CHAPTER 11&13 Sec A & B Chap 11 STATISTICAL PROCEDURES FOR PREDICTION AND CLASSIFICATION Chap 13 Factor Analysis.

CHAPTER 11&13 Sec A & B Chap 11 STATISTICAL PROCEDURES FOR PREDICTION AND CLASSIFICATION Chap 13 Factor Analysis

STATISTICAL PROCEDURES FOR PREDICTION AND CLASSIFICATION

Welcome & Agenda Exploratory Factor Analysis (EFA)Reliability Agenda:
Confirmatory Factor Analysis (CFA)Validity Structural Equation Modeling (SEM) Agenda: Discriminate Analysis (DA) Validity Statistical Basis for CFA Exploratory Factor Analysis Confirmatory Factor Analysis Overview of Structural Equation Modeling An introduction to AMOS and SEM Software The link between EFA and CFA Case Study Example of CFA

Discriminate Analysis (DA) Use Da in situations in which the test user wishes to use the predictor variables to classify examinees into various categories of selected criterion variable. In conducting such a validation study, the following 3 steps are executed: Next slide

Steps in Discriminate Analysis
A discriminant functional (Y) equation is determined which is the weighted linear combination of the predictor variables. A cutoff value for the discriminant function (Y) is determined to serve as the basis for the classification. The accuracy of the classifications made by using the discriminant function is assesse by means of the probability of misclassification. Steps in Discriminate Analysis

Discriminate Analysis (DA) We use DA to categorize participants (examinees) into 2 or more research groups. Use MANOVA with many groups. Ex. Next slide

Discriminate Analysis (DA) Ex1. The principle of an elementary school decides to separate the children with behavioral problem i.e. ADHD, ADD etc. from the children with no behavioral problem.

Discriminate Analysis (DA) Ex2. The supervisor of a psychiatric hospital decides to use different wards and separate the patients with affective/depresive d/o i.e., depression from the patients with psychiatric d/o i.e., schizophrenia. Next slide

Discriminate Analysis (DA)

Discriminate Function: The last column in the previous table is called Discriminate Function (DF) or (Y). It is a linear combination of all subtests Y= aX1+bX2+cX3+dX4 abcd are calculated weights W=Mean difference between the 2 groups (depressive and Schizophrenics W=μDep-μSch/σY With more than 2 groups we use ANOVA

Discriminate Analysis (DA) We use DA to describe a situation in which the test users wishes to use. We use the predictor variables (e.g. interview) to classify examinees into various categories of selected criterion variable (e.g. depressive, psychiatric). In doing so we follow 3 steps. Next slide

*3 Steps in conducting Discriminate Analysis (DA) 1. Calculate the value for Discriminate Function (Y). Y= aX1+bX2+cX3+dX4 2. Find a Cutoff Value for the Y (mean for the each group) which will serve as a basis for the classification. 3. Calculate Mean of Mean difference (W). W=[μDep-(μSch)]/2 *DA is used for the accuracy/validity of a classification.

SPSS  DA 1. Analyze  classify  Discriminate
2. Grouping Variable Psychiatric Group then define range>>>1 min and 2 max continue 3. Select inter independence together 4. independent variables are the predictors I, e., thinking disturbance, withdrawal, hostile, anxiety… Click on Classify  all group equal>>>separate group>>>display casewise results>>> summary table. OK selection variable is used when you want to do just one group (i.e psychiatrics).

Discriminate Function: (Y) The last column in the previous table is called Discriminate Function (Y). It is a linear combination of all subtests Y= aX1+bX2+cX3+dX4 a, b, c, and d are discriminant weights X1, X2, X3, and X4 are scores on the items If more than 2 groups we use ANOVA

Discriminate Function: (Y) W=Mean difference between the 2 groups (Y depressive and Y Schizophrenics). It is like Post Hoc Tests W= Standardized Mean Difference between the 2 groups W= [μSch +(μDep)]/2 σY = number of groups to determine the average of two means [μSch + (μDep)]/ Ex. next If more than 2 groups we use ANOVA

Mean score of schizophrenic patients for answering the depression related questions is 0.0 Mean score of schizophrenic patients for answering the schizophrenia related questions is 4 Mean score of depressed patients for answering the schizophrenia related questions is 1 Mean score of depressed patients for answering the depression related questions is 5

Calculate the Discriminant Analysis for the graduate Students who;
Next Example Calculate the Discriminant Analysis for the graduate Students who; 1=COMPLETED the PHD program & 2=DID NOT COMPLETE the PHD program

SPSS  DA SPSS data fileselect the graduate.save or graduate
1. Analyze  classify  Discriminate 2. Grouping Variable Category then define range>>>1 min and 2 max continue 3. Select inter independence together 4. independent variables are the predictors I, e., GRE scores, Letters of recommendations, etc … Click on Classify  all group equal>>>separate group>>>display casewise results>>> summary table. OK selection variable is used when you want to do just one group (i.e psychiatrics).

Select Graduate.sav From data files ****input****

Classification Resultsa Predicted Group Membership
Graduate.sav output Classification Resultsa 1=COMPLETED PHD, 2=DID NOT COMPLETE PHD Predicted Group Membership Total FINISH NOT FINISH Original Count 23 2 25 % 92.0 8.0 100.0 a. 92.0% of original grouped cases correctly classified.

Statistical Basis for CFA and Structural Equation Modeling
First let us review some familiar procedures that serve as the basis for these advanced statistical procedures. Correlation Bi-variate Regression, Y=bX+a or Y' = a + bX Multiple Regression, and Path Analysis.

Correlation Correlation is about the relationship between 2 variables, Predictor (X) and Criterion (Y) Ex. Relationship between Perfectionism and Depression Dep Y Per X

Correlation Correlation measures the strength and the direction of the relationship between two or more variables. A correlation has three components: The strength of the coefficient The direction of the relationship The form of the relationship The strength of the coefficient is indicated by the absolute value of the coefficient. The closer the value is to 1.0, either positive or negative, the stronger or more linear the relationship. The closer the value is to 0, the weaker or nonlinear the relationship.

Correlation The direction of coefficient is indicated by the sign of the correlation coefficient. A positive coefficient indicates that as one variable (X) increases, so does the other (Y). A negative coefficient indicates that as one variable (X) increases, the other variable (Y) decreases. The form of the relationship The form of the relationship is linear. In correlation variables are not identified as independent or dependent because the researcher is measuring the one relationship that is mutually shared between the two variables As a result, causality should not be implied with correlation.

Correlation Remember, the correlation coefficient can only measure a linear relationship. A zero correlation does not indicate no relationship. a coefficient of zero rules out linear relationship, but a curvilinear could still exist. The scatterplots below illustrate this point:

Partial Correlation Coefficient We use Partial Correlation Coefficient when we want to find out about the relationship between a Criterion and a Predictor and there is a second Predictor. Ex. The relationship between Churches, Crime, and Population. crime churches pop

Bi-Variate Regression Analysis
Bi-variate regression analysis extends correlation and attempts to measure the extent to which a predictor variable (X) can be used to make a prediction about a criterion measure (Y). E Bi-variate regression uses a linear model to predict the criterion measure. The formula for the predicted score is: Y bX + a

Regression (SEM) y x Y = bX +a e
*Bi-variant Regression: We have one X and one y Y = bX +a b a y x e σ²X σσ²aa

Bivariate Regression The components of the line of best fit Y' = bX + a a= the Y-intercept/Constant b= the slope X= Variable The Y-intercept is the average value of Y' when X is zero. The Y-intercept is also called constant. Because, this is the amount of Y that is constant or present when the influence of X is null (0). The slope (b) is average value of a one unit change in Y' for a corresponding one unit change in X. Thus, the slope represents the direction and intensity of the line.

Bivariate Regression Line of Best Fit: Y' = 0.204X + 2.635
If a= and b= 0.204 Line of Best Fit: Y' = 0.204X With this equation a predicted score may be made for any value of X within the range of data. Y-intercept Slope = 0.204 2.635

Multiple Regression *Multiple Regression Analysis R²
We use Multiple Regression Analysis for validity of 2 or more predictors Ex. next slide Y=bX1+bX2+a Y Regression Line X1 and X2

Multiple regression Ex. The association between Depression and 2 aspects of Personality (Perfectionism and OCPD). Next slide SEM per dep ocpd

Structural Equation Modeling
Regression Path Analysis, Confirmatory Factor Analysis, ANCOVA Structures , Latent Variable Models, Causal Modeling

Structural Equation Modeling (SEM)
*One of the advantages of SEM is that The variables can play a double role IV and DV which makes the SEM more useful. (Research Hypothesis) Perfectionism Depression OCPD E

Multiple Regression Analysis
Multiple regression analysis is an extension of bi-variate regression, in which several predictor variables (X) are used to predict one criterion measure (Y). In general, this method is considered to be advantageous; since seldom can an outcome measure be accurately explained by one predictor variable. 3 aspects of personality (OCPD, Narcissistic, Histrionic) and Depression E Y' = b1X1 +b2X2 +b3X3 + c

In Experimental Research we have IV and DV and investigate causality or cause and effect between the variables In Correlational Research we have Predictors and Criterion and investigate correlation or relationship between the variables. In SEM we have Exogenous and Endogenous variables and use Qualitative Causal Assumptions which is a combination of causality and correlation.

*Exogenous Variables are similar to IV or predictors . They are the predictor variables in the model. You will see an arrow pointing away from them (outward) X *Endogenous Variables: They are similar to DV or criterion. It is the variable that regresses on or being predicted by Exogenous Variables. You will see an arrow pointing towards them Y

e Exogenous Variables X
There are 2 different types of Exogenous Variables Observable Exogenous Variables Such as Xs Unobservable Exogenous Variables Such as errors (e) See AMOS ex X e

Endogenous Variables F1 Endogenous Variables
Such as a Factors (F1 or F2) Endogenous variables are unobservable except in a special occasion. See AMOS ex F1

Path Analysis Path Analysis is an extension of regression. In Path analysis the researcher is examining the ability of more than one predictor variable (X) to explain or predict multiple dependent variables (Y). As we enter into the first of our modeling procedures we must clarify some key terms 3 aspects of personality (OCPD, Narcissistic) and (Depression and Anxiety) X 1 Y Y 1 2 X 2 E E

Path Analysis *Path analysis is an extension of the regression model, used to test the fit of the correlation matrix against two or more causal models which are being compared by the researcher.

Path Analysis The model is usually depicted in a circle-and-arrow figure in which single-headed arrows indicate causation. A regression is done for each variable in the model as a dependent on others which the model indicates are causes.

Path Analysis The regression weights predicted by the model are compared with the observed correlation matrix for the variables, and a goodness-of-fit (GOF) statistic is calculated. The best-fitting of two or more models is selected by the researcher as the best model for advancement of theory.

Key Concepts and Terms *Path Model. A path model is a diagram relating independent/ intermediary, and dependent variables. Single arrows indicate causation between exogenous or intermediary variables and the dependent(s). Arrows also connect the error terms with their respective endogenous variables.

Key Concepts and Terms Double arrows indicate correlation between pairs of exogenous variables. Sometimes the width of the arrows in the path model are drawn in a width which is proportional to the absolute magnitude of the corresponding path coefficients (see next slide).

Path Coefficient/Path Weight.
Key Concepts and Terms Path Coefficient/Path Weight. A path coefficient is a standardized regression coefficient (beta) showing the direct effect of an independent variable on a dependent variable in the path model. Special case is a zero-order correlation coefficient.

Key Concepts and Terms

Key Concepts and Terms *Disturbance terms. The residual error terms, also called disturbance terms, reflect unexplained variance (the effect of unmeasured variables) plus measurement error..

Key Concepts and Terms Note that the dependent in each equation is an endogenous variable (in this case, all variables except age, which is exogenous). Note also that the independents in each equation are all the variables with arrows to the dependent.

Path Analysis key terms
Measured variables Exogenous variables Endogenous variables Direct Effects Indirect Effects Errors in Prediction E E

Definition of Terms Measured Variables
Variables that the researcher has observed and measured. In all diagrams, measured variables are depicted by squares or rectangles In path analysis, all variables are measured even unobservable variables such as errors E

e Exogenous Variables X
There are 2 different types of Exogenous Variables Observable Exogenous Variables Such as Xs Unobservable Exogenous Variables Such as errors (e) X e

Definition of Terms Observable Exogenous Variables
Exogenous variables are those for which the model makes no attempt to explain (measure). Just like IV and predictors. In this path analysis, two observable exogenous variables exist: X1 and X2. E

Unobservable Exogenous Variables or Errors
Errors in Prediction: As in any prediction model, errors in prediction always exist. Thus, Y1 and Y2 will have errors in prediction. In this path analysis, two unobservable exogenous variables exist: e1 and e2 e1 e2

Definition of Terms Endogenous Variables
Endogenous variables are those which the model attempts to explain(measure). Just like DV and criterion. In this path analysis, two endogenous variables exist: Y1 and Y2. E

Endogenous Variables F1 Items or Questions
There are 2 different types of Endogenous Variables Observable Endogenous Variables Such as selecting an item in your data set to substitute F1 Unobservable Endogenous Variables Such as naming your Factor (F1, F2) or outgoing, depression, antisocial, etc. Items or Questions F1

LATENT VARIBLES Latent variables are not measured directly such as Exogenous and Endogenous both have latent variables same as unobservable variables e F

Definition of Terms Direct Effects
Direct effects are those parameters that estimate the "direct" effect one variable has on another. These are indicated by the arrows that are drawn from one variable to another. In this model, four direct effects are measured E

Definition of Terms Indirect Effects
Indirect effects are those influences that one variable may have on another, that is mediated through a third variable. In this model, X1 and X2, have a direct effect on Y1 and an indirect effect on Y2 through Y1. X1 has direct effect on Y2 E

*One of the abilities of SEM is to measure the “Latent Variables” or variables that are not measured directly, such as factors (F)& errors (e). *Factor Analysis, Path Analysis, and Regression represent special cases of SEM.

*Steps in Performing SEM Analysis. SEM is a confirmatory technique. It is suited to theory testing and confirming a hypothesis (hypothesis testing) rather than theory development or exploratory techniques. *The correct model has 2 different kind of variables. They are Exogenous and Endogenous variables.

Latent Variables are placed in oval or circle, and Manifest (Main) Variables are placed in boxes in the Paths Diagram. The simplest example would be the classical linear regression Equation. Next slide

Path Analysis Path Analysis is an extension of regression.
In Path analysis the researcher is examining the ability of more than one predictor variable (X) to explain or predict multiple dependent/criterion variables (Y). Ex 1. Hypothesis The Association between personality (OCPD and Narcissistic) and mental disorder (Depression and Anxiety) X 1 Y Y 1 2 X 2 E E

Ex.1: Predicting the severity of crimes by level of education, age, and marital status

Example 1 from Sample IBM SPSS Data Set
TURN-ON IMOSCLICK ON FILE DATA FILE FILE NAME LOCAL DISK C PROGRAM FILESX86 (OR JUST PROGRAM FILE)IBM-> SPSS STATISTICS 20, 21, OR 25examples/SAMPLES ENGLISHRECIDIVISM or feeling goodOK CLICK ON VIEWVARIABLES IN DATA SETDRAG AND MOVE Create error for DV Click on Plugins and select (name unobserved variables). Click on View  Analysis properties estimate: put a check mark next to “estimate means and intercepts.” Output: check the top four on both columns Analyze  Calculate estimates click on top right blue arrow Click on VIEWTEXT OUTPUT Click “Title” on the left menu Type your name

Example 1 from Sample IBM SPSS Data Set (cAU)
TURN-ON AMOSCLICK ON FILE DATA FILE FILE NAME Computer LOCAL DISK C PROGRAM FILESX86 (OR JUST PROGRAM FILE)IBM-> SPSS STATISTICS 20, SAMPLES ENGLISHRECIDIVISM.save OK CLICK ON VIEW VARIABLES IN DATA SETDRAG AND MOVE Create error for DV by clicking on the rectangular image with a circle on top  Click on Plugins and select (name unobserved variables). Click on View  Analysis properties estimate: put a check mark next to “estimate means and intercepts.” Output: check the top four on both columns Analyze  Calculate estimates click on the left upper pointed brown arrow Click on VIEWTEXT OUTPUT Click “Title” on the left menu Type your name

Example 1 from Sample IBM SPSS Data Set
TURN-ON IMOSCLICK ON FILE DATA FILE FILE NAME click on file view data CLICK ON VIEWVARIABLES IN DATA SETDRAG AND MOVE Create error for DV Click on Plugins and select (name unobserved variables). Click on View  Analysis properties estimate: put a check mark next to “estimate means and intercepts.” Output: check the top four on both columns Analyze  Calculate estimates click on top right blue arrow Click on VIEWTEXT OUTPUT Click “Title” on the left menu Type your name

Ex. 3 samples from data files Path analysis spss
1. download data from “DATA FILES.” Download the helping1.save or just helping1.

Ex. 4 copy and paste

Using Structural Equation Modeling Software

Structural Equation Packages
Structural Equation Programs are based upon the principle of confirming models (validity). This principle is the opposite of data mining where the program explores (i.e., 'mines') the data for trends or patterns. (Use SPSS) The following graphic, describes the flow of information within Structural Equation Packages: Use AMOS 2. Data 1. Model SEM Package 3. Fit Measures 4. Parameter Estimates

Key Concepts and Terms Income Age Job satisfaction autonomy

Software Originally, Structural Modeling was based in Matrix Algebra and the software was difficult to understand and manipulate. Like many other statistical procedures, the advances in technology have now provided this technique to novice nä-vəs researcher.

The basis of the Measurement Model in SEM! Linking EFA to CFA
Factor Analysis Chap 13 The basis of the Measurement Model in SEM! Linking EFA to CFA

Factor Analysis Factor Analysis is about dimension reduction, or when we have too many items and want to reduce them.

Factor analysis can help the test developer
1. to eliminate or revise items that do not load on the predicted factor. 2. to identify whether test items appear to be measuring the same construct.

the following is TRUE of factor analysis
a. It represents a class of mathematical procedures. b. It represents a data reduction technique. c. It explains the extent to which a factor or factors explain test scores.

CHAPTER 13 FACTOR ANALYSIS
Factor Analysis is about Data reduction Raymond Cattell in his 16 PF questioner reduced 4500 questions and 200 factors into 185 related questions and 16 related factors (components).

Warmth (A) Descriptors of Low Range Primary Factor
Descriptors of High Range Impersonal, distant, cool, reserved, detached, formal, aloof Warmth (A) Warm, outgoing, attentive to others, kindly, easy-going, participating, likes people Concrete thinking, lower general mental capacity, less intelligent, unable to handle abstract problems Reasoning (B) Abstract-thinking, more intelligent, bright, higher general mental capacity, fast learner Reactive emotionally, changeable, affected by feelings, emotionally less stable, easily upset Emotional Stability (C) Emotionally stable, adaptive, mature, faces reality calmly Deferential, cooperative, avoids conflict, submissive, humble, obedient, easily led, docile, accommodating Dominance (E) Dominant, forceful, assertive, aggressive, competitive, stubborn, bossy Serious, restrained, prudent, taciturn, introspective, silent Liveliness (F) Lively, animated, spontaneous, enthusiastic, happy-go-lucky, cheerful, expressive, impulsive Expedient, nonconforming, disregards rules, self-indulgent Rule-Consciousness (G) Rule-conscious, dutiful, conscientious, conforming, moralistic, staid, rule bound Shy, threat-sensitive, timid, hesitant, intimidated Social Boldness (H) Socially bold, venturesome, thick-skinned, uninhibited Utilitarian, objective, unsentimental, tough minded, self-reliant, no-nonsense, rough Sensitivity (I) Sensitive, aesthetic, sentimental, tender-minded, intuitive, refined Trusting, unsuspecting, accepting, unconditional, easy Vigilance (L) Vigilant, suspicious, skeptical, distrustful, oppositional Grounded, practical, prosaic, solution oriented, steady, conventional Abstractedness (M) Abstract, imaginative, absent minded, impractical, absorbed in ideas Forthright, genuine, artless, open, guileless, naive, unpretentious, involved Privateness (N) Private, discreet, nondisclosing, shrewd, polished, worldly, astute, diplomatic Self-assured, unworried, complacent, secure, free of guilt, confident, self-satisfied Apprehension (O) Apprehensive, self-doubting, worried, guilt prone, insecure, worrying, self blaming Traditional, attached to familiar, conservative, respecting traditional ideas Openness to Change (Q1) Open to change, experimental, liberal, analytical, critical, free-thinking, flexibility Group-oriented, affiliative, a joiner and follower dependent Self-Reliance (Q2) Self-reliant, solitary, resourceful, individualistic, self-sufficient Tolerates disorder, unexacting, flexible, undisciplined, lax, self-conflict, impulsive, careless of social rules, uncontrolled Perfectionism (Q3) Perfectionistic, organized, compulsive, self-disciplined, socially precise, exacting will power, control, self-sentimental Relaxed, placid, tranquil, torpid, patient, composed low drive Tension (Q4) Tense, high energy, impatient, driven, frustrated, over wrought, time driven. Primary Factors and Descriptors in Cattell's 16 Personality Factor Model (Adapted From Conn & Rieke, 1994 Raymond Cattell's 16 Personality Factors

FACTOR ANALYSIS In Factor Analysis we take a large number of observable variables to measure an unobservable Variable (construct) Ex. (outgoing). Attending loud parties Appear comfortable interacting with others Outgoing Usually seen with others Talks a lot Outgoing (construct) can be predicted by the above Observable Variables

FACTOR ANALYSIS In Factor Analysis we take a large number of observable variables to measure an unobservable Variable (construct) Ex. (outgoing). Depressed mood Weight loss or weight gain Depression Insomnia or hypersomnia Fatigue/loss of energy Depression (construct) can be predicted by the above Observable Variables

FACTOR ANALYSIS Factor Analysis is used to identify the factor (outgoing) that is used to represent relationships among sets of interrelated variables e.g. the 4 variables.

FACTOR ANALYSIS *Four Basic Steps to Conduct a Factor Analysis
Calculate a Correlation Matrix (table) of all variables to be used in the analysis Extract factor (reduce the factors which is done by SPSS) Rotate factors to create a more understandable factor structure (done by SPSS) Interpret results

FACTOR ANALYSIS *Factor Loading The correlations of the items with the hypothesized factors (F1 and F2) are called Factor Loading

communality The percentage of the total test variance accounted for by factors that the test has in common with other tests in a factor analysis is the communality of the test.

FACTOR ANALYSIS *Communality Value: Communalities are designed to show the proportion of variance “σ²” that the factors contribute to explaining a particular variable. Each of the variables is initially assigned a communality value of 1.0 by SPSS.

FACTOR ANALYSIS Total Variance “σ²” explained: Occurs when SPSS selects the combination of variables whose shared correlations explains the greatest amount of the total variance.

FACTOR ANALYSIS *Eigenvalues: Eigenvalues are designed to show the proportion of variance accounted for by each factor. SPSS default is set to keep any factor with an Eigenvalue larger than If a factor has an Eigenvalue less than 1.00 it explains less variance “σ²” than an original variance, and is usually rejected.

FACTOR ANALYSIS *Scree Plot: It plot the eigenvalues on a bicoordinate plane. It determines the number of Factors (components).

Factor analysis Factor analysis Intercorrelations among items
Exploratory Reduce to the basic dimensions Confirmatory Test hypothesized structure Empirical emphasis, as in Empirical Criterion Keying.

For More Information In case you missed it: recorded version and slides available at Product questions? Call SPSS at or Please fill out the post event survey

Factor Analysis Factor analysis is a fundamental component of Structural Equation Modeling. Factor analysis explores the inter-relationships among variables to discover if those variables can be grouped into a smaller set of underlying factors. Next, we will review the components of Factor Analysis.

*Applications of Factor Analysis
Three primary applications of Factor Analysis include: 1-Explore data for patterns: Often a researcher is unclear if items or variables have a discernible/ reliable patterns (reliability). Factor Analysis can be done in an Exploratory fashion to reveal patterns among the inter-relationships of the items (reliability).

Three primary applications of Factor Analysis include: 2-Data Reduction: Factor analysis can be used to reduce a large number of variables into a smaller and more manageable number of factors. Factor analysis can create factor scores for each subject that represents these higher order variables. Raymond Cattell in his 16 PF reduced 4500 personality related questions into 187 questions and 16 related variables or factors.

Three primary applications of Factor Analysis include: 3-Confirm hypothesis of factor structure: In measurement research when a researcher wishes to validate a scale with a given or hypothesized factor structure, Confirmatory Factor Analysis is used (Validity).

Exploratory Factor Analysis
In exploratory factor analysis, the researcher is attempting to explore the relationships among items to determine if the items can be grouped into a smaller number of underlying factors. In this analysis, all items are assumed to be related to all factors. E

Exploratory Factor Analysis Key Terms
*Measured Variables observed variables or indicators *Unmeasured Variables unobserved variables latentvariables/FACTORS *Factor Loadings *Errors in Measurement E

*Measured Variables or Indicators As in path analysis, these variables are those that the researcher has observed or measured. In this example, they are the four items on the scale Note, they are drawn as rectangles or squares. E

*Unmeasured or Latent variables These variables are not directly measurable, rather the researcher only has indicators of these measures. These variables are more often the more interesting, but more difficult variables to measure (e.g., outgoing self-efficacy or self-esteem). In this example, the latent variables are the two factors. Note, they are drawn as circles. E

*Factor Loadings Measure the relationship between the items and the factors. Factors are interpreted primarily by examining the factor Loadings Factor loadings can be interpreted like correlation coefficients; ranging between -1.0 and +1.0. The closer the value is to 1.0, positive or negative, the stronger the relationship between the factor and the item. loadings can be both positive or negative. E

Factor Loadings The correlations between factors and tests are referred to as Factor Loadings Note the direction of the arrows; the factors are thought to influence the indicators, not vice versa. It means each item is being predicted by the factors. (we use this to group the items into a factor) E

*Errors in Measurement Each of the indicator variables has some error in measurement. The small circles with the E indicate the error. The error is composed of 'we know not what' or are not measured directly. These errors in measurement are considered the reliability estimates for each indicator variable. E

Confirmatory Factor Analysis
Confirmatory factor analysis, meets the third application of factor analysis: To confirm a hypothesized factor structure. Used as a validity procedure in measurement research. Confirmatory factor analysis differs from exploratory factor analysis in that for confirmatory analysis, a specific relationship between the items and the factors is confirmed. Certain items are hypothesized to go to given factors Not all items go to all factors

Comparison of EFA versus CFA
Exploratory Factor Analysis Confirmatory Factor Analysis E E

Confirmatory Factor Analysis
Note: In confirmatory factor analysis, only certain items are proposed to be indicators of each factor. The curved line indicates the relationship that could exist between the factors. Again, the errors in measurement are shown by the circles with the E. E

*Basic Steps for CFA *There are six basic steps to performing a CFA:
Define the factor model: (research hypotheses)The first thing you need to do is to precisely define the model you wish to test. This involves selecting the number of factors, and determining the nature of the loadings (correlation) between the factors and the measures. These loadings can be fixed at zero, fixed at another constant value, allowed to vary freely, or be allowed to vary under specified constraints (such as being equal to another loading in the model). Here you use the knowledge derived from your EFA to specify your CFA model. We will talk more about this step, next!

Basic Steps for CFA There are six basic steps to performing an CFA: Collect measurements: You need to measure your variables on the same (or matched) participants. Remember, you would not use the same data that was used for the EFA in the CFA. Obtain the correlation matrix: You need to obtain the correlations (or covariances) between each of your variables.

Basic Steps for CFA There are six basic steps to performing an CFA: (continued) Fit the model to the data: You will need to choose a method to obtain the estimates of factor loadings that were free to vary. The most common model-fitting procedure is Maximum likelihood estimation, which should probably be used unless your measures seriously lack multivariate normality.

Basic Steps for CFA Evaluate model adequacy:
There are six basic steps to performing an CFA: (continued) Evaluate model adequacy: When the factor model is fit to the data, the factor loadings (correlations) are chosen to minimize the discrepancy between the correlation matrix implied by the model and the actual observed matrix. The amount of discrepancy after the best parameters are chosen can be used as a measure of how consistent the model is with the data. This is where we use our fit statistics!

Basic Steps for CFA Compare with other models:
There are six basic steps to performing an CFA: (continued) Compare with other models: If you want to compare two models, one of which is a reduced form of the other, you can just examine the difference between their X2 statistics. Almost all tests of individual factor loadings can be made as comparisons of full and reduced factor models. In cases where you are not examining full and reduced models you can compare the Root Mean Square Error (variance)of Approximation (RMSEA), which is an estimate of discrepancy per degree of freedom in the model.

*Parsimony better to have a model with fewer paths (simple).
able to explain variables with fewer paths and fewer equations similar principle in multiple regression - more economical to be able to predict with fewer predictors I use the principle of Parsimony in my teaching method because we prefer the simplest useful explanations.

https://www. youtube. com/watch
“To be or not to be” is a Meta Analytic question. i.e., “ Is it good to do this thing or is it not good. Ex. “To drink or not to drink” or To smoke or not to smoke”

perform CMA Meta-Analysis Use Meta-Analysis for decision making
Use Meta-Analysis for the validity of research See PP#13 Advanced STSTS to perform CMA

*Why Use Structural Equation Modeling
Problems which need SEM: Many variables Continuous variables to be used in a prediction model Have various indicators of latent variables Want to compare groups? Categorical variables Multi-sample problem Does the model differ for males and females?

Questions!

CHAPTER 11&13 Sec A & B Chap 11 STATISTICAL PROCEDURES FOR PREDICTION AND CLASSIFICATION Chap 13 Factor Analysis.

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CHAPTER 11&13 Sec A & B Chap 11 STATISTICAL PROCEDURES FOR PREDICTION AND CLASSIFICATION Chap 13 Factor Analysis.

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