Introduction To Confirmatory Factor Analysis and Item Response Theory

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1 Introduction To Confirmatory Factor Analysis and Item Response Theory Lecture 23 May 3, 2005 Applied Regression Analysis Lecture #23-5/3/2005 Slide 1 of 21

2 Today s Lecture Confirmatory Factor Analysis. Today s Lecture Item Response Theory (Item Factor Analysis). Structural Equation Modeling. Lecture #23-5/3/2005 Slide 2 of 21

3 Example #1 Example #2 CFA Recall the following discussion of a proxy variable (from the lecture on 3/10): Often in nonexperimental research, variables are proxies for target variables in the regression equation. Such proxies are used because the target variable of interest is difficult to measure. Care should be taken in interpretation of such proxies so that they are interpreted as the variables they truly are, and not the variable they represent. Lecture #23-5/3/2005 Slide 3 of 21

4 Definition of proxy from Example #1 Example #2 CFA 1. A person authorized to act for another; an agent or substitute. 2. The authority to act for another. 3. The written authorization to act in place of another. Lecture #23-5/3/2005 Slide 4 of 21

5 In Practice From McDonald (1999). Test Theory: A Unified Treatment. Example #1 Example #2 CFA Imagine a psychologist is interested in determining an individual s satisfaction with life. In order to do so, the psychologist creates the following questionnaire: 1. I am satisfied with my life. 2. The conditions of my life are excellent. 3. In most ways my life is close to the ideal. 4. So far I have gotten the important things I want from life. 5. If I could live my life over, I would change almost nothing. Lecture #23-5/3/2005 Slide 5 of 21

6 In Practice Example #1 Example #2 CFA For each question, the respondent is asked to respond using a seven-point Likert scale: 7. Strongly agree 6. Agree 5. Slightly agree 4. Neither agree nor disagree 3. Slightly disagree 2. Disagree 1. Strongly disagree This short questionnaire has a total of five questions, all somewhat related to the fuzzy concept that is called satisfaction with life (note that this example comes indirectly from the work of Ed Diener). Lecture #23-5/3/2005 Slide 6 of 21

7 In Practice Example #1 Example #2 CFA For the researching interested in seeing how satisfaction with life is related to other measures, what can be done? There are five questions, each serving as a proxy for satisfaction with life. Should a single question be used as an independent or dependent variable in another analysis? Should the question that provides the best fit be used? Should some type of combination of the questions be used? Lecture #23-5/3/2005 Slide 7 of 21

8 In Practice: Example 2 Example #1 Example #2 CFA Each year, students of all grades are administered standardized tests in order to demonstrate achievement within a given subject domain. Take the following items from a test of fraction subtraction (Tatsuoka, 1990): For each item, a value of one is recorded if an examinee gives the correct response, otherwise a value of zero is recorded. Lecture #23-5/3/2005 Slide 8 of 21

9 In Practice: Example 2 Example #1 Example #2 CFA Because each of us has spent an enormous amount of time in school, we are led to believe that after a test is given, the will simply add the number of correct responses. For today s class, please remove this believe from your mind. Instead, consider each of the test items as being a proxy variable for knowledge of fractions or fraction achievement. Much as in Example #1, the question remains: how does one use the test items? Also of concern is the type of measurement for the variables of the fraction subtraction test, where each is dichotomously valued (or binary - 0/1) Lecture #23-5/3/2005 Slide 9 of 21

10 Enter: Factor Analytic Techniques Example #1 Example #2 CFA For both examples, confirmatory factory analytic techniques can be employed to: Confirm the measurement of some latent variable. Here latent refers to the concept for which each item was used as a proxy. Estimate the manner in which each proxy variable relates to the latent construct. Perhaps some proxy variables are better measures of a latent construct than are others. Reduce the number of variables (or dimensionality) by creating a new variable from a linear combination of the proxy variables. Determine how reliably new latent construct is measured by the combination of the proxy variables. Lecture #23-5/3/2005 Slide 10 of 21

11 Review Of Definitions Latent Variables Parameters Dichotomous Items Factor analytic techniques (and structural equation models in general) commonly make use of path diagrams. Recall from the lecture given on 3/8, the terms: Exogenous variable - a variable whose variability is assumed to be determined by causes outside of the causal model under consideration. Endogenous variable - a variable whose variability is to be explained by exogenous and other endogenous variables in the causal model. Lecture #23-5/3/2005 Slide 11 of 21

12 Linear Regression Model X 1 Definitions Latent Variables Parameters Dichotomous Items X 2 Y e Exogenous variables - X 1 and X 2. Endogenous variable - Y. Curved line with arrowheads at both ends - correlated variables (correlation is assumed and not modeled - always the case with correlated exogenous variables). Straight line - causal path. Lecture #23-5/3/2005 Slide 12 of 21

13 Linear Regression Model X 1 Definitions Latent Variables Parameters Dichotomous Items X 2 Y e For this diagram, the linear regression model is given by the following equation: Y ij = a + b 1 X 1 + b 2 X 2 + e ij Where Y, X 1, and X 2 are continuous variables, and e N(0, σ 2 e). Lecture #23-5/3/2005 Slide 13 of 21

14 More Definitions From Latent: Definitions Latent Variables Parameters Dichotomous Items 1. Present or potential but not evident or active: latent talent. 2. Pathology. In a dormant or hidden stage: a latent infection. 3. Biology. Undeveloped but capable of normal growth under the proper conditions: a latent bud. 4. Psychology. Present and accessible in the unconscious mind but not consciously expressed. Manifest: 1. Clearly apparent to the sight or understanding; obvious. Lecture #23-5/3/2005 Slide 14 of 21

15 Latent Variables/ F 1 Definitions Latent Variables Parameters Dichotomous Items X 1 X 2 X 3 e X1 e X2 e X3 The latent construct is denoted by the circle around the variable F 1. Each of the manifest (or observable) variables is a function of the latent construct (and error). Lecture #23-5/3/2005 Slide 15 of 21

16 Latent Variables/ F 1 Definitions Latent Variables Parameters Dichotomous Items X 1 X 2 X 3 e X1 e X2 e X3 X 1 = a X1 + b X1 F 1 + e X1 = µ X1 + λ X1 F 1 + e X1 X 2 = a X2 + b X2 F 1 + e X2 = µ X2 + λ X2 F 1 + e X2 X 3 = a X3 + b X3 F 1 + e X3 = µ X3 + λ X3 F 1 + e X3 Lecture #23-5/3/2005 Slide 16 of 21

17 Parameters X 1 = a X1 + b X1 F 1 + e X1 = µ X1 + λ X1 F 1 + e X1 Definitions Latent Variables Parameters Dichotomous Items µ X1 - Analogous to the intercept in regression: value of X 1 when F 1 is zero. λ X1 - Analogous to slope in regression: the loading of X 1 onto factor F 1. e X1 - The error associated with X 1 (part not explained by F 1 ). Lecture #23-5/3/2005 Slide 17 of 21

18 Dichotomous Items Definitions Latent Variables Parameters Dichotomous Items As logistic regression is to linear regression, item response theory (or item factor analysis) is to linear factor analysis. Instead of modeling X directly, we now use a link function. The most prevalent way of doing this (in educational measurement) is by using the logistic link function (IRT), where we now model the probability of a correct response as a function of the latent factor. A common way of doing this is the two-parameter logistic (2PL) item response model (often the latent variable F 1 is symbolized by θ): P(X ij = 1 F 1 ) = ea j(θ i b j ) 1 + e a j(θ i b j ) Lecture #23-5/3/2005 Slide 18 of 21

19 IRT Parameters P(X ij = 1 F 1 ) = ea j(θ i b j ) 1 + e a j(θ i b j ) Definitions Latent Variables Parameters Dichotomous Items a j - The discrimination parameter for item j. Controls the direction and speed of the increase from low predicted probabilities to high predicted probabilities. b j - The difficulty parameter for item j. Controls where the ogive increases along the θ scale. Lecture #23-5/3/2005 Slide 19 of 21

Final Thought Final Thought Next Class CFA and IRT are used for measurement of latent variables. CFA is commonly associated with continuously valued items.

20 Final Thought Final Thought Next Class CFA and IRT are used for measurement of latent variables. CFA is commonly associated with continuously valued items. IRT is the name of the CFA analog in educational measurement. Estimates of these models cannot be obtained directly via SPSS. Whole classes are devoted to both of these topics (usually treated as separate entities). This is the Final Final Thought of the course! Lecture #23-5/3/2005 Slide 20 of 21

21 Next Time As promised...cupcakes! Final exam discussion. ICES Forms. Final Thought Next Class Lecture #23-5/3/2005 Slide 21 of 21

Comparing IRT with Other Models

Comparing IRT with Other Models Lecture #14 ICPSR Item Response Theory Workshop Lecture #14: 1of 45 Lecture Overview The final set of slides will describe a parallel between IRT and another commonly used