CFA and More in R! Menglin Xu Department of Educational Studies (QREM)

Transcription

1 CFA and More in R! Menglin Xu Department of Educational Studies (QREM) 1

2 Outline Overview of lavaan and PISA data Data Screening in R, a brief overview Confirmatory factor analysis (CFA) One-factor CFA, continuous vs ordinal data Two-factor CFA Measurement Invariance Structural equation modeling (SEM) Structural model Mediation model 2

3 lavaan An R package aimed for latent variable analysis (Rosseel, 2012). Regression CFA Path analysis SEM Publicly downloadable Yield results comparable to Mplus (mimic= mplus ) 3

4 lavaan Basic operators & utilities (please refer to Rosseel, 2017 for more details). Syntax Descriptions y ~ x1 + x2 + x3 + x4 y is regressed on x1-x4 (works for both observed and latent Vs) F1 =~ y1 + y2 + y3 F1 is measured by y1-y3. y1 ~~ y2 error covariance of (y1, y2) y1 ~ 1 intercept of y1 cfa() analyzing measurement models, multiple group. sem() analyzing models with structural paths. 4

5 PISA 2015 Student Data_U.S. The Programme for International Student Assessment (PISA) Data retrieved from Coordinated by the Organization for Economic Co-operation and Development (OECD) Spanning 35 OECD countries/regions, PISA assesses skills in reading, maths, and science with the focus rotating every three years. Targeting at 15-year-old secondary school students Assesses multiple cognitive, social, and emotional well-being (e.g., selfefficacy, belief, engagement). N = 5712 (2854 males) in science_15.dat 5

6 Selected Variables omale (1: male; 0: female) oparental support (pa_sup1 pa_sup4), 4-point omotivation (mot1 mot9, 5 for enjoyment, 4 for instrumental), 4-point oscience efficacy (sci_eff1-sci_eff8), 4-point ohome educational resources (HEDRES), continuous oscience performance (PV1SCIE), continuous (M = 496, SD = 97.5) oall missing data is coded as 999.

7 Data Screening in R A Brief Practice 7

8 Set-up Load the R package lavaan install.packages("lavaan") library(lavaan) ## only need to install once ## load the package each time Define a working directory where the data is stored setwd("c:/users/xu.1384/documents/lavaan workshop") Read in data ## header=t: our data has variable names; na.string: our missingness is coded as 999. science <- read.table("science_15.dat", header=t, na.string=999) ## to display the first six lines of data head(science) ## everything looks OK? 8

9 Set-up ##in case the first column name is disordered names(science)[1] <- "male" 9

10 Data Management Basic summary dim(science) ## to display the dimension [1] summary(science) ## to produce summary information of each V sapply(science, function(x) sum(is.na(x))/length(x)) ## to obtain missing rate 10

11 Data Management To display variable distributions par(mfrow=c(2,2)) ## tell R to display graphs in 2*2 format pa <- science[, 2:5] ## to extract parental support items sapply(pa, function(x) hist(x)) ## to exhibit the histogram of the 4 Vs 11

12 Data Management How are the variables related with each other? pairs(pa, panel=panel.smooth) ## to show pairwise correlation plot 12

13 To get the bivariate correlations Data Management cor(pa, use="complete.obs", method="pearson") ## to show the correlations 13

14 Confirmatory Factor Analysis 14

15 Parental support (4 items) Item descriptions One-factor CFA Example pa_sup1: My parents are interested in my school activities. pa_sup2: My parents support my educational efforts and achievements. pa_sup3: My parents support me when I am facing difficulties at school. pa_sup4: My parents encourage me to be confident. CFA: a useful tool for measurement purposes. To test how well the 4 items represent parental support. 15

16 One-factor CFA Example R code for model fitting ## specify the one-factor CFA model, naming the latent factor to be pa_sup pa.model <- 'pa_sup =~ pa_sup1 +pa_sup2 + pa_sup3 + pa_sup4 ## fit the model, fill in the model, data, and estimator; model.pa <- cfa(model = pa.model, data = science, estimator = "MLR", mimic="mplus") ##Note. the naming in the left hand side of <- is flexible. ## to obtain the output for the analysis summary(model.pa, fit.measures =TRUE, standardized=true, rsquare=t) 16

17 One-factor CFA Example Yes, please refer to the Robust results The Robust column refers to the MLR estimates, use this one 17

18 One-factor CFA Example Estimates Scaling latent factor to be 1 Scaling both latent factor and DVs to be 1 18

19 One-factor CFA Example ## to flexibly extract fit indices of interest XX.scaled refers to MLR estimates fitmeasures(model.pa, c("cfi.scaled", "tli.scaled","rmsea.scaled","srmr","aic","bic")) ## to extract unstandardized parameter estimates parameterestimates(model.pa) ## to obtain the standardized factor loadings only, use the following two steps: std <- standardizedsolution(model.pa) std[std$op=="=~","est.std"] ## std$op== =~ means factor loadings [1]

20 When data is treated as ordinal 20

21 One-factor CFA_ordinal data ## check frequency distribution of each V first. sapply(pa, function(x) table(x)) To apply table(x) to all the Vs in pa 21

22 One-factor CFA_ordinal data ## fit the same model while treating the variables as ordinal type model.pa_cat <- cfa(model=pa.model, data=science, estimator = "WLSMV", mimic="mplus", ordered=c("pa_sup1","pa_sup2","pa_sup3","pa_sup4")) ## same procedures for summary and results extraction. summary(model.pa_cat, fit.measures=true, standardized=true, rsquare=t) fitmeasures(model.pa_cat, c("cfi.scaled", "tli.scaled","rmsea.scaled","srmr")) std <- standardizedsolution(model.pa_cat) ## to obtain standardized estimates std[std$op=="=~","est.std"] ## to extract standardized loadings [1]

23 One-factor CFA Questions so far? Highlights Basic data management Specify model in lavaan Specify the fitting function, if data is ordinal, use ordered= summary() for model fit and estimates Ways to extract specific information from output, e.g., fit indices, loadings. 23

24 Practice I Science efficacy (8 items, sci_eff1 sci_eff8). To create a efficacy data set hint: efficacy <- science[, 20:27] To make histogram plot for the 8 items Hint: sapply(efficacy, function(x) hist(x)) Please fit a one-factor CFA model for sci_eff treating data as continuous Does it fit well? All loadings significant? Please fit the same model while treating data as ordinal. Does it fit well? All loadings significant? Please compare selected model fit indices (cfi.scaled, tli.scaled, rmsea.scaled, srmr), and standardized loadings for the two approaches. Which one is better? 24

25 Item Descriptions for Efficacy sci_eff1 sci_eff2 sci_eff3 sci_eff4 sci_eff5 sci_eff6 sci_eff7 sci_eff8 To what extent I can Recognise the science question that underlies a newspaper report on a health issue. Explain why earthquakes occur more frequently in some areas than in others. Describe the role of antibiotics in the treatment of disease. Identify the science question associated with the disposal of garbage. Predict how changes to an environment will affect the survival of certain species. Interpret the scientific information provided on the labelling of food items. Discuss how new evidence can lead you to change your understanding about the possibility of life on Mars. Identify the better of two explanations for the formation of acid rain. 25

26 Sample Output Model fit comparison Standardized loading comparison 26

27 Two-factor CFA Example Science Motivation (9 items) Item descriptions mot1_enj I have fun when I am learning science mot2_enj I like reading about science topics. mot3_enj I am happy working on science topics. mot4_enj I enjoy acquiring new knowledge in science. mot5_enj I am interested in learning about science. Making an effort in my science subject(s) is worth it because this will help me in the mot6_int work I want to do later on What I learn in my science subject(s) is important for me because I need this for mot7_int what I want to do later on Studying my science subject(s) is worthwhile for me because what I learn will mot8_int improve my career prospects. mot9_int Many things I learn in my science subject(s) will help me to get a job. 27

28 Two-factor CFA Example R code for model fitting ## specify the two-factor CFA model, naming the latent factor to be enjoy & instru mot.2f <- 'enjoy =~ mot1_enj+mot2_enj+mot3_enj+mot4_enj+mot5_enj instru =~ mot6_int+mot7_int+mot8_int+mot9_int' ## fit the model, fill in the model, data, and estimator; model.mot_2f<-cfa(mot.2f, data=science, estimator = "MLR", mimic="mplus") ##Note. the naming in the left hand side of <- is flexible. ## to obtain the output for the analysis summary(model.mot_2f, fit.measures=true, standardized=true, rsquare=t) 28

29 Two-factor CFA Example Model fit Estimates What s the inter-factor correlation? 29

30 Practice II Assuming continuous data, fit a one-factor CFA model to the motivation data (9 items); Compare the one-factor vs two-factor CFA in terms of model fit and parameter estimates. Which one is better? Note. If estimator = ML, chi-square difference test between two nested models can be made by anova(model1, model2). For the practice, simply look at the respective output. 30

31 Sample Output fit indices of one-factor vs two-factor CFA 31

32 Other Models Could be Considered Bi-factor model: when it is posited that variables are explained by a single underlying construct, while there is uniqueness among groups of items. The general trait factor The uniqueness for intrinsic motivation The uniqueness for extrinsic motivation 32

33 Measurement Invariance (MI) 33

34 Measurement Invariance (MI) Group differences on cognitive, psychological, and social traits are of popular interest in social sciences, e.g., Gender, race, ses differences on academic achievement; Depression levels between clinical and non-clinical samples. Levels of achievement and psychological well-being across Grade 1, 2, 3,. It is assumed that scales are measuring the same construct across groups of interest/across time. How to test it? (Meredith, 1993). Configural invariance: same loading patterns Metric invariance: equal loadings Scalar invariance: equal loadings + intercepts Strict invariance: equal loadings + intercepts + error variances (optional) 34

35 Measurement Invariance (MI) Research question: is motivation (enjoy & instrumental) invariantly measured across gender? Configural Invariance: the two groups share the same loading patterns Male Female 35

36 Configural Invariance MI_1 <- cfa(mot.2f, data=science, group="male", estimator = "MLR", mimic="mplus") fitmeasures(mi_1, c("chisq.scaled","df","cfi.scaled","tli.scaled","rmsea.scaled","srmr","bic")) summary(mi_1,fit.measures=true, standardized=true) The fit indices refer to how good the model fits the data assuming the groups share the same factor structure 36

37 Metric Invariance Metric (Weak) Invariance: the two groups share the same structure + loadings Male Female λ 21 λ 31 λ 21 λ 31 λ 41 λ 41 λ 51 λ λ 72 λ 82 λ 92 λ 72 λ 82 λ 92 37

38 Metric Invariance MI_2 <- cfa(mot.2f, data=science, group="male", estimator = "MLR", mimic="mplus", group.equal=c("loadings")) fitmeasures(mi_2, c("chisq.scaled","df","cfi.scaled", "tli.scaled","rmsea.scaled","srmr","bic")) summary(mi_2,fit.measures=true, standardized=true) 38

39 Scalar Invariance Scale (Strong) Invariance: the two groups share the same structure + loadings + intercepts τ1 τ2 τ1 τ2 All the intercepts are equal across group τ9 τ9 39

40 Scalar Invariance MI_3 <- cfa(mot.2f, data=science, group="male", estimator = "MLR", mimic="mplus", group.equal=c("loadings", "intercepts")) fitmeasures(mi_3, c("chisq.scaled","df","cfi.scaled","tli.scaled","rmsea.scaled","srmr","bic")) summary(mi_3,fit.measures=true, standardized=true) Latent mean differences can 40be told

41 Strict Invariance Strict invariance: equal error variance on the basis of strong invariance. Not mandatory. MI_4 <- cfa(mot.2f, data=science, group="male", estimator = "MLR", mimic="mplus", group.equal=c("loadings", "intercepts","residuals")) fitmeasures(mi_4, c("chisq.scaled","df","cfi.scaled", "tli.scaled","rmsea.scaled","srmr","bic")) summary(mi_4,fit.measures=true, standardized=true) 41

42 Summary of MI Steps Fit and changes in fit Differences in the fit indices between adjacent models: less constrained constrained model Model χ 2 df CFI TLI RMSEA SRMR BIC Δχ 2 Δdf ΔCFI ΔTLI ΔRMSEA Configural Metric Scalar Strict e.g., this row is calcu Configural - Me Evaluation Criteria: a) anova(model1, model 2) can be used for χ 2 difference test if estimator = ML. When estimator = MLR, please refer to the Mplus website: χ 2 difference test is sensitive to sample size, so tentatively skipped. b) Differences in CFI, TLI <.01, RMSEA <.015 indicates nonsignificant change (Cheung and Rensvold, 2002; Chen, 2007) 42

43 Summary of MI Steps Implications of the MI results: Motivation has same meaning across gender. Latent means of enjoy & instru are comparable across gender. Observed means are also comparable across gender. What if MI is not satisfied at a certain step? => partial invariance (for details, please refer to Hirschfeld & von Brachel, 2014). 43

44 Practice III Please run a series of MI models for science efficacy (sci_eff1 sci_eff8, the one-factor CFA model) across gender. χ 2 difference test can be skipped right now. No worry about partial invariance, just stop anywhere failing MI. Could consider use the following table to help organize results. Model CFI TLI RMSEA SRMR BIC ΔCFI ΔTLI ΔRMSEA Configural Metric Scalar Strict 44

45 Output for Reference 45

46 Structural Equation Modeling 46

47 SEM Example Research question: what are the effects of gender (male), home educational resources (HEDRES), and motivation (enjoy & instru) on science performance (PV1SCIE)? Observed variables: male, HEDRES, PV1SCIE. Latent variables: enjoy, instru Structural part 47

48 lavaan code SEM Example ## specify the SEM model sem1 <- 'enjoy =~ mot1_enj+mot2_enj+mot3_enj+mot4_enj+mot5_enj instru =~ mot6_int+mot7_int+mot8_int+mot9_int PV1SCIE ~ HEDRES + male + enjoy + instru ## fit the model sem1 sem_1 <- sem(sem1, data=science, estimator = "MLR", mimic="mplus") ## get the selected fit indices fitmeasures(sem_1, c("chisq.scaled","df","cfi.scaled", "tli.scaled","rmsea.scaled","srmr","bic")) ## to get the output for model fit and parameter estimates summary(sem_1, fit.measures=true, standardized=true) 48

49 SEM Example Output Check the p value for estimates, anything nonsignificant? 49

50 Practice IV Please delete the nonsignificant predictor(s) for PV1SCIE, re-run the SEM model How does it fit? 50

51 SEM_Mediation Research question: what is the effect of HEDRES on PV1SCIE mediated by motivation factors? Direct effect: c Indirect effects: a1*b1 a2*b2 Total effect c + a1*b1+ a2*b2 a1 a2 c b1 b2 51

52 Lavaan code SEM_Mediation mediation <- 'enjoy =~ mot1_enj+mot2_enj+mot3_enj+mot4_enj+mot5_enj instru =~ mot6_int+mot7_int+mot8_int+mot9_int # direct effect PV1SCIE ~ c*hedres # mediator enjoy ~ a1*hedres PV1SCIE ~ b1*enjoy instru ~ a2*hedres PV1SCIE ~ b2*instru # indirect effect (a*b) a1b1 := a1*b1 a2b2 := a2*b2 # total effect total := c + (a1*b1)+ (a2*b2) # allow the two dimensions be correlated, not a default in sem() ### enjoy~~instru med <- sem(mediation, data=science, estimator = "MLR", mimic="mplus") fitmeasures(med, c("chisq.scaled","df","cfi.scaled", "tli.scaled","rmsea.scaled","srmr","bic")) In the fitting function, for bootstrap approach for se, use sem(mediation, data=science, mimic="mplus, se = "boot", bootstrap = 10000) summary(med, fit.measures=true, standardized=true, rsquare=t) 52

53 Output SEM_Mediation a)how does the model fit? b)how much variance in DV is explained? c)any interesting findings? 53

54 Practice V Please fit a mediation model as shown in the following graph: a c b Exogenous: HEDRES Mediator: science efficacy (latent) Outcome: PV1SCIE 54

55 Some References Cheung, G.W., Rensvold, R.B.(2002). Evaluating goodness-of-fit indexes for testing measurement invariance. Structural Equation Modeling, 9, Chen, F.F. (2007). Sensitivity of goodness of fit indexes to lack of measurement invariance. Structural Equation Modeling. 14, Hirschfeld, G., & von Brachel, R.(2014). Multiple-Group confirmatory factor analysis in R A tutorial in measurement invariance with continuous and ordinal indicators. Practical Assessment, Research and Evaluation, 19(7), Meredith, W. (1993). Measurement invariance, factor analysis and factorial invariance. Psychometrika, 58(4), Organization for Economic Co-Operation and Development (OECD). Country Note: Key Findings from PISA 2015 for the United States; OECD Publishing: Paris, France, 2016; Available online: Rosseel, Y. (2012). lavaan: an R package for structural equation modeling. Journal of Statistical Software, 48, Rosseel, Y. (2017). The lavaan tutorial. Retrieved from 55

56 Thanks for joining us! Questions & Feedback 56