LISREL Workshop at M3 Conference 2015

Transcription

1 LISREL Workshop at M3 Conference 2015 Karl G Jöreskog Norwegian Business School & Uppsala University May 11, 2015 Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

2 Ways of Working with LISREL Use LISREL s Graphical Interface (GUI) Use Syntax Files PRELIS Syntax SIMPLIS Syntax LISREL Syntax Other Syntax Syntax for Generalized Linear Models Syntax for Multilevel Analysis Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

3 Learning by Examples Real Examples and Real Data Learning How to Do it with LISREL Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

4 Topics to be Covered Creating the LISREL system data file (LSF file) Exploratory and Confirmatory Factor Analysis Estimating Latent Mean Differences Estimating Latent Growth Curves from Data with Missing Values Latent Variable Scores Multilevel Analysis Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

5 Example: Holzinger & Swineford Two Schools Data In this example we learn 1 how to import data into a LISREL system data file.lsf 2 how to graph and screen the data 3 how to divide the data into two groups (schools) 4 how to do exploratory factor analysis in one school 5 how to do confirmatory factor analysis in the other school 6 how to deal with non-normality 7 how to modify the model by means of path diagrams 8 how to interpret chi-square differences 9 how to estimate and test latent variable differences Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

6 Holzinger & Swineford (1939) collected data on twenty-six psychological tests administered to seventh- and eighth-grade children in two schools in Chicago: the Pasteur School and the Grant-White School. The Pasteur School had students whose parents had immigrated from Europe, mostly France and Germany. The students of the Grant-White School came from middle American white families. Nine of these tests are selected for this example. The nine tests are (with the original variable number in parenthesis): VISPERC Visual Perception (V1) CUBES Cubes (V2) LOZENGES Lozenges (V4) PARCOMP Paragraph Comprehension (V6) SENCOMP Sentence Completion (V7) WORDMEAN Word meaning (V9) Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

7 ADDITION Addition (V10) COUNTDOT Counting dots (V12) S-C CAPS Straight-curved capitals (V13) The nine test scores are preceded by four variables: SCHOOL 0 for Pasteur School; 1 for Grant-White School GENDER 0 for Boy; 1 for Girl AGEYEAR Age in years BIRTHMON Birthmonth: 1 = January, 2 = Februray = December The data file is an SPSS data file hsschools.sav. There are no missing values. Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

8 Importing, Graphing and Screening the Data It is important to understand what you can do before you learn to measure how well you seem to have done it. (Tukey,1977). Importing hsschools.sav to hsschools.lsf Defining the attributes of the data Variable Type: Ordinal or Continuou Missing Values Histogram: Plot WORDMEAN Bar Chart: Plot AGEYEAR Scatter Plot: Plot WORDMEAN on VISPERC Three-Dimensional Bar Chart: Plot AGEYEAR on SCHOOL Box-Whisker Plot: Plot WORDMEAN on SCHOOL Data Screening Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

9 The Box-Whisker Plot As implemented in LISREL the Box and Whisker plot gives the most important characteristics of the distributions of the continuous variable for each value of the categorical variable. The box indicates the area between lower and upper quartile so that 25% of the observations is below the bottom border of the box and 25% of the observations is above the top border of the box. The border line between the two colored areas in the box is the mean. If the areas above and below the mean are of different sizes, this indicates that the distribution of the observations is skewed. The two unfilled circles are the largest and the smallest value in the data. The wiskers are located ±2 IQR, where IQR is the interquartile range, i.e. the distance between the upper and lower quartile which is the length of the box. Observations that are outside of the whiskers are sometimes called outliers. Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

10 Splitting the Data into Groups Selecting Cases Selecting or Deleting Variables Tests of Univariate and Multivariate Normality Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

11 Exploratory Factor Analysis of the Pasteur School Sample Number of Factors Rotation Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

12 Confirmatory Factor Analysis of the Grant-White School Sample The distinction between exploratory and confirmatory factor analysis is illustrated in the following figures Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

13 To illustrate confirmatory factor analysis we use the data on the nine psychological variables for the Grant White school. For simplicity here we will call the data npv.lsf and use the short form npv in all files. Bare in mind that this refer to the Grant White school sample. It is hypothesized that these variables have three correlated common factors: visual perception here called Visual, verbal ability here called Verbal and speed here called Speed such that the first three variables measure Visual, the next three measure Verbal, and the last three measure Speed. A path diagram of the model to be estimated is shown in the next slide. Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

14 Confirmatory Factor Analysis Model for Nine Psychological Variables Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

15 Estimating and Testing the Model by Maximum Likelihood (ML) Estimating and Testing the Model by Robust Maximum Likelihood (RML) Evaluating the Fit Modifying the Model Estimating and Testing the Modified Model by Robust Maximum Likelihood (RML) Evaluating Chi-Square Differences Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

16 Parameter Estimates, Normal Standard Errors, and Robust Standard Errors Parameter Parameter Standard Errors Factor Loadings Estimates Normal Robust VISPERC on Visual λ (x) CUBES on Visual λ (x) LOZENGES on Visual λ (x) PARCOMP on Verbal λ (x) SENCOMP on Verbal λ (x) WORDMEAN on Verbal λ (x) ADDITION on Speed λ (x) COUNTDOT on Speed SCCAPS on Speed Factor Correlations Verbal vs Visual φ Verbal vs Speed φ Verbal vs Speed φ λ (x) λ (x) Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

17 Modified Confirmatory Factor Analysis Model for Nine Psychological Variables Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

18 Example: Stability of Alienation Wheaton, et al. (1977) report on a study concerned with the stability over time of attitudes such as alienation, and the relation to background variables such as education and occupation. Data on attitude scales were collected from 932 persons in two rural regions in Illinois at three points in time: 1966, 1967, and The variables used for the present example are the Anomia subscale and the Powerlessness subscale, taken to be indicators of Alienation. This example uses data from 1967 and 1971 only. The background variables are the respondent s education (years of schooling completed) and Duncan s Socioeconomic Index (SEI). These are taken to be indicators of the respondent s socioeconomic status (Ses). The data file is stability.lsf. Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

19 The research questions are: Has the feeling of alienation increased or degreased between 1967 and 1971? Is the feeling of alienation stable over time? Stability is indicated by the variance of alienation. If the variance has changed much between 1967 and 1971 it means that for some people the feeling of alienation has increased while it has decreased for other people. Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

20 The model to be considered here is shown in Slide 22. For Models A and B we estimate this model as covariance strcture. Model A we assume uncorrelated measurement errors. For Model B we assume that the measurement errors on ANOMIA67 and ANOMIA71 and the mesurement errors on POWERL67 and POWERL71 are correlated as in Slide 22. For Model C we add a mean structure to estimate the latent mean difference between Alien71 and Alien67. Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

21 The error terms of ANOMIA and POWERL are assumed to be correlated over time to take specific factors into account. The four one-way arrows on the right side represent the measurement errors in ANOMIA67, POWERL67, ANOMIA71, and POWERL71, respectively. The two-way arrows on the right side indicate that some of these measurement errors are correlated. The covariance between the two error terms for each variable can be interpreted as a specific error variance. Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

22 Stability of Alienation Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

23 Using the notation y 1 = ANOMIA67, y 2 = POWERL67, y 3 = ANOMIA71, y 4 = POWERL71, x 1 = EDUC, x 2 = SEI, ξ = Ses, η 1 = Alien67, η 2 = Alien71, the model is y 1 y 2 y 3 y 4 ( η1 = η 2 ( x1 x 2 ) = τ (y) 1 τ (y) 2 τ (y) 3 τ (y) 4 ) ( = ( 0 0 β 0 + τ (x) 1 τ (x) 2 ) ) ( η1 1 0 λ λ 2 ( 1 + η 2 ) + λ 3 ( γ1 ( η1 η 2 ) + ) ( δ1 ξ + γ 2 ) ξ + ) δ 2 ( ζ1 ζ 2 ɛ 1 ɛ 2 ɛ 3 ɛ 4 ) (1) (2) (3) Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

24 Estimating a Specific Variance The unique part of a measurement (the error terms δ i and ɛ i ) usually consists of two components: a specific factor s i and a pure random measurement error e i. These are indistinguishable, unless the measurements are designed in such a way that they can be separately identified (panel designs and multitrait-multimethod). The error terms δ i and ɛ i are often called measurement errors even though it is widely recognized that these terms may also contain a specific factor. Since ANOMIA67 and ANOMIA71 are the same measurement they may contain a specific factor s, so that the error terms ɛ 1 = s + e 1, (4) ɛ 3 = s + e 3, (5) where e 1 and e 3 are pure random measurement error. Then Cov(ɛ 1, ɛ 3 = Var(s). (6) Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

25 Intercepts and Means in SIMPLIS The variable CONST is a variable which is equal to 1 for every case. This variable is always available in SIMPLIS; it need not be in the data. It is used to estimate an intercept term or a mean. For example, Y = CONST X is used to specify the regression of Y on X: Y = α + γx. α is the coefficient of CONST just like γ is the coefficient of X. One can also use CONST to estimate a mean. For example, Y = CONST will estimate the mean of Y as the coefficient of CONST. Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

26 Analysis of Longitudinal Data with Missing Values Examples of Longitudinal Data Psychology The develepment of cognitive abilities in children Economy Growth of nations or sales of companies Sociology Change in crime rates across communities Time Intervals: Dayly, Weekly, Monthly, or Annually Unit of Analysis: Individuals, Businesses, Regions, Countries Regardless of the subject area or the time interval, social and behavioral scientists have a keen interest in describing and explaining the time trajectories of their variables. (Bollen & Curran (2006, p. 1) Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

27 Types of Incomplete Data Unit nonresponse Subject attrition Item nonresponse Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

28 Mechanisms of Nonresponse MCAR Missing completely at random MAR Missing at random MNAR Missing not at random Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

29 Probability of Missing Data MCAR Pr(z ij = missing) does not depend on any variable in the data. MAR Pr(z ij = missing) may depend on other variables in the data but not on z ij. Example: A missing value of a person s income may depend on his/her age and education but not on his/her actual income. MNAR Pr(z ij = missing) depends on z ij. Example: In a questionnaire people with higher income tend not to report their income. Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

30 Solutions to Incomplete Data Listwise deletion Pairwise deletion Imputation by matching Multiple imputation EM MCMC Direct analysis (FIML) Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

31 Maximum Likelihood with Missing Values Let z be a random vector of order k with a multivariate normal distribution with mean vector µ and covariance matrix Σ both of which are functions of a parameter vector θ. Suppose we have a sample of N independent observations of z. If there are missing data we assume they ara missing at random. Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

32 Let z i of order k i be the subset of z observed in unit i. Then z i N(µ i, Σ i ), where µ i and Σ i consist of the rows and columns of µ and Σ for which there are observations in z i. The logarithm of the likeliihod for the total sample is ln L = 1 2 N i=1 {ln(2π) + ln Σ i + tr[σ 1 i (z i µ i )(z i µ i ) ]} Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

33 Instead of maximizing ln L, maximum likelihood estimates of θ can be obtained by minimizing the function F (θ) = N i=1 {ln Σ i + trσ 1 i G i }, where G i = (z i µ i )(z i µ i ). The first and second derivatives of F (θ) required for the minimization can be obtained in the usual way. The deviance for the saturated model can be obtained by minimizing with respect to µ and Σ. N i=1 {ln Σ i + trσ 1 i G i }, Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

34 The Linear Curve Model y it = a i + b i T t + e it (7) i = 1, 2,..., N persons (8) T t = Time at occasion t = 1, 2,..., n i (9) a i = α + u i (10) ( ui b i = β + v i (11) ) N(0, Φ) (12) v i e it N(0, σe) 2 (13) y it = (α + βt t ) + (u i + v i T t + e it ) (14) Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

35 Trajectory for Person 1 y Time Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

36 Fitted Regression Line for Person 1 y Time Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

37 Fitted Regression Lines for Four Persons y Time The thick line is the average regression line. Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

38 A Quadratic Curve Model y it = a i + b i T t + c i T 2 t + e it (15) i = 1, 2,..., N persons (16) T t = Time at occasion t = 1, 2,..., n i (17) a i = α + u i (18) b i = β + v i (19) u i v i w i c i = γ + w i (20) N(0, Φ), e it N(0, σ 2 e) (21) y it = (α + βt t + γt 2 t ) + (u i + v i T t + w i T 2 t + e it ) (22) Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

39 The Linear Curve Model with Covariate y it = a i + b i T t + e it (23) i = 1, 2,..., N persons (24) T t = Time at occasion t = 1, 2,..., n i (25) a i = α + γ a z i + u i (26) b i = β + γ b z i + v i (27) z i = Covariate observed on person i (28) ( ) ui N(0, Φ) (29) v i e it N(0, σe) 2 (30) y it = (α + βt t + γ a z i + γ b T t z i ) + (u i + v i T t + e it ) (31) Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

40 PD of the Linear Curve Model with Covariate y 1 y 2 y 3 y a b z Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

41 Example: Treatment of Prostate Cancer A medical doctor offered all his patients diagnosed with prostate cancer a treatment aimed at reducing the cancer activity in the prostate. The severity of prostate cancer is often assessed by a plasma component known as prostate specific antigen (PSA), an enzyme that is elevated in the presence of prostate cancer. The PSA level was measured regularly every three months. The data contains five repeated measurements of PSA. The age of the patient is also included in the data. Not every patient accepted the offer initially and several patients chose to enter the program after the first occasion. Some patients, who accepted the initial offer, are absent at some later occasions for various reasons. Thus there are missing values in the data. Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

42 The aim of this study is to answer the following questions: What is the average initial PSA value? Do all patients have the same initial PSA value? Is there an overall effect of treatment. Is there a decline of PSA values over time, and, if so, what is the average rate of decline? Do all patients have the same rate of decline? Does the individual initial PSA value and/or the rate of decline depend on the patient s age? This is a typical example of repeated measurements data, the analysis of which is sometimes done within the framework of multilevel analysis. It represents the simplest type of two-level model but it can also be analyzed as a structural equation model, see Bollen & Curran (2006). In this context it illustrates a mean and covariance structure model estimated from longitudinal data with missing values. Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

43 Two Ways of Organizing the Data As multilevel data As longitudinal data Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

44 PSA Data in Multilevel Form Patient Months PSA Age Patients 9 and 10 are missing at 3 months Patient 15 is missing at 3 and 6 months Patient 16 is missing at 0, 3, and 12 months Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

45 PSA Data in Longitudinal Form Patient PSA0 PSA3 PSA6 PSA9 PSA12 Age Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

46 The data file for this example is psavar.lsf, where missing values are shown as If the data is imported from an external source which already have a missing value code, the missing values will show up in the lsf file as , which is the global missing data code in LISREL. Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

47 In this kind of data it is inevitable that there are missing values. For example, a patient may be on vacation or ill or unable to come to the doctor for any reason at some occasion or a patient may die and therefore will not come to the doctor after a certain occasion. It is seen in that Patients 9 and 10 are missing at 3 months Patient 15 is missing at 3 and 6 months Patient 16 is missing at 0, 3, and 12 months In the following analysis it is assumed that data are missing at random (MAR), although there may be a small probability that a patient will be missing because his PSA value is high. Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

48 The model to be estimated is y it = a i + b i T t + e it (32) i = 1, 2,..., N individuals (33) T t = Time at occasion t = 1, 2,..., n i (34) a i = α + γ a z i + u i (35) b i = β + γ b z i + v i (36) z i = Covariate observed on individual i (37) ( ) ui N(0, Φ) (38) v i e it N(0, σ 2 e) (39) Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

49 An interpretation of this is as follows. Each patient has his own linear growth curve 2, represented by (32) which is the regression of y it on time with intercept a i and slope b i varying across patients. In principle, the intercepts a i and slopes b i could all be different across patients. It is of interest to know if the intercepts and/or the slopes are equal across patients. The four cases are illustrated in the next Figure. If there is variation in intercepts and/or the slopes across patients, one is interested in whether a covariate z i (in this case age) can predict the intercept and/or the slope. 2 In general, the growth curves are not restricted to be linear, but can be quadratic, cubic, or other types of functions of time, see Jöreskog, Sörbom, Du Toit, & Du Toit (2003). Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

50 Four Cases of Intercepts and Slopes Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

51 Linear Growth Curve for psavar Data Raw Data from File psavar.lsf Latent Variables: a b Relationships PSA0 = 1*a 0*b PSA3 = 1*a 3*b PSA6 = 1*a 6*b PSA9 = 1*a 9*b PSA12 = 1*a 12*b a b = CONST Equal Error Variances: PSA0 - PSA12 Path Diagram End of Problem Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

52 Linear Growth Curve with Covariate for psavar Data Raw Data from File psavar.lsf Latent Variables: a b Relationships PSA0 = 1*a 0*b PSA3 = 1*a 3*b PSA6 = 1*a 6*b PSA9 = 1*a 9*b PSA12 = 1*a 12*b a b = CONST Age Let the Errors on a and b correlate Equal Error Variances: PSA0 - PSA12 Path Diagram End of Problem Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

53 Example: Training of Air Traffic Controllers Kanfer & Ackerman (1989) analyzed data on 141 U.S. Air Force enlisted personnel who carried out a computerized air traffic controller task developed by Kanfer and Ackerman. 3 The subjects were instructed to accept planes into their hold pattern and land them safely and efficiently on one of four runways, varying in length and compass directions, according to rules governing plane movements and landing requirements. For each subject, the success of a series of between three and six 10-minute trials was recorded. The measurement employed was the number of correct landings per trial. A measure of cognitive ability was also obtained from each controller at the begin of the trials. 3 Permission for SSI to use the copyrighted raw data was provided by R. Kanfer and P.L. Ackerman. The data remain the copyrighted property of Ruth Kanfer and Phillip L. Ackerman. Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

54 In this example we learn 1 how to analyze longitudinal data with missing values 2 alternative ways to deal with missing values 3 how to plot and interpret a box-plot 4 how to decide the shape of the growth curve 5 how to estimate and test so called latent growth curves using FIML Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

55 Questions 1 Should the learning curve be linear, quadratic, cubic, or other? 2 Do the shape of the curve differ between controllers? 3 Do the coefficients describing the learning curve depend on cognitive ability? Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

56 Box Whisker Plot of Air Controller Data Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

57 A Cubic Fitted to Three Controllers Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

58 Latent Variable Scores There are several methods available for estimating latent variable scores 4, e.g., Lawley & Maxwell (1971, Chapter 8) or Bartholomew & Knott (1999, pp ). The two most commonly used methods for estimating latent variable scores are the regression method of Thomson (1939) and the Bartlett method (Bartlett, 1937). In LISREL we use the procedure of Anderson & Rubin (1956) as described in Jöreskog (2000). This procedure has the advantage of producing latent variable scores that have the same covariance matrix as the latent variables themselves. The results reported here are based on latent variable scores estimated by the Anderson & Rubin method. This section describes how latent variable scores can be obtained with LISREL 9 and illustrates their use with one example. 4 In classical exploratory factor analysis these are usually called factor scores Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

59 Example: Panel Model for Political Democracy Bollen (1989, p. 17) presents a panel model of political democracy and industrialization in 75 countries. The model is shown in the path diagram in the following Figure Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

60 The variables in the model are y 1 Freedom of press 1960 y 2 Freedom of political opposition 1960 y 3 Fairness of elections 1960 y 4 Effectiveness of legislature 1960 y 5 Freedom of press 1965 y 6 Freedom of political opposition 1965 y 7 Fairness of elections 1965 y 8 Effectiveness of legislature 1965 x 1 GNP per capita 1960 x 2 Energy consumtion per capita 1960 x 3 Percentage of labor force in industry 1960 η 1 Democracy in 1960 (Latent variable Dem60) η 2 Democracy in 1965 (Latent variable Dem65) ξ Level of industrialization in 1960 (Latent variable Indus) Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

61 y 1 y 4 are taken as indicators of the latent variable Dem60 (Democracy 1960) and y 5 y 8 are taken as indicators of the latent variable Dem65 (Democracy 1965). x 1 x 3 are taken as indicators of the latent variable Indus (Industrialization 1960). Data on y 1 x 3 are available for 75 developing countries. The datafile is POLIDEM.LSF. The following SIMPLIS syntax file (BA1a.SPL) will estimate scores on Demo60, Demo65, and Indus for each country in file POLIDEMnew.LSF. Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

62 Industrialization-Democracy Example Raw Data from file POLIDEM.LSF Latent Variables: Dem60 Dem65 Indus Relationships: Y1= 1*Dem60 Y2-Y4 = Dem60 Y5 = 1*Dem65 Y6-Y8 = Dem65 X9 = 1*Indus X10-X11 = Indus Dem60 = Indus Dem65 = Dem60 Indus Set Dem60 -> Y2 = Dem65 -> Y6 Set Dem60 -> Y3 = Dem65 -> Y7 Set Dem60 -> Y4 = Dem65 -> Y8 Let the errors of Y5 and Y1 be correlated let the errors of Y6 and Y2 be correlated Let the errors of Y7 and Y3 be correlated Let the errors of Y8 and Y4 be correlated LSFfile POLIDEMnew.LSF Path Diagram End of Problem Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

63 The variables y 1 y 4 are the same variables as y 5 y 8 measured at two points in time. So Bollen & Arminger (1991) assume that their loadings on η 1 and η 2 are the same. This is specified by the lines Set Dem60 -> Y2 = Dem65 -> Y6 Set Dem60 -> Y3 = Dem65 -> Y7 Set Dem60 -> Y4 = Dem65 -> Y8 The loadings of y 1 and y 5 are set to 1. Furthermore, they assume that the measurement errors of corresponding y-variables are correlated. This is specified by the lines Let the errors of Y5 and Y1 be correlated let the errors of Y6 and Y2 be correlated Let the errors of Y7 and Y3 be correlated Let the errors of Y8 and Y4 be correlated Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

64 After this run is completed the file POLIDEMnew.LSF contains the following variables Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 X1 X2 X3 Dem60 Dem65 Indus It would be useful to know which country each row belongs to. Unfortunately, this information is not available. However, one can construct a variable COUNTRY which runs from 1 to 75 5 This is done with the following PRELIS syntax file (BA2.PRL) which at the same time constructs scores on another latent variable Diff = Dem65 - Dem60. SY=POLIDEMnew.LSF New COUNTRY=TIME New Diff=Dem65-Dem60 CO ALL Select COUNTRY Dem60 Dem65 Diff OU RA=DEMDIFF.LSF 5 The variable TIME is always available in PRELIS. It assigns values 1, 2,..., N to the cases. It is intended mainly for time series. Hence its name TIME. Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

65 The file DEMDIFF.LSF contains the following variables COUNTRY Dem60 Dem65 Diff With this file one can do various things to find out which countries have most democracy or which countries increased or decreased their democracy between 1960 and For example, do a bivariate line plot of Diff against COUNTRY. This shows that country 2 increased democracy most and country 30 had the largest decrease. This can also be seen by sorting Diff in descending order. This shows that country 2 has a Diff value of 1.69 and country 30 has a Diff value of These are the best and worst countries. The second best and second worst countries are the countries 22 and 34. These have Diff values of 1.39 and -1.38, respectively. Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

66 Example: Attitudes to Drinking and Driving (DRINK) In this example we learn 1 how to test each part of a model one at a time 2 how to build the model successively by adding one part at a time 3 how to estimate and test each model using SIMPLIS syntax 4 how to evaluate the fit of the final model 5 how to estimate the final model using LISREL syntax Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

67 A study designed to determine the predictors of drinking and driving behavior used the theory of planned behavior of Ajzen (1991) to setup the model shown in the path diagram in Figure 68. The latent variables shown in the figure are as follows: Attitude attitude toward drinking and driving Norms social norms pertaining to drinking and driving Control percieved control over drinking and driving Intention intention to drink and drive Behavior drinking and driving behavior Attitude is measured by five indicators X 2 -X 5, Norms is measured by three indicators X 6 -X 8, Control is measured by four indicators X 9 -X 12, Intention is measured by two indicators Y 1 and Y 2, and behavior is measured by two indicators Y 3 and Y 4. Data on 756 male drivers aged years is given in file drinkdata.lsf. Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

68 Figure: Conceptual Path Diagram for Attitudes to Drinking and Driving Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

69 This example illustrates a possible strategy of analysis. Begin by testing the measurement model for Attitude, see file drink11a.spl. Drinking and Driving Testing the Measurement Model for Attitude Raw Data from File drinkdata.lsf Latent Variables Attitude Relationships X1-X5 = Attitude Path Diagram End of Problem Note that although the data file drinkdata.lsf contains many variables, LISREL automatically selects the subset of variables used in the model. Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

70 To test the measurement model for Attitude and Norms simultaneously, add Norms in the list of Latent Variables and add the line, see file drink12a.spl. X6-X8 = Norms To test the measurement model for Attitude, Norms, and Control simultaneously, add Control in the list of Latent Variables and add the line, see file drink13a.spl. X9-X12= Control Finally, to test the measurement model for all latent variables simultaneosly, add Intention and Behavior in the list of Latent Variables and add the two lines, see file drink14a.spl. Y1-Y2 = Intention Y3-Y4 = Behavior Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

71 If any of these analysis shows a large modification index for an indicator, the measurement model must be reconsidered and modified. For example, suppose there is a large modification index for the path from Attitude to X8. This might mean that X8 is not entirely an indicator of Norms but to some extent also a measure of Attitude. If this idea makes sense then the model should be modified by letting X8 be a composite measure of both Norms and Attitude. One can now test the full model in Figure 68 by adding the two lines, see file drink15a.spl. Intention = Attitude - Control Behavior = Intention defining the structural relationships among the latent variables Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

72 The full SIMPLIS command file is now Drinking and Driving Raw Data from File drinkdata.lsf Latent Variables Attitude Norms Control Intention Behavior Relationships Y1-Y2 = Intention Y3-Y4 = Behavior X1-X5 = Attitude X6-X8 = Norms X9-X12= Control Intention = Attitude - Control Behavior = Intention Robust Estimation Path Diagram End of Problem Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73

73 The output gives the following goodness-of-fit statistics Degrees of Freedom for (C1)-(C2) 96 Maximum Likelihood Ratio Chi-Square (C1) (P = ) Browne s (1984) ADF Chi-Square (C2_NT) (P = ) which suggests that the model does not fit. However, this is a test of exact fit. According to Browne & Cudeck (1993) one can use the following fit measures to test if the model fits approximately Root Mean Square Error of Approximation (RMSEA) Percent Confidence Interval for RMSEA ( ; ) P-Value for Test of Close Fit (RMSEA < 0.05) 1.00 which suggests that the model fits at least approximately. Karl G Jöreskog ( ) LISREL Workshop at M3 Conference 2015 May 11, / 73