Fun With Structural Equation Modellingin Psychological Research: Fun With Structural Equation Modelling in Psychological Research Jeremy Miles IBS, Derby University


Slide2: Structural Equation Modelling Analysis of Moment Structures Covariance Structure Analysis Analysis of Linear Structural Relationships (LISREL) Covariance Structure Models Path Analysis


Normal Statistics: Normal Statistics Modelling process What is the best model to describe a set of data Mean, sd, median, correlation, factor structure, t-value Data Model


SEM: SEM Modelling process Could this model have led to the data that I have? Model Data


Slide5: Theory driven process Theory is specified as a model Alternative theories can be tested Specified as models Data Theory A Theory B


Ooohh, SEM Is Hard: Ooohh, SEM Is Hard It was. Now its not Jöreskog and Sörbom developed LISREL Matrices: lx qd ly qe Y F b G Variables: X Y h x z Intercepts: t k


The Joy of Path Diagrams: The Joy of Path Diagrams Variable Causal Arrow Correlational Arrow


Doing “Normal” Statistics: Doing 'Normal' Statistics x y Correlation


Slide9: Doing 'Normal' Statistics x y T-Test


Slide10: Doing 'Normal' Statistics x1 y One way ANOVA (Dummy coding) x2 x3


Slide11: Doing 'Normal' Statistics x1 y Two- way ANOVA (Dummy coding) x2 x1 * x2


Slide12: Doing 'Normal' Statistics x y Regression x x


Slide13: Doing 'Normal' Statistics MANOVA x1 x2 y1 y2 y3


Slide14: Doing 'Normal' Statistics ANCOVA x y z


Slide15: etc . . .


Identification: Identification Often thought of as being a very sticky issue Is a fairly sticky issue The extent to which we are able to estimate everything we want to estimate


Slide17: X = 4 Unknown: x


Slide18: x = 4 y = 7 Unknown: x, y


Slide19: x + y= 4 x - y = 1 Unknown: x, y


Slide20: x + y = 4 Unknown: x, y


Slide21: Things We Know Things We Want to Know = x=4 x + y = 4, x - y = 2 Just identified Can never be wrong 'Normal' statistics are just identified


Slide22: Things We Know Things We Want to Know andlt; x + y = 7 Not identified Can never be solved


Slide23: Things We Know Things We Want to Know andgt; x + y = 4, x - y = 2, 2x - y = 3 over-identified Can be wrong SEM models are over-identified


Identification: Identification We have information (Correlations, means, variances) 'Normal' statistics Use all of the information to estimate the parameters of the model Just identified All parameters estimated Model cannot be wrong


Over-identification: Over-identification SEM Over-identified The model can be wrong If a model is a theory Enables the testing of theories


Parameter Identification: Parameter Identification x - 2 = y x + 2 = y Should be identified according to our previous rules it’s not though There is model identification there is not parameter identification


Sampling Variation and c2: Sampling Variation and c2 Equations and numbers Easy to determine if its correct Sample data may vary from the model Even if the model is correct in the population Use the c2 test to measure difference between the data and the model Some difference is OK Too much difference is not OK


Simple Over-identification: Simple Over-identification x y Estimate 1 parameter -just-identified x y Estimate 0 parameters -over-identified


Example 1: Example 1 Rab = 0.3, N = 100 Estimate = 0.3, SE = 0.105, C.R. = 2.859 The correlation is significantly different from 0 a b


Slide30: Model Tests the hypothesis that the correlation in the population is equal to zero It will never be zero, because of sampling variation The c2 tells us if the variation is significantly different from zero a b


Example 2: Example 2 Test the model Force the value to be zero Input parameters = 1 Parameters estimated = 0 The model is now over-identified and can therefore be wrong a b


Slide32: The program gives a c2 statistic The significance of difference between the data and the model Distributed with df = known parameters - input parameters c2 = 9.337, df = 1 - 0 = 1, p = 0.002 So what? A correlation of 0.3 is significant?


Hardly a Revelation : Hardly a Revelation No. We have tested a correlation for significance. Something which is much more easily done in other ways But We have introduced a very flexible technique Can be used in a range of other ways


Testing Other Than Zero: Testing Other Than Zero Estimated parameters usually tested against zero Reasonable? Model testing allows us to test against other values c2 = 2.3, n.s. Example 3


Example 4: Comparing correlations: Example 4: Comparing correlations 4 variables mothers' sensitivity mothers' parental bonding fathers' sensitivity fathers' parental bonding Does the correlation differ between mothers and fathers?


Slide36: M S M PB F PB F S 0.5 0.3 0.1 0.1 0.2 0.2


Slide37: Example 4a analyse with all parameters free 0 df, model is correct Example 4b fix FS-FPB and MS-MPB to be equal. See if that model can account for the data


Slide38: M S M PB F PB F S dave dave c2 = 1.82, df = 1 p = 0.177 dave = 0.41 (s.e. 0.08)


Latent Variables: Latent Variables The true power of SEM comes from latent variable modelling Variables in psychology are rarely (never?) measured directly the effects of the variable are measured Intelligence, self-esteem, depression Reaction time, diagnostic skill


Measuring a Latent Variable: Measuring a Latent Variable Latent variables are drawn as ellipses hypothesised causal relationship with measured variables Measured variable has two causes latent variable 'other stuff' random error


Slide41: x = t + e Reliability is: the square root of proportion of variance in x that is accounted the correlation between x and e Measured True Score Error


Identification and Latent Variables: Identification and Latent Variables 1 measured variable not (even close to) identified 4 measured variables 6 known, 4 estimated model is identified


Slide43: Need four measured variables to identify the model Need to identify the variance of the latent variable fix to 1


Why oh why oh why?: Why oh why oh why? Why bother with all these tricky latent variables? 2 reasons unidimensional scale construction attenuation correction


Unidimensionality: Unidimensionality Correlation matrix c2 = 3.65, df = 2, p = 0.16 1.00 0.68 1.00 0.73 0.63 1.00 0.68 0.63 0.69 1.00


Attenuation Correction: Attenuation Correction Why bother? Gets accurate measure of correlation between true scores Why bother theories in psychology are ordinal attenuation can only cause relationships to lower


The Multivariate Case: The Multivariate Case Much more complex and unpredictable x1 y1 x2 y2 a c d e b


Some More Models: Some More Models Multiple Trait Multiple Method Models (MTMM) Temporal Stability Multiple Indicator Multiple Cause (MIMIC)


MTMM: MTMM Multiple Trait more than one measure Multiple Method using more than one technique Variance in measured score comes from true score, random error variance, and systematic error variance, associated with the shared methods


What?: What? Example 6 (From Wothke, 1996) Three traits Getting along with others (G) Dedication (D) Apply learning (L) Three methods Peer nomination (PN) Peer Checklist (PC) Supervisor ratings (SC)


Matrix: Matrix 1 .524 1 .241 .403 1 .071 .102 -.018 1 .022 .096 .018 .435 1 .076 .102 .100 .342 .347 1 .136 .132 .061 .243 .203 .100 1 -.028 .168 .135 .093 .209 .042 .461 1 -.054 .162 .252 .053 .108 .108 .294 .280 1 g.pn d.pn l.pn g.pc d.pc l.pc g.sc d.sc l.sc


Analysis: Analysis g l d


Temporal Stability: Temporal Stability Usually sum the items correlate them BUT items may not be unidimensional relationship will be attenuated due to measurement error relationship will be inflated, due to correlated error


Slide54: L1 X3.1 X4.1 X5.1 X2.1 X1.1 L2 X3.2 X4.2 X5.2 X2.2 X1.2 Corrects for attenuation But - correlated errors may be a problem


Slide55: Added correlated errors Example 7b L1 X3.1 X4.1 X5.1 X2.1 X1.1 L2 X3.2 X4.2 X5.2 X2.2 X1.2


MIMIC Model: MIMIC Model 'Conventional wisdom' in psychological measurement is that a latent variable is the cause of the measured variables Assumption is made (implicitly) in many types of measurement Bollen and Lennox (1989) not necessarily the case


Value of a Car: Value of a Car Causes type, size, age, rustiness no reason they should, or should not, be correlated Effects assessment of value by people who know


Level of Depression: Level of Depression Questionnaire items causes or effects? been feeling unhappy and depressed? been having restless and disturbed nights? found everything getting 'on top' of you? MIMIC


Example 8: MIMIC: Example 8: MIMIC L1 c1 c2 c3 y4 y1 LY1 LY2 y2 y3 y5 y6 y7 y8


Concluding remarks: Concluding remarks Given a taster some may be too simple? Much more to say no time to say it See further reading (Books and WWW)


Further Info: Further Info SEMNET - email list semnet@bama.ua.edu (messages) listserv@ bama.ua.edu (leave) http://www.gsu.edu/~mkteer/semfaq.html the semnet FAQ


Books: Books See web page http://ibs.derby.ac.uk/~jeremym/fun/fun/index.htm


References: References See web page http://ibs.derby.ac.uk/~jeremym/fun/fun/index.htm