)] ) }
= k 2xt / [1 + (k-1) 2
xt ]
= Spearman-Brown formula
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RELIABILITY FROM SEM
• RELIABILITY OF THE COMPOSITE IS OBTAINABLE FROM THE LOADINGS:
= K/(K-1)[1 - 1/ ]
• example 2x = .8 , K=11
= 11/(10)[1 - 1/8.8 ] = .975
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SEM MODELING OF PARALLEL FORMS
• PROC CALIS COV CORR MOD;
• LINEQS
• X1 = L1 F1 + E1,
• X2 = L1 F1 + E1,
• …
• X10 = L1 F1 + E1;
• STD E1=THE1, F1= 1.0;
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TAU EQUIVALENCE
• ITEM TRUE SCORES DIFFER BY A CONSTANT:
i = j + k
• ERROR STRUCTURE UNCHANGED AS TO EQUAL VARIANCES, INDEPENDENCE
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TESTING TAU EQUIVALENCE
• ANOVA: TREAT AS A REPEATED MEASURES SUBJECT X ITEM DESIGN:
• PROC VARCOMP;CLASS ID ITEM;
• MODEL SCORE = ID ITEM;
• LOW VARIANCE ESTIMATE CAN BE TAKEN AS EVIDENCE FOR PARALLELISM (UNLIKELY TO BE EXACTLY ZERO
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CONGENERIC MODEL
• LESS RESTRICTIVE THAN PARALLEL FORMS OR TAU EQUIVALENCE:– LOADINGS MAY DIFFER– ERROR VARIANCES MAY DIFFER
• MOST COMPLEX COMPOSITES ARE CONGENERIC:– WAIS, WISC-III, K-ABC, MMPI, etc.
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x1
x1
e1
x2
e2
x2
(x1 , x2 )= x1 * x2
x3
e3
x3
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COEFFICIENT ALPHA
xx’ = 1 - 2E /2
X
• = 1 - [2i (1 - ii )]/2
X ,
• since errors are uncorrelated = K/(K-1)[1 - (s2
i )/ s2X ]
• where X = xi (composite score)
s2i = variance of subtest xi
sX = variance of composite
• Does not assume knowledge of subtest ii
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COEFFICIENT ALPHA- NUNNALLY’S COEFFICIENT
• IF WE KNOW RELIABILITIES OF EACH SUBTEST, i
N = K/(K-1)[s2i (1- rii )/ s2
X ]
• where rii = coefficient alpha of each subtest
• Willson (1996) showed N
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SEM MODELING OF CONGENERIC FORMS
MPLUS EXAMPLE: this is an example of a CFA
DATA: FILE IS ex5.1.dat;
VARIABLE: NAMES ARE y1-y6;
MODEL: f1 BY y1-y3;
f2 BY y4-y6;
OUTPUT: SAMPSTAT MOD STAND;
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x1
x1
e1
x2
e2
x2
XiXi = 2xi + s2
i
x3
e3
x3
s1
NUNNALLY’S RELIABILITY CASE
s2
s3
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x1
x1
e1
x2
e2
x2
Specificities can be
misinterpreted as a correlated
error model if they are
correlated or a second factor
x3
e3
x3
s
CORRELATED ERROR PROBLEMS
s3
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x1
x1
e1
x2
e2
x2
Specificieties can be
misinterpreted as a
correlated error model
if specificities are
correlated or are a
second factor
x3
e3
x3
CORRELATED ERROR PROBLEMS
s3
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SEM MODELING OF CONGENERIC FORMS- CORRELATED ERRORS
MPLUS EXAMPLE: this is an example of a CFA
DATA: FILE IS ex5.1.dat;
VARIABLE: NAMES ARE y1-y6;
MODEL: f1 BY y1-y3;
f2 BY y4-y6;
y4 with y5;
OUTPUT: SAMPSTAT MOD STAND;
specifies residuals of previous model, correlates them
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MULTIFACTOR STRUCTURE
• Measurement Model: Does it hold for each factor?– PARALLEL VS. TAU-EQUIVALENT VS.
CONGENERIC
• How are factors related?
• What does reliability mean in the context of multifactor structure?
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SIMPLE STRUCTURE
• PSYCHOLOGICAL CONCEPT:
• Maximize loading of a manifest variable on one factor ( IDEAL = 1.0 )
• Minimize loadings of the manifest variables on all other factors ( IDEAL = 0 )
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SIMPLE STRUCTURE
Example
SUBTEST FACTOR1 FACTOR2 FACTOR3
A 1 0 0
B 1 0 0
C 0 1 0
D 0 1 0
E 0 0 1
F 0 0 1
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MULTIFACTOR ANALYSIS
• Exploratory: determine number, composition of factors from empirical sampled data– # factors # eigenvalues > 1.0 (using squared
multiple correlation of each item/subtest i with
the rest as a variance estimate for 2xi
– empirical loadings determine structure
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MULTIFACTOR ANALYSISTITLE:this is an example of an exploratory
factor analysis with continuous factor
indicators
DATA: FILE IS ex4.1.dat;
VARIABLE:NAMES ARE y1-y12;
ANALYSIS:TYPE = EFA 1 4;
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MULTIFACTOR MODEL WITH THEORETICAL
PARAMETERS
MPLUS EXAMPLE: this is an example of a CFA
DATA: FILE IS ex5.1.dat;
VARIABLE: NAMES ARE y1-y6;
MODEL: f1 BY [email protected] [email protected] [email protected];
f2 BY [email protected] [email protected] [email protected];
f1 with [email protected];
OUTPUT: SAMPSTAT MOD STAND;
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1
x1
x11
e1
x2
e2
x22
x3
e3
x31
MINIMAL CORRELATED FACTOR STRUCTURE
2
x4e4
x42
12
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FACTOR RELIABILITY• Reliability for Factor 1:
= 2(x11 * x31 ) / (1 + x11 * x31 )(Spearman-Brown for Factor 1 reliability
based on the average inter-item correlation
• Reliability for Factor 2:
= 2(x22 * x42 ) / (1 + x22 * x42 )
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FACTOR RELIABILITY• Generalizes to any factors- reliability is
simply the measurement model reliability for the scores for that factor
• This has not been well-discussed in the literature– problem has been exploratory analyses produce
successively smaller eigenvalues for factors due to the extraction process
– second factor will in general be less reliable using loadings to estimate interitem r’s
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FACTOR RELIABILITY• Theoretically, each factor’s reliability should be
independent of any other’s, regardless of the covariance between factors
• Thus, the order of factor extraction should be independent of factor structure and reliability, since it produces maximum sample eigenvalues (and sample loadings) in an extraction process.
• Composite is a misnomer in testing if the factors are treated as independent constructs rather than subtests for a more global composite score (separate scores rather than one score created by summing subscale scores)
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CONSTRAINED FACTOR MODELS
• If reliabilities for scales are known independent of the current data (estimated from items comprising scales, for example), error variance can be constrained:
• s2ei = s[1 - i ]
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x1
x1
e1
x2
e2
x2
x3
e3
x3
CONSTRAINED SEM- KNOWN RELIABILITY
sx3 [1- 3 ]1/2
sx1 [1- 1 ]1/2 sx2 [1- 2 ]1/2
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CONSTRAINED SEM-KNOWN RELIABILITY
MPLUS EXAMPLE: this is an example of a CFA with known error unreliabilities
DATA: FILE IS ex5.1.dat;
VARIABLE: NAMES ARE y1-y6;
MODEL: f1 BY y1-y3;
f2 BY y4-y6;
OUTPUT: SAMPSTAT MOD STAND;
similar statement for each item
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SEM Measurement Procedures
• 1. Evaluate the theoretical measurement model for ALL factors (not single indicator variables included)
• Demonstrate discriminant validity by showing the factors are separate constructs
• Revise factors as needed to demonstrate- drop some manifest variables if necessary and not theoretically damaging
• Ref: Anderson & Gerbing (1988)