Class Introduction and Overview; Review of ANOVA, Regression, and Psychological Measurement

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1 Class Introduction and Overview; Review of ANOVA, Regression, and Psychological Measurement Introduction to Structural Equation Modeling Lecture #1 January 11, 2012 ERSH 8750: Lecture 1

2 Today s Class Introduction and overview of the course Syllabus information Review of prerequisites Analysis of Variance (ANOVA) Regression Psychological Measurement Introduction to Mplus Conducted in Room 228 of Aderhold Hall (3 rd hour of today) ERSH 8750: Lecture 1 2

3 A Note About The Room Change Introduction to SEM: From A Z ERSH 8750: Lecture 1 3

4 Today s Data Set To introduce and motivate SEM, and to review some prerequisites, we will make use of an example data set Data come from a sample of 200 (79 men and 121 women) participants in a study of the role of self esteem and motivation on achievement test scores Participant responded to three tests/surveys: 20 item achievement test (each item was scored right/wrong) 5 item motivation survey (each item used a 9 point Likert scale for responses; 1 9 in integers) 5 item self esteem survey (each item used a 9 point Likert scale for responses; 1 9 in integers) The researchers were interested in the effects of motivation and self esteem on achievement ERSH 8750: Lecture 1 4

5 Data File Setup The data file (exampledata.xls) has the following variables: ID identification number for each respondent AchievementScore Total score for achievement items SelfEsteemScore Total score for self esteem items MotivationScore Total score for motivation items SelfEsteem1 SelfEsteem5 Self esteem item responses Motivation1 Motivation5 Motivation item responses Achievement1 Achievement20 Achievement item responses Note: In order to use this file with Mplus, you must: Save it as a comma delimited file (.csv) Remove the first row containing the variable names ERSH 8750: Lecture 1 5

6 ANALYSIS OF VARIANCE REVIEW ERSH 8750: Lecture 1 6

7 ANOVA Refresher As a refresher on ANOVA, lets imagine that we are interested in the factors that have an effect on work output (denoted by Y) We design a two factor study where work output may be affected by: Lighting of the workplace High or Low Temperature Cold or Warm This experimental design is known as a 2 Way ANOVA ERSH 8750: Lecture 1 7

8 ANOVA Model Here is the 2 x 2 Factorial design: Low Lighting High Lighting Cold Temperature Warm Temperature The ANOVA model for a respondent s work output is ERSH 8750: Lecture 1 8

9 ANOVA Model The ANOVA model allows us to test for the presence of: The grand mean, A main effect associated with Temperature (A t ) A main effect associated with Lighting (B l ) An interaction effect associated with Temperature and Lighting (AB) tl ERSH 8750: Lecture 1 9

10 Linear Model Assumptions In the ANOVA and Regression (otherwise known as the general linear model or GLM), we assume that the error terms are Independent Normally distributed 0, for one observation Therefore, is a parameter that must be estimated We will come to call this a residual variance ERSH 8750: Lecture 1 10

11 ANOVA with Dummy Coded Variables The ANOVA model can also be re written using two dummy coded variables D rt and D rl Becomes a linear model (i.e., regression model) D rt D rt =0 for respondents in cold temperature condition D rt =1 for respondents in warm temperature condition D light D rl =0 for respondents in low lighting condition D rl =1 for respondents in high lighting condition ERSH 8750: Lecture 1 11

12 ANOVA with Dummy Coded Variables The ANOVA model then becomes: D rl = 0 Low Lighting D rl = 1 High Lighting D rt = 0 Cold Temperature D rt = 1 Warm Temperature ERSH 8750: Lecture 1 12

13 ANOVA Effects Explained β 0 is the mean for the cold and low light condition (reference group) The intercept β t is the change of the mean when comparing cold to warm temperature for a business with low lights (Simple Main Effect) β l is the change of the mean when comparing low to high lights for a business with a cold temperature (Simple Main Effect) β t*l is additional mean change that is not explained by the shift in temperature and shift and lights, when both occur (2 Way Interaction) Respondents from in the same condition have the same predicted value ERSH 8750: Lecture 1 13

14 ANOVA and our Example To further demonstrate the ANOVA model, we will now use our example data to determine if there are gender differences in achievement Because Mplus does not dummy code variables for us, we will use the following code for gender: 0 = male 1 = female Our gender ANOVA model would then be: ERSH 8750: Lecture 1 14

15 ANOVA Results The results from Mplus (note: we will replicate this analysis in lab) Here: Analysis Input File: sem12ersh8750_l01_m01.inp Represents the predicted score of achievement when all predictors are zero In this case, predicted value for GENDER = 0 (a male) Represents the change in the predicted achievement score for a 1 unit increase of the predictor Here, the predictor can either be 0 (a male) or 1 (a female) Gives the difference between achievement scores for the genders ERSH 8750: Lecture 1 15

16 More ANOVA Results Using the model parameters, we can find that the predicted achievement score for men was: The predicted achievement score for women was: But was the difference significant? The p value listed is not from your typical ANOVA test It is from a Wald test (we will learn about this in week 3) Because we have only one level for our IV, we can tell that women are significantly different from men Something to think about: how would we know for IVs with more than one level? ERSH 8750: Lecture 1 16

17 Even More ANOVA Results The residual variance was estimated to be: This represents the amount of variance in achievement scores not explained by gender To determine what the overall model would be, we would have to fit a model without gender and compare residual variances shown below: Gender explained:... Or 10% of the variance of achievement ERSH 8750: Lecture 1 17

18 REVIEW OF REGRESSION ERSH 8750: Lecture 1 18

19 Regression Review Although the ANOVA model presented previously was phrased as a regression model, we will now discuss the finer points of regression analysis We will now build upon our model with gender, adding self esteem and motivation as additional dependent variables: This is what is often called an analysis of covariance model (ANCOVA) where the slope for self esteem and for motivation is equal across genders ERSH 8750: Lecture 1 19

20 Regression Results Predicted value of achievement for: Men (GENDER = 0) With self esteem = 0 With motivation = 0.561(ns) Indicates that the intercept for women (achievement when self esteem and motivation are zero) is.561 points lower Is now not significant.201 (ns) The slope for self esteem for every one point increase in self esteem score, achievement goes up.201 (holding gender and motivation constant).320 (p = 0.002) The slope for motivation for every one point increase in motivation score, achievement goes up.320 (holding self esteem and gender constant) ERSH 8750: Lecture 1 20

21 More Regression Results The estimated residual variance was Therefore, adding self esteem and motivation to the model helped explain an additional... 13% of the variance in achievement score or So are you satisfied with this result? We should check the assumption of equal slopes within group ERSH 8750: Lecture 1 21

22 Investigating Equal Slopes To investigate the assumption of equal slopes, we need to test the interaction between: Gender and motivation Gender and self esteem Conditional on gender, the regression lines are: For males (GENDER = 0): For females (GENDER = 1): ERSH 8750: Lecture 1 22

23 Model Results Difference in slope for self esteem for females Not significant meaning equal slopes seems appropriate for self esteem Difference in slope for motivation for females Not significant meaning equal slopes seems appropriate for motivation ERSH 8750: Lecture 1 23

24 Not Done Yet The ANCOVA model indicated: A significant relationship between motivation and achievement Equal slopes within gender But non significant relationships between gender and achievement and between self esteem and achievement P value was (close to significance) Given our sample size of 200, it makes sense to remove one or more non significant effects Will increase power for remaining IVs Therefore, we will remove gender Self esteem was closer to significant (although choice is somewhat arbitrary) ERSH 8750: Lecture 1 24

25 Model without Gender The new model is then: Now, self esteem is significant So we think we may be done, right? Or are we? Can you think of anything we may have to worry about with this analysis? ERSH 8750: Lecture 1 25

26 REVIEW OF MEASUREMENT (RELIABILITY) ERSH 8750: Lecture 1 26

27 Reliability The general linear model (regression/anova combo) assumed that all variables were measured without error Surveys, questionnaires, and tests are all psychological instruments that are imprecise Adding up scores is a very naïve statistical model so the scores all have measurement error Related to the measurement error of a test score is the reliability of a test The GLM assumes Reliability = 1 for all variables ERSH 8750: Lecture 1 27

28 Measurement Models Measurement models can be divided into two families of models based on response format alone: Continuous responses :: Confirmatory Factor Models Both of these families fall under a larger framework: Generalized Linear Latent and Mixed Models Provide measurement models for other types of responses Other relevant families we will be discussing: Structural Equation Models :: provides estimates of correlations amongst latent variables in measurement models Path Analysis :: simultaneous regression amongst multiple observed variables ERSH 8750: Lecture 1 28

29 Differences Among Measurement Models Fundamental difference is in unit of analysis CTT :: unit of analysis is the entire test Sum of items = latent trait estimate Positives: Can always be done; No need for advanced computing Negatives: Restrictive assumptions; limited generalizability CFA (and, actually, item response theory IRT) :: unit of analysis is the item Model how item response relates to latent trait Different models for different types of item response formats Provides a framework for testing adequacy of measurement models Each family of models has a different name for the trait: CTT :: True Score (T) CFA :: Factor Score (F) IRT :: Ability (commonly); Theta (θ) ERSH 8750: Lecture 1 29

30 Classical Test Theory Basics In CTT, the test is the unit of analysis: True score T: best estimate of latent trait, mean over infinite replications of the test Error e: mean of zero, uncorrelated with T Variance of test scores: Goal is to quantify reliability :: proportion of test variance accounted for by true score variance: Items are assumed to be exchangeable (they all count the same) More items means higher reliability, regardless of type ERSH 8750: Lecture 1 30

31 More Classical Test Theory Error is a unitary construct in CTT Error variance has been quantified in various ways Goal is to reduce error variance as much as possible Standardization of testing conditions (reduces confounds) Aggregation of additional items (errors should cancel out) Items are exchangeable Followed by generalizability theory to decompose error e.g., rater variance, person variance, time variance Because each of our variables in our final analysis is a scale, we must see how reliable each happens to be ERSH 8750: Lecture 1 31

32 Classical Test Theory In CTT, the TEST is the unit of analysis: Y total = T + e True score T: Best estimate of latent trait : Mean over infinite replications Error e: Expected value (mean) of 0, expected to be uncorrelated with T e s are supposed to wash out over repeated observations So the expected value of T is Y total In terms of observed test score variance: Observed variance = true variance + error variance Goal is to quantify reliability Reliability = true variance / (true variance + error variance) True Score Y total error Reliability calculation is conducted on sums across items (so type of item is not relevant), but will require assumptions about the items?? ERSH 8750: Lecture 1 32

33 Conceptualizing Reliability CLASSICAL TRUE SCORE MODEL: Y total = True Score + error The expected value (predicted value) of a person s test score is their true score If a test is reliable, then a given person should get pretty much the same score over repeated replications (except for random error processes) We can t measure everybody a multiple times (cost? retest effects?) So, we can conceptualize reliability as something that pertains to a sample of persons instead by writing it in terms of variances Var(Y) = Var(T) + Var(e) = Var(T) + Var(e) + 2Cov(T, e) = Var(T) + Var(e) Reliability = Var(T) / Var(Y) Proportion of variance due to true score out of total variance ERSH 8750: Lecture 1 33

34 How Only Two Scores Give Us a Reliability Coefficient in CTT Y 1 = T + e 1 Y 2 = T + e 2 CTT assumptions to calculate reliability: Same true score (T) observed at both times e 1 and e 2 are uncorrelated with each other and T e 1 and e 2 have same variance Y 1 and Y 2 have same variance,,,,,,, Same as: Reliability of Y =Var(True) / Var(Y) We express unobservable true score variance in terms of the correlation between the two total scores and the variance of the total scores (assumed to be the same across tests) We now have an index of how much of the observed variance is true (if we believe all the assumptions) ERSH 8750: Lecture 1 34

35 Reliability Measured by Alpha For quantitative items (items with a scale although used on categorical items), this is Cronbach s Alpha Or Guttman Cronbach alpha (Guttman 1945 > Cronbach 1951) Another reduced form of alpha for binary items: KR 20 Alpha is described in multiple ways: Is the mean of all possible split half correlations Is expected correlation with hypothetical alternative form of the same length Is lower bound estimate of reliability under assumption that all items are tau equivalent (more about that later) As an index of internal consistency Although nothing about the index indicates consistency! ERSH 8750: Lecture 1 35

36 Where Alpha Comes From The sum of the item variances is given by: Var(I 1 ) + Var(I 2 ) + Var(I 3 ). + Var(I k ) (just the item variances) The variance of the sum of the items is given by the sum of ALL the item variances and covariances: Var(I 1 + I 2 + I 3 ) = Var(I 1 ) + Var(I 2 ) + Var(I 3 ) + 2Cov(I 1,I 2 ) + 2Cov(I 1,I 3 ) + 2Cov(I 2,I 3 ) Where does the 2 come from? Covariance matrix is symmetric Sum the whole thing to get to the variance of the sum of the items I 1 I 2 I 3 I 1 σ 1 2 σ 12 σ 13 I 2 σ 21 σ 2 2 σ 23 I 3 σ 31 σ 32 σ 3 2 ERSH 8750: Lecture 1 36

37 Cronbach s Alpha Covariance Version: k = # items Numerator reduces to just the covariance among items Sum of the item variances Var(X) + Var(Y) = Var(X) + Var(Y) just the item variances Variance of total Y (the sum of the items) Var(X+Y) = Var(X) + Var(Y) + 2Cov(X,Y) PLUS covariances So, if the items are related to each other, the variance of the total Y item sum should be bigger than the sum of the item variances How much bigger depends on how much covariance among the items the primary index of relationship ERSH 8750: Lecture 1 37

38 Kuder Richardson (KR) 20: Alpha for Binary Items KR20 is actually the more general form of alpha From Equation 20 in 1937 paper: k = # items p = prop. passing q = prop. failing Numerator again reduces to covariance among items Sum of the item variances (sum of pq) is just the item variances Variance of the sum of the items has the covariance in it, too So, if the items are related to each other, the variance of the total sum should be bigger than the sum of the item variances How much bigger depends on how much covariance among the items the primary index of relationship ERSH 8750: Lecture 1 38

39 Reliability for Motivation Scale To calculate the reliability for motivation (assumed to be quantitative or continuous variables) we need the covariance matrix We will discuss covariances in more detail next week For now, let s just consider these to be indices of association that have a scale Luck for us, Mplus will estimate covariance matrices More on how, what, and why in Week #3 Reliability for self esteem and achievement (although not shown) can be calculated similarly ERSH 8750: Lecture 1 39

40 Variances and Covariances of Motivation Scale ERSH 8750: Lecture 1 40

41 Calculating Alpha for Motivation Scale The sum of the variances is: The sum of TWO times the covariances is: So, the total variance is The estimate of reliability for the motivation scale is: Therefore the motivation scale is not perfectly reliable Violates the assumption of the GLM/Regression ERSH 8750: Lecture 1 41

42 MOTIVATING THE USE OF STRUCTURAL EQUATION MODELNG ERSH 8750: Lecture 1 42

43 Not 100% Reliable: What s the Big Deal? When the GLM (regression/anova/all other forms) are used with variables that are not measured perfectly, several deleterious things can happen: The effects may be biased Depends on variable type (i.e., how continuous are your measures) The standard errors may be biased Happens any time Therefore the effect sizes are biased Basically, any conclusions you make can be drawn into question when you have not reliable variables used The issue is with just how reliable is reliable enough ERSH 8750: Lecture 1 43

44 The Answer Don t Use Aggregates USE SEM Structural Equation Modeling seeks to determine the relationship between Latent constructs only Latent our example: Achievement, Motivation, and Self Esteem Latent and observed constructs From our example: how does gender factor into the model? Complex relationships between latent constructs and observed variables Does motivation mediate the relationship between self esteem and achievement? Do either/both mediate the relationship between gender and achievement? SEM is a generalization of linear modeling using observed and latent (sometimes called random) variables I tend to think of SEM as a part of a bigger picture you will see that SEM people think everything is part of SEM ERSH 8750: Lecture 1 44

45 Path Diagram of Our Regression Example A common way of depicting models in SEM is with a path diagram :: a pictorial representation of the statistical model Observed variables: Squares Latent variables: Circles Direct effects: Arrows with one head Indirect effects: Arrows with two heads Achievement Self Esteem Path Diagram of Regression Result Motivation Error ERSH 8750: Lecture 1 45

46 A More Accurate Path Diagram A3 A4 A2 A1 Categorical Observed Variables SE1 SE2 SE3 SE4 SE5 Confirmatory Factor Analysis A5 A6 A7 A8 A9 A10 Achievement Self Esteem Structural Equation Modeling (AKA Generalized Linear Mixed Model) Continuous Latent Variables A11 A12 A13 A14 A15 A16 A17 Item Response Theory (AKA Item Factor Analysis) Motivation M1 M2 M3 M4 M5 Continuous Observed Variables A18 A19 A20 ERSH 8750: Lecture 1 46

47 The (Really) Big Picture Statistical distributions are what drive the process Each distribution is described by a set of parameters Think of the normal distribution (mean and variance) Each of the lines represents model parameters The statistical distribution of the boxes and circles are described by the model parameters Model parameters provide constraints to the statistical distribution parameters Reduce complexity of model Provide for meaningful inference A model is bound by distributions assumed and, hence, the number of possible parameters We will learn statistics and path models Both are needed to be good at SEM ERSH 8750: Lecture 1 47

48 Up Next Quick Mplus tutorial Head to Room 228 and share the Windows Machines ERSH 8750: Lecture 1 48

49 INTRODUCTION TO MPLUS ERSH 8750: Lecture 1 49

50 The Mplus Statistical Package Mplus provides a general latent variable modeling framework that allows for combinations of: Continuous or categorical latent variables Continuous, categorical, count, nominal or censored data Mplus is commercial software that is available on Windows machines in the Aderhold computer labs Mplus is also available for purchase: Available at From $195 to $350 (student) ERSH 8750: Lecture

51 Mplus Data File Input Format Mplus input files must be ASCII text based (so not binary) Text based file formats: *.txt, *.dat, *.csv Not text based file formats: *.xlsx, *.sas7bdat, *.sav The easiest way to get data files into Mplus is to use freeformatting (some type of delimiter between columns) I prefer comma delimited files and will only use those in this course Cannot start with variable names in first row of data Typically, I store data in Excel and save as a comma delimited file Save As *.csv (then click OK to the first question) (then click YES to the second) Ignore the warning (click NO) to re save when closing the Excel Workbook ERSH 8750: Lecture 1 51

52 Mplus Syntax Conventions Most syntax must have a semi colon end each line (;) Exceptions: TITLE section, comments, and continuing lines Comments are denoted with an exclamation point (!) Syntax is organized by sections; headings of sections end with colons (:) TITLE:, DATA:, VARIABLE:, DEFINE:, and MODEL: are what we use this week Syntax cannot exceed 90 characters per row ( ) Mplus input files are typically saved with the extension *.inp Mplus output files are typically saved with the extension *.out Both are ASCII text (i.e., you can open with text editors) The default location for the data file and output file are the folder containing the input file ERSH 8750: Lecture 1 52

53 Mplus TITLE Section The TITLE section contains the label of the analysis You can type whatever you want here it will appear verbatim at the top of the output file You do not have to terminate this section with a semi colon This section is optional ERSH 8750: Lecture 1 53

54 Mplus DATA Section The DATA section is where data files are defined Define the name (and path if different from input file folder) by using the command: FILE = mydata.csv POTENTIAL MISTAKES: Data must be numeric if not errors happen First row of data should not contain variable names This section is NOT OPTIONAL ERSH 8750: Lecture 1 54

55 Mplus VARIABLE Section The Mplus VARIABLE section defines the names of the variables in the data file, variable types, and variables in your analysis NAMES = provides variable names Names cannot be more than 8 characters Lists of variables can be created (i.e., X1 X10 makes 10 variables) By default all variables listed in the NAMES section are assumed to be part of the analysis USEVARIABLE = provides names of variables used in the analysis (optional) IDVARIABLE = provides the ID variable name (optional) This section is NOT OPTIONAL ERSH 8750: Lecture 1 55

56 Mplus DEFINE Section The DEFINE section is where new variables are created To test our equal slopes hypothesis we created interaction variables by the following syntax: ERSH 8750: Lecture 1 56

57 Mplus MODEL Section The MODEL section is where you define the model The only models we ran this week were GLMs These models use the ON statement (ON = REGRESSION) achieve ON selfest motivate And an empty model to figure out the covariance matrix of the MOTIVATION items This used the WITH statement (WITH = COVARIANCE) motiv1 motive5 WITH motive1 motive5 ERSH 8750: Lecture 1 57

58 CONCLUDING REMARKS ERSH 8750: Lecture 1 58

59 Wrapping Up Today we covered the structure of the course, a review of the prerequisites, and an introduction to Mplus We will not come to lab again during class (office hours are held in this room, though) Please note: Homework Assignment #1 is posted on the course website and is due next Wednesday ERSH 8750: Lecture 1 59

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