Diagrams Representing Relationships between … · Web viewCorrelations or covariances between variables are represented by double-headed arrows. "Cor / Cov" Arrow Author Michael

Path Diagrams, EFA, CFA, and Amos(You should have read the EFA chapter in T&F and should read the CFA chapter.)Observed Variable

A variable whose values are observable.

Examples: IQ Test scores (Scores are directly observable), GREV, GREQ, GREA, UGPA, Minnesota Job Satisfaction Scale, Affective Commitment Scale, Gender, Questionnaire items.

Latent Variable

A variable, i.e., characteristic, presumed to exist, but whose values are NOT observable. A Factor in Factor Analysis literature. A characteristic of people that is not directly observable.

Intelligence, Depression, Job Satisfaction, Affective Commitment, Tendency to display affective state

No direct observation of values of latent variables is possible. Brain states? Brain chemistry?

Indicator

An observed variable whose values are assumed to be related to the values of a latent variable.

Reflective Indicator

An observed variable whose values are partially determined by, i.e., are influenced by or reflect, the values of a latent variable. For example, responses to Conscientiousness items are assumed to reflect a person’s Conscientiousness.

Formative Indicator

An observed variable whose values partially determine, i.e., cause or form, the values of a latent variable.

Exogenous Variable (Ex = Out)

A variable whose values originate from / are caused by influences outside the model, i.e., are not explained within the theory with which we’re working. That is, a variable whose variation we don’t attempt to explain or predict by whatever theory we’re working with. Causes of exogenous variable originate outside the model. Exogenous variables can be observed or latent.

Endogenous variable (En ~~ In)

A variable whose values are explained within the theory with which we’re working. We account for all variation in the values of endogenous variables using the constructs of whatever theory we’re working with. Causes of endogenous variables originate within the model.

CFA, Amos - 1 Printed on 5/8/2023

Basic EFA, CFA, SEM Path Analytic NotationObserved variables are symbolized by squares or rectangles.

Latent Variables are symbolized by Circles or ellipses.

Correlations or covariances between variables are represented by double-headed arrows.

"Causal" or "Predictive" or “Regression” relationships between variables are represented by single-headed arrows


ObservedVariable

103 84121 76 . . . 97 81

106 78115 80. . . 93 83

101 90128 72 . . . 93 80

103 84121 76 . . . 97 81

ObservedVariable A

"Cor / Cov"Arrow

ObservedVariable B

LatentVariable A

LatentVariable B

"Cor / Cov"Arrow

106 78115 80. . . 93 83

104 79114 79. . . 92 81

"Causal"Arrow

ObservedVariable

LatentVariable

"Causal"Arrow

ObservedVariable

LatentVariable

LatentVariable

"Causal"Arrow

LatentVariable

ObservedVariable

ObservedVariable

"Causal"Arrow

Latent Variable

Values of individuals on latent variables are not observable, hence the dimmed text.

Exogenous Observed Variables

Exogenous variable connect to other variables in the model through either a “causal” arrow or a correlation

Exogenous Latent Variables

Exogenous latent variables also connect to other variables in the model through either a “causal” arrow or a correlation

Endogenous Observed Variables - Endogenous Latent Variable

Endogenous variables connect to other variables in the model by being on the “receiving” end of one or more “causal” arrows. Specifically, endogenous variables are typically represented as being “caused” by 1) other variables in the theory and 2) random error. Thus, 100% of the variation in every endogenous variable is accounted for by either other variables in the model or random error. This means that random error is an exogenous latent variable in SEM diagrams. Random error is a catch-all concept representing all “other” things that are affecting the endogenous variable.

Summary statistics associated with symbols

Our SEM program, Amos, prints means and variances above and to the right. Typically the mean and variance of latent variables are fixed at 0 and 1 respectively, although there are exceptions to this in advanced applications.


"Causal"Arrow

ObservedVariable

ObservedVariable

"Causal"Arrow

LatentVariable Latent

Variable

ObservedVariable

"Causal"Arrow "Causal"

ArrowLatent

Variable

Randomerror

"Correlation"Arrow

"Correlation"Arrow

ObservedVariable

Mean, Variance

LatentVariable

Mean, Variance

B or

"Causal"Arrowr or Covariance

"Correlation"Arrow

Randomerror

Path Diagrams of Analyses We’ve Done Previously

Following is how some of the analyses we’ve performed previously would be represented using path diagrams.

1. Simple correlation between two observed variables.

2. Simple correlations between three observed variables.

3. Simple regression of an observed dependent variable onto one observed independent variable.

4. Multiple Regression of an observed dependent variable onto three observed independent variables.


B or eP511GGRE-Q

rVA

rQArVQ

GRE-AGRE-QGRE-V

rVQ

GRE-QGRE-V

GRE-Q P511G eBQ or Q

GRE-V

UPGA

BV or V

BU or U

Note that the endogenous variable is caused in part by catch-all influences.

Note that the endogenous variable is caused in part by catch-all influences.

Path Analyses of ANOVAs

Since ANOVA is simply regression analysis, the representation of ANOVA in SEM is merely as a regression analysis. The key is to represent the differences between groups with group coding variables, just as we did in 513 and in the beginning of 595 . . .

1) Independent Groups t-testThe two groups are represented by a single, dichotomous observed group-coding variable. It is the independent variable in the regression analysis.

2) One Way ANOVAThe K groups are represented by K-1 group-coding variables created using one of the coding schemes (although I recommend contrast coding). They are the independent variables in the regression analysis. If contrast codes are used, the correlations between all the group coding variables are 0, so no arrows between them need be shown.

3) Factorial ANOVA.Each factor is represented by G-1 group-coding variables created using one of the coding schemes. The interaction(s) is/are represented by products of the group-coding variables representing the factors. Again, no correlations between coding variables need be shown if contrast codes are used.


eDependentVariable

Dichotomous variable representing the two groups

eDependentVariable

. . . . .(K-1)th Group-coding contrast code variable.

2nd Group-coding contrast code variable.

1st Group-coding contrast code variable

eDependentVariable

Interaction

Interaction

Interaction

Interaction

2st Factor

2st Factor

1st Factor

1st Factor

Note: Contrast codes were used so Group-coding variables are uncorrelated.

Note: Contrast codes should be used to make sure the group-coding variables uncorrelated (assuming equal sample sizes.)

Path Diagrams representing Exploratory Factor Analysis1) Exploratory Factor Analysis solution with one factor.

The factor is represented by a latent variable with three or more observed indicators. (Three is the generally recommended minimum no. of indicators for a factor.)

Note that factors are exogenous. Indicators are endogenous. It’s a rule that all of the variance of endogenous variables must be accounted for in the diagram. Thus, each indicator must have an error or sometimes called residual latent variable to account for the variance in it not accounted for by the factor.

2) Exploratory Factor Analysis solution with two orthogonal factors.

For exploratory factor analysis, each variable is required to load on all factors. Of course, the hope is that the loadings will be substantial on only some of the factors and will be close to 0 on the others, but the loadings on all factors are estimated, even if they’re close to 0.

The loadings of items are conceptualized in two group. First , are the primary loadings - those connecting a factor to the indicators we believe it should be connected to.

The second group is secondary or cross loadings. These are the loadings that connect a factor to all the “other” variables – the variables that should not be connected to the factor even though they’re required to be estimated.

Let’s assume that Obs 1, 2, and 3 are thought to be primary indicators of F1 and 4,5,6 are thought to be the primary indicators of F2.

In the figure below, the primary loadings of each factors are in black with the cross loadings in red.

The lack of an arrow between the two factors means that they will be estimated as being orthogonal, i.e., independent


F2

F1

e6

e5

e4

e3

e2

e1

Obs 6

Obs 5

Obs 4

Obs 3

Obs 2

Obs 1

Obs 1

Obs 2

Obs 3

e1

e2

e3

F

Orthogonal factors represent uncorrelated aspects of behavior.

Note what is assumed in the above figure: There are two independent characteristics of people – F1 and F2. Each one influences responses to all six items, although it is hoped that F1 influences primarily the first 3 items and that F2 influences primarily the last 3 items.

If Obs 1 thru Obs 3 are one class of behavior and Obs 4 thru Obs 6 are a second class, then if the loadings “fit” the expected pattern – loadings of 1,2, and 3 on F1 are large, while those of 4, 5, and6 on F2 are large, with all cross loadings small,, this would be evidence for the existence of two independent dispositions – that represented by F1 and that represented by F2.

3) Exploratory Factor Analysis solution with two oblique factors.

Each factor is represented by a latent variable with three or more indicators. The obliqueness of the factors is represented by the fact that there IS an arrow connecting the factors.

Again, in exploratory factor analysis, all indicators load on all factors, even if the loadings are close to zero.

This solution is potentially as important as the orthogonal solution, although in general, I think that researchers are more interested in independent dispositions than they are in correlated dispositions.

But discovering why two dispositions are separate but still correlated is an important and potentially rewarding task.


F2

F1

e9

e8

e7

e6

e5

e4

Obs 6

Obs 5

Obs 4

Obs 3

Obs 2

Obs 1

Diagram of EFA model of NEO-FFI Big Five 60 item questionnaire.

(From Biderman, M. (2014). Against all odds: Bifactors in EFAs of Big Five Data. Part of symposium: S. McAbee & M. Biderman, Chairs. Theoretical and Practical Advances in Latent Variable Models of Personality. Conducted at the 29th annual conference of The Society for Industrial and Organizational Psychology; Honolulu, Hawaii, 2014.

Crossloadings are in red.


Cross loadings are red’d.

It took me about 2d hours to draw this figure.

Path Diagrams vs the Table of Loadings.(Cross loadings are in red in both representations.)

Whew – there are tons of cross loadings, most of them near 0. Can’t they just be assumed to be zero? This kind of thinking leads to Confirmatory Factor Analysis Models.


Pattern Matrixa

Factor1 2 3 4 5

ne1 -.025 -.001 -.056 -.067 .678ne2 .086 .002 .021 -.026 .268ne3 .126 -.023 .133 .229 .260ne4 .054 .014 .112 .184 .492ne5 -.115 -.045 -.069 -.269 .624ne6 .042 .206 -.200 .085 .563ne7 .194 -.168 .046 -.203 .166ne8 .342 -.127 .087 .299 .432ne9 .270 .046 .092 .419 .223ne10 -.017 -.138 -.105 .135 .327ne11 .209 -.270 .009 .102 .278ne12 .140 -.170 -.103 -.070 .483na1 -.100 -.193 .081 .518 .249na2 .347 -.153 -.076 .421 -.139na3 .091 -.123 -.050 .560 .125na4 -.139 .034 -.011 .504 -.099na5 .298 .073 .033 .335 .188na6 .335 .157 -.070 .353 .031na7 .019 -.177 .067 .231 .346na8 .053 .029 -.114 .543 .292na9 .163 .115 -.120 .319 -.211na10 -.057 -.123 .167 .594 .271na11 .028 .012 .055 .471 -.227na12 .027 -.210 -.075 .534 -.022nc1 .092 -.429 -.090 .183 .008nc2 .019 -.580 -.049 .055 -.086nc3 -.030 -.376 -.037 .011 -.140nc4 -.093 -.406 .052 .156 .028nc5 .026 -.716 -.025 -.156 .057nc6 .146 -.476 .100 .241 -.052nc7 -.154 -.694 -.070 -.121 .109nc8 .092 -.528 .019 -.017 .110nc9 .040 -.573 .044 -.050 .094nc10 .021 -.720 .016 -.103 .044nc11 .035 -.551 -.067 .148 .005nc12 .065 -.628 .035 -.011 .018ns1 .501 .072 .145 -.231 -.031ns2 .544 -.119 .132 -.044 .027ns3 .653 .037 -.026 -.069 -.118ns4 .664 .025 -.141 .006 .189ns5 .660 .082 -.031 .200 .047ns6 .677 -.053 -.104 .011 .046ns7 .658 .035 .027 -.069 .011ns8 .552 .068 -.078 .265 .012ns9 .724 -.093 .134 -.021 .027ns10 .662 -.041 -.068 -.009 .220ns11 .563 -.266 .035 .030 -.132ns12 .611 -.082 -.177 .075 -.003no1 -.146 .334 .267 -.003 -.103no2 .030 .366 .136 .061 .087no3 -.014 .044 .661 -.040 .046no4 .116 .037 .142 -.119 -.008no5 -.010 -.090 .731 .102 -.166no6 .029 .168 .217 .124 .207no7 -.064 .025 .207 .082 -.082no8 .024 .233 .240 -.101 -.197no9 -.008 -.012 .822 .049 -.123no10 .052 .135 .536 .026 -.033no11 -.026 -.111 .560 -.041 .110no12 .021 .131 .615 -.174 .090Extraction Method: Maximum Likelihood. Rotation Method: Oblimin with Kaiser Normalization.a. Rotation converged in 14 iterations.

Confirmatory vs Exploratory Factor AnalysisIn Exploratory Factor Analysis, as discussed above, the loading of every item on every factor is estimated. The analyst hopes that some of those loadings will be large and some will be small. An EFA two-orthogonal-factor model is represented by the following diagram.

As mentioned above there are arrows (loadings) connecting each variable to each factor. We have no hypotheses about the loading values – we’re exploring – so we estimate all loadings and let them lead us. Generally, EFA programs do not allow you to specify or fix loadings to pre-determined values.

In contrast to the exploration implicit in EFA, a factor analysis in which some loadings are fixed at specific values is called a Confirmatory Factor Analysis. The analysis is confirming one or more hypotheses about loadings, hypotheses representing by our fixing them at specific (usually 0) values.

Below is shown one common confirmatory analysis model – one in which the primary loadings are represented but the secondary, aka, cross loadings are set to 0. In the diagram, a loading or correlatin that has been set to 0 is simply left out of the diagram.


F2

F1

e6

e5

e4

e3

e2

e1

Obs 6

Obs 5

Obs 4

Obs 3

Obs 2

Obs 1

Exploratory

Confirmatory

E6

E5

E4

E3

E2

E1

F1

F2

Obs 6

Obs 5

Obs 4

Obs 3

Obs 2

Obs 1

The Identification Problem Skip in 2017Consider the simple regression model . . .

Quantities which can be computed from the data . .

Mean of the X variable Variance of the X variableMean of the Y variable. Variance of the Y variable.

Correlation of Y with X

Quantities in the model .

Remember that in path diagrams, all the variance in every endogenous variable must be accounted for. For that reason, the path diagram includes a latent “Other factors” or “Error of measurement” or “Residual” variable, labeled “E” in the above diagram..

Mean of X Mean of EVariance of X Variance of E

Intercept of X->Y regressionSlope of X->Y regressionCorrelation of E with Y

Whoops! There are 5 quantities in the data but 7 in the model.

There are too few quantities in the data.

The model is underidentified. – not identified enough - there aren't enough quantities from the data to identify each model value.


Mean, Variance

Mean, Variance

E

YXY = a + b*X

Corr(E,Y)

Note: Mean and variance of Y are not separately identified in the model because they are assumed to be completely determined by Y’s relationship to X and to E.

Dealing with underidentification . . . Skip in 2017

Solution 10) The mean of E is always assumed to be 0.1) Fix the variance of E to be 1.

So in this regression model, the path diagram will be

In this case, there are 5 quantities in the model that must be estimated – mean of X, variance of X, intercept of equation, slope of equation, and correlation of E with Y. There are also 5 quantities that can be estimated from the observed data.

The model is said to be “just identified” or “completely identified”. This means that every estimable quantity in the model corresponds in some way to one quantity obtained from the data.

Or,

Solution 20) The mean of E is always assumed to be 0.1) Fix covariance of E with Y at 1.

Underidentified models: Cannot be estimated.

Just identified models: Every model quantity is a function of some data quantity. But no parsimony.

Overidentified models: There are more data quantities than model quantities. It is said that you then have “degrees of freedom” in your model. This is good. Relationships are being explained by fewer model quantities than there are data quantities. This is parsimonious – what science is all about.


Y = a + b*X

Y=a+b*X

X Y

E

Mean, Variance

0, 1

1

rEY

0, Variance

Mean, Variance

E

YX

Technical Issue: Identification in CFA models –

For mathematical reasons, we have to make a few assumptions about which values are to be estimated in CFA models. Here’s a typical CFA two-factor model.

Assumption 1. Insuring that the “Residuals” part of the model – that involving the “E”s – is identified.

We always assume all “E” means = 0.

Solution 1. Fix all “E” variances to 1.

or

Solution 2. Fix all E O loadings to 1 and estimate variances of “E”s.


0, Variance

0, Variance

0, Variance

0, Variance

0, Variance

0, Variance

0,1

0,1

0,1

0,1

0,1

0,1E6

E5

E4

E3

E2

E1

F1

F2

Obs 6

Obs 5

Obs 4

Obs 3

Obs 2

Obs 1

E6

E5

E4

E3

E2

E1

F1

F2

Obs 6

Obs 5

Obs 4

Obs 3

Obs 2

Obs 1

1

1

1

1

1

1

I recommend

this.

Assumption B. Insuring that the Factors part of the CFA is identified

Solution 1. Fix one of the loadings for each factor at 1 and estimate all factor variances

Or

Solution 2. Fix the variance of each factor at 1 and estimate all factor loadings.

The main point for us – what the factor analysis gives us.

The above models tell us that the variation in 6 observed variables – Obs1, Obs2, Obs3, Obs4, Obs5, Obs5 - is due to variation in just two internal characteristics – F1 and F2.

So we have explained why there is variation in the observed variables – because of variation in F1 and F2.

We have also explained why the variation in Obs1 to Obs3 is unrelated to the variation in Obs4 to Obs6 – because F1 and F2 are uncorrelated.


1

1

E6

E5

E4

E3

E2

E1

F1

F2

Obs 6

Obs 5

Obs 4

Obs 3

Obs 2

Obs 1 I recommend this, although some

examples below will use the above

method.

1

1

E6

E5

E4

E3

E2

E1

F1

F2

Obs 6

Obs 5

Obs 4

Obs 3

Obs 2

Obs 1The item whose loading is fixed is called the reference item.

Examples

1. Fixing all variances.

2. Fixing residual loadings but Factor variances

3. Fixing residual loadings and factor loadings.


1

1

1

1

11

1

1

1

1

1

1

1

1

E6

E5

E4

E3

E2

E1

F1

F2

Obs 6

Obs 5

Obs 4

Obs 3

Obs 2

Obs 1

1

1

1

1

1

1

1

1

E6

E5

E4

E3

E2

E1

F1

F2

Obs 6

Obs 5

Obs 4

Obs 3

Obs 2

Obs 1

E6

E5

E4

E3

E2

E1

F1

F2

Obs 6

Obs 5

Obs 4

Obs 3

Obs 2

Obs 11

1

My favorite.

Factors vs Summated Scales

Recall that X = T + E as was discussed briefly in PSY 5130

. An observed score is comprised of both the True score and error of measurement.

This means that X – the variable we’re measuring – is partly error. There will be some variation in X that has nothing to do with our theory - it’s just random variation, clouding our results.

The basic argument for using latent variables is that the relationships between latent variables are closer to the “true score” relationships than can be found in any existing analysis. They eliminate the noise.

If we compute the average of 10 Conscientiousness items to form a scale score, for example, that scale score includes the errors associated with each of the items averaged.

Here’s a path diagram representing a scale score . . .Assume the items are Conscientiousness items.


Item X7e7

e8

Item X3

Item X4

Item X5

Item X6

e3

e4

e5

e6

e1

e2

Scale score C+Junk

ee

ee2

e4

Item X8

Item X9

Item X10

e9

Item X1

Item X2

e10

A scale score contains everything – pure content (the Ts) plus error of measurement.

So any correlation involving a scale score is contaminated by the error of measurment contained in the scale score.e ee3

e1q

But if we create a C factor, the factor represents only the C present across all the items, and none of the error that also contaminates the item.

The errors affecting the items are treated separately, rather than being lumped into the scale score.

The result is that the latent variable, C, in the diagram below is a purer estimate of conscientiousness than would be a scale score. Its correlations with other variables will not be contaminated by errors of measurement.

From Schmidt, F. (2011). A theory of sex differences in technical aptitude and some supporting evidence. Perspectives on Psychological Science, 6, 560-573.“Prediction 3 was examined at both the observed and the construct levels. That is, both observed score and true score regressions were examined. However, from the point of view of theory testing, the true score regressions provide a better test of the theoretical predictions, because they depict processes operating at the level of the actual constructs of interest, independent of the distortions created by measurement error.”


C

Item 7

Item 8

Item 9

Item 10

e

e

e

e

Item 3

Item 4

Item 5

Item 6

e

e

e

e

Item 1

Item 2

e

e

GPA

So, what do we get from factor analyses?

1. Reduction in dimensionality

As discussed in the first lecture on factor analysis – the first purpose is to reduce the complexity of the problem, allowing us to focus on a few factors rather than multiple observed variables.

2. Uncontaminated measurement

Factors estimated from appropriate models give us estimates of characteristics will be unaffected by measurement error.

The way to access a factor’s values for individual persons is is through factor scores. Right now, only Mplus makes that easy to do. I expect more programs to provide such capabilities soon.

3. Purer relationships

As stated above, relationships among latent variables are free from the “noise” of errors of measurement, so if two characteristics are related, their factor correlations will be farther from 0 than will be their scale correlations.

That is – assess the relationship with scale scores. Assess the same relationship using a latent variable model. The r from the latent variable analysis will usually be larger than the r from the analysis involving scale scores.

4. Sanity and advancement of the science

The sometimes random-seeming mish-mash of correlations will be a little less random-seeming when the effects of errors of measurement on relationships and on measurement of characteristics are taken out.


Example illustrating the differences between scale score correlations and latent variable correlations.

From Biderman, M. D., McAbee, S. T., Chen, Z., & Nguyen, N. T. (2015). Assessing the Evaluative Content of Personality Questionnaires Using Bifactor Models. MS submitted for publication.

The correlations are those between three different measures of affect. For each measure, the scale was represented by a single summated score on the left and by a factor on the right.

Table 3Correlations of Measures of Affect

Summated Scales Latent Variables

RSE PANAS Dep. RSE PANAS Dep.Measured along with the NEO-FFI-31

NEO-FFI-3

RSE

PANAS .81 1.00

Depression - -.76 -.79 -.84 -1.00

Measured along with HEXACO-PI-R2

HEXACO-PI-R

RSE .

PANAS . .74 . .80

Depression -.78 -.73 -.87 -.82

Note. 1N=317, 2N=788. Dep. = depression. Values on diagonals are reliabilities of Scales and factor determinacies of latent variables.

As an aside, note that even the scale scores were very highly correlated, raising the issue of whether the three scales might all be measuring the same affective characteristic.


Programming with diagrams: Introduction to Amos

Amos is an add-on program to SPSS that performs confirmatory factor analysis and structural equation modeling.

It is designed to emphasize a visual interface and has been written so that virtually all analyses can be performed by drawing path diagrams.

It also contains a text-based programming language for those who wish to write programs in the command language.

The Amos drawing toolkit with functions of the most frequently used tools.


Observed variable tool

Tool to draw latent variables

Tool to draw correlation arrows

Tool to draw regression arrows

Tool to select a single object

Tool to put text on the diagram

Tool to select all objects in diagram

Tool to deselect all objects in diagram

Tool to copy an object Tool to erase an object

Tool to move an object

Tool to tell Amos to run the diagram.

Other programsLISRELEQSAmosMplus – the best

LAVAAN

Creating an Amos analysis

1. Open Amos Graphics.1b. File -> New2. File -> Data Files . . . (Because you have to connect the path diagram to a data file.)

3. Specify the name of the file that contains the raw or summary data.a. Click on the [File Name] button.b. Navigate to the file and double-click on it.c. Click on the [OK] button.

In this example, I opened a file called IncentiveData080707.sav

4. Draw the desired path diagram using the appropriate drawing tools.

The example below is a simple correlation analysis.

6. Save the model. File -> Save As...7. To run Amos, click on the

button.

8. Click on to see output.


5. Name the variables by right-clicking on each object. And choosing “Object Properties . . .”

Amos Details

For most of the analyses you’ll perform using Amos, you should get in the habit of doing the following . . .

View -> Analysis Properties -> Estimation

Check “Estimate means and intercepts”

View -> Analysis Properties -> Output

Check “Standardized estimates”Check “Squared multiple correlations”


Doing old things in a new way: Analyses we’ve done before, now performed using Amos Skip in 2017The data used for this example are the valdatnm data in the htlm2\5510\datafiles folder. We’ll simply look at the output here. Later, we’ll focus on the menu sequences needed to get this output.

a. SPSS analysis of the correlation of FORMULA with P511G

Correlations

b. Amos Input Path Diagram - Input Parameter Values

(Note, I told Amos to estimate means for this analysis.)

c. Amos Output Path Diagram - Unstandardized (Raw) coefficients

c. Amos Path Diagram - Standardized coefficients


All variables are exogenous.

The mean and variance of Formula.

The mean and variance of p511g.

The covariance of p511g and Formula.

The correlation of p511g and Formula.

Means and variances of standardized variables are not displayed, since they are 0 and 1 respectively.

Simple Regression Analysis: SPSS and Amos 2017 skipThe data used here are the VALDAT data.

a. SPSS Version 10 output

GET FILE='E:\MdbT\P595\Amos\valdatnm.sav'..REGRESSION /MISSING LISTWISE /STATISTICS COEFF OUTS R ANOVA /CRITERIA=PIN(.05) POUT(.10) /NOORIGIN /DEPENDENT p511g /METHOD=ENTER formula .

RegressionVariables Entered/Removedb

FORMULAa . EnterModel1

VariablesEntered

VariablesRemoved Method

All requested variables entered.a.

Dependent Variable: P511Gb.

Model Summary

.480a .230 .220 4.725E-02Model1

R R SquareAdjus tedR Square

Std. Error ofthe Es timate

Predic tors : (Cons tant), FORMULAa.

ANOVAb

5. 005E-02 1 5. 005E-02 22. 420 . 000a

. 167 75 2. 233E-03

. 217 76

RegressionResidualTot al

Model1

Sum ofSquares df Mean Square F Sig.

Predict ors: (Const ant ) , FORMULAa.

Dependent Var iable: P511Gb.

Coeffi ci entsa

. 496 .078 6. 361 .0003. 004E-04 .000 .480 4. 735 .000

(Const ant )FORMULA

Model1

B St d. Error

Unst andardizedCoef f icients

Beta

St andardized

Coef f icients

t Sig.

Dependent Variable: P511Ga.


b. Amos Input Path Diagram - Input parameter values

c. Amos Output Path Diagram - Unstandardized (Raw) coefficients; Means not estimated


The model is underidentified unless you fix the value of one parameter. Fix either the variance of the latent error variable to 1 or the regression weight to 1. Here, the variance has been fixed.

Note that the fixed parameter values were not changed.

For what it's worth, the estimated unstandardized (raw score) relationship of p511g to the “other factors” latent variable.

The estimated unstandardized (raw score) relationship of p511g .to Formula - the slope, to 2 decimal places.

Variance of formula.

d. Amos Output Path Diagram - Standardized coefficients

(View/Set -> Analysis Properties -> Output to get Amos to print Standardized estimates what a pain!!)

Note that .482 + .882 = 1. All of variance of p511g has been accounted for.

We say that the formula and the error partition the total variance of p511g.


Correlation of p511g with latent “other factors”..= sqrt(1-r2)=sqrt(1-.482) = sqrt(1-.23) = sqrt(.77)=.88

Squared multiple correlation of dependent variable (p511g) with predictor (only formula in this example).

Correlation of p511g with formula.

Two IV Regression Example - SPSS and Amos 2017 skipThe data here are the VALDATnm data. UGPA and GREQ are predictors of P511G.

a. SPSS output.GET FILE='G:\MdbT\P595\P595AL09-Amos\valdatnm.sav'.DATASET NAME DataSet1 WINDOW=FRONT.REGRESSION /DESCRIPTIVES MEAN STDDEV CORR SIG N /MISSING LISTWISE /STATISTICS COEFF OUTS R ANOVA /CRITERIA=PIN(.05) POUT(.10) /NOORIGIN /DEPENDENT p511g /METHOD=ENTER ugpa greq .

Regression[DataSet1] G:\MdbT\P595\P595AL09-Amos\valdatnm.sav

Correlations

1.000 .225 .322

.225 1.000 -.262

.322 -.262 1.000

. .025 .002

.025 . .011

.002 .011 .

77 77 77

77 77 77

77 77 77

p511g

ugpa

greq

p511g

ugpa

greq

p511g

ugpa

greq

PearsonCorrelation

Sig. (1-tailed)

N

p511g ugpa greq

Variables Entered/Removed b

greq, ugpa a . EnterModel1

Variables EnteredVariablesRemoved Method

All requested variables entered.a.

Dependent Variable: p511gb.

Model Summary

.455a .207 .185 .04828Model1

R R Square Adjusted R SquareStd. Error ofthe Estimate

Predictors: (Constant), greq, ugpaa.

Coefficients a

.571 .069 8.228 .000

.048 .016 .332 3.098 .003

.000 .000 .410 3.817 .000

(Constant)

ugpa

greq

Model1

B Std. Error

Unstandardized Coefficients

Beta

StandardizedCoefficients

t Sig.

Dependent Variable: p511ga.


b. Amos Input Path Diagram - Input parameters.

c. Amos Output Path Diagram - Unstandardized (Raw) coefficientsI forgot to check “Estimate means and intercepts, so no means are printed.)


Raw Regression coefficient relating p511g to residual effects.

Raw partial regression coefficient relating p511g to ugpa

The variance of the (unobserved) error latent variable must be specified at 1.

Note that if the IVs are correlated, you must specify that they are correlated. Otherwise, Amos will perform the analysis assuming they're uncorrelated.

Covariance of ugpa and greq.

Variance of ugpa

Raw partial regression coefficient relating p511g to GREQ to 2 decimal places.

d. Amos Output Path Diagram - Standardized coefficients.

Note that .332 + .412 + .892 = 1.07 > 1.0. This is because r2s partition variance only when variables are uncorrelated.e. Amos Text Output - Details of input and minimization (Early version of Amos without p values)Chi-square = 0.000Degrees of freedom = 0Probability level cannot be computedMaximum Likelihood Estimates----------------------------Regression Weights: Estimate S.E. C.R. Label ------------------- -------- ------- ------- -------

p511g <----- ugpa 0.048 0.015 3.140 p511g <---- error 0.047 0.004 12.329 p511g <----- greq 0.000 0.000 3.869

Standardized Regression Weights: Estimate-------------------------------- --------

p511g <----- ugpa 0.332 p511g <---- error 0.891 p511g <----- greq 0.410

Covariances: Estimate S.E. C.R. Label ------------ -------- ------- ------- -------

ugpa <-----> greq -8.537 3.861 -2.211

Correlations: Estimate------------- --------

ugpa <-----> greq -0.262

Variances: Estimate S.E. C.R. Label ---------- -------- ------- ------- -------

error 1.000 ugpa 0.134 0.022 6.164 greq 7897.622 1281.163 6.164

Squared Multiple Correlations: Estimate------------------------------ --------

p511g 0.207


Standardized partial regression coefficients , sometimes called betas..

Correlation of ugpa and greq.

Multiple R2.

SQRT(1-R2)=sqrt(1-.21) = sqrt(.79)=.89

Note – No overall test of significance of R2.This test is available in the ANOVA box in SPSS.

Oneway Analysis of Variance Example - SPSS and Amos 2017 skip

The data for this example follow. They're used to introduce the 595 students to contrast coding. The dependent variable is Job Satisfaction (JS). The research factor is Job, with three levels. It is contrast coded by CC1 and CC2.

The data for this example are in ‘MdbT\P595\Amos\ OnewayegData.sav’

ID JS JOB CC1 CC2

1 6 1 .667 .000 2 7 1 .667 .000 3 8 1 .667 .000 4 11 1 .667 .000 5 9 1 .667 .000 6 7 1 .667 .000 7 7 1 .667 .000 8 5 2 -.333 .500 9 7 2 -.333 .500 10 8 2 -.333 .500 11 9 2 -.333 .500 12 10 2 -.333 .500 13 8 2 -.333 .500 14 9 2 -.333 .500 15 4 3 -.333 -.500 16 3 3 -.333 -.500 17 6 3 -.333 -.500 18 5 3 -.333 -.500 19 7 3 -.333 -.500 20 8 3 -.333 -.500 21 2 3 -.333 -.500

a. SPSS Oneway output.

OnewayANO VA

JS

40. 095 2 20. 048 5. 930 . 011

60. 857 18 3. 381

100. 952 20

Bet weenG r oups

Wit hin G r oups

Tot al

Sum ofSquar es df M ean Squar e F Sig.


The rule for forming a contrast variable between two sets of groups is

1st Value = No. of groups in 2nd set / Total no. of groups.

2nd Value= - No. of groups in 1st set / Total no. of groups.

3rd Value = 0 for all groups to be excluded.

So, 1st Value of CC1 = 2 / 3 = .667.

2nd Value of CC1 = - 1 / 3

1st Value of CC2 = 1 / 2 = .5

2nd Value of CC2 = -1 / 2 = -..5

3rd Value of CC2 = 0 to exclude Job 1.

b. SPSS Regression Output.

regression variables = js cc1 cc2 /dependent = js /enter.

RegressionVariables Entered/Removed b

CC2, CC1 a . EnterModel1

VariablesEntered

VariablesRemoved Method

All reques ted variables entered.a.

Dependent Variable: JSb.

Mo d e l Su m m a ry

.6 3 0 a .3 9 7 .3 3 0 1 .8 3 8 7Mo d e l1

R R Sq u a reAd j u s te d R

Sq u a reStd . Erro r o fth e Es ti ma te

Pre d i c to rs : (Co n s ta n t ), CC2 , CC1a .

ANO VA b

40. 095 2 20. 048 5. 930 . 011a

60. 857 18 3. 381

100. 952 20

Regr ession

Residual

Tot al

M odel1

Sum ofSquar es df M ean Squar e F Sig.

Pr edict or s: ( Const ant ) , CC2, CC1a.

Dependent Var iable: JSb.

Coef f i ci ent sa

6. 952 . 401 17. 326 . 000

1. 357 . 851 . 292 1. 594 . 128

3. 000 . 983 . 559 3. 052 . 007

( Const ant )

CC1

CC2

M odel1

B St d. Er r or

Unst andar dized Coef f icient s

Bet a

St andar dizedCoef f icient s

t Sig.

Dependent Var iable: JSa.


c. Amos Input Path Diagram.

d. Amos Output Path Diagram - Unstandardized (Raw) Coefficients

e. Amos Output Path Diagram - Standardized Coefficients


This was prepared using Amos 3.6. I chose the "Estimate means" option. This was not required, but it caused means to be displayed.

Mean and variance.

Note that .292 + .562 + .782 = 1.01 ~~ 1.r2s partition variance since the variables are all independent.

Intercept

Multiple R2

Note that the correlation between group coding variables must be estimated. It's zero here because they're contrast codes, but estimate it anyway.

f. Amos Text Output – ResultsView -> Text Output


Results continued . . .


Note that AMOS does not provide a test of the null hypothesis that in the population, the multiple R = 0. This test is provided in the ANOVA box in SPSS.

Documents

Diagrams Representing Relationships between … · Web viewCorrelations or covariances between variables are represented by double-headed arrows. "Cor / Cov" Arrow Author Michael