Transcript Psykologisen mittarin tilastollinen analysointi

Structural Equation Modelling
Jouko Miettunen, PhD
Department of Psychiatry
University of Oulu
e-mail: [email protected]
1
Topics of this presentation

Background
 Factor
analyses
 Regression analyses



Theory
Modeling with AMOS
References
2
Structural Equation Modeling


Based on factor analysis
First studies by Karl E. Jöreskog
and Dag Sörbom in 1970’s
 “LISREL



–models”
Combination of factor analysis and
regression analysis
Continuous and discrete predictors
and outcomes
Relationships among measured or
latent variables
3
Structural Equation Modeling
(SEM) is a generalization of many
techniques including
•Regression Analysis
•Path Analysis
•Discriminant Analysis
•Canonical Correlation
•Confirmatory Factor Analysis
4
Regression analysis
Multiple Regression Analysis
or Path Analysis
5
Exploratory factor analysis
6
Confirmatory factor analysis
7
An example of SEM model with measurement and structural part (modified from Byrne 2001).
rectangles = measured variables
ovals = latent variables
circles = error terms
Model may also include e.g. doubleheaded arrows to indicate fixed
(typically to 1) or estimated
correlations between error terms.
8
Variables in SEM

Exogenous variables = independent

Endogenous variables = dependent

Observed variables = measured

Latent variables = unobserved
9
Sample size



15 cases per predictor in a standard
ordinary least squares multiple
regression analysis.
Researchers may go as low as five cases
per parameter estimate in SEM analyses,
but only if the data are perfectly wellbehaved
Usually 5 cases per parameter is
equivalent to 15 measured variables.
Bentler and Chou (1987), Stevens (1996)
10
Phases of SEM

Theoretical model
 Drawing
the model
 Including constraints




Model identification
Estimation
Fit of the model
Improving model
11
Model identification





P is # of measured variables
DF = [P*(P+1)]/2- (# of estimated
parameters)
If DF>0 model is over identified
If DF=0 model is just identified
If DF<0 model is under identified
12
Model identification

Scaling the latent variable
One fixed nonzero loading
 For causal factors, fixed factor variance
 For caused factors, fixed factor disturbance
(residual)


Sufficient number of indicators

At least 2-3 indicators
Whose errors are uncorrelated
 Whose inter-correlations should be statistically
significant
 Product of the correlations should be positive


For more see

http://davidakenny.net/cm/identify.htm
13
Calculating degrees of freedom
error2
error4
error11
error12
error17
error5
error1
error3
error8
error10
error6
error15
error7
error16
error9
error18
error13
error19
error14
error20
Degrees of freedom = [P*(P+1)]/2 - (number of estimated parameters)
= [20*(20+1)]/2 – (20+20+3) = 210 – 43 = 167
P=number of measured variables
14
Constraining parameters
 Typically some of the factor loadings
are constrained or fixed to be zero.
 For each factor it is also necessary to
fix one loading to the value one in order
to give the latent factor an interpretable
scale.
 One solution is to fix the variance of all
factors to one and then estimate all factor
loadings.
15
Estimation methods
Maximum Likelihood Estimation (MLE)
which assumes multivariate normal data
 reasonable sample size, e.g. about 200
observations.

Asymptotically Distribution Free (ADF)
Continuous (or ordinal) but not normal data
 Also known as WLS (weighted least squares)
 Large sample sizes

Unweighted Least Squares (ULS)
 Non-normal data
16
Model testing

Test statistics
 Chi-square
test
 Akaike’s Information Criteria (AIC,
CAIC)
 Root Mean Square Error Of
Approximation (RMSEA)
 Goodness of Fit Index (GFI, AGFI)
 CFI
 Tucker-Lewis Index (TLI)
17
Measures of fit

Chi-square test (X2)
 Should
be non-significant (p>0.05)
 Absolute index
 Not appropriate with a large sample
size, rejects (p<0.05) model too easily

X2/df (relative X2)
 df
= degrees of freedom
 Should be > 3
18
Measures of fit



GFI (Goodness of Fit Index)
AGFI (Adjusted GFI)
IFI (Increment Fit Index)
 Values
between 0-1
 Recommended criteria vary, e.g.
 >0.90
(”adequate”)
 >0.95 (”good”)
19
Relative measures



Compare to baseline model
Normed Fit Index (NFI)
Non-Normed Fit Index (NNFI)
= Tucker-Lewis Index (TLI)

Comparative Fit Index (CFI)
 Values
are between 0-1
 Recommended criteria vary, e.g.
 >0.90
(”adequate”)
 >0.95 (”good”)
20
Adjusted measures



Are related to number of parameters
RMR (Root Mean square Residual)
RMSEA (Root Mean Square Error of
Approximation)
 Values
are between 0-1
 ”Adequate”, if <0.08 (or <0.10)
 ”Good”, if <0.05 (or 0.06)
21
Measures for model comparisons

Akaike’s Information Criteria (AIC)
Consistent AIC (CAIC)
Bayes Information Criteria (BIC)

Better model has lower value


22
Modification indices
 If the fit of a model is not adequate,
you can delete non-significant
parameters and add other parameters
that will improve the fit.
 The value given is the minimum
amount that the chi-square statistic is
expected to decrease if the
corresponding parameter is freed.
 Theoretical justification is needed.
23
Modification indexes
AMOS text output
E.g. if error terms eps2 and eps4 were
allowed to correlate, Chi square statistics
would be 13.161 units lower. Degrees of
freedom would decrease by one.
 Make changes to the model only if justified
by the theory

24
SEM Software
 LISREL (www.ssicentral.com)
 EQS (www.mvsoft.com)
 AMOS (www.spss.com/amos)
 Mplus (www.statmodel.com)
 SAS (PROC Calis)
For more software and a general guide
to SEM resources see web page at
http://www.hawaii.edu/sem/sem.html
25
SEM analyses in AMOS

AMOS Graphics
 draw SEM graphs
 runs SEM models using graphs

AMOS Program Editor
 runs SEM models using syntax
26
SEM Assumptions in AMOS


Continuously and Normally
Distributed Endogenous
Variables
Unlike AMOS, Mplus
software can handle noncontinuous variables
(www.statmodel.com)
27
Reading Data into AMOS


File  Data Files
The following dialog appears:
28
Model drawing in AMOS
Latent variable
Measured variable
Latent measurement error
29
AMOS -software tools
Variable names
Constraining parameters
30
Icons in AMOS
31
Performing the analysis in AMOS
To run the program, click
32
Presentation of model results in AMOS

Text output

Graphics output

Examples later
33
Example I
Social Perception as a Mediator of the Influence of Early Visual
Processing on Functional Status in Schizophrenia
The authors used SEM to test whether one
aspect of social cognition (social perception)
mediates relations between visual perception
and functional status in patients with
schizophrenia (N=75).
SEM supported social perception as a
mediator of relations between early visual
processing and functional status in
schizophrenia.
Direct relationship between early visual
processing and functional status was
significant in a model that did not include social
perception but was not significant in the
mediation model that included social
perception.
Sergi et al. Am J Psychiatry 2006:163:448-54
34
Basic model with no mediator
Chi-square= 20.60, df=13, p=0.08; CFI= 0.87; RMSEA=0.09
35
Model with mediator
Chi-square=22.02, df=18, N=75, p=0.23, CFI=0.95, RMSEA=0.06
36
Example II
Confirmatory Factor Analysis of the Psychopathy Checklist: Screening
Version in Offenders With Axis I Disorders
One hundred forty-nine inpatients within a maximum
security psychiatric facility were assessed with the
Psychopathy Checklist: Screening Version. Within
the total sample, 68% had a psychotic disorder and
30% met criteria for psychopathy. Using CFA, the
authors tested the 2-, 3- and 4-factor models.
Results indicated good fit for each model, with the 4factor model showing best overall fit. SEM was used
to determine which psychopathy factors predicted 6month follow-up of inpatient aggression. The 2-, 3-,
and 4-factor models, respectively, accounted for 16%,
27%, and 31% of the variance in aggression.
Hill et al. Psychol Assessm 2004:16:90-5.
37
Confirmatory Factor Analysis results
38
Example III
A Longitudinal Model of Social Contact, Social Support,
Depression, and Alcohol Use
The longitudinal relations among social contact,
perceived social support, depression, and alcohol use
were examined. A random sample of 1,192 adults.
Results revealed that (a) social contact was positively
related to perceived social support; (b) perceived
social support was negatively related to depression;
and (c) depression was positively related to alcohol
use for 1 of 2 longitudinal lags.
There was partial support for the feedback hypothesis
that increased alcohol use leads to decreased
contact with family and friends.
Peirce et al. Health Psychol 2000:19:28-38.
39
40
Example IV
Risk and Protective Factors for Substance Use Among African
American High School Dropouts
Risk and protective factors that predict substance use were
investigated with 318 youths.
A conceptual model linking positive family relationships and
religious involvement to youths’ substance use and
conventional peer affiliations through a positive life
orientation was examined with SEM.
Positive life orientation fully mediated the influence of family
relationships on conventional peer affiliations. Religious
involvement directly predicted conventional peer affiliations
and positive life orientation. Conventional peer affiliations
mediated the other variables’ influence on substance use.
Kogan et al. Psychol Addict Behav 2005:19:382-91.
41
42
43
Example V


Instrument measuring alexithymia: TAS-20
Data from Northern Finland 1986 Birth Cohort
(NFBC 1986), 15-16 year follow-up
Large data (N=6668)
 20 likert -scales (1-5) items
 Some are normally distributed, some not



We will test a three-factor model which has been
found in adult samples
We compare results to a 31 year follow up data of
an earlier cohort (Northern Finland 1966 Birth
Cohort, NFBC 1966)
44
Toronto Alexithymia Scale -20
Item
Question
1
I am often confused about what emotion I am feeling
2
It is difficult for me to find the right words for my feelings
3
I have physical sensations that even doctors don’t understand
4*
I am able to describe my feelings easily
5*
I prefer to analyze problems rather than just describe them
6
When I am upset, I don’t know if I am sad, frightened, or angry
7
I am often puzzled by sensations in my body
8
I prefer to just let things happen rather than to understand why they turn out that
way
9
I have feelings that I can’t quite identify
10*
Being in touch with emotions is essential
11
I find it hard to describe my feelings more
12
People tell me to describe my feelings more
13
I don’t know what’s going on inside me
14
I often don’t know why I am angry
15
I prefer talking to people about their daily activities rather than their feelings
16
I prefer to watch “light” entertainment shows rather than psychological dramas
17
It is difficult for me to reveal my innermost feelings, even to close friends
18*
I can feel close to someone, even in moments of silence
19*
I find examination of my feelings useful in solving personal problems
20
Looking for hidden meanings in movies or plays distracts from their enjoyment
* These variables were revised in analyses
45
Theoretical model
Joukamaa ym. 2001, Miettunen 2004
46
Text output
Unstandardized regression weights
Estimate = Estimate of regression weight
S.E. = Standard Error
C.R. = Critical Ratio
- If >1.96 then p<0.05
Estimate
S.E.
C.R.
P
tas01
<---
F1
1,000
tas03
<---
F1
,642
,020
32,239
***
tas06
<---
F1
1,038
,028
37,065
***
tas07
<---
F1
,895
,022
40,184
***
tas09
<---
F1
1,201
,027
43,816
***
tas13
<---
F1
1,098
,025
43,881
***
tas14
<---
F1
1,144
,030
37,842
***
tas02
<---
F2
1,000
das04
<---
F2
,734
,021
35,374
***
tas11
<---
F2
,798
,021
38,320
***
tas12
<---
F2
,734
,023
31,282
***
tas17
<---
F2
,799
,025
31,935
***
das05
<---
F3
1,000
tas08
<---
F3
,435
,059
7,333
***
das10
<---
F3
1,934
,094
20,583
***
tas15
<---
F3
1,589
,090
17,754
***
tas16
<---
F3
,816
,067
12,225
***
das18
<---
F3
1,863
,091
20,472
***
das19
<---
F3
2,050
,097
21,047
***
tas20
<---
F3
,867
,064
13,554
***
Label
47
Variances
Estimate = Estimate of variance
S.E. = Standard Error
C.R. = Critical Ratio
- If >1.96 then p<0.05
Estim
ate
S.E.
C.R.
P
F1
,379
,015
25,839
***
F2
,514
,019
27,234
***
F3
,082
,007
11,442
***
e1
,545
,011
47,952
***
e3
,523
,010
52,030
***
e6
,874
,017
50,303
***
e7
,480
,010
48,588
***
e9
,580
,013
45,398
***
e13
,481
,011
45,322
***
e14
,987
,020
49,934
***
e2
,552
,014
40,917
***
e4
,669
,014
48,952
***
e11
,599
,013
46,940
***
e12
,970
,019
50,878
***
e17
1,082
,021
50,619
***
e5
,625
,012
52,086
***
e8
1,112
,020
54,762
***
e10
,560
,013
42,154
***
e15
1,127
,022
50,872
***
e16
1,123
,021
53,976
***
e18
,556
,013
43,032
***
e19
,417
,012
35,687
***
e20
,937
,017
53,586
***
48
Standardized regression weights
Correlations:
NFBC
1966
NFBC
1986
F1
<-->
F2
,648
,793
F1
<-->
F3
,253
-,111
F2
<-->
F3
,589
,210
NFBC
1966
NFBC
1986
tas01
<---
F1
,69
,64
tas03
<---
F1
,47
,48
tas06
<---
F1
,57
,56
tas07
<---
F1
,63
,62
tas09
<---
F1
,70
,70
tas13
<---
F1
,75
,70
tas14
<---
F1
,59
,58
tas02
<---
F2
,79
,69
das04
<---
F2
,70
,54
tas11
<---
F2
,61
,59
tas12
<---
F2
,47
,47
tas17
<---
F2
,66
,48
das05
<---
F3
,27
,34
tas08
<---
F3
,34
,13
das10
<---
F3
,50
,60
tas15
<---
F3
58
,39
tas16
<---
F3
,47
,22
das18
<---
F3
,36
,58
das19
<---
F3
,55
,67
tas20
<---
F3
,49
,25
49
Summary of goodness of fit statistics (NFBC 1986)
Model
NPAR
CMIN
DF
P
CMIN/DF
RMR
GFI
AGFI
PGFI
Default
model
43
4751,46
167
,000
28,452
,067
,922
,901
,733
Model
NFI
Delta1
RFI
rho1
IFI
Delta2
TLI
rho2
CFI
PRATIO
PNFI
PCFI
Default
model
,821
,797
,826
,802
,826
,879
,722
,726
Model
NCP
LO 90
HI 90
FMIN
F0
LO 90
HI 90
Default
model
4584,455
4363,23
4812,932
,783
,756
,719
,793
Model
RMSEA
LO 90
HI 90
PCLOSE
ECVI
LO 90
HI 90
MEC
VI
Default
model
,067
,066
,069
,000
,797
,761
,835
,797
Model
AIC
BCC
BIC
CAIC
HOELTER
.05
HOELTER
.01
Default
model
4837,455
4837,75
5126,019
5169,019
254
272
NFBC 1966
• GFI = 0.935, AGFI = 0.918, RMSEA = 0.061
Recommended criteria
• GFI, AGFI > 0.95 (good), >0.90 (adequate)
•RMSEA < 0.05/0.06 (good), <0.08/0.10 (adequate)
50
Graphics output
e1
e3
e6
,41
tas01
,23
tas03
e7
,32
tas06
,48
e9
,39
tas07
,56 ,62
e13
,48
tas09
e14
,49
tas13
,33
R2
tas14
,70
,70
,58
,64
,12
Regression coefficient (R)
F1
,01
,34
-,11
tas17
,39
,59
e2
tas02
Model statistics
tas16
F2
,58
e18
,45
,69
Chi-square = 4751,455
df = 167
p = ,000
e16
,34
,67 das18
,21
G-of-F = ,922
Adj. G-of-F = ,901
RMS = ,067
e15
,22
,54
das04
,48
tas15
,05
F3
,47
tas11
e10
,15
,48
,29
e4
das10
,59
tas12
,35
e11
,35
,79
,22
e12
e8
tas08
,12
,23
e17
e5
das05
,25
das19
e19
,06
tas20
e20
51
Multiple group analysis




You can test equality/invariance of the
factor loadings for two separate groups
1) test the model to both groups
separately to check the entire model
2) the same model by multiple group
analysis
Need to have 2 separate data files for
each group.
Byrne. Structural Equation Modeling, 11, 272-300, 2004
52
Handling Missing data in SEM




Listwise or pairwise
Mean substitution
Regression methods
Expectation Maximization (EM) approach
Best methods

Full Information Maximum Likelihood
(FIML)

Multiple imputation (MI)
53
Checking for normality
Assessment of normality
Variable
min
max
skew
c.r.
kurtosis
c.r.
IDM
1.182 3.727
.381
4.649
.496
3.025
SEX1
1.000 2.000
.182
2.222
-1.967
-11.997
FRBEHB1 1.000 6.000
-.430
-5.245
-.778
-4.748
ISSUEB1
1.000 4.000
-.431
-5.259
-1.387
-8.462
SXPYRC1 2.000 7.000
-.937
-11.436
-.715
-4.360
-3.443
-6.149
Multivariate
Critical ratio of +/- 2 for skewness and kurtosis
statistical significance of NON-NORMALITY
Multivariate kurtosis >10  Severe Non-normality
54
Handling non-continuous data:
Bootstrapping
 Bootstrapping generates an
estimate of the sampling
distribution from the available data
and computes the p-values and
construct confidence intervals.
 Bootstrapping is useful for
estimating standard errors for
statistics with complex
distributions, for which there is no
practical approximate
 Bootstrapping in AMOS assumes
multivariate normality
55
Handling non-continuous data:
Bootstrapping


The “population” in nonparametric
bootstrapping is merely the
researcher’s sample
If the researcher’s sample is small,
unrepresentative, or the observations
are not independent, resembling
from it can magnify the effects of
these features
56
Problems in interpreting SEM
 Statistical assumptions and required
sample sizes are needed to have
confidence in the results
 Misrepresentation of causal
relationships. Most applications of
SEM are on non-experimental data
but many nevertheless interpret the
final model as causal.
57
References




Barrett. Structural Equation Modelling:
Adjudging model fit. Pers Indiv Diff, In
press.
Bentler & Chou. Practical issues in
structural modeling. Sociological Methods
and Research 16(1): 78-117, 1987.
Bentler & Stein. Structural equation models
in medical research. Stat Methods Med Res
1: 159–181, 1992.
Bollen. Structural equations with latent
variables. John Wiley & Sons, Inc, New
York, 1989.
58
References




Byrne. Structural Equation Modeling with
AMOS: Basic Concepts, Applications, and
Programming. Lawrence Erlbaum
Associates, Inc., 2001
Finch & West. The investigation of
personality structure: statistical models. J Res
Pers 31: 439–485, 1997.
MacCallum & Austin. Applications of
structural equation modeling in
psychological research. Annu Rev Psychol
51: 201–226, 2000.
De Stavola et al. Statistical issues in life
course epidemiology. Am J Epidemiol 163:
84-96, 2006.
59
References

Stevens. Applied multivariate statistics for
the social sciences. Mahwah, NJ: Lawrence
Erlbaum Publishers, 1996.
Wolfle. The introduction of path analysis to
the social sciences, and some emergent
themes: an annotated bibliography. Struct
Equation Model 10(1):1-34, 2003.

More references etc. in internet.






www.statmodel.com
www.spss.com/amos
http://www.upa.pdx.edu/IOA/newsom/semrefs.ht
m
http://amosdevelopment.com/AmosCitations.htm
http://www2.chass.ncsu.edu/garson/pa765/structur
.htm
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