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Introduction to Path Analysis and Structural Equation Modelling
with AMOSDaniel Stahl
Biostatistics and Computing
Last week: Family Tree of SEM
BivariateCorrelation
MultipleRegression
PathAnalysis
StructuralEquationModeling
FactorAnalysis
ExploratoryFactorAnalysis
ConfirmatoryFactorAnalysis
Depression
Pain
Function
Model 1
Depression
Pain
Function
Model 2
Last week: Path analysis
Pain
Item 1
Error 1
1
1
Item 2
Error 2
1
Item 3
Error 3
1
Function
Item 1
Error 1
Item 2
Error 2
Item 3
Error 3
1
111
Depression
Item 1 Error 1
Item 2 Error 2
Item 3 Error 3
11
1
1
Last week: Path analysis with latent constructs = Structural Equation Modelling
Pain
Item 1
Error 1
1
1
Item 2
Error 2
1
Item 3
Error 3
1
Function
Item 1
Error 1
Item 2
Error 2
Item 3
Error 3
1
111
Depression
Item 1 Error 1
Item 2 Error 2
Item 3 Error 3
11
1
1
Measurement model
Structural model
Last week:
1. Observed variables
2. Unobserved variables
3. Drawing latent variable (draws latent variable and items)
4. Drawing path (causal relationship �regression)
5. Draw covariances (correlation, no direction)
6. Unique variable (error variable, add e.g. to each dependent var
7. List variables (open data file first, then drag and drop variabl
8. Select one object, select all, deselect
9. Move object
10.Delete
11.Select data file
12.Analysis properties (choose statistics)
13.Calculate estimates (starts the analysis)
14.View test (see results)
15.Copy graph in clipboard
16.Save
1 2 3
4 5 6
8 8 8
9 10
11 12 13
14 15 16
7
2
Last week: SEM diagram symbols
1
1
1 1
Observed variable
Observed variable with measurement error (endogenous variable)
Latent variable with items (observed variables)
1Latent variable with disturbance or error
Unidirectional path (“regression”)
Correlation between variables
Reciprocal relation between variables
depresspain
Simple linear regresion
Error depression
1
depresspain
Correlation
depress
pain
function
error
1
Multiple linear regression
Last week: Correlation and Regression as AMOS path models
Endogenous variables have got error variances variables pointing at them = unexplained variance
Today
• Variance, covariance, correlation and regression coefficients
• SEM/path analysis is based on covariance matrix
• The Logic of Model Testing in SEM• Model fit and model comparisons• Simple latent trait models
Exercise
• Do a similar analysis as last week with data file: PATH-INGRAM.sav.
• The data are from: Ingram, K. L., Cope, J. G., Harju, B. L., & Wuensch, K. L. (2000). Applying to graduate school: A test of the theory of planned behavior. Journal of Social Behavior and Personality, 15, 215-226.
• Ajzen’s theory of planned behavior was used to predict student’s intentions and application behaviour (to graduate school) from their attitudes, subjective norms and perceived behavioural control.
Five Variables (derived from questionnaires)
• Perceived Behavioural Control (PBC) • Subjective norm• Attitude• Intention (to apply to college)• Behaviour (applications)
Ajzen’s theoretical model
• PBA, subjective norm and attitude influence intention• PBA, subjective norm and attitude correlate with each
other• Intention influences Behaviour• PBA also influences Behaviour
3
Exercise
• Draw the path diagram for the model• (Conduct a path analysis with a series of multiple
regression analyses using SPSS.)• (Calculate the standardised indirect effects using the
standardised estimates from the regression analysis.)• Check your results using AMOS• Use a bootstrap analysis to evaluate the indirect effect.• Remove some indirect effects and compare the results
with the theoretical model.
Path diagram for AMOS
PerceivedBehaviorControl
Attitude
Subjective Norm Intention Behavior
e1
1
e2
1
Results
PerceivedBehaviorControl
Attitude
Subjective Norm
.60
Intention
.34
Behavior
-.13
.09
.81
.35
.51
.47
.67
.34e1 e2
Standardized Indirect Effects
.000-.044.033.282Behavior
IntentPBCSubNormAttitude
Standardized Regression Weights:
.005.555.092.336PBC<---Behavior
.013.548.075.350Intent<---Behavior
.002.985.596.807Attitude<---Intent
.430.314-.118.095SubNorm<---Intent
.293.126-.352-.126PBC<---Intent
PUpperLowerEstimateParameter
95%CI: (0.08-0.5) (-0.04-0.13) (-0.14-0.04)
P: 0.007 0.339 0.277
Main results: AMOS output
• How do we know that this model fits the data well?
• Are there better models? How can we compare two or more models?
• First we need to know a little bit about covariances…
Variances, Covariances and Correlations
SEM and path analysis are based on variances and covariances .
Variance is a measure of the dispersion of data. It indicates how values are spread around the mean.
Covariance is a measure of the covariation between two variables, such as pain and function.
4
• Variance is defined as:
1
)(
)( 1
2
−
−=∑=
N
XX
XVar ii
Example: scores of pain and function
group2.521.510.5
scor
e
120
110
100
90
80
70
60
50
40
30
Centering
• We centre both variables = subtracting the mean from each individual score.
• We get a mean of 0 but same distribution around the mean and same variance
• Removes the constant in a regression and makes life easier for us.
group2.521.510.5
c_sc
ore
30
20
10
0
-10
-20
-30
Variance of pain and function
Descriptive Statistics
50 49.45 4.482 20.08450 74.65 12.530 157.01250 .00 4.482 20.08450 .00 12.530 157.01250
painfunctionc_painc_functionValid N (listwise)
N MeanStd.
Deviation Variance
Scatterplot of pain versus body function (not centred)
Scatterplot of pain versus body function (centred)
c_pain1050-5-10
c_fu
nctio
n
30
20
10
0
-10
-20
-30
R Sq Linear = 0.33
5
Covariance
• The covariance is an unstandardised measure of association between two variables.
• measure of the degree to which x and y vary together
1
)()(
),( 1
−
−−=∑=
N
YYXX
YXCovi
ii
In our example the covariance between pain and function is 32.
This means if pain increases by 1 unit of variance (Variance of pain = 20), function will increase by 32:
c_pain1050-5-10
c_fu
nctio
n
30
20
10
0
-10
-20
-30
R Sq Linear = 0.33
32
Covariance matrix• The covariance of variable X with itself is :
• Covariance matrix of pain and function:
• (= Variance/Covariance matrix)
)(1
)()(
),( 1 XVarN
XXXX
XXCovi
ii
=−
−−=∑=
15732Function
3220Pain
FunctionPain
Corrrelation
• The correlation coefficient r is a standardised measure of the association between two variables with a range from [-1,+1]:
• -1 = perfect negative association • 0 = no association (random)• +1 = perfect positive association
• If we standardise X and Y to new variables Zx and Zy to have a mean of 0 and a variance of 1 , then the Cov(Zx, Zy)=Corr(X,Y).
===)(*)(
),cov(
)var(*)var(
),cov(),(
YsdXsd
YX
YX
YXYXCorr
Correlation
• In our example the correlation between pain and function is 0.455.• The interpretation is:
If pain increases by 1 standard deviation (=sqrt(Var)), then body function increases by 0.455 SD.
• SD of pain = 4.5 and SD of function = 12.5• If pain increases by 4.5 then function will increase by
0.575*12.5=7.2
Regressoin coefficient b
• If we assume that variable X (pain) influences variable Y (function), then we can describe the relationship by a regression equation:
We can estimate the regression coefficient b by (based on the least squared method):
xby
c
xbcy
*
0
*
==
+=
)(
),(ˆXVar
YXCovb =
6
)(
),(ˆPainVar
FunctionPainCovb = 605.1
1.20
25.32ˆ ==b
Coefficients a
5.8E-012 1.466 .000 1.0001.605 .330 .574 4.858 .000
(Constant)c_pain
Model1
B Std. Error
UnstandardizedCoefficients
Beta
StandardizedCoefficients
t Sig.
Dependent Variable: c_functiona.
15732Function
3220Pain
FunctionPainV/CV
ExampleWhat’s the regression coefficient b for the regression equation: function=b*pain?
Question• If we assume that function influences pain what will be b?
(pain = b*function)
157Function
3220Pain
FunctionPainV/CV
205.0157
25.32
)(
),(
)(
),(ˆ ====functionVar
functionpainCov
YVar
YXCovb
Coefficients a
-3E-012 .524 .000 1.000.205 .042 .574 4.858 .000
(Constant)c_function
Model1
B Std. Error
UnstandardizedCoefficients
Beta
StandardizedCoefficients
t Sig.
Dependent Variable: c_paina.
Extension to Multiple Regression
• In the case of multiple predictors (X1 and X2), b becomes a partial regression coefficient:
• Y=c+ b1 X1+ b2 X2
• Depression=c+ b1*Function + b2*Pain
• b1 is the average change in Y per unit change in X1 with X2 held constant.
• b1 is the average change in depression score per unit change in body function with pain held constant.
Partial regression coefficient is obtained by removing (partialing out) the correlation with other predictors.
The Point: Partial regression coefficients can be viewed as functions of operations on a correlation or covariance matrix .
ε+−+= functionpaincdepression *523.0*061.0
Optional
• How to calculate the partial correlation (partial covariance) and partial beta for standardised data simply from the correlation matrix (= covariance matrix for standardised data).
Optional: Partial correlation
– Partial correlation can be calculated just from the correlation matrix.
– Similar partial regression coefficients can be obtained from the Variance/Covariance matrix.
22|)1()1(
*
yzxz
yzxzxyzxy
rr
rrrr
−−
−=
2,
2,
,,,|,
)1()1(
*
functdeprfunctpain
functdeprfunctpaindeprpainfunctdeprpain
rr
rrrr
−−
−=
Optional: Example of partial correlation
• Find the correlation between pain and depression while partialling out the effect of function.
18.08.0
145.0
])421.0(1[*])455.0(1[
)421.0(*)455.0(337.022|, =−=
−−−−
−−−=functdeprpainr
Correlationmatrix
1 .337** -.455**.337** 1 -.421**
-.455** -.421** 1
paindepressfunction
pain depress function
**.
)1)(1(
*
,2
,2
,,,|,
functdeprfunctpain
functdeprfunctpaindeprpainfunctdeprpain
rr
rrrr
−−
−=
Correlations
1.000 .180. .028
0 146.180 1.000.028 .146 0
CorrelationSignificance (2-tailed)dfCorrelationSignificance (2-tailed)df
pain
depress
Control Variablesfunction
pain depress
7
Optional: Standardised partial beta
221ˆ
:data d)transforme-(z edstandardis withmodel Regression
ZZZ xy ββ +=
22,1
2,12,1,1
1 xx
xxxyxyx
r
rrr
−−
=β 22,1
2,11,2,2 1 xx
xxxyxyx
r
rrr
−−
=β
Partial standardized regression coefficients can be obtained from the correlation coefficients:
Conclusion: Variance/Covariance Matrix
• Knowing the Variance/Covariance matrix of our variables allows us to estimate our regression as well as our path and structural equation models.
• � All information for SEM is in the Variance/Covariance matrix
• We do not need the original data set to do our path analyses or SEMs.
• Raw data will be converted in a covariance matrix (and we can also just import a covariance or a correlation matrix into AMOS).
• (Sample size is needed for statistical tests)• Because the analyses are based on the covariances , SEM
is also called covariance structure analysis.
Correlation matrix
• If we use the correlation matrix instead of the covariance matrix, we would obtain the standardised beta coefficients.
• Knowing the variances of the original data allows to calculate the unstandardised regression coefficients form the correlation matrix.
• In general, correlation matrices may lead to imprecise parameter estimates and standard errors in complex models and variance/covariance matrices are preferred.
Aim of SEM
• The aim of SEM is often to find a model with the smallest necessary number of parameters which still adequately describes the observed covariance structure (‘reconstructed’ or ‘estimated’ covariance matrix based upon our theoretical model should resemble the observed covariance matrix).
• How do we know that a model is a good model and fits the data?
The Logic of Model Testing in SEM1. We start with our observed covariance or correlation matrix
for X1, X2, and X3:X1 X2 X3
X1 1.0 r12 r13X2 1.0 r23X3 1.0
1. We hypothesize a model to test: e.g. : X1 � X2 � X3
2. This model can be represented by the following equations:• ŕ12 = r12 (direct effect of X1 on X2) • ŕ13 = r23r12 (indirect effect via X2)• ŕ23 = r23 (direct effect of X2 on X3)
3. ŕij represent ‘reconstructed’ or ‘estimated’ correlations (covariances) based upon the theoretical model.
4. We compare the reconstructed with the observed correlations (covariances).
Observed correlation matrix (= covariance matrix with standardised variables)
1.000-.421.337depress
1.000-.455function
1.000pain
depressfunctionpainpain
function
depress
0,
error
0,
error2
1
1
Our hypothesized model:
8
pain
.21
function
.18
depress
error
-.46 error2
-.42
This model can be represented by the following correlations (equations):
ŕ(PF)= -0.46 (direct effect of Pain on Function)
ŕ(FD) = r(FD)r(PF)= -0.46* -0.42 = 0.192(indirect effect of pain on depression via function)
ŕFD = rFD = -0.42 (direct effect of function on depression)
All variances are 1 (standardized).
‘Reconstructed’ or ‘estimated’ correlations based upon our theoretical model:
1.000-.421.192depress
1.000-.455function
1.000pain
depressfunctionpain
1.000-.421.337depress
1.000-.455function
1.000pain
depressfunctionpain
1.000-.421.192depress
1.000-.455function
1.000pain
depressfunctionpain
• Observed Correlations in Data
• Reconstructed Correlations based upon Path model
• Is the expected correlation matrix very different from the observed matrix (the more similar the better the model)?
• �e.g. χ2 goodness of fit test: χ2 = Σ(rij(o) – ŕij(e))2/ŕij(e))
• Next step: If our model variance/covariance matrix fits the observed data well, we can calculate the path regression coefficients and error variances from the matrix.
• In our simple example (without partial coefficients) correlation coefficients and standardised betas are identical.
• Again, all we need is the covariance matrix!
pain
.21
function
.18
depress
error
-.46 error2
-.42
Conclusion
� Path coefficients can be estimated using multiple regression methods (standardized partial coefficients) based upon a given model and can be used to “reconstruct” the correlation (or covariance) matrix.
� The “estimated” (reconstructed or expected) correlations can be compared with the observed correlations and a chi-square will show whether it fits.
� Important: a non-significant chi-square denotes good fit: The more similar observed and expected correlations, the smaller the Chi-square, the better the model.
model) eddf(saturat model) sizeddf(hypothe n
freedom of degrees n with
)(exp
)](exp)([ 22
=
−= ∑
ectedr
ectedrobservedr
ij
ijijχ
Our model is significant different from the observed covariance matrix. Therefore, it does not adequately describe the observed relationships in our data!
Chi Square (χ2) test for association
• most commonly used model fit statistics
• χ2 calculates the degree of discrepancy between the theoretically expected values vs. the empirical data
• The larger the discrepancy (independence), the sooner χ2 becomes significant
• Because we are dealing with a measure of misfit, the p-value for χ2
should be larger than .05 (at least 0.2) to decide that the theoretical model fits the data!
• However, overall χ2 is a poor measure of fit and with large n, χ2 will generally be significant.
• χ2 test can only be used to compare nested models (i.e.., identical but Model 2 deletes at least one parameter found in Model 1),
• Many other measures of model fit, each with their own assumptions and limitations, are developed.
9
Factor AnalysisAddresses these questions• Can the covariances or correlations between a set of
observedvariables be explained in terms of their relationshipwith a smaller number of of unobservable latent variables?• i.e does the correlation between each pair of observed
variablesresult from their mutual association with the latentvariables; i.e. is the partial correlation between any pair ofobserved variables, given the values of the latent variables,approximately zero?
Coefficient estimation methods
Estimation methods to minimize the discrepancy between the obtained covariance matrix and the covariance matrix implied by the model:
• Maximum Likelihood (ML)• Unweighted Least square method (ULS)• General Least Squared Method (GLS)• Scale free least square method (SLS)• Asymptotically distribution free (ADF)
(related to weighted least square method)
Parameter estimation methods
NNNNYInformation criteria measures (AIC/BIC)
YNNYYInference (chi2)
Standardise variables
Best precision if assumption met
1.5*p(p+1) (p=observed
variables)
>100>100>100>100Minimum sample size
YYNYYInvariance of scale
NNNYYAssumption of multinormal distribution
ADFSLSULSGLSMLMaximum Likelihood Estimation
• ML most commonly used method• Objective of maximizing the likelyhood of a parameter is to find a value that
maximizes the joint probability density function - = maximizes the likleyhoodof having observed the data (=finding parameters that make the predicted covariance matrix as similar as possibble to the observed onecovariancematrix).
• ML in AMOS assumes multivariate normal distribution and continuous data• = each variable should be normally distributed for each value of each other
variable, specifically ML requires multinormality of endogenous variables • often robust to violations of multinormal distribution and continuous data • Good method for missing data if missing at random (MAR): AMOS uses Full
ML• If ordinal data are used, they should have at least five categories and
skewness and kurtosis should be small (values in the range of -1 to +1 or -1.5 to +1.5).
• AMOS 7 can handle ordinal data “but” needs Bayesian estimation methods.
• Same visual inspection as for multiple regression: look at histogramms, QQ plots, Boxplots
• Test of overall model fit– Chi2 Test– Bollen Stein bootstrap
• Test of individiaul paramter estimates (regression coefficients or covariances):– C.R.: Critical ratio. The critical ratio is the
parameter estimate divided by an estimate of its standard error (z or t value).
– Bootstrap CI and Tests
10
• Simulation studies (see Kline 1998: 209) suggest that under conditions of severe non-normality of data, SEM parameter estimates (ex., path estimates) are still fairly accurate but corresponding significance coefficients are too high.
• � use bootstrap tests if possible
What is bootstrappingBootstrapping is a way of estimating standard error, confidence
intervals and significance based not on assumptions of normality but on empirical resampling with replacement of the data.
It has minimum assumptions: It is merely based on the assumption that the sample is a good representation of the unknown population
� We assume that the observed data resemble the true distribution.� generates information on the variability of parameter estimates or of
fit indexes based on the empirical samples, not on assumptions about probability theory of normal distributions.
� Bootstrapping in SEM still requires moderately large samples.� If data are multivariate normal, MLE will give less biased estimates.
However, if data lack multivariate normality, bootstrapping gives less biased estimates.
• The general bootstrap algorithm• 1. Generate a sample of size n from your data set with replacement.• 2. Compute your paramter of interest ˆ θ for this bootstrap sample • (e.g. do a SEM analysis and get a regression coefficent b)• � For each random sample we get a different parameter estimate. • 3. Repeat steps 1 and 2, 1000 time.• By this procedure we end up with 1000 bootstrap values ˆ θ∗ = (ˆθ1, ˆθ2, . . . , ˆθ1000 ). • Sort the bootstrap values from smallest to largest.• Using the sorted ˆθ1, ˆθ2, . . . , ˆθ1000 values, find the 2.5% ile value and the 97.5% ile
value. • Or simply the 25th and 975th observations from the sorted 1000 values.• Calculate the standard deviation of the 1000 ˆθ1’s. This is the estimate of the
standard error (se) of your parameter estimate b.• An approximate 2 sided 95% confidence interval is: B ± 1.96(SE). • Test statistic z=B/SE. If z>1.96, then p<0.05
Frequency distribution of the B's calculated using Bootstrap samples
0
50
100
150
200
250
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Value of Bi
Freq
uen
cy
Maximum Likelihood Estimation
• Violations of assumptions will cause an inflation of parameter estimates and chi2 values.
• �Lack of normality will cause a type I error (inflated chi2 values could lead to think that the model does not fit well)!
• Bollen Stein bootstrap or Satorra-Bentler (not in AMOS). adjusted chi2 can be used for inference if normal distribution assumption is violated.(but: see www.statmodel.com/chidiff.shtml)
• Use ADF estimation as a robust alternative.• Use other goodness of fit measures to evaluate models• Violations will tend to underestimate standard errors of
parameter estimates, which causes regression paths or covariances are found to be statistically significant more often than they should be (Type II error).
• � If possible use bootstrap tests and confidence intervals.
Goodness of fit measure based on predicted vs. observed covariance matrix
Not the most robust method, sample size dependent
Compare model with saturated, full model (e.g. all paths) or compare two nested models
P>0.2Chi2 Test
CriteriaGoodness of fit measure
Not in AMOSCorrects for kurtosis and bias
P>0.2Satorra Bentel corrected chi2
CMIN/DF in AMOS
Chi2 /degress of freedom
≤ 2.5Relative chi2
Chi2/df
robust method, does not assume multinormaldistribution, sample size dependent
Compare model with saturated, full model (e.g. all paths) or compare two nested models
P>0.2Bollen Stein bootstrap
11
Goodness of fit comparisons between models
Similar interpretation as CFI:0.9= model is 90% away from saturated model
0.9=90% of covariation in data can be reproduced by given model
Similar to NFI but penalizes for model complexity and little affected by sample size,
≥ 0.9-0.95
TLI is the Tucker-Lewis coefficient(=non-normed fit index (NNFI))
Similar to CFI but more robust but underestimates fit if N is small
≥0.9Normed Fit index (NFI)
>0.9Incremental Fit (IFI)
Least affected by sample size
≥0.9Comparative fit index (CFI)
Goodness of fit measure
Goodness of fit measures based on predicted and observed covariance matrix but penalizes for lack of parsimony
(overfitting)
PRATIO * CFI
PRATIO * NNFI
PRATIO * BBI
Df model/df independence
Discrepancy per df (
Smaller is better-Parsimonious CFI
(PCFI)
Smaller is better-Parsimonious NFI
(PNFI)
Not in AMOS (?), >0.9Parsimony Index
Parsimony ratio (PRATIO)
Least affected by sample size, penalizes for lack of parsimony, compare different models (but information criteria better)
≤ 0.5Root mean square error of approximation (RMSE, RMR)
Goodness of fit measure
Goodness of fit measures based on information theory• Measures can be used to compare non-nested and
nested model, penalize for model complexity (overfitting), all need ML estimation
?Penalizes for sample size and lack of parsimony, penalty greater than AIC and BIC
May select too parsimonious models
Consistent AIC(CAIC)
Smallest model is most likely the true model among the set of models
Penalizes for sample size and lack of parsimony
May select too parsimonious models
Bayesian IC,(BIC)
Smallest AIC of a set of model is most likely the best model (not the true model)
Penalizes for lack of parsimony
May select more complex models
Aikaike’s IC (AIC)
Idea/PhilosophySelection criteriaProblemMeasure
Evaluation of fit of model structures
Model fit criteria do not specify which parts of the model may not fit.
Compare observed and model covariance:• Evaluate residuals matrix (residual covariance): should
be <0.1 better<0.05• Standardised residuals Critical Ratio: Estimate divided by its standard error. • If data are a random sample variables and multinormal
distributed, estimates with critical ratios more than 1.96 are significant at the .05 level and suggests important contribution to model. Use bootstrap methods with small sample sizes and/or nonnormal distribution.
Implied Covariances (Group number 1 - Default model)
pain function depress pain 4.079
function -.397 .186 depress .259 -.121 .448
Residual Covariances (Group number 1 - Default model)
pain function depress pain .000
function .000 .000 depress .197 .000 .000
Standardized Residual Covariances (Group number 1 - Default model)
pain function depress pain .000
function .000 .000 depress 1.740 .000 .000
Observed (implied) covariance matrix and residual matrices
Residual matrices suggests that covariance depression and pain is not adequately modelled
4.93, 4.08
pain
3.10
function
3.25
depress-.65
0, .37
error
1
-.10
0, .15
error2
1
x=\agfi
Critical ratios (C.R.)
Regression Weights: (Group number 1 - Default model)
***-5.642.116-.652function<---depress
***-6.223.016-.097pain<---function
PC.R.S.E.Estimate
12
Exercise
• Check the model fit of Ajzen’s theoretical model• Try to find a better model (remove unnecaissary paths,
include necaissary path based e.g. on chi2 tests for nested models, residual covariance matrix, critical ratio, AIC, RMSE…)
Nested model comparison
• Amos examines every pair of models in which one model of the pair can be obtained by constraining the parameters of the other (e.g. removing one path = setting b to 0).
• For every such pair of "nested" models, a likelihood ratio chi2 test can be performed to see if the constrained model is not significantly better (=fits similar good as the more complex model).
• For non-nested models use AIC (or BIC). The smallest AIC is the best model. (see “Model selection course” or handouts on Biostatistics webpage for details)
Nested model comparison• Right click on path you want to restrict to 0 (=“delete”).• Enter name for path in “regression weight” box.
Nested model comparison
Alternate between output of default and model 2
13
Nested model comparison• Double-click on “XX-Default model” box• Click “New” to create a new model and label it.• Constraint the parameter (your path in this example) to 0
(see box): sn_int=0 , you can add more than 1 constraint
Nested model comparison
• Go to “analysis properties”• Then run the model
Output I: Nested model comparison
• There was no significant difference between model 2 (without path sn� int) and model 1 (with the path):
• chi2(1df) =0.935, p=0.334. this suggests that the path from subjective norm to intention is not necaissary.
• Amos also reports the changes in the fit measures, NFI, TLI, RFI and IFI, They all increase which suggests a better fit.
Output II: nested models comparisons
• Select View/Set, Analysis Properties, Output tab and check "Tests for normality and outliers."
Assessment of normality (Group number 1)
5.3054.760Multivariate
6.2852.5227.8681.5794.4001.000depress
5.0792.039-7.484-1.5023.000.938function
-2.013-.808-.334-.0679.0001.000pain
c.r.kurtosisc.r.skewmaxminVariable
Quick introduction: How to define latent variables in AMOS:
Example: simple factor analysis: • Can we reduce the three variables into one factor (latent
variable) without loosing too much information?• = Factor analysis with Maximum likelihood estimation
14
Latent variable in AMOS• Use button on top left (#3) to create a latent variable• Move variables into item boxed• Name error variable as e1,e2,e3• Name latent variable as “well-being”• Tick “standard estimates”, “squared correlations” and
“factor scores weight” in analysis property box
Well-being
pain
e1
1
1
depress
e2
1
function
e3
1
Factor analysis SPSS: SEM analysis AMOS:Communalities
.233 .365
.204 .312
.288 .568
pain
depress
function
Initial Extraction
Extraction Method: Maximum Likelihood.
Factor Matrix a
.604
.558
-.754
paindepress
function
1
Factor
Extraction Method: Maximum Likelihood.
1 factors extracted. 4 iterations required.a.
Well-being
.36
pain
e1
.60
.31
depress
e2
.56
.57
function
e3
-.75
• Latent variable models are a broad subclass of latent structure models . They postulate some relationship between the statistical properties of observable variables (or "manifest variables", or "indicators") and latent variables. A special kind of statistical analysis corresponds to each kind of the latent variable models.
Analysis of
Latent class analysis
Latent trait analysis
Categorical/(Ordinal)
Latent profile analysis
Factor analysis models
MetricalManifest
Categorical
Metrical
Latent
Checking assumptions of ML with AMOS
• Skewness and curtosis (peakedness of distribution) for each parameter should be within +/- 2
• Mardia's statistic measures of the degree to which the assumption of multivariate normality has been violated.
• Mardia's measure is based on functions of skewnessand kurtosis and should be less than 3 to assume the assumption of multivariate normality is met.
• Large values of Mardia’s value may suggest some multivariate outliers in dataset
• (Barbara G. Tabachnick and Linda S. Fidell (2001). Using Multivariate Statistics, Fourth Edition. Needham Heights, MA: Allyn & Bacon.)
• Multivariate outlier: an extreme combination, like juvenile with a high income. Observations farthest from the centroid under assumptions of multinormality.
•• Mahalanobis distance is the most common measure
used for multivariate outliers.• the higher Malanobis d-squared distance for a case, the
more likely to be a outlier under assumptions of normality.
• The cases are listed in descending order of Mahalanobisd2. Check if cases with the highest d-squared as possible (but not necaissarily) outliers. Consider cases as outliers if the MD are well seperated from other M. distances (Arbuckle and Wothke1999) .
• ML estimation requires indicator variables with multivariate normal distribution and valid specification of the model;
• Ordinal variables are widely used in practice: If ordinal data are used, they should have at least five categories and not be strongly skewed.
15
Assumption of parametric tests
• Check assumptions as in any multivariable or multivariate analysis, see:– http://www2.chass.ncsu.edu/garson/pa765/assumpt.htm– Brian Everitt and Graham Dunn (2001) Applied
multivariate data analysis. Arnold– Barbara G. Tabachnick and Linda S. Fidell (2001).
Using Multivariate Statistics, Fourth Edition. Needham Heights, MA: Allyn & Bacon.)
