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EPSY 651: Structural Equation Modeling I

EPSY 651: Structural Equation Modeling I. Where does SEM fit in Quantitative Methodology?. Draws on three traditions in mathematics and science: Psychology (Spearman, Kelley, Thurstone, Cronbach, etc. Sociology (Wright)

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EPSY 651: Structural Equation Modeling I

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  1. EPSY 651: Structural Equation Modeling I

  2. Where does SEM fit in Quantitative Methodology? Draws on three traditions in mathematics and science: Psychology (Spearman, Kelley, Thurstone, Cronbach, etc. Sociology (Wright) Agriculture and statistics: (Pearson, Fisher, Neymann, Rao, etc.) Largely due to Jöreskog in 1960s & 1970s Map below shows its positioning

  3. MANIFEST MODELING • Classical statistics within the parametric tradition • Canonical analysis subsumes most methods as special cases

  4. LATENT MODELING • Psychological concept of “FACTOR” is central to latent modeling: unobserved directly but “indicated” through observed variables • Emphasis on error as individual differences as well as problem of observation (measurement) rather than “lack of fit” conception in manifest modeling

  5. STRUCTURAL EQUATION MODELING PURPOSES • MODEL real world phenomena in social sciences with respect to • POPULATIONS • ECOLOGIES • TIME

  6. SEM PROCEDURE • FOCUS ON DECOMPOSITION OF COVARIANCE MATRIX: xy = (x,y,2x,2y,xy) + (ex,ey, exy) x =  +  y = By + x + e

  7. TESTING in SEM • SEM tests A PRIORI (theoretically specified) MODELS • SEM has potential to consider model revisions • SEM is not necessarily good for exploratory modeling

  8. SEM COMPARISONS • SEM can COMPARE Ecologies or Populations for identical models or • Simultaneously compare multiple groups or ecologies with each having unique models • Statistical testing is available for all parts of all models as well as overall model fit

  9. CORRELATION

  10. Karl Pearson (1857-1936. (exerpted from E S Pearson, Karl Pearson: An Appreciation of some aspects of his life and works, Cambridge University Press, 1938).

  11. Pearson Correlation n  (xi – xx)(yi – yy)/(n-1) rxy = i=1_____________________________ = sxy/sxsy sx sy =  zxizyi/(n-1) = COVARIANCE / SD(x)SD(y)

  12. COVARIANCE • DEFINED AS CO-VARIATION COVxy = Sxy • “UNSTANDARDIZED CORRELATION” • Distribution is statistically workable • Basis of Structural Equation Modeling (SEM) is constructing models for covariances of variables

  13. correlation covariance  1 – r2 se = standard deviation of errors .364 (40) .932(.955) SAT Math Calc Grade error Figure 3.4: Path model representation of correlation between SAT Math scores and Calculus Grades

  14. Path Models • path coefficient -standardized coefficient next to arrow, covariance in parentheses • error coefficient- the correlation between the errors, or discrepancies between observed and predicted Calc Grade scores, and the observed Calc Grade scores. • Predicted(Calc Grade) = .00364 SAT-Math + .5 • errors are sometimes called disturbances

  15. e X Y X Y Y X a b c Figure 3.2: Path model representations of correlation

  16. BIVARIATE DATA • 2 VARIABLES • QUESTION: DO THEY COVARY? • IF SO, HOW DO WE INTERPRET? • IF NOT, IS THERE A THIRD INTERVENING (MEDIATING) VARIABLE OR EXOGENOUS VARIABLE THAT SUPPRESSES THE RELATIONSHIP? OR MODERATES THE RELATIONSHIP

  17. IDEALIZED SCATTERPLOT • POSITIVE RELATIONSHIP Y Prediction line X

  18. IDEALIZED SCATTERPLOT • NEGATIVE RELATIONSHIP 95% confidence interval around prediction Y Prediction line Y. X. X

  19. IDEALIZED SCATTERPLOT • NO RELATIONSHIP Y Prediction line X

  20. SUPPRESSED SCATTERPLOT • NO APPARENT RELATIONSHIP Y MALES Prediction lines FEMALES X

  21. MODEERATION AND SUPPRESSION IN A SCATTERPLOT • NO APPARENT RELATIONSHIP Y MALES Prediction lines FEMALES X

  22. IDEALIZED SCATTERPLOT • POSITIVE CURVILINEAR RELATIONSHIP Y Quadratic prediction line Linear prediction line X

  23. Hypotheses about Correlations

  24. Hypotheses about Correlations • One sample tests for Pearson r • Two sample tests for Pearson r • Multisample test for Pearson r • Assumptions: normality of x, y being correlated

  25. One Sample Test for Pearson r • Null hypothesis:  = 0, Alternate   0 • test statistic: t = r/ [(1- r2 ) / (n-2)]1/2 with degrees of freedom = n-2

  26. One Sample Test for Pearson r • ex. Descriptive Statistics for Kindergarteners on a Reading Test (from SPSS) • Mean Std. Deviation N • Naming letters .5750 .3288 76 • Overall reading .6427 .2414 76 • Correlations • Naming Overall • Naming letters 1.000 .784** • Sig. (1-tailed) . .000 • N 76 76 • Overall reading .784** 1.000 • Sig. (1-tailed) .000 . • N 76 76 • ** Correlation is significant at the 0.01 level (1-tailed).

  27. One Sample Test for Pearson r Null hypothesis:  = c, Alternate   c • test statistic: z = (Zr - Zc )/ [1/(n-3)]1/2 where z=normal statistic, Zr = Fisher Z transform

  28. Fisher’s Z transform • Zr = tanh-1 r = (1/2) ln[(1+  r  ) /(1 -  r |)] • This creates a new variable with mean Z and SD 1/1/(n-3) which is normally distributed

  29. Non-null r example • Null: (girls) = .784 • Alternate: (girls)  .784 Data: r = .845, n= 35 • Z (girls=.784) = 1.055, Zr(girls=.845)=1.238 z = (1.238 - 1.055)/[1/(35-3)]1/2 = .183/(1/5.65685) = 1.035, nonsig.

  30. Two Sample Test for Difference in Pearson r’s • Null hypothesis: 1 = 2 • Alternate hypothesis 1  2 • test statistic: z =( Zr1 - Zr2 ) / [1/(n1-3) + 1/(n2-3)]1/2 where z= normal statistic

  31. Example • Null hypothesis: girls = boys • Alternate hypothesis girls  2boys • test statistic: rgirls = .845, rboys = .717 ngirls = 35, nboys = 41 z = Z(.845) - Z(.717) / [1/(35-3) + 1/(41-3)]1/2 = ( 1.238 - .901) / [1/32 + 1/38] 1/2 = .337 / .240 = 1.405, nonsig.

  32. Multisample test for Pearson r • Three or more samples: • Null hypothesis: 1 = 2 = 3 etc • Alternate hypothesis: some i  j • Test statistic: 2 = wiZ2i - w.Z2w which is chi-square distributed with #groups-1 degrees of freedom and wi = ni-3, w.= wi , and Zw = wiZi /w.

  33. Example Multisample test for Pearson r Nonsig.

  34. Multiple Group Models of Correlation • SEM approach models several groups with either the SAME or Different correlations: boys xy = a X y girls xy = a X y

  35. Multigroup SEM • SEM Analysis produces chi-square test of goodness of fit (lack of fit) for the hypothesis about ALL groups at once • Other indices: Comparative Fit Index (CFI), Normed Fit Index (NFI), Root Mean Square Error of Approximation (RMSEA) • CFI, NFI > .95 means good fit • RMSEA < .06 means good fit

  36. Multigroup SEM • SEM assumes large sample size, multinormality of all variables • Robust as long as skewness and kurtosis are less than 3, sample size is probably > 100 per group (200 is better), or few parameters are being estimated (sample size as low as 70 per group may be OK with good distribution characteristics)

  37. Multiple regression analysis

  38. Multiple regression analysis • The test of the overall hypothesis that y is unrelated to all predictors, equivalent to • H0: 2y123… = 0 • H1: 2y123… = 0 • is tested by • F = [ R2y123… / p] / [ ( 1 - R2y123…) / (n – p – 1) ] • F = [ SSreg / p ] / [ SSe / (n – p – 1)]

  39. Multiple regression analysis SOURCE df Sum of Squares Mean Square F x1, x2… p SSreg SSreg / p SSreg/ p SSe /(n-p-1) e (residual) n-p-1 SSe SSe / (n-p-1) total n-1 SSy SSy / (n-1)

  40. Multiple regression analysis predicting Depression LOCUS OF CONTROL, SELF-ESTEEM, SELF-RELIANCE

  41. SSreg ssx1 SSy SSe ssx2 Fig. 8.4: Venn diagram for multiple regression with two predictors and one outcome measure

  42. Type I ssx1 SSx1 SSy SSe SSx2 Type III ssx2 Fig. 8.5: Type I contributions

  43. Type III ssx1 SSx1 SSy SSe SSx2 Type III ssx2 Fig. 8.6: Type IIII unique contributions

  44. Multiple Regression ANOVA table SOURCE df Sum of Squares Mean Square F (Type I) • Model 2 SSreg SSreg / 2 SSreg / 2 • SSe / (n-3) • x1 1 SSx1 SSx1 / 1 SSx1/ 1 • SSe /(n-3) • x2 1 SSx2  x1 SSx2  x1SSx2  x1/ 1 • SSe /(n-3) • e n-3 SSe SSe / (n-3) • total n-1 SSy SSy / (n-3)

  45. PATH DIAGRAM FOR REGRESSION  = .5 X1 .387 r = .4 Y e X2  = .6 R2 = .742 + .82 - 2(.74)(.8)(.4)  (1-.42) = .85

  46. Depression e .471 .4 LOC. CON. -.317 DEPRESSION SELF-EST R2 = .60 -.186 SELF-REL

  47. Shrinkage R2 • Different definitions: ask which is being used: • What is population value for a sample R2? • R2s = 1 – (1- R2)(n-1)/(n-k-1) • What is the cross-validation from sample to sample? • R2sc = 1 – (1- R2)(n+k)/(n-k)

  48. Estimation Methods • Types of Estimation: • Ordinary Least Squares (OLS) • Minimize sum of squared errors around the prediction line • Generalized Least Squares • A regression technique that is used when the error terms from an ordinary least squares regression display non-random patterns such as autocorrelation or heteroskedasticity. • Maximum Likelihood

  49. Maximum Likelihood Estimation • Maximum likelihood estimation • There is nothing visual about the maximum likelihood method - but it is a powerful method and, at least for large samples, very preciseMaximum likelihood estimation begins with writing a mathematical expression known as the Likelihood Function of the sample data. Loosely speaking, the likelihood of a set of data is the probability of obtaining that particular set of data, given the chosen probability distribution model. This expression contains the unknown model parameters. The values of these parameters that maximize the sample likelihood are known as theMaximum Likelihood Estimatesor MLE's.  Maximum likelihood estimation is a totally analytic maximization procedure. • MLE's and Likelihood Functions generally have very desirable large sample properties:  • they become unbiased minimum variance estimators as the sample size increases • they have approximate normal distributions and approximate sample variances that can be calculated and used to generate confidence bounds • likelihood functions can be used to test hypotheses about models and parameters  • With small samples, MLE's may not be very precise and may even generate a line that lies above or below the data pointsThere are only two drawbacks to MLE's, but they are important ones:  • With small numbers of failures (less than 5, and sometimes less than 10 is small), MLE's can be heavily biased and the large sample optimality properties do not apply • Calculating MLE's often requires specialized software for solving complex non-linear equations. This is less of a problem as time goes by, as more statistical packages are upgrading to contain MLE analysis capability every year.

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