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`Mplus for Windows: An Introduction and OverviewAlan C. Acock Department of HDFS Oregon State University 7/2008 Topics 1. Using Mplus 1.1. Launching Mplus 1.2. Input and Output Windows 1.3. Reading Data and Outputting Sample Statistics 1.4. Defaults 1.5. Commands 2. Exploratory Factor Analysis 2.1. Exploratory Factor Analysis with Continuous Variables 2.2. Comparing Solutions 2.3. Exploratory Factor Analysis with Categorical Variables 2.4. Comparing Two Solutions 2.5. Comparison of Categorical and Continuous solutions 3. Confirmatory Factor Analysis 3.1. Confirmatory Factor Analysis with Continuous Variables 3.2. Output and Interpretation 3.2.1. Missing value summary 3.2.2. Model fit 3.2.3. Model results 4. Exploratory Factor Analysis as Alternative to CFA 5. Path Analysis 5.1. Model and Program 5.2. Indirect Effects 6. Putting it Together--Structural Equation Model and CFA 6.1. Output and Interpretation 6.2. Interpretation of Modification Indices 7. Putting it Together--SEM with Both CFA and EFA 8. Summary 2 2 2 5 6 6 8 8 10 10 11 14 16 17 19 19 20 21 25 29 29 31 31 36 36 37 42Introduction to Mplus, Alan C. Acock1Section 1: Using Mplus1.1. Launching Mplus1.2. Input and Output Windows· The window shown above is the input window. · You write Mplus programs in this window to read the data to be analyzed and to specify your model of interest. · You then save your Mplus program and select Run Mplus from the Mplus menu to submit your program to the Mplus engine for processing.  File  Open Open ex1.inp. This is located at c:\Mplus Examples\ex1.inp.Introduction to Mplus, Alan C. Acock2We will utilize files from the Mplus manual for many of our examples. These typically involve simulated data. Sometimes we well assign hypothetical variable names to make these somewhat realistic. The Manual itself does not provide a substantial about of explanation of the examples and the specific output so we hope the discussion of them here will be useful at a later time when you are trying to read the manuals.Here is one screen of the data in ex1.dat (The ANALYSIS: command here is not needed in the current version of Mplus) · We have labeled missing values with a -9. Easiest to pick one value that will work for all variables--can be any number or a dot. · Notice we have one observation, case 13, that has a missing value on all variables. · The data happens to be in a fixed format. · Could be comma delimited, cvs file from Excel. · Could be free format, other formats possible, but more complicatedIntroduction to Mplus, Alan C. Acock3Here is the confirmatory factor analysis model we are estimatingWe will explain the program in a moment, but for now we will just run it to see how the interface works.  MplusIntroduction to Mplus, Alan C. Acock4Run Mplus Or, you can click the Run icon. Once Mplus has finished processing your command program, it opens an output window. · The output window first displays your Mplus program. · Below the Mplus program are the Mplus model results. · If there is an error in your Mplus program or you want to modify your Mplus program in any way (e.g., to fit a different model to the data), you must return to the input window and you can then modify the previous commands, save the modified command file, and run Mplus once again to obtain new output.1.3. Mplus Command StructureAfter you have launched Mplus, you may build a command file. There are nine sets of Mplus commands (ususally only a few of these are used, but some have numerous subcommands) : 1. 2. 3. 4. 5. 6. 7. 8. 9. TITLE: (optional unless you want to know what the file is intended to do) DATA: (required), VARIABLE: (required), DEFINE: (some data transformations are available) SAVEDATA: (used for specialized applications) ANALYSIS: (for special analyses such as EFA MODEL: (a series of equations) OUTPUT: (many options are available) MONTECARLO: (used for simulations, power analysis)Rules: 1. 2. 3. 4. 5. 6. 7. All commands (Title, Data, etc.) must begin on a new line. All command names must be followed by a colon. For e.g., Title: Once you enter the colon, the key word becomes blue. Semicolons separate command options--similar to SAS. The records in the input setup must be no longer than 80 columns. They can contain upper and/or lower case letters and tabs. Only variable names are case sensitive. (Y1 and y1 are different variables)5Introduction to Mplus, Alan C. Acock1.4. Defaults· The current version of Mplus assumes that you either have no missing values or are using full information maximum likelihood estimation and assuming missing values are missing at random (MAR) · Parameters such as loadings can be fixed, e.g., many loadings are fixed at 0.0 in the CFA models because the item should not load on the factor. There is no path from F1 to Y4 in our figure. · Fixed parameters can be &quot;freed,&quot; meaning you will estimate them. We could add a path from to Y4 or let E1 be correlated with E4 · Fixed parameters are required to stay at a specified value, such as 0.0. · All free parameters are put into a vector and iterations change values of these free parameters, until the model's fit is optimal. · Unless we tell it otherwise, Mplus will fix the first indicator's loading at 1.0 as the reference indicator. For example, F1Y1 and F2Y4 have fixed loadings of 1.0 by default.1.5. CommandsThe TITLE command allows you to specify a title that Mplus will print on each page of the output file. · This can go on and on for many lines and usually should. · Everything is a Title until a command name appears at the start of a new line. · I like to put the file name as the first line of a title. The DATA command specifies where Mplus will locate the data, the format of the data, and the names of variables. At present, Mplus will read the following file formats: · tab-delimited text, · space-delimited text, and · comma-delimited text. · The input data file may contain records in free field format or fixed format. · If you are using data stored in another form (e.g., Stata, SAS, SPSS, or Excel), you will need to convert it to one of the formats with which Mplus can work before you read it into Mplus. SAS and SPSS require you to write a file out as a fixed format ASCII file.Introduction to Mplus, Alan C. Acock6If you have the data in Stata you can use stata2mplus to set things up for you. You can obtain it using findit stata2mplus and install the program. Here is the Stata session: stata2mplus using &quot;I:\flash\HDFS630\mplus\classnsfh&quot;, replace   This creates two files: · classnsfh.inp that will run a basic analysis in Mplus and · classnsfh.dat, a comma delimit ASCII file that Mplus can read with all missing values coded/recoded as -9999. 1,1,1,1,2,1,1,1,1,1,1,1,2,1,1 3,2,2,3,2,3,2,2,2,2,2,2,2,2,2 · · · 1,1,1,-9999,1,2,2,1,2,2,2,2,2,1,1 3,3,3,3,3,3,3,3,3,3,3,3,3,3,3 2,2,1,2,1,2,2,1,1,2,1,-9999,1,1,1   The DATA command tells Mplus where the data is stored. · If you store the program and the data in the same folder, you don't need to include the path. · Recommended to make a separte folder for each project such. · A long file reference can exceed the character limits per line in Mplus. · Mplus uses all available data by default. If you want to use listwise deletion you must specify this under the Data command. Listwise = on; The VARIABLE command names variables. · These must be in the identical order to the way Stata/SAS/SPSS wrote the data file. (common mistake) · Mplus variable names may not have more than 8 characters. Change variable names to be 8 characters or less or you will get error messages. · Variable names are case sensitive. Must be consistent (common mistake) The ANALYSIS command tells Mplus what type of analysis to perform. · Many analysis options are available. · Some of these such as Type = EFA make additional commands unnecessary.Introduction to Mplus, Alan C. Acock7SECTION 2: Exploratory Factor Analysis2.1. EFA with Continuous VariablesTITLE: efa1.inp This is an example of an exploratory factor analysis with continuous factor indicators FILE IS &quot;c:\Mplus Examples\efa1.dat&quot;; NAMES ARE y1-y12; TYPE = EFA 1 4; ESTIMATOR = ml; ROTATION = Geomin; sampstat;DATA: VARIABLE: ANALYSIS: ! ! OUTPUT:· The Type = EFA 1 4 tells Mplus to perform an exploratory factor analysis. · The 1 and 4 following the EFA specification tells Mplus to generate all possible factor solutions between and including 1 and 4. · Finally, the ESTIMATOR = ml option has Mplus use the maximum likelihood estimator to perform the factor analysis and compute a chi-square goodness of fit test that the number of hypothesized factors is sufficient to account for the correlations among the six variables in the analysis. · This has an exclamation mark in front of it which makes it green. Anything green is a comment and is ignored by the program. This subcommand is not necessary because maximum likelihood estimation is the default. · Mplus uses the geomin rotation which is oblique as its default. More traditional rotations such as varimax are available. See help for a listing of options. · We do not need a MODEL: command because the EFA 1 4 takes care of this. One useful feature of Mplus is its ability to handle non-normal input data. · Recall that the default ml estimator assumes that the input data are distributed joint multivariate normal. · If you have reason to believe that this assumption has not been met and your sample is reasonably large (e.g., n  200), you may substitute mlm or mlmv in place of ml on the ESTIMATOR = line. · The mlm option provides a mean-adjusted chi-square model test statistic whereas theIntroduction to Mplus, Alan C. Acock8· mlmv option produces a mean and variance adjusted chi-square test of model fit. · SEM users who are familiar with Bentler's EQS software program should also note that the mlm chi-square test and standard errors are equivalent to those produced by EQS in its ML;ROBUST method. You may also add the OUTPUT command following the ANALYSIS and MODEL commands. The OUTPUT command is used to specify optional output. For this example the keyword sampstat tells Mplus to include sample statistics as part of its printed output. OUTPUT: sampstat ;You can use Mplus' Help menu to get a listing of all the options available for each command. You might try this to see what OUTPUT options are available. Mplus produces the · Sample correlations, · Root Mean Square Error of Approximation (RMSEA), and the · Chi-square test of the one, two, three, and four factor models. · Standard errors and z-tests for loadings and correlations of factors. As you can see from the results, shown below, the chi-square test is statistically significant, so the null hypothesis that a single factor fits the data is rejected; more factors are required to obtain a non-significant chi-square. Since the Chi-square test is: · Sensitive to sample size (such that large samples often return statistically significant chi-square values) and · Non-normality in the input variables. Mplus also provides the Root Mean Square Error of Approximation (RMSEA) statistic. The RMSEA is not as sensitive to large sample sizes. According to Hu and Bentler (1999), RMSEA values below .06 indicate satisfactory model fit. Kline indicates a .08 is acceptable. Run the program and interpret the results.Introduction to Mplus, Alan C. Acock92.2. Comparing two SolutionsYou can test whether the adding additional factors significantly improves the fit to the data.Model Model Model Model 1 2 3 4 chi-square chi-square chi-square chi-square (54 (43 (33 (24 degrees degrees degrees degrees of of of of freedom) freedom) freedom) freedom) = 1052.089; p &lt; .001 = 723.022; p &lt; .001 = 341.268; p &lt; .001 = 25.799; p not sign.Is model 4 better than model 3. Model 3 chi-square (33 degrees of freedom) = 341.268 Model 4 chi-square (24 degrees of freedom) = 25.799 Difference chi-square (9 degrees of freedom) = 315.469; p &lt; .001 This is significant at the .05 level With Stata, you can get the probability when this is not in a table . display 1-chi2(9,315.469) 0 This is obviously less than .05. Often you can't use tables for chi-square because you have lots of degrees of freedom and tables only show significance levels for relatively few degrees of freedom. Estimate the model and interpret the results.2.3. EFA with Categorical OutcomesFor the purposes of illustration, suppose that you recode each variable into a replacement variable where all six variables' values at the median or below are assigned a categorical value of 1.00 and all values above the median assigned a value of 2.00. · For categorical variables, Mplus automatically recodes the lowest value to zero with subsequent values increasing in units of 1.00. · While the four underlying latent factors remain continuous, the six categorical observed variables' response values are now ordered dichotomous categories.Introduction to Mplus, Alan C. Acock10You may use the program that appeared in the initial exploratory factor analysis example, with the following modifications, and the new data file that contains the categorical variables ex4.2.dat, as shown below. TITLE: ex4.2.inp This is an example of an exploratory factor analysis with categorical factor indicators It uses weighted least squares estimation It computes tetrachoric correlations and does the Factor analysis on them. The RMSEA and chi-square Values are reported. FILE IS ex4.2.dat; VARIABLE: NAMES ARE u1-u12; CATEGORICAL ARE u1-u12; ANALYSIS: TYPE = EFA 1 4; ESTIMATOR = MLR; You tell Mplus which variables are categorical with the CATEGORICAL subcommand of the VARIABLE command, like this: CATEGORICAL ARE u1 ­ u2 ; You should also change the ESTIMATOR option for the ANALYSIS command. · The default estimator for categorical variables is weighted least squares · I have used MLR, Maximum Likelihood Robust. This uses a default 7 integration points and is extremely slow to converge. It makes it possible to compare models using a likelihood ratio test.DATA:2.4. Comparing Two SolutionsSelected output from the analysis appears below. Notice that the categorical nature of the data precludes computation of the descriptive model fit statistics such as the RMSEA, though Mplus does produce the familiar chi-square test of overall model fit.Introduction to Mplus, Alan C. Acock11Factors 1 2 3 4Df 24 35 54Pearson Log Chi-Square LikelihoodParameter s Estimated -4056.167 1.118 24 -3967.769 1.128 35 No Convergences with Three Factors -3812.608 1.000 54Scaling FactorAIC 8161.334 8005.458 1733.216We can use the log likelihood values to estimate a chi-square difference test. We will compare the 4 factor solution to the three factor solution: Notation: .cd is the difference test scaling correction factor (estimated below) .p0 is number of parameters in simpler/nested model (2 factor solution here) .c0 is the scaling correction factor for the MLR likelihood value (based on comparing MLR to ML likelihood values .p1 is the number of parameters in the bigger model (4 factor solution here .c1 is the scaling correction factor for the MLR likelihood value (based on comparing the MLR to the ML likelihood values .df0 is the degrees of freedom for the simpler/nested model .df1 is the degrees of freedom for the bigger model .TRd is the computed chi-square difference tests with df1 ­ df0 degrees of freedom .L0 is likelihood value for simpler/nested model (2 factor solution) .L1 is likelihood value for bigger model (4 factor solution) For 2 factor solution: L0 = -3967.760, c0 = 1.118, df0 = 4060, p0 = 35 parameters For 4 factor solution: L1 = -3812.608, c1 = 1.000, df1 = 4041, p1 = 54 parameters Computed difference test correction, cd .cd = (p0×c0 - p1×c1)/(p0 ­ p1 = (35×1.128 ­ 54×1.000)/(35-54) = .764 Compute Chi-Square Difference Test, TRd TRd = -2(L0 ­ L1)/cd TRd = -2(-3967.769 - -3812.608)/.764 = -2(3967.769 + 3812.608)/.764 = 406.18Introduction to Mplus, Alan C. Acock 12Chi-square (19) = 406.18, p &lt; .001 Thus, the 4 factor solution does significantly better than the 2 factor solution If you use Weighted Least Squares (WLSM) with categorical data you get a RMESA to help compare the models and can do a chi-square test as described at http://www.statmodel.com/chidiff.shtml This approach has no convergence problem with the three factor solution and is dramatically faster than MLR solution. We will use it. For the 3 Factor solution we obtain this fit information. Chi-Square Test of Model Fit Value Degrees of Freedom P-Value Scaling Correction Factor for MLR CFI/TLI CFI 0.753 TLI 0.506 Number of Free Parameters 33 RMSEA (Root Mean Square Error Of Approximation) Estimate 0.129 SRMR (Standardized Root Mean Square Residual) Value 0.095 And for the 4 Factor solution we obtain this fit information Chi-Square Test of Model Fit Value Degrees of Freedom P-Value Scaling Correction Factor for MLR CFI/TLIIntroduction to Mplus, Alan C. Acock 13308.143* 33 0.0000 0.55722.741* 24 0.5351 0.402CFI 1.000 TLI 1.003 Number of Free Parameters 42 RMSEA (Root Mean Square Error Of Approximation) Estimate 0.000 SRMR (Standardized Root Mean Square Residual) Value 0.024 Run this to get the results and interpret the four factor solution: 2.5.ComparisonofCategoricalandContinuousSolutions  Onewayofevaluatingtheefficacyofacategoricalfactoranalysisisitsabilitytoreproduce thefactorsobtainedwhenthedataiscontinuous.Theprogramex4.1.inpestimatesthe samefactorsforthe12itemswhentheyarecontinuousandwecancomparetheresults. Thelowloadingsoneachfactorarealllowwhetherwehavethecontinuousvariablesor havedichotomizedthevariables.Thehighloadsareallfairlyclosematches. First, here is the result when the variables are continuous:GEOMIN ROTATED LOADINGS 1 2 3 ________ ________ ________ 0.637 0.008 0.074 0.808 0.022 -0.005 0.631 -0.042 -0.058 0.027 0.646 -0.002 -0.029 0.760 -0.023 0.010 0.674 0.030 -0.006 0.003 0.734 -0.040 0.002 0.727 0.049 -0.007 0.707 -0.037 0.006 -0.010 0.004 0.013 0.001 0.035 -0.036 0.008 GEOMIN FACTOR CORRELATIONS 1 2 ________ ________ 1.000 -0.039 1.000 0.007 0.029 -0.002 -0.121 4 ________ -0.021 0.041 -0.028 -0.018 0.017 -0.012 0.018 -0.016 -0.001 0.692 0.791 0.658Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y10 Y11 Y123 ________ 1.000 -0.0284 ________1 2 3 41.000Introduction to Mplus, Alan C. Acock14Andherearetheresultsforthecategoricalsolution:GEOMIN ROTATED LOADINGS 1 ________ U1 0.628 U2 0.938 U3 0.690 U4 0.072 U5 -0.130 U6 0.015 U7 0.026 U8 -0.036 U9 0.017 U10 -0.066 U11 -0.005 U12 0.110 2 ________ -0.004 0.042 -0.073 0.705 0.805 0.602 -0.014 -0.003 0.034 -0.017 0.040 -0.069 3 ________ 0.098 -0.012 -0.019 -0.120 0.018 0.089 0.805 0.720 0.669 0.057 -0.056 0.022 4 ________ -0.061 0.070 -0.036 -0.005 0.016 -0.045 0.009 -0.029 0.025 0.654 0.872 0.6241 2 3 4GEOMIN FACTOR CORRELATIONS 1 2 ________ ________ 1.000 -0.026 1.000 0.025 0.032 -0.029 -0.1503 ________ 1.000 -0.0594 ________1.000Item 1 2 3 4 5 6 7 8 9 10 11 12F1 (con) .637 .808 .631F1 (cat) .628 .938 .690F2 (con)F2 (cat)F3 (con)F3 (cat)F4 (con)F4 (cat).646 .760 .674.705 .805 .602 .734 .727 .707 .805 .720 .669 .692 .791 .658 .654 .872 .624There are several notes worth keeping in mind when you perform exploratory factor analysis with categorical outcome variables.Introduction to Mplus, Alan C. Acock 15·· ···Although one or more of the observed variables may be categorical, any latent variables in the model are assumed to be continuous (this is a property of the exploratory factor analysis model. The analysis specification and interpretation of the output is the same whether one, a subset, or all observed variables are categorical. Categorical observed variables may be dichotomous or ordered categorical outcomes of more than two levels), but nominal level observed variables with more than two categories may not be used in the analysis as outcome variables using this strategy. Sample size requirements are somewhat more stringent than for continuous variables; typically you want a minimum of 200 cases (preferably more) to perform any analysis with categorical outcome variables. Mplus provides standard errors and z-tests for all loadings and correlations.SECTION 3: Confirmatory Factor AnalysisWhat if you had an a priori hypothesis that the visual perception (Y1), cubes (Y2), and lozenges (Y3) variables belonged to a single factor whereas the paragraph (Y4), sentence (Y5), and word meaning (Y6) variables belonged to a second factor? The diagram shown below illustrates the model visually.Introduction to Mplus, Alan C. Acock16You can test this hypothesized factor structure using confirmatory factor analysis, as shown in the next section.3.1. CFA with Continuous Variables       TITLE: ex1.inpCFA with continuous factor indicators There are Missing values DATA: FILE IS &quot;c:\mplus examples\ex1.dat&quot; ; VARIABLE: NAMES ARE y1-y6; MISSING ARE all (-9) ; MODEL: ! OUTPUT: f1 BY y1-y3; f2 BY y4-y6; f1 WITH f2; sampstat standardized residual patterns mod(3.84); When Mplus sees EFA it sets up the relationship in a certain way, but in a CFA, Mplus needs you to provide a MODEL: to tell it how to set up the relationships that you wish to confirm). · The model is general in the sense that o You must define what parameters are estimated; o All other parameters are assumed to be fixed. o Fixed parameters are either zero or some value you set. · Under VARIABLE we have defined what code is used to represent missing values. · You do not need an ANALYSIS section, since we use the MODEL section to specify the model and are going with the default analysis. o This assumes full information maximum likelihood. o To do listwise deletion we would specify this in the DATA command · The MODEL command allows you to specify the parameters of your model. o The first line of the MODEL command shown above defines a latent factor for the first factor.Introduction to Mplus, Alan C. Acock 17o The BY keyword (an abbreviation for &quot;measured by&quot;) is used to define the latent variables; o The latent variable name appears on the left-hand side of the BY keyword whereas the measured variables appear on the right-hand side of the BY keyword. o Mplus will fix the loading for the first indicator at 1.0 unless you tell it otherwise. Put the &quot;best&quot; indicator first. · Similarly, in the second line of the MODEL: command a latent factor called verbal has three indicators: Y1, Y2, and Y3. The third line of MODEL: command uses the WITH keyword to correlate the F1 latent factor with the F2 latent factor. By With   Measured by Correlated withWe do not need F1 with F2 because that is the default. If we wanted to see how the model did with these fixed we would add the line F1 with [email protected] ; Finally, the OUTPUT command contains an added keyword, standardize. This option instructs Mplus to output standardized parameter estimate values in addition to the default unstandardized values. Selected output from the analysis appears below. Why is one loading fixed at 1.0? The default fixes the unstandardized loading of the first item after BY at 1.0 This has to do with model identification. In exploratory factor analysis the variance of the factor (latent variable) is fixed at 1.0 by the program. Given this, the program estimates the loadings. You need to set a variance for the latent variable because the size of the loadings are scaled from the size of the variance. Setting the variance of the latent variable (factor) at 1.0 solves this problem and you get standardized loadings. This is possible with Mplus, but Mplus suggests a more general approach in which you fix one of the loadings of each latent variable (factor) at 1.0. Why is this more general? Suppose you wanted to compare boys and girls on these tests. Girls and boys might differ in the loadings because some skills may be more central to girls overall skills than they are to boys. Studying marital satisfaction weIntroduction to Mplus, Alan C. Acock 18might find that emotional support is more central to marital satisfaction among wives than it is among husbands. At the same time (and here comes the answer), one group might be more variable than another. We might find that girls not only have higher verbal skills than boys, but that they are either more homogeneous or more heterogeneous in these skills. An intervention that not only improves the mean outcome, but does so in a way that makes the distribution more homogeneous is preferred. In some cases we are interested in the variances of the latent variables as an important topic and we could not study that if we fixed the variance at 1.0. Regardless of which item you pick to fix the loading at 1, the standardized solution will always be the same because that solution rescales the variance of the latent variable to be 1 and the fully standardized solution also rescales the variance of each indicator to be 1. We should pick the strongest indicator at 1.0. This makes the results less confusing to readers because all of the loadings will be less than 1.0. If you fixed a weak indicator at 1.0, an indicator that was twice as strong would have a loading of 2.0 and that would be confusing to readers. You do not need to fix the loadings at 1, any number will identify the model equally well.3.2.1 Output and Interpret3.2.1 Missing value summarySUMMARY OF DATA Number of patterns SUMMARY OF MISSING DATA PATTERNS MISSING DATA PATTERNS 1 2 3 4 5 6 x x x x x x x x x x x x x x x x x x x x x x x x x x x x 7 x x x x x 8 x x x x x 8Y1 Y2 Y3 Y4 Y5 Y6MISSING DATA PATTERN FREQUENCIES Pattern Frequency Pattern Frequency 1 473 4 1 2 15 5 1Pattern 7 8Frequency 2 3Introduction to Mplus, Alan C. Acock19336 0.1001COVARIANCE COVERAGE OF DATA Minimum covariance coverage value PROPORTION OF DATA PRESENT Covariance Coverage Y1 Y2 ________ ________ 0.994 0.990 0.996 0.992 0.994 0.984 0.986 0.992 0.994 0.960 0.962 Covariance Coverage Y6 ________ 0.966Y3 ________ 0.998 0.988 0.996 0.964Y4 ________Y5 ________Y1 Y2 Y3 Y4 Y5 Y60.990 0.990 0.9600.998 0.966Y63.2.2. Model FitTHE MODEL ESTIMATION TERMINATED NORMALLY TESTS OF MODEL FIT Chi-Square Test of Model Fit Value Degrees of Freedom P-Value3.895 8 0.8665Chi-Square Test of Model Fit for the Baseline Model Value 589.067 Degrees of Freedom 15 P-Value 0.0000 CFI/TLI CFI TLI 1.000 1.013 -4850.279 -4848.331 19 9738.558 9818.597 20Loglikelihood H0 Value H1 Value Information Criteria Number of Free Parameters Akaike (AIC) Bayesian (BIC) Introduction to Mplus, Alan C. AcockSample-Size Adjusted BIC (n* = (n + 2) / 24)9758.290RMSEA (Root Mean Square Error Of Approximation) Estimate 0.000 90 Percent C.I. 0.000 Probability RMSEA &lt;= .05 0.995 SRMR (Standardized Root Mean Square Residual) Value 0.0150.0273.2.3. Model result We are usually interested in the fully standardized results but the unstandardized results appear first. MODEL RESULTS--Unstandardized Estimate F1 Y1 Y2 Y3 F2 Y4 Y5 Y6 F2 F1 Intercepts Y1 Y2 Y3 Y4 Y5 Y6 WITH -0.033 -0.017 0.030 0.037 -0.022 -0.012 0.066 0.053 0.063 0.063 0.063 0.065 0.058 0.059 -0.621 -0.267 0.478 0.590 -0.336 -0.209 1.120 0.534 0.790 0.633 0.555 0.737 0.835 0.263 BY 1.000 1.032 0.869 0.000 0.129 0.105 999.000 7.972 8.316 999.000 0.000 0.000 BY 1.000 1.123 1.019 0.000 0.098 0.088 999.000 11.430 11.532 999.000 0.000 0.000 S.E. Est./S.E. Two-Tailed P-ValueIntroduction to Mplus, Alan C. Acock21Variances F1 F2 Residual Variances Y1 Y2 Y3 Y4 Y5 Y60.912 0.786 1.041 0.803 1.012 1.287 0.861 1.0770.125 0.138 0.095 0.100 0.095 0.123 0.112 0.0987.308 5.677 10.977 8.044 10.612 10.449 7.664 10.9920.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000STANDARDIZED MODEL RESULTS STDYX Standardization Estimate F1 Y1 Y2 Y3 F2 Y4 Y5 Y6 F2 F1 Intercepts Y1 Y2 Y3 Y4 Y5 WITH -0.039 -0.012 0.021 0.026 -0.015 -0.009 0.062 0.045 0.045 0.045 0.045 0.045 -0.622 -0.267 0.478 0.590 -0.336 -0.209 0.534 0.790 0.633 0.555 0.737 0.835 BY 0.616 0.702 0.596 0.046 0.047 0.045 13.498 14.916 13.112 0.000 0.000 0.000 BY 0.683 0.767 0.695 0.035 0.034 0.035 19.573 22.537 20.011 0.000 0.000 0.000 S.E. Est./S.E. Two-Tailed P-ValueIntroduction to Mplus, Alan C. Acock22Y6 Variances F1 F2 Residual Variances Y1 Y2 Y3 Y4 Y5 Y6 STDY Standardization0.051 1.000 1.000 0.533 0.411 0.517 0.621 0.507 0.6450.045 0.000 0.000 0.048 0.052 0.048 0.056 0.066 0.0541.120 999.000 999.000 11.174 7.868 10.699 11.057 7.677 11.8870.263 999.000 999.000 0.000 0.000 0.000 0.000 0.000 0.000Estimate F1 Y1 Y2 Y3 F2 Y4 Y5 Y6 F2 F1 Intercepts Y1 Y2 Y3 Y4 Y5 Y6 WITH -0.039 -0.012 0.021 0.026 -0.015 -0.009 0.051 BY 0.616 0.702 0.596 BY 0.683 0.767 0.695S.E. 0.035 0.034 0.035 0.046 0.047 0.045 0.062 0.045 0.045 0.045 0.045 0.045 0.045Est./S.E. 19.573 22.537 20.011 13.498 14.916 13.112 -0.622 -0.267 0.478 0.590 -0.336 -0.209 1.120Two-Tailed P-Value 0.000 0.000 0.000 0.000 0.000 0.000 0.534 0.790 0.633 0.555 0.737 0.835 0.263Introduction to Mplus, Alan C. Acock23Variances F1 F2 Residual Variances Y1 Y2 Y3 Y4 Y5 Y61.000 1.000 0.533 0.411 0.517 0.621 0.507 0.6450.000 0.000 0.048 0.052 0.048 0.056 0.066 0.054999.000 999.000 11.174 7.868 10.699 11.057 7.677 11.887999.000 999.000 0.000 0.000 0.000 0.000 0.000 0.000Each unstandardized estimate represents the amount of change in the outcome variable as a function of a single unit change in the variable causing it. · Different measures often have different scales, so you will often find it useful to examine the standardized coefficients when you want to compare the relative strength of associations across observed variables that are measured on different scales. · Mplus provides two standardized coefficients. The first, labeled StdYX, standardizes based on latent and observed variables' variances. This standardized coefficient represents the amount of standardized change in an outcome variable per standard deviation unit of a predictor variable. · Finally, the r-square output illustrates the amount of variance accounted for in the indicators. As is the case with exploratory factor analysis of continuous outcome variables, you may want to use the mlm or mlmv estimators in lieu of the default ml estimator if your input data are not distributed joint multivariate normal by using the ESTIMATOR = option on the ANALYSIS command. The mlm option provides a mean-adjusted chisquare model test statistic whereas the mlmv option produces a mean and variance adjusted chi-square test of model fit; both options also induce Mplus to produce robust standard errors displayed in the model results table that are used to compute z tests of significance for individual parameter estimates.Introduction to Mplus, Alan C. Acock24SECTION 4: Exploratory Factor Analysis as an Alternative to CFAMost often, when doing a CFA, a researcher uses modification indexes to modify the matrix by allowing some fixed parameters to be free. · We may allow an item load on more than one factor or · We may allow two items to have correlated errors. When we change a model this way it is no longer confirmatory, but exploratory. We are combining the modification indexes with our own judgement to change the model. · In Mplus 5.1 and EFA alternative was introduced that can challenge CFA. · The rotation will find the optimal solution and this will be a better fit than we can do by looking at a few indexes and using our own judgment. · However, if the optimal solution makes no sense, then we have a different problem. · Suppose you had two latent variables measured for both the husband and wife. · Alternatively, you might think of these as two latent variables measured for the same person, but at two times, say one year apart. · You believe that the first three measures are indicators of the first factor and the second three are indicators of the second factor. · However, it is usually unreasonable to assume that all the cross loadings are 0.000. You expect them to be small, butIntroduction to Mplus, Alan C. Acock25there is no necessity to say they must be exactly zero. · Consider andolescents who have three beliefs about the certainty that they will be caught and three beliefs about the severity of punishment if they are caught. You measure them at two time points, at age 15 and again at age 17. Discuss how this is different from a CFA model What results would support your thinking? 1. The three beliefs about the certainty of being caught (Y1 ­ Y3) would load strongly on factors 1 (measured the first year) and the same 3 measured a year later (Y7 ­ Y9), but have weak loadings on factors 2 (measured the first year) and 4 (measured a year later). 2. Conversely, the beliefs about the severity of punishment (Y4 ­ Y6; Y10 ­ Y12) should load strongly on factors 2 and 4 but should have relatively weak loadings on factors 1 and 3. 3. The loadings of Y1 ­ Y6 should be identical to the corresponding loadings of Y7 ­ Y12 4. The errorst E1 ­ E6 should be correlated with the corresponding errors in E7 ­ E12. Mplus calls this exploratory factor analysis because we are not fixing values at particular values, but clearly we are putting enormous constraints on the model.Mplus VERSION 5.1 MUTHEN &amp; MUTHEN 06/30/2008 6:14 PM INPUT INSTRUCTIONS TITLE: example3.inp this is an example of an EFA at two timepoints with factor loading invariance and correlated residuals across time FILE IS example3.dat; NAMES ARE y1-y12; f1-f2 BY y1-y6 (*t1 1); f3-f4 BY y7-y12 (*t2 1); f3-f4 WITH f1-f2; y1-y6 PWITH y7-y12; TECH1 STANDARDIZED; ML GEOMIN COVARIANCE OBLIQUEDATA: VARIABLE: MODEL:OUTPUT:Estimator Rotation Row standardization Type of rotation THE MODEL ESTIMATION TERMINATED NORMALLYTESTS OF MODEL FITIntroduction to Mplus, Alan C. Acock26Chi-Square Test of Model Fit Value Degrees of Freedom P-Value 43.990 42 0.3873Chi-Square Test of Model Fit for the Baseline Model Value Degrees of Freedom P-Value CFI/TLI CFI TLI Loglikelihood H0 Value H1 Value Information Criteria Number of Free Parameters Akaike (AIC) Bayesian (BIC) Sample-Size Adjusted BIC (n* = (n + 2) / 24) 48 18888.807 19091.108 18938.753 -9396.403 -9374.409 0.998 0.997 1265.442 66 0.0000RMSEA (Root Mean Square Error Of Approximation) Estimate 90 Percent C.I. Probability RMSEA &lt;= .05 SRMR (Standardized Root Mean Square Residual) Value 0.027 0.010 0.000 1.000 0.032MODEL RESULTS Estimate F1 Y1 Y2 Y3 Y4 Y5 Y6 F2 Y1 BY 0.052 0.058 0.898 0.369 BY 0.744 0.896 0.726 0.014 -0.098 0.013 0.062 0.072 0.055 0.039 0.060 0.034 11.959 12.523 13.103 0.370 -1.619 0.398 0.000 0.000 0.000 0.712 0.106 0.690 S.E. Est./S.E. Two-Tailed P-ValueIntroduction to Mplus, Alan C. Acock27Y2 Y3 Y4 Y5 Y6 F3 Y7 Y8 Y9 Y10 Y11 Y12 Y7 Y8 Y9 Y10 Y11 Y12 F3 F1 F2 F4 F1 F2 F3 F1 Y7 Y8 · WITH WITH BY-0.016 0.005 0.734 0.908 0.749 0.744 0.896 0.726 0.014 -0.098 0.013 0.052 -0.016 0.005 0.734 0.908 0.749 0.414 0.310 0.299 0.289 0.494 0.451 0.397 0.128 ·0.053 0.018 0.061 0.072 0.064 0.062 0.072 0.055 0.039 0.060 0.034 0.058 0.053 0.018 0.061 0.072 0.064 0.066 0.067 0.069 0.070 0.085 0.069 0.061 0.058-0.304 0.256 12.082 12.626 11.760 11.959 12.523 13.103 0.370 -1.619 0.398 0.898 -0.304 0.256 12.082 12.626 11.760 6.241 4.592 4.364 4.116 5.823 6.571 6.463 2.2200.761 0.798 0.000 0.000 0.000 0.000 0.000 0.000 0.712 0.106 0.690 0.369 0.761 0.798 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.026F4BYF2 Y1 Y2 ·WITH WITH WITHSTANDARDIZED MODEL RESULTS STDYX Standardization Estimate F1 Y1 Y2 Y3 Y4 Y5 Y6 BY 0.584 0.707 0.569 0.011 -0.077 0.010 S.E. 0.043 0.049 0.038 0.030 0.048 0.026 Est./S.E. 13.475 14.449 15.086 0.370 -1.614 0.398 Two-Tailed P-Value 0.000 0.000 0.000 0.712 0.107 0.690Introduction to Mplus, Alan C. Acock28F2Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y10 Y11 Y12BY0.041 -0.013 0.004 0.568 0.720 0.570 0.586 0.728 0.605 0.011 -0.079 0.0110.046 0.042 0.014 0.042 0.051 0.043 0.046 0.045 0.040 0.030 0.049 0.027 0.046 0.043 0.015 0.041 0.054 0.040 0.059 0.0630.898 -0.304 0.256 13.617 14.120 13.177 12.690 16.053 15.166 0.370 -1.621 0.398 0.897 -0.304 0.256 13.651 13.443 14.556 6.820 4.7580.369 0.761 0.798 0.000 0.000 0.000 0.000 0.000 0.000 0.711 0.105 0.691 0.369 0.762 0.798 0.000 0.000 0.000 0.000 0.000F3BYF4 Y7 Y8 Y9 Y10 Y11 Y12 F3 F1 F2 · ·BY 0.041 -0.013 0.004 0.565 0.730 0.586 WITH 0.403 0.301 · 18:14:23 18:14:24 00:00:01Beginning Time: Ending Time: Elapsed Time:MUTHEN &amp; MUTHEN 3463 Stoner Ave. Los Angeles, CA 90066 Tel: (310) 391-9971 Fax: (310) 391-8971 Web: www.StatModel.com Support: [email protected] Copyright (c) 1998-2008 Muthen &amp; MuthenSECTION 5: Path Analysis4.1. Model and ProgramIntroduction to Mplus, Alan C. Acock29TITLE: DATA: VARIABLE: MODEL:OUTPUT:MODEL indirect: y2 ind x1; y2 ind x2; y2 ind x3; y3 ind x1; y3 ind x2; y3 ind x3; standardized mod(3.84);ex3.11 This is an example of a path analysis with continuous dependent variables FILE IS ex3.11.dat; NAMES ARE y1-y3 x1-x3; y1 y2 ON x1 x2 x3; y3 ON y1 y2 x2;Introduction to Mplus, Alan C. Acock305.2. Indirect EffectsThe MODEL INDIRECT: subcommand estimates indirect effects for you · You get the Total indirect effect that combines as many specific indirect effects as there are in the model · Specific indirect effects of x1 go y3 include o x1  y1  y3 o x1  y2  y3 · Tests of significant for both specific and total indirect effects Estimate and interpret the output:SECTION 6: Putting it Together--Structural Equation Model with CFAInterpret the figure. Notice indirect effects.Introduction to Mplus, Alan C. Acock31TITLE:DATA: VARIABLE: MODEL:OUTPUT:example2cfa This is an example of a SEM with CFA factors with continuous factor indicators And Indirect Effects FILE IS example2.dat; NAMES ARE y1-y12; f1 BY y1-y3; f2 by y4-y6; f3 by y7-y9; f4 BY y10-y12; f3 ON f1-f2; f4 ON f3; MODEL INDIRECT: f4 ind f1; f4 ind f2; standardized mod(3.84)6.1. Output and InterpretationMplus VERSION 5.1 MUTHEN &amp; MUTHEN 06/30/2008 8:12 PM SUMMARY OF ANALYSIS Number Number Number Number Number of of of of of groups observations dependent variables independent variables continuous latent variables 1 500 12 0 4Observed dependent variables Continuous Y1 Y2 Y3 Y7 Y8 Y9 Continuous latent variables F1 F2 F3Y4 Y10 F4Y5 Y11Y6 Y12Estimator Information matrix Maximum number of iterations Convergence criterion Maximum number of steepest descent iterationsML OBSERVED 1000 0.500D-04 20Introduction to Mplus, Alan C. Acock32TESTS OF MODEL FIT Chi-Square Test of Model Fit Value 53.492 Degrees of Freedom 50 P-Value 0.3417 Chi-Square Test of Model Fit for the Baseline Model Value 4600.240 Degrees of Freedom 66 P-Value 0.0000 CFI/TLI CFI 0.999 TLI 0.999 Loglikelihood H0 Value -6483.831 H1 Value -6457.085 Information Criteria Number of Free Parameters 40 Akaike (AIC) 13047.662 Bayesian (BIC) 13216.247 Sample-Size Adjusted BIC 13089.284 (n* = (n + 2) / 24) RMSEA (Root Mean Square Error Of Approximation) Estimate 0.012 90 Percent C.I. 0.000 0.032 Probability RMSEA &lt;= .05 1.000 SRMR (Standardized Root Mean Square Residual) Value 0.019 MODEL RESULTS F1 BY Estimate 1.000 1.103 0.942 1.000 1.006 1.023 1.000 0.894 0.902 1.000 S.E. 0.000 0.062 0.058 0.000 0.057 0.060 0.000 0.021 0.021 0.000 Est./S.E. 999.000 17.881 16.346 999.000 17.691 17.064 999.000 41.937 42.479 999.000 Two-Tailed P-Value 999.000 0.000 0.000 999.000 0.000 0.000 999.000 0.000 0.000 999.000F2Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y10BYF3BYF4BYIntroduction to Mplus, Alan C. Acock33F3 F4 F2Y11 Y12 F1 F2 F3 F1ON ON WITH0.734 0.684 0.640 0.912 0.546 0.297 0.599 0.618 0.367 0.296 0.412 0.400 0.340 0.392 0.183 0.191 0.181 0.240 0.183 0.213 0.525 0.5650.028 0.028 0.069 0.074 0.032 0.038 0.061 0.064 0.033 0.033 0.033 0.034 0.031 0.034 0.019 0.017 0.017 0.027 0.017 0.018 0.049 0.04926.424 24.405 9.271 12.399 16.975 7.767 9.766 9.717 11.044 8.946 12.309 11.640 10.888 11.370 9.799 11.268 10.812 8.746 10.791 11.998 10.636 11.4880.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000Variances F1 F2 Residual Variances Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y10 Y11 Y12 F3 F4STANDARDIZED MODEL RESULTS STDYX Standardization Estimate F1 Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 BY 0.787 0.843 0.751 0.779 0.805 0.789 0.948 0.934 0.938 0.023 0.020 0.025 0.023 0.021 0.022 0.006 0.007 0.007 34.084 41.362 30.614 34.055 37.480 35.305 153.038 131.246 136.242 S.E. Est./S.E.Two-Tailed P-Value 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000F2BYF3BYIntroduction to Mplus, Alan C. Acock34F4Y10 Y11 Y12 F1 F2 F3BY0.902 0.869 0.835 0.388 0.561 0.680 0.488 1.000 1.000 0.380 0.289 0.437 0.393 0.352 0.378 0.101 0.128 0.120 0.186 0.244 0.302 0.322 0.5380.013 0.014 0.017 0.039 0.036 0.027 0.043 0.000 0.000 0.036 0.034 0.037 0.036 0.035 0.035 0.012 0.013 0.013 0.023 0.025 0.028 0.031 0.03770.200 59.982 50.003 10.057 15.610 24.795 11.343 999.000 999.000 10.450 8.397 11.862 11.018 10.197 10.713 8.572 9.598 9.287 8.010 9.698 10.828 10.421 14.4330.000 0.000 0.000 0.000 0.000 0.000 0.000 999.000 999.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000F3 F4 F2ON ONWITH F1 Variances F1 F2 Residual Variances Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y10 Y11 Y12 F3 F4TOTAL, TOTAL INDIRECT, SPECIFIC INDIRECT, AND DIRECT EFFECTS STANDARDIZED TOTAL, TOTAL INDIRECT, SPECIFIC INDIRECT, AND DIRECT EFFECTS STDYX Standardization Two-Tailed Estimate S.E. Est./S.E. P-Value Effects from F1 to F4 Total Total indirect Specific indirect F4 F3 F1 Effects from F2 to F4 Total Total indirect Specific indirect 0.263 0.263 0.028 0.028 9.269 9.269 0.000 0.0000.263 0.382 0.3820.028 0.029 0.0299.269 12.999 12.9990.000 0.000 0.000Introduction to Mplus, Alan C. Acock35F4 F3 F20.3820.02912.9990.000MODEL MODIFICATION INDICES Minimum M.I. value for printing the modification index M.I. E.P.C. Std E.P.C. BY Statements F3 BY Y1 5.980 0.103 0.131 WITH Statements Y3 WITH Y2 6.126 0.091 0.091 Y4 WITH Y3 5.405 0.053 0.053 Y5 WITH Y3 5.265 -0.049 -0.049 Y8 WITH Y2 5.695 -0.037 -0.037 Y9 WITH Y6 4.801 0.035 0.035 Beginning Time: Ending Time: Elapsed Time: 20:12:48 20:12:49 00:00:01 3.840 StdYX E.P.C. 0.134 0.260 0.130 -0.132 -0.155 0.1336.2. Interpretation of Modification Indices· We could reduce Chi-square, which now is Chi-square(50) = 53.492, by about 5.265 if we allowed the error term for Y5 to be correlated with the error term for Y3. · The correlation of the two errors would be about -.132--does this make sense? · We would do these one at a time · We would only do it if it made sense. Say Y5 and Y3 are pen and pencil tests and all the others are face to face interviews. There might be a method effect that we could incorporate as an error term · We might not have much to gain even if there is a big modification index if the fit is already good. · New Chi-square would be approximately Chi-square(49) = 53.492 ­ 5.265. A reduction in Chi-square of 5.265 with one degree of freedom would be highly significant. Not much need to improve on a CFI = .997; RMSEA = .012;Introduction to Mplus, Alan C. Acock36SECTION 7: Putting it Together--Structural Equation Model with EFA &amp; CFAWe may have a situation where we are sufficiently confident to have F3 and F4 represented by a CFA model, but not that confident about F1 and F2 for which we want to do an EFA.Here are the program and results: The (*1) in the Model line for f1-f2 by y1-y6 (*1); is included so Mplus knows this is an EFA set. We expect y1-y3 to have strong loadings on f1 and weak loadings on f2. We expect y4-y6 to have weak loadings on f1 and strong loadings on f2. Still, we are not sufficiently confident of this to impose the restriction that these loadings are exactly 0.000.TITLE: example2.inp This is an example of a SEM with 37Introduction to Mplus, Alan C. AcockDATA: VARIABLE: MODEL:OUTPUT:EFA and CFA factors with continuous factor indicators FILE IS example2.dat; NAMES ARE y1-y12; f1-f2 BY y1-y6 (*1); f3 BY y7-y9; f4 BY y10-y12; f3 ON f1-f2; f4 ON f3; MODEL INDIRECT: f4 ind f1; f4 ind f2; Standardized mod(3.84)Mplus VERSION 5.1 MUTHEN &amp; MUTHEN 06/30/2008 8:32 PM TESTS OF MODEL FIT Chi-Square Test of Model Fit Value 51.353 Degrees of Freedom 46 P-Value 0.2720 Chi-Square Test of Model Fit for the Baseline Model Value 4600.240 Degrees of Freedom 66 P-Value 0.0000 CFI/TLI CFI 0.999 TLI 0.998 Loglikelihood H0 Value -6482.762 H1 Value -6457.085 Information Criteria Number of Free Parameters 44 Akaike (AIC) 13053.524 Bayesian (BIC) 13238.966 Sample-Size Adjusted BIC 13099.308 (n* = (n + 2) / 24) RMSEA (Root Mean Square Error Of Approximation) Estimate 0.015 90 Percent C.I. 0.000 0.034 Probability RMSEA &lt;= .05 1.000 SRMR (Standardized Root Mean Square Residual) Value 0.018 MODEL RESULTSIntroduction to Mplus, Alan C. Acock38F1F2Y1 Y2 Y3 Y4 Y5 Y6 Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y10 Y11 Y12 F1 F2 F3BYEstimate 0.751 0.858 0.736 0.036 -0.028 0.002 0.034 -0.002 -0.008 0.763 0.810 0.802 1.000 0.894 0.902 1.000 0.734 0.684 0.493 0.721 0.546 0.479 1.000 1.000 0.376 0.290 0.406 0.408 0.329 0.393 0.183 0.191 0.181 0.240 0.183 0.213S.E. 0.048 0.042 0.045 0.051 0.049 0.004 0.045 0.016 0.035 0.050 0.048 0.041 0.000 0.021 0.021 0.000 0.028 0.028 0.058 0.057 0.032 0.053 0.000 0.000 0.034 0.035 0.034 0.035 0.033 0.035 0.019 0.017 0.017 0.027 0.017 0.018Est./S.E. 15.608 20.467 16.353 0.711 -0.568 0.627 0.755 -0.150 -0.220 15.367 16.837 19.461 999.000 41.937 42.479 999.000 26.424 24.405 8.461 12.752 16.975 9.094 999.000 999.000 11.064 8.239 11.817 11.742 10.046 11.073 9.796 11.269 10.812 8.746 10.791 11.998Two-Tailed P-Value 0.000 0.000 0.000 0.477 0.570 0.530 0.450 0.881 0.826 0.000 0.000 0.000 999.000 0.000 0.000 999.000 0.000 0.000 0.000 0.000 0.000 0.000 999.000 999.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000BYF3BYF4BYF3 F4 F2ON ONWITH F1 Variances F1 F2 Residual Variances Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y10 Y11 Y12Introduction to Mplus, Alan C. Acock39F3 F40.527 0.5650.049 0.04910.644 11.4880.000 0.000STANDARDIZED MODEL RESULTS STDYX Standardization Estimate F1 Y1 Y2 Y3 Y4 Y5 Y6 Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y10 Y11 Y12 F1 F2 F3 BY 0.764 0.848 0.758 0.036 -0.028 0.002 0.034 -0.002 -0.008 0.756 0.825 0.787 0.948 0.934 0.938 0.902 0.869 0.835 0.386 0.565 0.680 0.479 0.008 0.031 0.007 0.074 0.071 0.068 0.044 0.050 0.056 S.E. 0.037 0.024 0.033 0.051 0.050 0.003 0.046 0.015 0.036 0.037 0.035 0.023 0.006 0.007 0.007 0.013 0.014 0.017 0.043 0.038 0.027 0.053 0.045 0.045 0.045 0.045 0.045 0.045 0.045 0.045 0.045 Est./S.E. 20.741 34.915 23.068 0.711 -0.568 0.627 0.755 -0.150 -0.220 20.282 23.257 33.668 153.043 131.230 136.226 70.200 59.982 50.002 8.914 14.919 24.796 9.094 0.183 0.688 0.146 1.657 1.590 1.528 0.983 1.115 1.252Two-Tailed P-Value 0.000 0.000 0.000 0.477 0.570 0.530 0.450 0.881 0.826 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.855 0.491 0.884 0.098 0.112 0.126 0.326 0.265 0.211F2BYF3BYF4BYF3 F4 F2ON ON WITHF1 Intercepts Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9Introduction to Mplus, Alan C. Acock40Y10 Y11 Y12 Variances F1 F2 Residual Variances Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y10 Y11 Y12 F3 F4 R-SQUARE Latent Variable F3 F40.008 0.028 0.025 1.000 1.000 0.390 0.283 0.431 0.401 0.341 0.378 0.101 0.128 0.120 0.186 0.244 0.302 0.323 0.5380.045 0.045 0.045 0.000 0.000 0.037 0.036 0.038 0.036 0.036 0.036 0.012 0.013 0.013 0.023 0.025 0.028 0.031 0.0370.170 0.616 0.554 999.000 999.000 10.510 7.793 11.385 11.149 9.459 10.467 8.570 9.599 9.287 8.009 9.698 10.828 10.432 14.4330.865 0.538 0.580 999.000 999.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 Two-Tailed P-Value 0.000 0.000Estimate 0.677 0.462S.E. 0.031 0.037Est./S.E. 21.887 12.398STANDARDIZED TOTAL, TOTAL INDIRECT, SPECIFIC INDIRECT, AND DIRECT EFFECTS STDYX Standardization Two-Tailed Estimate S.E. Est./S.E. P-Value Effects from F1 to F4 Total 0.263 0.031 8.353 0.000 Total indirect 0.263 0.031 8.353 0.000 Specific indirect F4 F3 F1 0.263 0.031 8.353 0.000 Effects from F2 to F4 Total Total indirect Specific indirect F4 F3 0.384 0.384 0.030 0.030 12.590 12.590 0.000 0.000Introduction to Mplus, Alan C. Acock41F20.3840.03012.5900.000 3.840 StdYX E.P.C. 0.187 0.336 -0.133 0.135 -0.133 -0.157 0.133MODEL MODIFICATION INDICES Minimum M.I. value for printing the modification index M.I. E.P.C. Std E.P.C. BY Statements F3 BY Y1 6.537 0.144 0.184 WITH Statements Y3 WITH Y2 5.262 0.115 0.115 Y4 WITH Y1 4.954 -0.052 -0.052 Y4 WITH Y3 5.288 0.055 0.055 Y5 WITH Y3 4.367 -0.049 -0.049 Y8 WITH Y2 5.716 -0.037 -0.037 Y9 WITH Y6 4.853 0.036 0.036 Beginning Time: Ending Time: Elapsed Time: 20:32:33 20:32:33 00:00:00Section 8: SummaryThis provides a brief introduction to Mplus. We have not covered any of the statistical theory underlying Mplus, but this should be enough for you to read the Manual and follow more complex explications of Mplus and SEM. Key things to remember: 1. BY means measured by and is the path (loading) between latent variables and their indicators. 2. ON is the structural path between variables. In last example, F4 depends ON F3, F3 depends ON both F1 and F2. 3. WITH means correlated with. Two uses include: a. For exogenous variables WITH means the exogenous variables are correlated. In last example, F1 is correlated WITH F2. b. For indicators WITH means the errors/residuals are correlated. In last examples, the modification indices suggest we might correlate the error for Y3 WITH the error for Y5. Additional introductory content is available at: http://www.ats.ucla.edu/stat/mplus/Introduction to Mplus, Alan C. Acock 42`

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