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# Introduction to Structural Equation Modeling

My advanced stats lecture notes on Introduction to Structural Equation Modeling.

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### Introduction to Structural Equation Modeling

1. 1. ©drtamil@gmail.com 2020 Introduction to SEM Dr Azmi M Tamil
2. 2. ©drtamil@gmail.com 2020 Disclaimer • I am not a trainer for SEM. I just introduce the topic for my postgraduate students, as part of a course module in Advanced Statistics. • These notes are partially based on Prof Mohd. Ayub Sadiq@Lin Naing’s 2011 lecture notes on AMOS. • Those using SEM in their thesis are advised to attend other workshops specifically for SEM.
3. 3. ©drtamil@gmail.com 2020 My Reference A Beginner's Guide to Structural Equation Modeling. 4th edition. Randall E. Schumacker, Richard G. Lomax
5. 5. ©drtamil@gmail.com 2020 Latent versus Measured Variables • Latent variables – also known as construct or factors. They are usually not directly observable or measurable. • Latent variables are indirectly observed or measured or inferred from a set of items or questions that we posed to respondents. • The set of questions tend to be focused on the latent trait that we want to measure such as the level of fitness or the level of mental health.
6. 6. ©drtamil@gmail.com 2020 Covariance versus Correlation When comparing data samples from different populations, • covariance is used to determine how much two random variables vary together. Range from negative infinity to positive infinity. • Whereas correlation is used to determine when a change in one variable can result in a change in another. Range from negative 1 to positive 1. • Both covariance and correlation measure linear relationships between variables. Scatter diagram is similar for both.
7. 7. ©drtamil@gmail.com 2020 Co-variance • Co-variance - the value of the product of the deviations (variance) of two variates from their respective means. • Variance – deviations of a single variate from the mean.
8. 8. ©drtamil@gmail.com 2020 • Covariance is a measure of how much two random variables vary together. It’s similar to variance, but where variance tells you how a single variable varies, co variance tells you how two variables vary together. – Xij & Xik are the random variables (of X & Y) – xj = is the expected value (the mean μ) of the random variable X. – xk= is the expected value (the mean μ) of the random variable Y. – N = the number of items in the data set. • Look at the formula, it is like variance, but not squared. https://www.statisticshowto.datasciencecentral.com/covariance/
9. 9. ©drtamil@gmail.com 2020 2015. Randall E. Schumacker, Richard G. Lomax. A Beginner's Guide to Structural Equation Modelling. 4th ed.
10. 10. ©drtamil@gmail.com 2020 Pearson’s Correlation • measures the degree of linear association between two interval scaled variablese.g., between height and weight. • r lies between -1 and 1. Values near 0 means no (linear) correlation and values near ± 1 means very strong (linear) correlation.
11. 11. ©drtamil@gmail.com 2020 • Direction of correlation; • r 2 =coefficient of determination. • Coefficient of determination = the portion of variability in one of the variables that can be accounted for by variability in the second variable. Pearson’s Correlation Positive and LinearNegative and Linear No correlation
12. 12. ©drtamil@gmail.com 2020 History of SEM
13. 13. ©drtamil@gmail.com 2020 Correlation & Factor Analysis • Sir Francis Galton came out with the concept of co- relation when studying height of sons & their fathers. He died in 1911. • 1896 – Karl Pearson developed the correlation formula. • 1904 – Charles Spearman used correlation to develop factor analysis technique. If a set of many items correlated with one another, the responses could be summed to yield a score. That set of items is called a “construct”. Factor analysis is used to create a measurement instrument or a construct.
14. 14. ©drtamil@gmail.com 2020 CFA – to test a construct • Confirmatory Factor Analysis (CFA) is done to test a construct. • The pioneers of CFA were Howe (1955), Anderson & Rubin (1956) and Lawley (1958). • Karl Gustav Joreskog published the first article on CFA in 1969 and helped develop the first CFA software programme (LISREL - linear structural relations). • Factor analysis create measurement instruments. CFA is used to test the theoretical construct of these instruments.
15. 15. ©drtamil@gmail.com 2020 Path Models uses Correlation & Regression • A biologist, Sewell Wright developed the path model between 1918 till 1934, which uses correlation and regression, to draw complex relationships between observed variables. He used it for models of animal behaviour. • In 1950s, econometricians such as H. Wold used it for simultaneous equation modelling. • Sociologists such as D. Duncan & H.M. Blalock also used it for the same purpose in the 1960s. • Path analysis involves solving a set of simultaneous regression equations that theoretically establish the relationship among the observed variables in the path models.
16. 16. ©drtamil@gmail.com 2020 SEM combines Path & CFA • SEM essentially combine path models with CFA since SEM incorporates both latent and observed variables. • This model was initially known as the JKW model due to the work of; – Karl Gustav Joreskog (1973) – Ward Keesling (1972) and – David Wiley (1973) • Since analysis was done using LISREL, it became known as LInear Structural RELations model in 1973. Now it is better known as SEM.
17. 17. ©drtamil@gmail.com 2020 Do we need to do SEM? • Not all postgrad research need to do SEM. • Usually only for those that need to validate their model or questionnaire will use SEM. • But the dependent variable (outcome) must be interval or ratio. – Interval scales are numeric scales in which we know not only the arrangement order, but also able to quantify the differences between the values. – A ratio variable, has all the properties of an interval variable, and also has a clear definition of what is “0”.
18. 18. ©drtamil@gmail.com 2020 2015. Randall E. Schumacker, Richard G. Lomax. A Beginner's Guide to Structural Equation Modelling. 4th ed. (Kindly read “On the Theory of Scales of Measurement”; S.S. Stevens, 1946 to better understand the difference of nominal, ordinal, interval and ratio data.)
19. 19. ©drtamil@gmail.com 2020 Ordinal or Interval? • We have seen attempts by many researchers to claim that their ordinal Likert scale is really interval data. Such as by using a Likert-10 scale or Likert-100 scale. But since the respondents are unable to differentiate the difference in their answers, such efforts are futile. • Computer applications are unable to differentiate between ordinal and interval data unless defined by user. Some software such as PRELIS, will determine the variable to be ordinal if it has lesser than 15 distinct scale points (< 15 categories). • A 15-point criterion allows Pearson correlation coefficient to vary between +1.0. Otherwise the range will only be between +0.5.
20. 20. ©drtamil@gmail.com 2020 Sample Size for SEM • Ding, Velicer & Harlow (1995) stated 100 to 150 is the minimum satisfactory sample size. • Boonsma (1983) recommended 400. • Textbooks suggest either 10 or 20 per variable. • Bentler & Chou (1987) suggested 5 per variable is sufficient for normally distributed data. And 10 per variable for others. • Most published research used 250 to 500 subjects.
21. 21. ©drtamil@gmail.com 2020 Missing Data • It is usually not possible to run SEM if there are missing data issues. Therefore please correct it before analysis using such methods. 2015. Randall E. Schumacker, Richard G. Lomax. A Beginner's Guide to Structural Equation Modelling. 4th ed.
22. 22. ©drtamil@gmail.com 2020 2015. Randall E. Schumacker, Richard G. Lomax. A Beginner's Guide to Structural Equation Modelling. 4th ed. Please explore data to ensure linearity
23. 23. ©drtamil@gmail.com 2020 Why do SEM? Hox, Joop & Bechger, Timo. (1999). An Introduction to Structural Equation Modeling. Family Science Review. 11. http://joophox.net/publist/semfamre.pdf
24. 24. ©drtamil@gmail.com 2020 Why do SEM? • Combine regression with factor analysis (latent) • Confirmatory factor analysis (if using SPSS, only EFA is available) • Regression models • Complex path models
25. 25. ©drtamil@gmail.com 2020 Why do SEM? • Confirmatory factor analysis
26. 26. ©drtamil@gmail.com 2020 Why do SEM? • Regression models
27. 27. ©drtamil@gmail.com 2020 Why do SEM? • Complex path models
28. 28. ©drtamil@gmail.com 2020 SEM provides • Convenient framework for statistical analysis – Factor analysis – Regression Analysis – Discriminant Analysis – Canonical Correlation • Often visualized by a graphical path diagram.
29. 29. ©drtamil@gmail.com 2020 Why more researchers do SEM?
30. 30. ©drtamil@gmail.com 2020 1. Why SEM getting popular? 2015. Randall E. Schumacker, Richard G. Lomax. A Beginner's Guide to Structural Equation Modelling. 4th ed.
31. 31. ©drtamil@gmail.com 2020 2. Why SEM getting popular? 2015. Randall E. Schumacker, Richard G. Lomax. A Beginner's Guide to Structural Equation Modelling. 4th ed.
32. 32. ©drtamil@gmail.com 2020 3. Why SEM getting popular? 2015. Randall E. Schumacker, Richard G. Lomax. A Beginner's Guide to Structural Equation Modelling. 4th ed.
33. 33. ©drtamil@gmail.com 2020 4. Why SEM getting popular? 2015. Randall E. Schumacker, Richard G. Lomax. A Beginner's Guide to Structural Equation Modelling. 4th ed.
34. 34. ©drtamil@gmail.com 2020 Why AMOS? • AMOS has a student evaluation version of AMOS 5.01, which does not expire. • It used to be available; but not anymore. • Since the aim is just to introduce postgrad students to SEM, I’ll stick to the legal demo version. • Google for “SPSS Amos Trial download version” https://www- 01.ibm.com/marketing/iwm/iwmdocs/tnd/data/web/ en_US/trialprograms/G556357A25118V85.html • Please download and install.
35. 35. ©drtamil@gmail.com 2020 Basic Rules in AMOS • Square box – measured variable • Oval box – unmeasured variable • Single headed arrow – regression • Double headed arrow - correlation knowledge value satisfaction performance e1 1
36. 36. ©drtamil@gmail.com 2020 Spatial VISPERCe11 1 CUBESe2 1 LOZENGESe3 1 Verbal paragraphe4 SENTENCEe5 WORDMEANe6 1 1 1 1 Basic Rules in AMOS • Square box – measured variable • Oval box – unmeasured variable • Single headed arrow – regression • Double headed arrow - correlation
37. 37. ©drtamil@gmail.com 2020 Regression models Lin Naing 2011. Structural Equation Modeling with SPSS AMOS Workshop Lecture Notes. 21-23 October 2011.
38. 38. ©drtamil@gmail.com 2020 How to do Regression Models https://youtu.be/rNQw5RkGA6g
39. 39. ©drtamil@gmail.com 2020 Un- stan dard ised 4.11 knowledge 7.68 value 9.42 satisfaction performance 3.98 6.89 3.54 1.03 .98 1.12 24.26 e1 1
40. 40. ©drtamil@gmail.com 2020 Un- stan dard ised 4.11 knowledge 7.68 value 9.42 satisfaction performance 3.98 6.89 3.54 1.03 .98 1.12 24.26 e1 1
41. 41. ©drtamil@gmail.com 2020 Un- stan dard ised 4.11 knowledge 7.68 value 9.42 satisfaction performance 3.98 6.89 3.54 1.03 .98 1.12 24.26 e1 1 Mean Std. Deviation Variance knowledge 19.73 2.035 4.14 value 19.95 2.781 7.73 satisfaction 20.02 3.080 9.49 performance 59.76 8.913 79.44 Descriptive Statistics
42. 42. ©drtamil@gmail.com 2020 knowledge value satisfaction .69 performance .71 .81 .57 .23 .31 .39 e1 Stan dard ised
43. 43. ©drtamil@gmail.com 2020 knowledge value satisfaction .69 performance .71 .81 .57 .23 .31 .39 e1 Stan dard ised
44. 44. ©drtamil@gmail.com 2020 knowledge value satisfaction .69 performance .71 .81 .57 .23 .31 .39 e1 Stan dard ised
45. 45. ©drtamil@gmail.com 2020 Complicated Path Lin Naing 2011. Structural Equation Modeling with SPSS AMOS Workshop Lecture Notes. 21-23 October 2011.
46. 46. ©drtamil@gmail.com 2020 How to do Complicated Path https://youtu.be/JbElfduMufA
47. 47. ©drtamil@gmail.com 2020 Confirmatory Factor Analysis Lin Naing 2011. Structural Equation Modeling with SPSS AMOS Workshop Lecture Notes. 21-23 October 2011.
48. 48. ©drtamil@gmail.com 2020 Data for this lesson. • https://wp.me/p4mYLF-sV • Download AMOS-Grnt_fem.sav • Delete all the value labels using SPSS. 2015. Randall E. Schumacker, Richard G. Lomax. A Beginner's Guide to Structural Equation Modelling. 4th ed.
50. 50. ©drtamil@gmail.com 2020 Modification Indices Parameters
51. 51. ©drtamil@gmail.com 2020 How to display M.I. in AMOS ----------------------------------------- To copy and paste to AMOS using "title" ----------------------------------------- Chi-square (df) = cmin (df); P value (>=0.05) = p; Relative Chi-Sq (<=2) = cmindf; GFI(>=0.95) = gfi; AGFI(>=0.9) = agfi; CFI(>=0.9) = cfi; Pratio = pratio; RMSEA(<=0.08) = rmsea. (format) View the video to see how to insert it.
52. 52. ©drtamil@gmail.com 2020 How to do CFA https://youtu.be/9PYBbg36iOs
54. 54. ©drtamil@gmail.com 2020 Modification Index
55. 55. ©drtamil@gmail.com 2020 Shortcut for CFA Using Plugins in AMOS Why draw the boxes when you can just copy and paste from SPSS?
56. 56. ©drtamil@gmail.com 2020 Download & Copy The Plugins • http://statwiki.kolobkreations.com/index.php?title=Plugins • MasterValidity - This plugin produces an HTML file with a correlation table of constructs, including the square root of the AVE on the diagonal, the CR and the AVE, as well as the less used MSV and MaxR. It also provides some interpretation and indication of validity issues. When validity issues occur, it also provides some recommendations. References for validity thresholds are provided. • ModelBias - This plugin automates the tedious job of testing the a model for specific bias or common method bias by running multiple contrained and unconstrained models through chi-square difference tests. The output is an HTML file that includes a table of the results, as well as interpretation, recommendations, and a reference. • ModelFit - This plugin creates an HTML file with all the relevant model fit measures, their thresholds, and an interpretation, as well as references for the suggested thresholds. • PatternMatrixBuilder - This plugin automates the tedious job of creating a CFA from a pattern matrix. You can paste a pattern matrix from SPSS into the plugin window and it will automatically generate your model for you. All you have to do after that is to rename the latent factors appropriately.
60. 60. ©drtamil@gmail.com 2020 How To Use The Plugins • https://www.youtube.com/user/Gaskination/ search?query=plugin • Let us re-try the previous CFA exercise using this plugin.
61. 61. ©drtamil@gmail.com 2020 Data for this lesson. • https://wp.me/p4mYLF-sV • Download AMOS-Grnt_fem.sav • Delete all the value labels using SPSS. 2010. Randall E. Schumacker, Richard G. Lomax. A Beginner's Guide to Structural Equation Modelling. 3rd ed.
63. 63. ©drtamil@gmail.com 2020 Validation of Questionnaire Using AMOS Azmi Mohd Tamil
64. 64. ©drtamil@gmail.com 2020 The Questionnaire Validation of Questionnaire Using AMOS
65. 65. ©drtamil@gmail.com 2020 Sample of an instrument to measure QOL
66. 66. ©drtamil@gmail.com 2020 Basic Concept • In psychometry, an instrument (i.e. questionnaires) with items are created to measure a latent trait that is not usually measurable in the normal physical way. • For example, what if we want to measure physical fitness? So we come up with items that is related to measuring physical fitness. • The following are sample measures of fitness in ascending difficulty;
67. 67. ©drtamil@gmail.com 2020 Sample Measures of Fitness Physical Activity 1=Limit ed a lot 2=A bit limited 3=Not limited Bathing or dressing yourself Bending, kneeling or stooping Lifting or carrying groceries Walking one block Climb one flight of stairs Walking several blocks Climb several flight of stairs Moderate activities such as moving a table Walking a mile (1.6 km) or more Vigorous activities such as running or strenuous sports. i n c r e a s i n g d i f f • Minimum score 10 – very unfit, maximum score 30 – very fit. • These items combined to measure a single factor/latent trait -> Fitness.
68. 68. ©drtamil@gmail.com 2020 Sample of a Fitness Score Physical Activity 1=Limit 2=A bit 3=Not Score Bathing or dressing yourself  3 Bending, kneeling or stooping  2 Lifting or carrying groceries  2 Walking one block  3 Climb one flight of stairs  2 Walking several blocks  2 Climb several flight of stairs  1 Moderate activities such as moving a table  2 Walking a mile (1.6 km) or more  1 Vigorous activities such as strenuous sports.  1 Total Score 19 i n c r e a s i n g d i f f • Cut-off=(Maximum-Minimum)/2+Minimum=(30-10)/2+10=20. • Score of 19 is below the cut-off point. Is the respondent unfit? • The reliability of a `composite score’ can be checked by Cronbach α.
69. 69. ©drtamil@gmail.com 2020 Aim of the current exercise • To assess the psychometric properties of the Mental & Physical Scale using AMOS. • To ensure continuity, I shall use the same dataset as the one we did with EFA and Rasch. • 10 items on Physical Function. • 5 items on Mental Function.
70. 70. ©drtamil@gmail.com 2020 15 Variables, 108 Complete Dataset.
71. 71. ©drtamil@gmail.com 2020 Physical Scores
72. 72. ©drtamil@gmail.com 2020 Scoring for Physical “Limited A Lot” “Limited A Little” “Not Limited At All” 1 2 3 Higher score indicate higher physical capability. • Respondents scored 1 to 3 only.
73. 73. ©drtamil@gmail.com 2020 Physical Function Questions 1 Vigorous activities, such as running, lifting heavy objects, participating in strenuous sports 2 Moderate activities, such as moving a table, pushing a vacuum cleaner, bowling, or playing golf 3 Lifting or carrying groceries 4 Climbing several flights of stairs 5 Climbing one flight of stairs 6 Bending, kneeling, or stooping 7 Walking more than a mile 8 Walking several blocks 9 Walking one block 10 Bathing or dressing yourself
74. 74. ©drtamil@gmail.com 2020 Physical Scores Score 1 2 3 4 5 6 7 8 9 10 1 37 22 9 13 10 12 26 20 21 11 2 61 49 24 52 15 38 52 34 44 14 3 24 47 86 55 92 68 39 63 52 93 Total 122 118 119 120 117 118 117 117 117 118 Higher Score, Better Physical Function • Minimum score 10 – very unfit, maximum score 30 – very fit. • These items combined to measure a single factor/latent trait -> Fitness. Response Limited a lot Limited a bit Not limited
75. 75. ©drtamil@gmail.com 2020 Mental Scores
76. 76. ©drtamil@gmail.com 2020 Scoring for Mental • Negative 1, 4, 5, 8. Positive 2, 3, 6, 7, 9. • The 5 questions used in this exercise are the 5 positive ones. The scores are not reversed. • So higher score indicate better mental health. All the time Most time A good bit Some time A little time None at all 1 2 3 4 5 6 1 Did you feel full of pep? 2 Have you been a very nervous person? 3 Have you felt so down in the dumps that nothing could cheer you up? 4 Have you felt calm and peaceful? 5 Did you have a lot of energy? 6 Have you felt downhearted and blue? 7 Did you feel worn out? 8 Have you been a happy person? 9 Did you feel tired? + + +
77. 77. ©drtamil@gmail.com 2020 Mental Function Score 1 2 3 4 5 1 2 2 2 1 6 2 6 6 8 2 7 3 17 6 31 13 30 4 48 35 9 48 15 5 27 27 36 27 31 6 12 37 26 19 22 Total 112 113 112 110 111 Items Nervous In Dump Blue Worn Out Tired Response All the time Most time A good bit Some time A little time None at all • Minimum score 5 – poor mental health, maximum score 30 – good mental health. • These items combined to measure a single factor/latent trait -> Mental Health.
78. 78. ©drtamil@gmail.com 2020 VALIDATION OF INSTRUMENT Exploratory Factor Analysis
79. 79. ©drtamil@gmail.com 2020 Data for this lesson. • https://wp.me/p4mYLF-sV • Download PF-MH-AMOS.sav and open in SPSS. Delete all the labels, otherwise the plugins will not work. Save the file and open AMOS. • Then follow these instructions -> Next slide.
81. 81. ©drtamil@gmail.com 2020 Analyze->Dimension Reduction>Factor
82. 82. ©drtamil@gmail.com 2020 Analyze->Dimension Reduction>Factor
83. 83. ©drtamil@gmail.com 2020 • KMO need to be 0.6 or higher. • Bartlett’s Test significant means that there is more than one dimension. Factor Analysis Amount of variance from 15 items, extracted by each factor is called ‘eigenvalue’ of each factor.
84. 84. ©drtamil@gmail.com 2020 Factor Analysis - Options • Easier to view the factor loadings of each component once we removed the smaller values. • Both PF & MH clearly split into two groups.
85. 85. ©drtamil@gmail.com 2020 Copy Pattern Matrix Into AMOS
86. 86. ©drtamil@gmail.com 2020 Automatically Drawn CFA
87. 87. ©drtamil@gmail.com 2020 Manually Drawn CFA
91. 91. ©drtamil@gmail.com 2020 Run Model Fit Measures
92. 92. ©drtamil@gmail.com 2020 Run Master Validity Plugin
93. 93. ©drtamil@gmail.com 2020 Conclusion • CFA is used to test the theoretical construct of these instruments. • Based on Model Fit Measures & Master Validity, the theoretical construct of these instruments have been tested and found excellent and acceptable. • Further analysis could be done by following examples in the attached video.

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My advanced stats lecture notes on Introduction to Structural Equation Modeling.

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