By Hui Bian Office For Faculty Excellence

Fall 2011

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Regression models: only observed variables are modeled. only the dependent variable in regression has an error term. Independent

variables are assumed to be modeled without error. The partial coefficient for any independent variable controls for all other

independents, whether or not an actual causal effect is plausible. Path models:

only observed variables without latent variables. Unlike regression models but like structural equation models, independents

can be both causes and effects of other variables. Only the endogenous variables in path models have error terms. Exogenous

variables in path models are assumed to be measured without error. Partial coefficients are calculated using only the independents in a direct path

to the endogenous variable.

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AMOS output Standardized regression weights: Structural or path coefficients in SEM. Standardized

estimates are used, for instance, when comparing direct effects on a given endogenous variable in a single-group study.

Indicator variable regression weights. By convention, the indicator variables should have standardized regression weights of .7 or higher on the latent variable they represent.

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AMOS output: Communalities. The Squared Multiple Correlation is the communality estimate for

an indicator variable. The communality measures the percent of variance in a given indicator variable explained by its latent variable (factor) and may be interpreted as the reliability of the indicator.

If a variable has low theoretic importance and a low communality, it may be targeted for removal in the model-modification.

The communality is equal to the squared standardized regression weight. This is why communalities are sometimes defined as the squared factor loadings, where loadings are defined as the standardized regression weights.

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AMOS output Unstandardized regression weights: are based on raw

data or covariance matrixes. When comparing across groups (across samples) and

groups have different variances, unstandardized comparisons are preferred.

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AMOS output The critical ratio and significance of path coefficients.

When the critical ratio (CR) is > 1.96 for a regression weight, that path is significant at the .05 level or better (that is, its estimated path parameter is significant). In the p-value column, three asterisks (***) indicate significance smaller than .001.

The critical ratio and the significance of factor covariances. The significance of estimated covariances among the latent variables are assessed in the same manner: if CR > 1.96, the factor covariance is significant.

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Purpose of this exercise is to show you how AMOS estimates parameters in multiple regression.

Data used is from Schumacker and Lomax (2004).

We have three predictors and one dependent variable.

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Path diagram

Draw path diagram using AMOS: File > New Three independent variables: IV1-IV3 (observed) One dependent variable: DV No latent variables in this model Three independent variables are correlated

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The single-headed arrows represent linear dependencies. For example, the arrow leading from IV1 to DV indicates that DV scores depend, in part, on IV1.

The variable error is enclosed in a circle because it is not directly observed. Error represents much more than random fluctuations in DV scores due to measurement error. 10

Model identification The variance of a variable, and any regression

weights associated with it, depends on the units in which the variable is measured.

Error is an unobserved variable, there is no natural way to specify a measurement unit for it. Assigning an arbitrary value to a regression weight associated with error can be thought of as a way of indirectly choosing a unit of measurement for error. 11

Model identification It is impossible to estimate the regression weight and

variance for the regression of DV on error, There is just not enough information.

We can solve this identification problem by fixing either the regression weight applied to error in predicting DV, or the variance of the error variable itself, at an arbitrary, nonzero value. Let’s fix the regression weight at 1. This will yield the same estimates as conventional linear regression.

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Model identification Every unobserved variable presents this

identification problem, which must be resolved by imposing some constraint that determines its unit of measurement.

Changing the scale unit of the unobserved error variable does NOT change the overall model fit.

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Fix regression weight Right-click the arrow that points from error to DV and

choose Object Properties from the pop-up menu.

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Type 1 inside Regression weight box, which is under Parameters tab.

Before run data analysis Go to View > Analysis Properties > Click Output tab

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Text outputs

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4 sample variances and 6 sample covariances, for a total of 10 sample moments, or use p (p+1)/2, we have 4 observed variables, then the number of sample distinct value is equal to 4(4+1)/2 = 10.

3 regression paths, 4 model variances, and 3 model covariances, for a total of 10 parameters that must be estimated. Hence, the model has zero degrees of freedom.

Such a model is often called saturated or just-identified.

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Text output

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The standardized regression weights and the correlations are independent of the units in which all variables are measured; therefore, they are not affected by the choice of identification constraints.

Text output

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Squared multiple correlations are independent of units of measurement. Amos displays a squared multiple correlation for each endogenous variable.

Graphics output (unstandardized estimates)

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Graphic output (standardized estimates)

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Correlations

Conclusion In this example, IV1, IV2, and IV3 account for 69% of

the variance of DV. IV1 and IV3 are significant predictors.

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Path analysis model Only focus on relationships of multiple observed

variables Analysis of several regression equations

simultaneously. Use the same idea of model fitting and testing as any

SEM. Data used is still from Schumacker and Lomax’s book:

A beginner’s guide to structural equation modeling (2004).

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The research question is whether the specified model is supported by the sample data?

Path diagram (please try to draw and identify this model)

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Chi-square test is not significant

Unstandardized graphic output

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Standardized graphic output

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Choices of model fit indexes Reporting CMIN, RMSEA, and one of the baseline fit

measures. If there is model comparison, also report one of the

parsimony measures and one the information theory measures.

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Model fit

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The closer RMR is to 0, the better the model fit. Rule of thumb: RMR should be < .10, or .08, or .06, or .05 or even .04.

Model fit

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Rule of thumb: a value of the RMSEA of about 0.05 or less would indicate a close fit of the model in relation to the degrees of freedom.

Model fit

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NFI values above .95 are good. RFI, IFI, TLI, and CFI values close to 1 indicate a very good fit.

Model fit Chi-square: χ2 = 1.25, df = 3, p = .74 Root-mean-square error of approximation (RMSEA): it

is equal to 0.00 (<.05 is acceptable) Goodness-of-fit index (GFI): .997 (>.95 is acceptable)

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Model fit

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Path diagram

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This example: 4 tests: knowledge, value, satisfaction, and performance. Each test was randomly split into two halves, and each half was scored separately.

Measurement model The portion of the model that specifies how the

observed variables depend on the unobserved, or latent, variables is sometimes called the measurement model.

The current model has four distinct measurement submodels.

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The scores of the two split-half subtests, 1knowledge and 2knowledge, are hypothesized to depend on the single underlying latent variable, knowledge.

According to the model, scores on the two subtests may still disagree, owing to the influence of measurement errors.

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Measurement model (e.g.) 1knowledge and 2knowledge are called indicators of the

latent variable knowledge.

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Measurement model

Structural model The portion of the model that specifies how the latent

variables are related to each other is sometimes called the structural model.

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Structural model

Model identification It is necessary to fix the unit of measurement of each

unobserved variable by suitable constraints on the parameters.

Find a single-headed arrow leading away from each unobserved variable in the path diagram, and fix the corresponding regression weight to an arbitrary value such as 1.

If there is more than one single-headed arrow leading away from an unobserved variable, any one of them will do.

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Text output

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The hypothesis that current Model is correct is accepted.

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Standardized regression weights

Reliability estimates

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The purpose of confirmatory factor analysis is to test hypothesis about a factor structure. The theories come first. The model is derived from the theory. The model is tested for consistency with observed

data.

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Diagram of CFA model

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Two-factor model: spatial ability and verbal ability Three observed variables measure each construct. The relationship between the factor and its indicator

is represented by a factor loading. The measurement error represents other variation

for a particular observed variable. The variance of measurement error is estimated.

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Model summary

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Regression weights

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Path diagram with standardized estimates displayed.

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Squared multiple correlations

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The squared multiple correlations can be interpreted as follows: To take wordmean as an example: 71% of its variance is accounted for by verbal ability. The remaining 29% of its variance is accounted for by the

unique factor e6. If e6 represented measurement error only, we could say

that the estimated reliability of wordmean is 0.71. 0.71 is an estimate of a lower-bound on the reliability of

wordmean.

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Model fit Chi-square: χ2 = 7.85, df = 8, p = .45 Root-mean-square error of approximation (RMSEA): it

is equal to 0.00 (<.05 is acceptable) Goodness-of-fit index (GFI): .966 (>.95 is acceptable)

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AMOS allows us to compare multiple samples across the same measurement instrument or multiple population groups (e.g., males vs. females).

We are going to use the data from IBM SPSS company (the previous data for CFA).

We want to test the equality of the factor loadings for two separate groups of school children, girls and boys.

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Before testing measurement invariance across groups, we need test individual mode first.

If consistency is found, then we will proceed to do multiple groups testing.

The goal of testing for measurement invariance is to determine if the same SEM model is applicable across groups.

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The general procedure is to test measurement invariance between the unconstrained model for all groups combined, then for a model with constrained parameters (parameters are constrained to be equal between the groups).

If the chi-square difference statistic is not significant between the original and constrained models, then we conclude that the model has measurement invariance across groups.

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Which parameters are constrained to be equal? The selection of parameters to constrain depends on our research questions. Invariant factor loadings Invariant structural relations among latent variables

If lack of measurement invariance is found, the meaning of the latent construct is shifting across groups.

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First draw a diagram for a single group

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By default, Amos Graphics assumes that both groups have the same path diagram, so the path diagram does not have to be drawn a second time for the second group.

Select Manage Groups from the Analyze menu. Name the first group Girls.

Click on the New button to add a second group to the analysis. Name this group Boys.

Click the New button successively to add additional groups as needed.

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Click

Type Boys

Select data sets: Use of the Grouping Variable and Group Value buttons. Select the Grouping Variable > identify the grouping

variable within a database > Click the Group Value button > select which value of the grouping variable represents the group of interest.

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File > Data files

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Open data files

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We will name the variances, covariances, and regression weights in both the Girls and Boys models.

We will name the parameters in Girls’ model first. Use the Object Properties dialog box. Uncheck the box

for “All groups”, so you can give the variances different names in the two groups.

To name these parameters for the Boys model, highlight “boys” and go through the same procedure as before, use different names for variances, covariance, and factor loadings.

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There is a good way to name parameters. Go to Plugins > Click Name Parameters.

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Check Covariances, Regression weights, and Variances

We give parameters different names for Boys group. Here is a example:

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Make sure All groups is NOT

checked

New name for Boys group

For Girls group

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For Boys group

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Double-click on the Default Model label shown on the left side of the path diagram window.

The Manage Models window is open.

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You can rename the default model as something meaningful (we name it Original model).

Click New, a new model that imposes a set of equality constraints on the default model such that the unstandardized factor loadings are equal across boys' and girls' groups (we name this model Equal loading model).

Identify the four pairs relevant factor loadings of interest in the girls group and the boys group. By double-clicking on c1 and then double-clicking on c2.

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Manage models

Model summary of two models

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Model parameters for Girls

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Model parameters for Boys

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Model fit: we can use CFI, NCP, and GFI because they are independent of model complexity and sample size.

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Model comparison

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The Chi-square difference of two models is 18.292-16.480 = 1.812.

The results from this model comparison (Chi-square = 1.812 with 4 DF, p =.77 ) suggest that imposing the additional restrictions of four equal factor loadings across the gender groups did not result in a statistically significant worsening of overall model fit.

AMOS assumes that the baseline model (our original model) is true. The model (equal loading model) that specifies a group-invariant factor pattern, is supported by the sample data.

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Another way to do multiple group analysis The first step: set up groups (give names for each

group). Go to analyze > Manage Groups The second step: open data. Go to File > Data Files.

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Go to Analyze > Multiple-Group Analysis

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Four different models are obtained.

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Four models test invariances of measurement weights, structural covariances, and measurement residuals.

Model for Girls (AMOS automatically assigns names for each parameter).

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Model for Boys

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Model summary: original model (unconstrained)

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Model summary: Measurement weights model

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Model summary: Structural covariances model

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Model summary: Measurement residuals model

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Model fit

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Model comparisons

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If we find non-invariance across groups, the next step is to know what is causing this within the model. Usually, start with the factor loadings. Then, test structural weights.

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Nested model comparisons work by imposing a constraint or set of multiple constraints on a starting or less restricted model to obtain a more restricted final model.

Example: we want to compare the equality of factor loadings with a CFA model.

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We want to test the equality of the Cubes factor loading and the Sentence factor loading, as well as the Lozenges factor loading and the Wordmean factor loading.

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Test: w1 = w3 w2 = w4

Next, double-click on the section of the AMOS diagram window labeled Default Model.

Manage Models window is open.

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Model comparison

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Conclusion: The nested model comparison that assesses the worsening

of overall fit due to imposing the two restrictions on the original model shows a statistically significant chi-square value of 12.795 with 2 DF, resulting in a probability value of .002.

That the two models differ indicates that constraining the parameters in the default model to obtain the equal loadings model results in a substantial worsening of overall model fit.

Therefore, we reject the equal factor loadings model in favor of the original model.

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The bootstrap technique It is a resampling procedure Multiple subsamples of the same size as the parent

sample are drawn randomly from the original data. Parameter estimates are computed for each

subsample.

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When we use bootstrapping Data fail to meet the assumption of multivariate

normality. Presence of excessive kurtosis. Data are from a moderately large sample.

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Example: path diagram

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Assess multivariate normality: Go to View > Analysis Properties > Check Test for normality and outliers.

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Assess multivariate normality

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The multivariate kurtosis value of 13.167 is Mardia's coefficient. Critical ratio (c.r.)values of 1.96 or less mean there is non-significant kurtosis. Values of 7.979 > 1.96 mean there is significant non-normality.

Assess multivariate normality

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1.Malanobis d-squared distance for a case, the more it is improbably far from the solution centroid under assumptions of normality. 2.The cases are listed in descending order of d-square. 3.We may consider the cases with the highest d-squared to be outliers and might delete them from the analysis. 4.This should be done with theoretical justification (ex., rationale why the outlier cases need to be explained by a different model). 5.After deletion, it may be the data will be found normal by Mardia's coefficient when model fit is re-run.

Run bootstrapping: Go to View > Analysis properties

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Bootstrap ML estimates 1. The first column (SE) is Bootstrap

estimate of the standard error fro the parameter.

2. The second column (SE-SE)is standard error of bootstrap standard error itself.

3. The third column (Mean): is the mean parameter estimate computed across 500 subsamples.

4. The fourth column (Bias): represents the difference between the original mean estimate and bootstrap mean estimate.

5. The fifth column (SE-Bias): standard error of the bias estimate.

Bootstrap confidence intervals

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1. It is bias-corrected confidence interval.

2. If the range does not include zero, that the hypothesis of the parameter is equal to zero is rejected

It is used to study change. Latent growth analysis on individual and group

levels. The measurements are taken 3 or more times

(longitudinal data). Intercept The initial value, the average or mean of the outcome we

are interested in. Think about this: for each individual in the study,

everybody has an intercept of a certain value.

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Slope How much the curve grows over time, an average or

mean rate of growth. Each individual has a slope.

Goal of LGM Understand the average change. Understand individual variation in change.

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Example: a longitudinal study with 4-time points data set. We want to know the change of alcohol drinking over time. a28, b28, c28, and d28 are variables we measured. The regression weights from intercept to measured

variables are fixed to 1. In this way, we establish the initial level of alcohol drinking.

The path values from slope to measured variables are also fixed at a set of continuous values (time intervals).

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Diagram

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Fixing the values from the slope is how we identify model growth.

Parameters Mean and variance of intercept: Mean intercept is the

average start value. The variance of intercept reflects the variation of individual start value.

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Parameters Mean and variance of slope: Mean slope is the average of

rate of change. The variance of the slope reflects the extent to which individuals have different rates of change.

Covariance: to test whether individuals who start higher (higher intercepts) also change at a faster rate (higher slope).If such a relationship exists, we expect the covariance to be significant.

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From AMOS, choose Plugins > Growth Curve Model.

Enter the number of measures for the number of time points.

In this example, we would enter 4 for the number of time points .

Choose View > Analysis Properties > Estimation tab. Check the Estimate Means and Intercepts check box.

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Model identifications Right-click on the latent variable circles (labeled ICEPT

and SLOPE by AMOS) and select Object Properties. Remove the 0 constraints on the means. Fix the

variance to zero for ICEPT and SLOPE. Right-click on each of the 4 error variance circles and

select Object Properties. Fix their mean values to 0 and set their variance values to 1.00.

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For each of the paths connecting the error circles to the observed variables, replace the original value of 1.00 with the new parameter name.

Add two new error terms to the ICEPT and SLOPE latent variables.

For the newly-created error terms, fix their mean values to 0 and their variance values to 1.00.

Next, replace the 1.00 values for the path arrows connecting errors to ICEPT and Slope. Name the newly freed parameters.

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Remove the covariance double-headed arrow between ICEPT and SLOPE.

Place the covariance double-headed arrow between two new error terms and give a new name to the covariance.

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Text output

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Text output

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1. The mean intercept value of 1.35 indicates that the average starting amount of alcohol drinking was 1.35 units. 2.The mean slope value was .11. It means the average rate of change is .11 units. 3.The correlation between the intercepts and the slopes was 2.09. 4. The means were statistically significant when tested with the null hypothesis that their true values are zero in the population from which this sample was drawn.

Diagram with estimated parameters

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The intercept indicates a statistical significant mean alcohol use at the initial level (i.e., at baseline) and the slope mean indicates a significant average increase, via a liner functional form.

This alcohol use in adolescents is expected to increase by .11 each studied time period, beginning with an average score of 1.35.

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We also want to know the extent to which adolescents in the sample vary around their group average (mean) trajectories in alcohol use.

This can be evaluated by looking at the variances. The corresponding variances ( .57 for intercept and

.30 for slope) are statistically significant, indicating significant individual variability in the initial level and rate of change (growth) in alcohol use across the four waves of measurement.

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Hoyle, R. H. (1995). Structural equation modeling: Concepts, issues, and applications. Thousand Oaks, CA: Sage Publications, Inc.

Raykov, T. & Marcoulides, G. A. (2000). A first course in structural equation modeling. Mahwah, NJ: Lawrence Erlbaum Associates, Inc.

Schumacker, R. E. & Lomax, R. G. (2004). A beginner’s guide to structural equation modeling. Mahwah, NJ: Lawrence Erlbaum Associates, Inc.

AMOS 19.o user’s guide. IBM SPSS, Chicago.

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