Goodness of fit assessment of item response theory models

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Goodness of fit assessment of item response theory models"

Transcription

1 Goodness of fit assessment of item response theory models Alberto Maydeu Olivares University of Barcelona Madrid November 1, 014

2 Outline Introduction Overall goodness of fit testing Two examples Assessing goodness of approximation Fine tuning the model: identifying misfitting items Final remarks

3 3 Introduction Statistical modeling involves finding a model that may have generated the data What is (absolute) goodness of fit testing? o Testing whether the fitted model may have generated the observed data Why is this important? o Because inferences drawn on poorly fitting models are misleading How misleading? o It depends on a number of factors, among them the discrepancy between the true and fitted model

4 4 Why is fit important? 3 df 3 df 3 df 1 df 1 df 1 df 0 df

5 5 Outline Introduction Overall goodness of fit testing

6 6 How to assess goodness of fit in IRT models IRT are models for multivariate categorical data o The goodness of fit of IRT models is assessed using goodness of fit statistics for categorical data, such as Pearson s X statistic o There is nothing really special about testing IRT models

7 7 Pearson s X Notation: o N = number of respondents o n = number of items o K = number of response alternatives (number of categories) o C = K n = number of possible response patterns (c = 1,, C) o p c = observed proportion of a response pattern o c = probability of a response pattern o = vector of item parameters (not latent traits!) o q = number of item parameters where ˆ ˆ, ˆ o For the MLE, X ( p ˆ ) ˆ 1 X N N ˆ c c p ˆ D p ˆ, (1) c c p are the cell residuals, and D ˆ diag ˆ d Cq1

8 8 Two small examples A 1PL (a Rasch model where the latent trait is treated as a random effect) applied to a 15 item test measuring mathematical proficiency in Chilean adults o n = 15; N = 3,000; K = ; q = 16; C = 15 = 3,768; df = 3,751 Samejima s graded model applied to the 7 items of the short version of the PROMIS depression scale o n = 7; N = 767; K = 4; q = 35; C = 4 7 = 16,384; df = 16,348 Problem: o (asymptotic) p values for X are only accurate when the expected frequencies of every pattern are large ( Nc 5 is the usual rule of thumb) o In large models (large C) that s impossible to accomplish o p values for X are only accurate in models with a small number of possible response patterns (300?) In IRT we are interested in much larger models than C = 300, how do we go about testing them?

9 9 Limited information goodness of fit testing There have been a number of proposals, most of them involve limited information test statistics Key references o Reiser (1996, Psychometrika) o Cai, Maydeu Olivares, Coffman & Thissen (British J. of Math and Stat Psych, 006) o Maydeu Olivares & Joe (005, J. American Stat Assoc; 006, Psychometrika)

10 10 What is limited information goodness of fit testing? Multivariate categorical data admits (at least) two representations cells moments Y 1 Y 0 1 Y Y 0 1 (1)(1) 1 (1) (1)() 1 () (1) 1 There is a one to one relationship between them All formulae in categorical data analysis can be written using cells or moments 1 ˆ ˆ ˆ ˆ p p p D pˆ ˆ ˆ 1 p p X N N, () Pearson s X is a quadratic form in all moments up to order n

11 11 What is limited information goodness of fit testing? 1 ˆ ˆ ˆ ˆ p p p D pˆ ˆ ˆ 1 p p X N N A limited information test statistic is ˆ 1 p p ˆ , (3) L N (4) For chi square distributed statistics degrees of freedom = number of statistics number of item parameters Degrees of freedom for Samejima s model are positive when univariate and bivariate moments are used (r = ) o Limited information testing requires using at least univariate and bivariate moments

12 1 Moments or margins? cells moments Y 1 Y 0 1 Y Y 0 1 (1)(1) 1 (1) (1)() 1 () (1) 1 Univariate moments = univariate probabilities that do not involve category 0 A researcher interested in univariate testing can use (1) (1) () o All univariate moments: 1,, o All univariate margins:,,,, (0) 1 (1) 1 o The relationship is one to one It is convenient to use moments because they are mathematically independent o Fewer o Their covariance matrix is of full rank Moments up to order r is equivalent to use all r way margins Bivariate information = univariate and bivariate moments (0) (1) ()

13 13 Pros and cons of limited information testing Suppose bivariate information is used PROS: 1) Accurate (asymptotic) p values can be obtained with N = 100 o Asymptotic approximation to the limited information statistic requires that 4 way frequencies are large For r way test, r way frequencies must be large ) If misfit is located in the bivariate margins, higher power is obtained CON: If misfit is not located in the bivariate margins, no power

14 14 Overall limited information test statistics Consider the quadratic form in univariate and bivariate moments Q N p ˆ Wˆ p ˆ. (5) Q is asymptotically distributed as a mixture of independent chi square variates. When Ŵis chosen so that ΣWΣWΣ ΣWΣ (6) where N is the asymptotic covariance matrix of p ˆ, Q is asymptotically chi square Two ways to satisfy (6) o Choose Ŵ such that is a generalized inverse of W: M of M O and Joe (005,006) M N ˆ ˆ p C p ˆ, X -X D ( D X D ) DX C, (7) o Choose Ŵ such that W is a generalized inverse of : R of Reiser (1996) R N p ˆ Σˆ p ˆ, (8)

15 15 Overall limited information test statistics Third alternative: 1 Use an easily computable weight matrix ˆ ˆ 1 W ˆ,diag, p values by adjusting Q by its asymptotic mean and variance 1 o With ˆ diagˆ I and compute W and binary data, this is Cai et al. (006) and Bartholomew and Leung (00) o Within the context of SEM, these are the Satorra Bentler corrections Degrees of freedom: o M : number of statistics number of parameters o R : estimated rank of ˆΣ (integer) o MV: df tr tr WΣ ˆˆ WΣ ˆˆ (a real number) MV corrected statistics can be transformed so that they have the same number of degrees of freedom as M

16 16 The covariance matrix of the bivariate residuals (moments) Acov( ˆ ) (9) Notation: o = (asymptotic) covariance matrix of univariate and bivariate proportions (that exclude category 0) o = derivatives of univariate and bivariate moments with respect to item parameters o Acov( ˆ) = (asymptotic) covariance matrix of item parameter estimates 1 For multinomial ML (marginal ML), Acov( ˆ ) ˆ

17 17 Estimating the covariance matrix of the item parameter estimates Expected information, E o First order derivatives for all possible patterns o It can only be computed for small models Cross products information, XP o First order derivatives for observed patterns Observed information, O o First and second order derivatives for observed patterns 1 1 Sandwich (aka robust) information, S O XP O Crossing the different estimators of the covariance matrix of the item parameter estimates with the different choices of bivariate statistics, there are many possible choices of test statistics!

18 18 Outline Introduction Overall goodness of fit testing Two examples

19 19 Two numerical examples A 1PL applied to Chilean mathematical proficiency data o n = 15; N = 3,000; K = ; q = 16; C = 15 = 3,768; df = 3,751 o Samejima s graded model applied to the short version of the PROMIS anxiety scale o n = 7; N = 767; K = 4; q = 35; C = 4 7 = 16,384; df = 16,348 PROMIS anxiety Chilean mathematical proficiency stat value df RMSEA value df p RMSEA M R Y L Y L

20 0 Type I errors, N = 300 K =, n = 10 K = 5, n = 10 stat =.01 =.05 =.10 =.01 =.05 =.10 M Y Y O XP L L O XP R O XP L N p p, ˆ diag ˆ 1 ˆ ˆ ˆ 1 ˆ Y N p p, (10)

21 Computational issues For large unidimensional models, computing the goodness of fit statistic can take longer than the estimation of the item parameters and their SEs The number of univariate and bivariate moments is s = nn ( 1) nk ( 1) ( K 1) o For n = 0 and K = 5, s = 3,10; o For n = 30 and K = 7, s = 15,840 (probably unfeasible) When data is nominal, there is no solution When data is ordinal one can use instead of a quadratic form in residual univariate and bivariate moments a quadratic form in residual (multinomial) means and cross products The number of means and cross products is s = o One can test much larger models nn ( 1) 1

22 Means and cross products for multinomial variables 0 Pr 0 ( 1) Pr 1 EY Y K Y K, (11) i i i i i i ij E YY i j 00Pr Yi 0, Yj 0 ( K 1) ( K 1) Pr Y K 1, Y K 1 i j i i j j, (1) with sample counterparts k i y i (the sample mean), and kij yy i j / (the sample cross product), respectively, where y i denotes the observed data for item i. N For our previous 3 example, the elements of are 1PrY 1 (1) 1Pr 1 Pr 1 (1) () EY EY Y Y EYY 11Pr Y1, Y 1 1Pr Y1, Y. (1)(1) (1)() (13)

23 3 Testing large models for ordinal data, M ord Using these statistics one can construct a goodness of fit statistic for ordinal data M N k C k, ord ˆ ˆ ˆ ord X -X D ( DX D ) D X C, (14) ord ord ord ord r ord ord ord ord It is just like M except that M ord uses as summaries statistics that further concentrate the information o If the misfit is in the direction where the information has been concentrated, M ord will be more powerful than M It is good practice to check the rank of D or D ord in applications. If the rank of these matrices is unstable, the estimated statistic is not reliable o The model is empirically underidentified from the information used for testing M ord cannot be computed when K is large and n is small due to lack of df, use M M ord = M for binary data

24 4 Outline Introduction Overall goodness of fit testing Two examples Assessing goodness of approximation

25 5 Assessing the goodness of approximation Finding a well fitting model is more difficult as the number of variables increase o If I generate a 1 variable dataset and I ask you what statistical model I used to generate the data, it will be easier for you than if I give you a 10 variable dataset In our experience, finding a well fitting model is more difficult as the number of categories increase o Are our models for polytomous data worse than our models for binary data? o Are personality/attitudinal/patient reported data more difficult to model than educational data?

26 6 Assessing the goodness of approximation I would argue that finding a well fitting model in IRT is harder than in FA o In IRT we model all moments (univariate, bivariate, trivariate,, n variate) o In FA we only model univariate and bivariate moments If we reject the fitted model, how do we judge how close we are to the true and unknown model? o I.e., how do we judge the goodness of approximation of our model?

27 7 Bivariate RMSEA and full information RMSEA The Root Mean Squared Error of Approximation is the standard procedure for assessing goodness of approximation in SEM A RMSEA can be constructed based on Pearson s X (RMSEA n ), M (RMSEA ), or any other statistic with known asymptotic distribution For members of the RMSEA r family r T 0 0 T 0 r r Crr r df r r M r ˆ r (15) df r RMSEA will generally be larger than RMSEA n because M is generally more powerful than X and statistics with higher power have a higher M / df ratio. r r

28 8 Choice of cutoff values for RMSEA and RMSEA n in binary IRT models A cut off of RMSEA 0.05 separates unidimensional from twodimensional models A cut off RMSEA n 0.03 separates unidimensional from twodimensional models

29 9 RMSEA n, RMSEA, and RMSEA ord RMSEA n is based on X There is a limit in the size of models for which X can be computed o Its sampling distribution will only be well approximated in small models (just as the sampling distribution of X ) o It is not possible to offer an overall cut off as population values of RMSEA n depend on the number of variables and the number of categories

30 30 RMSEA n, RMSEA, and RMSEA ord RMSEA is based on M o It can be computed for larger models o Its sampling distribution can be well approximated in large models o One can obtain confidence intervals and a test of close fit o It is possible to offer an overall cut off RMSEA ord is based on M ord o It can be computed for even larger models o Its sampling distribution can be well approximated in large models o One can obtain confidence intervals and a test of close fit BUT o It is not possible to offer an overall cut off as population values of RMSEA ord depend on the number of variables and the number of categories

31 31 Effect of the number of variables and categories on RMSEA and RMSEA ord RMSEA adjusted by the number of categories is stable when misfit is small RMSEA ord decreases as n and K increase. o It is easier to obtain a low RMSEA ord for large n and K

32 3 Goodness of approximation when the model is large If the model is so large that RMSEA cannot be computed and we should not use RMSEA ord because we cannot offer an overall cutoff, what do we do? The RMSEAs are ill designed to assess pure goodness of approximation o They cannot be interpreted substantively They are a weighted sum of residuals divided by df The RMSEAs by construction mix goodness of approximation and model selection How can we measure pure goodness of approximation? We want to measure the magnitude of the misfit (effect size) o In substantively interpretable units o Such that we can offer an overall cut off valid for any n and K

33 The Standardized Root Mean Squared Residual (SRMSR) A more appropriate name is the Squared Root Mean Residual Correlation o It is only valid for binary or ordinal data (just as the RMSEA ord ) r ˆ ij ij SRMSR (16) nn ( 1)/ i j r ij is just the product moment (Pearson) correlation and ˆ ˆ ˆ ij i i ˆ ij. (17) ˆ ˆ ˆ ˆ ii i jj j where the means ( i and j ) and the cross product ij were given in (11) and (1), and ii is ii E Yi 0 Pr Yi 0 ( Ki 1) Pr Yi Ki 1. (18) 33

34 Relationship between RMSEA and SRMR 34

35 Relationship between RMSEA adjusted by the number of categories and SRMR 35

36 36 Proposal of cut off values for assessing close fit in IRT criterion RMSEA SRMR adequate fit close fit excellent fit 0.05 / (K 1) 0.07 / (K 1) Criteria for adequate fit and close fit are essentially identical to those proposed by Browne and Cudeck (1993) in the context of SEM Criteria for excellent and close fit are equal for binary data It is possible (but cumbersome) to compute CIs and tests for the SRMR o It is best used as a goodness of fit index When used as a goodness of fit index, the SRMR can be computed for models of any size o It only requires computing means and cross products under the IRT model

37 37 Goodness of fit statistics are only summary measures It is possible that the model fits well overall but that it fits very poorly some items o It is good practice to inspect item and item pairs statistics and to report the largest observed statistics If the model misfits we wish to locate the source of the misfit

38 38 Outline Introduction Overall goodness of fit testing Two examples Assessing goodness of approximation Fine tuning the model: identifying misfitting items

39 39 Identifying misfitting items in IRT One could compute Pearson s statistic for every item and pair of items. For a pair of items we can write where X N p D p (19) ˆ ˆ ˆ 1 ij ij ij ij ij ij p, ˆ are vectors of dimension K. ij ij o Unfortunately X does not follow an asymptotic chi square distribution when applied to a subtable As in the case of the overall goodness of fit statistics we can use M N ˆ ˆ ˆ R ˆ ˆ ˆ ij N ij ij ij ij ij p C p, 1 ij ij ij ij ij ij or a mean and variance corrected X ij Cˆ Dˆ Dˆ Δˆ Δˆ Dˆ Δˆ Δˆ D ˆ (0) ij ij ij ij ij ij ij ij ij p Σ p. (1)

40 40 Identifying misfitting items in IRT Alternatively, we could use z statistics for the residual cross products z ord k ˆ ˆ ij ij kij ij SE ˆ / N k ˆ ij ij ord. () For all statistics (but M ij ) the (asymptotic) covariance matrix of the residuals for the pair of items is needed. This is, for multinomial ML, and for the z statistics we need where v is the 1 K vector Σ D ππ Δ Δ. (3) 1 ij ij ij ij ij ( ij) ij ˆ ord ij ˆ v Σ v, (4) v 00,01, 0 ( K 1),,( K 1) 0,( K 1) 1,,( K 1) ( K 1). (5)

41 Identifying misfitting items in IRT: Type I errors for a PL 41

42 Identifying misfitting items: Power for Samejima s model 4

43 43 Assessing the source of misfit in the PROMIS anxiety data Item Average A 90% CI for RMSEA is (0.09; 0.040); SRMSR = o The model provides a close fit to the data z ord statistics are displayed above the diagonal, residual correlations below the diagonal. z ord statistics significant at the 5% level with a Bonferroni adjustment have been boldfaced.

44 44 Outline Introduction Overall goodness of fit testing Two examples Assessing goodness of approximation Fine tuning the model: identifying misfitting items Final remarks

45 45 Recommendations For overall goodness of fit use M o Mean and variance diagonally weighted statistic may be more powerful to detect the presence of mixtures and guessing It requires the estimation of the covariance matrix of the item parameter estimates using the observed information matrix For assessing the magnitude of the misfit use RMSR (only with binary or ordinal data) For assessing goodness of approximation (taking into account model parsimony) use the RMSEA o Cut off values for the RMSR and RMSEA are available

46 46 Recommendations Report largest statistics for item pairs o Residual correlations and z statistics for binary and ordinal data o Mean and variance adjusted X statistics for polytomous nominal data They require the estimation of the covariance matrix of the item parameter estimates

47 47 Concluding remarks Model data assessment in IRT can now be performed as well (if not better) than in SEM Most often, our models will be misspecified to some extent o Research is needed to investigate the practical implications of different levels of misspecification The theory presented here is completely general o Applicable to any model for multivariate discrete data (e.g., cognitive diagnostic models) Better IRT models (or other measurement models) are needed o Particularly for polytomous data

Goodness of fit assessment of item response theory models. Alberto Maydeu-Olivares. University of Barcelona. Author note

Goodness of fit assessment of item response theory models. Alberto Maydeu-Olivares. University of Barcelona. Author note GOF OF IRT MODELS 1 Goodness of fit assessment of item response theory models Alberto Maydeu-Olivares University of Barcelona Author note This research was supported by an ICREA-Academia Award and grant

More information

Indices of Model Fit STRUCTURAL EQUATION MODELING 2013

Indices of Model Fit STRUCTURAL EQUATION MODELING 2013 Indices of Model Fit STRUCTURAL EQUATION MODELING 2013 Indices of Model Fit A recommended minimal set of fit indices that should be reported and interpreted when reporting the results of SEM analyses:

More information

A model-based approach to goodness-of-fit evaluation in item response theory

A model-based approach to goodness-of-fit evaluation in item response theory A MODEL-BASED APPROACH TO GOF EVALUATION IN IRT 1 A model-based approach to goodness-of-fit evaluation in item response theory DL Oberski JK Vermunt Dept of Methodology and Statistics, Tilburg University,

More information

Comparing the Fit of Item Response Theory and Factor Analysis Models

Comparing the Fit of Item Response Theory and Factor Analysis Models Structural Equation Modeling, 8:333 356, 20 Copyright Taylor & Francis Group, LLC ISSN: 070-55 print/532-8007 online OI: 0.080/07055.20.58993 Comparing the Fit of Item Response Theory and Factor Analysis

More information

A Cautionary Note on Using G 2 (dif) to Assess Relative Model Fit in Categorical Data Analysis

A Cautionary Note on Using G 2 (dif) to Assess Relative Model Fit in Categorical Data Analysis MULTIVARIATE BEHAVIORAL RESEARCH, 4(), 55 64 Copyright 2006, Lawrence Erlbaum Associates, Inc. A Cautionary Note on Using G 2 (dif) to Assess Relative Model Fit in Categorical Data Analysis Albert Maydeu-Olivares

More information

Lecture #2 Overview. Basic IRT Concepts, Models, and Assumptions. Lecture #2 ICPSR Item Response Theory Workshop

Lecture #2 Overview. Basic IRT Concepts, Models, and Assumptions. Lecture #2 ICPSR Item Response Theory Workshop Basic IRT Concepts, Models, and Assumptions Lecture #2 ICPSR Item Response Theory Workshop Lecture #2: 1of 64 Lecture #2 Overview Background of IRT and how it differs from CFA Creating a scale An introduction

More information

Simple Linear Regression Inference

Simple Linear Regression Inference Simple Linear Regression Inference 1 Inference requirements The Normality assumption of the stochastic term e is needed for inference even if it is not a OLS requirement. Therefore we have: Interpretation

More information

REINVENTING GUTTMAN SCALING AS A STATISTICAL MODEL: ABSOLUTE SIMPLEX THEORY

REINVENTING GUTTMAN SCALING AS A STATISTICAL MODEL: ABSOLUTE SIMPLEX THEORY REINVENTING GUTTMAN SCALING AS A STATISTICAL MODEL: ABSOLUTE SIMPLEX THEORY Peter M. Bentler* University of California, Los Angeles *With thanks to Prof. Ke-Hai Yuan, University of Notre Dame 1 Topics

More information

Notes for STA 437/1005 Methods for Multivariate Data

Notes for STA 437/1005 Methods for Multivariate Data Notes for STA 437/1005 Methods for Multivariate Data Radford M. Neal, 26 November 2010 Random Vectors Notation: Let X be a random vector with p elements, so that X = [X 1,..., X p ], where denotes transpose.

More information

Goodness-of-Fit Testing

Goodness-of-Fit Testing Goodness-of-Fit Testing A Maydeu-Olivares and C García-Forero, University of Barcelona, Barcelona, Spain ã 2010 Elsevier Ltd. All rights reserved. Glossary Absolute goodness of fit The discrepancy between

More information

Least Squares Estimation

Least Squares Estimation Least Squares Estimation SARA A VAN DE GEER Volume 2, pp 1041 1045 in Encyclopedia of Statistics in Behavioral Science ISBN-13: 978-0-470-86080-9 ISBN-10: 0-470-86080-4 Editors Brian S Everitt & David

More information

Multivariate Normal Distribution

Multivariate Normal Distribution Multivariate Normal Distribution Lecture 4 July 21, 2011 Advanced Multivariate Statistical Methods ICPSR Summer Session #2 Lecture #4-7/21/2011 Slide 1 of 41 Last Time Matrices and vectors Eigenvalues

More information

DEPARTMENT OF ECONOMICS. Unit ECON 12122 Introduction to Econometrics. Notes 4 2. R and F tests

DEPARTMENT OF ECONOMICS. Unit ECON 12122 Introduction to Econometrics. Notes 4 2. R and F tests DEPARTMENT OF ECONOMICS Unit ECON 11 Introduction to Econometrics Notes 4 R and F tests These notes provide a summary of the lectures. They are not a complete account of the unit material. You should also

More information

Simple Second Order Chi-Square Correction

Simple Second Order Chi-Square Correction Simple Second Order Chi-Square Correction Tihomir Asparouhov and Bengt Muthén May 3, 2010 1 1 Introduction In this note we describe the second order correction for the chi-square statistic implemented

More information

Chi Square for Contingency Tables

Chi Square for Contingency Tables 2 x 2 Case Chi Square for Contingency Tables A test for p 1 = p 2 We have learned a confidence interval for p 1 p 2, the difference in the population proportions. We want a hypothesis testing procedure

More information

Lecture - 32 Regression Modelling Using SPSS

Lecture - 32 Regression Modelling Using SPSS Applied Multivariate Statistical Modelling Prof. J. Maiti Department of Industrial Engineering and Management Indian Institute of Technology, Kharagpur Lecture - 32 Regression Modelling Using SPSS (Refer

More information

Chi-square test Testing for independeny The r x c contingency tables square test

Chi-square test Testing for independeny The r x c contingency tables square test Chi-square test Testing for independeny The r x c contingency tables square test 1 The chi-square distribution HUSRB/0901/1/088 Teaching Mathematics and Statistics in Sciences: Modeling and Computer-aided

More information

Weighted Least Squares

Weighted Least Squares Weighted Least Squares The standard linear model assumes that Var(ε i ) = σ 2 for i = 1,..., n. As we have seen, however, there are instances where Var(Y X = x i ) = Var(ε i ) = σ2 w i. Here w 1,..., w

More information

Psychology 405: Psychometric Theory Homework on Factor analysis and structural equation modeling

Psychology 405: Psychometric Theory Homework on Factor analysis and structural equation modeling Psychology 405: Psychometric Theory Homework on Factor analysis and structural equation modeling William Revelle Department of Psychology Northwestern University Evanston, Illinois USA June, 2014 1 / 20

More information

Variance of OLS Estimators and Hypothesis Testing. Randomness in the model. GM assumptions. Notes. Notes. Notes. Charlie Gibbons ARE 212.

Variance of OLS Estimators and Hypothesis Testing. Randomness in the model. GM assumptions. Notes. Notes. Notes. Charlie Gibbons ARE 212. Variance of OLS Estimators and Hypothesis Testing Charlie Gibbons ARE 212 Spring 2011 Randomness in the model Considering the model what is random? Y = X β + ɛ, β is a parameter and not random, X may be

More information

MATHEMATICAL METHODS OF STATISTICS

MATHEMATICAL METHODS OF STATISTICS MATHEMATICAL METHODS OF STATISTICS By HARALD CRAMER TROFESSOK IN THE UNIVERSITY OF STOCKHOLM Princeton PRINCETON UNIVERSITY PRESS 1946 TABLE OF CONTENTS. First Part. MATHEMATICAL INTRODUCTION. CHAPTERS

More information

Inferential Statistics

Inferential Statistics Inferential Statistics Sampling and the normal distribution Z-scores Confidence levels and intervals Hypothesis testing Commonly used statistical methods Inferential Statistics Descriptive statistics are

More information

Applications of Structural Equation Modeling in Social Sciences Research

Applications of Structural Equation Modeling in Social Sciences Research American International Journal of Contemporary Research Vol. 4 No. 1; January 2014 Applications of Structural Equation Modeling in Social Sciences Research Jackson de Carvalho, PhD Assistant Professor

More information

The Latent Variable Growth Model In Practice. Individual Development Over Time

The Latent Variable Growth Model In Practice. Individual Development Over Time The Latent Variable Growth Model In Practice 37 Individual Development Over Time y i = 1 i = 2 i = 3 t = 1 t = 2 t = 3 t = 4 ε 1 ε 2 ε 3 ε 4 y 1 y 2 y 3 y 4 x η 0 η 1 (1) y ti = η 0i + η 1i x t + ε ti

More information

ANSWERS TO EXERCISES AND REVIEW QUESTIONS

ANSWERS TO EXERCISES AND REVIEW QUESTIONS ANSWERS TO EXERCISES AND REVIEW QUESTIONS PART FIVE: STATISTICAL TECHNIQUES TO COMPARE GROUPS Before attempting these questions read through the introduction to Part Five and Chapters 16-21 of the SPSS

More information

It is important to bear in mind that one of the first three subscripts is redundant since k = i -j +3.

It is important to bear in mind that one of the first three subscripts is redundant since k = i -j +3. IDENTIFICATION AND ESTIMATION OF AGE, PERIOD AND COHORT EFFECTS IN THE ANALYSIS OF DISCRETE ARCHIVAL DATA Stephen E. Fienberg, University of Minnesota William M. Mason, University of Michigan 1. INTRODUCTION

More information

Descriptive Statistics

Descriptive Statistics Descriptive Statistics Primer Descriptive statistics Central tendency Variation Relative position Relationships Calculating descriptive statistics Descriptive Statistics Purpose to describe or summarize

More information

Evaluating the Fit of Structural Equation Models: Tests of Significance and Descriptive Goodness-of-Fit Measures

Evaluating the Fit of Structural Equation Models: Tests of Significance and Descriptive Goodness-of-Fit Measures Methods of Psychological Research Online 003, Vol.8, No., pp. 3-74 Department of Psychology Internet: http://www.mpr-online.de 003 University of Koblenz-Landau Evaluating the Fit of Structural Equation

More information

Chapter 5 Analysis of variance SPSS Analysis of variance

Chapter 5 Analysis of variance SPSS Analysis of variance Chapter 5 Analysis of variance SPSS Analysis of variance Data file used: gss.sav How to get there: Analyze Compare Means One-way ANOVA To test the null hypothesis that several population means are equal,

More information

Introduction to Multivariate Models: Modeling Multivariate Outcomes with Mixed Model Repeated Measures Analyses

Introduction to Multivariate Models: Modeling Multivariate Outcomes with Mixed Model Repeated Measures Analyses Introduction to Multivariate Models: Modeling Multivariate Outcomes with Mixed Model Repeated Measures Analyses Applied Multilevel Models for Cross Sectional Data Lecture 11 ICPSR Summer Workshop University

More information

Sample Size in Factor Analysis: The Role of Model Error

Sample Size in Factor Analysis: The Role of Model Error Multivariate Behavioral Research, 36 (4), 611-637 Copyright 2001, Lawrence Erlbaum Associates, Inc. Sample Size in Factor Analysis: The Role of Model Error Robert C. MacCallum Ohio State University Keith

More information

Writing about SEM/Best Practices

Writing about SEM/Best Practices Writing about SEM/Best Practices Presenting results Best Practices» specification» data» analysis» interpretation 1 Writing about SEM Summary of recommendation from Hoyle & Panter, McDonald & Ho, Kline»

More information

Overview Classes. 12-3 Logistic regression (5) 19-3 Building and applying logistic regression (6) 26-3 Generalizations of logistic regression (7)

Overview Classes. 12-3 Logistic regression (5) 19-3 Building and applying logistic regression (6) 26-3 Generalizations of logistic regression (7) Overview Classes 12-3 Logistic regression (5) 19-3 Building and applying logistic regression (6) 26-3 Generalizations of logistic regression (7) 2-4 Loglinear models (8) 5-4 15-17 hrs; 5B02 Building and

More information

Logit Models for Binary Data

Logit Models for Binary Data Chapter 3 Logit Models for Binary Data We now turn our attention to regression models for dichotomous data, including logistic regression and probit analysis. These models are appropriate when the response

More information

11/20/2014. Correlational research is used to describe the relationship between two or more naturally occurring variables.

11/20/2014. Correlational research is used to describe the relationship between two or more naturally occurring variables. Correlational research is used to describe the relationship between two or more naturally occurring variables. Is age related to political conservativism? Are highly extraverted people less afraid of rejection

More information

Ordinal Regression. Chapter

Ordinal Regression. Chapter Ordinal Regression Chapter 4 Many variables of interest are ordinal. That is, you can rank the values, but the real distance between categories is unknown. Diseases are graded on scales from least severe

More information

Multivariate Analysis of Variance (MANOVA): I. Theory

Multivariate Analysis of Variance (MANOVA): I. Theory Gregory Carey, 1998 MANOVA: I - 1 Multivariate Analysis of Variance (MANOVA): I. Theory Introduction The purpose of a t test is to assess the likelihood that the means for two groups are sampled from the

More information

Basic Statistcs Formula Sheet

Basic Statistcs Formula Sheet Basic Statistcs Formula Sheet Steven W. ydick May 5, 0 This document is only intended to review basic concepts/formulas from an introduction to statistics course. Only mean-based procedures are reviewed,

More information

SAS Software to Fit the Generalized Linear Model

SAS Software to Fit the Generalized Linear Model SAS Software to Fit the Generalized Linear Model Gordon Johnston, SAS Institute Inc., Cary, NC Abstract In recent years, the class of generalized linear models has gained popularity as a statistical modeling

More information

Example: Credit card default, we may be more interested in predicting the probabilty of a default than classifying individuals as default or not.

Example: Credit card default, we may be more interested in predicting the probabilty of a default than classifying individuals as default or not. Statistical Learning: Chapter 4 Classification 4.1 Introduction Supervised learning with a categorical (Qualitative) response Notation: - Feature vector X, - qualitative response Y, taking values in C

More information

STATISTICA Formula Guide: Logistic Regression. Table of Contents

STATISTICA Formula Guide: Logistic Regression. Table of Contents : Table of Contents... 1 Overview of Model... 1 Dispersion... 2 Parameterization... 3 Sigma-Restricted Model... 3 Overparameterized Model... 4 Reference Coding... 4 Model Summary (Summary Tab)... 5 Summary

More information

Statistics Graduate Courses

Statistics Graduate Courses Statistics Graduate Courses STAT 7002--Topics in Statistics-Biological/Physical/Mathematics (cr.arr.).organized study of selected topics. Subjects and earnable credit may vary from semester to semester.

More information

Why Taking This Course? Course Introduction, Descriptive Statistics and Data Visualization. Learning Goals. GENOME 560, Spring 2012

Why Taking This Course? Course Introduction, Descriptive Statistics and Data Visualization. Learning Goals. GENOME 560, Spring 2012 Why Taking This Course? Course Introduction, Descriptive Statistics and Data Visualization GENOME 560, Spring 2012 Data are interesting because they help us understand the world Genomics: Massive Amounts

More information

SPSS and AMOS. Miss Brenda Lee 2:00p.m. 6:00p.m. 24 th July, 2015 The Open University of Hong Kong

SPSS and AMOS. Miss Brenda Lee 2:00p.m. 6:00p.m. 24 th July, 2015 The Open University of Hong Kong Seminar on Quantitative Data Analysis: SPSS and AMOS Miss Brenda Lee 2:00p.m. 6:00p.m. 24 th July, 2015 The Open University of Hong Kong SBAS (Hong Kong) Ltd. All Rights Reserved. 1 Agenda MANOVA, Repeated

More information

A Introduction to Matrix Algebra and Principal Components Analysis

A Introduction to Matrix Algebra and Principal Components Analysis A Introduction to Matrix Algebra and Principal Components Analysis Multivariate Methods in Education ERSH 8350 Lecture #2 August 24, 2011 ERSH 8350: Lecture 2 Today s Class An introduction to matrix algebra

More information

Mixture Models for Understanding Dependencies of Insurance and Credit Risks

Mixture Models for Understanding Dependencies of Insurance and Credit Risks Mixture Models for Understanding Dependencies of Insurance and Credit Risks Emiliano A. Valdez, PhD, FSA, FIAA University of New South Wales Brighton-Le-Sands, Sydney 11-12 May 2006 Emiliano A. Valdez

More information

The effect of varying the number of response alternatives in rating scales: Experimental evidence from intra-individual effects

The effect of varying the number of response alternatives in rating scales: Experimental evidence from intra-individual effects Behavior Research Methods 009, 4 (), 95-308 doi:0.3758/brm.4..95 The effect of varying the number of response alternatives in rating scales: Experimental evidence from intra-individual effects Alberto

More information

Algebra 1 Course Information

Algebra 1 Course Information Course Information Course Description: Students will study patterns, relations, and functions, and focus on the use of mathematical models to understand and analyze quantitative relationships. Through

More information

Statistical Foundations: Measures of Location and Central Tendency and Summation and Expectation

Statistical Foundations: Measures of Location and Central Tendency and Summation and Expectation Statistical Foundations: and Central Tendency and and Lecture 4 September 5, 2006 Psychology 790 Lecture #4-9/05/2006 Slide 1 of 26 Today s Lecture Today s Lecture Where this Fits central tendency/location

More information

Chapter 11: Two Variable Regression Analysis

Chapter 11: Two Variable Regression Analysis Department of Mathematics Izmir University of Economics Week 14-15 2014-2015 In this chapter, we will focus on linear models and extend our analysis to relationships between variables, the definitions

More information

Probability and Statistics

Probability and Statistics CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 0 Probability and Statistics Kristel Van Steen, PhD 2 Montefiore Institute - Systems and Modeling GIGA - Bioinformatics ULg kristel.vansteen@ulg.ac.be

More information

Recall this chart that showed how most of our course would be organized:

Recall this chart that showed how most of our course would be organized: Chapter 4 One-Way ANOVA Recall this chart that showed how most of our course would be organized: Explanatory Variable(s) Response Variable Methods Categorical Categorical Contingency Tables Categorical

More information

Association Between Variables

Association Between Variables Contents 11 Association Between Variables 767 11.1 Introduction............................ 767 11.1.1 Measure of Association................. 768 11.1.2 Chapter Summary.................... 769 11.2 Chi

More information

Missing Data: Part 1 What to Do? Carol B. Thompson Johns Hopkins Biostatistics Center SON Brown Bag 3/20/13

Missing Data: Part 1 What to Do? Carol B. Thompson Johns Hopkins Biostatistics Center SON Brown Bag 3/20/13 Missing Data: Part 1 What to Do? Carol B. Thompson Johns Hopkins Biostatistics Center SON Brown Bag 3/20/13 Overview Missingness and impact on statistical analysis Missing data assumptions/mechanisms Conventional

More information

Department of Economics

Department of Economics Department of Economics On Testing for Diagonality of Large Dimensional Covariance Matrices George Kapetanios Working Paper No. 526 October 2004 ISSN 1473-0278 On Testing for Diagonality of Large Dimensional

More information

Sample Size Determination

Sample Size Determination Sample Size Determination Population A: 10,000 Population B: 5,000 Sample 10% Sample 15% Sample size 1000 Sample size 750 The process of obtaining information from a subset (sample) of a larger group (population)

More information

Package EstCRM. July 13, 2015

Package EstCRM. July 13, 2015 Version 1.4 Date 2015-7-11 Package EstCRM July 13, 2015 Title Calibrating Parameters for the Samejima's Continuous IRT Model Author Cengiz Zopluoglu Maintainer Cengiz Zopluoglu

More information

Confidence Intervals for the Difference Between Two Means

Confidence Intervals for the Difference Between Two Means Chapter 47 Confidence Intervals for the Difference Between Two Means Introduction This procedure calculates the sample size necessary to achieve a specified distance from the difference in sample means

More information

Analysis of Microdata

Analysis of Microdata Rainer Winkelmann Stefan Boes Analysis of Microdata With 38 Figures and 41 Tables 4y Springer Contents 1 Introduction 1 1.1 What Are Microdata? 1 1.2 Types of Microdata 4 1.2.1 Qualitative Data 4 1.2.2

More information

Introduction to Matrix Algebra

Introduction to Matrix Algebra Psychology 7291: Multivariate Statistics (Carey) 8/27/98 Matrix Algebra - 1 Introduction to Matrix Algebra Definitions: A matrix is a collection of numbers ordered by rows and columns. It is customary

More information

Multinomial and Ordinal Logistic Regression

Multinomial and Ordinal Logistic Regression Multinomial and Ordinal Logistic Regression ME104: Linear Regression Analysis Kenneth Benoit August 22, 2012 Regression with categorical dependent variables When the dependent variable is categorical,

More information

UNDERSTANDING THE TWO-WAY ANOVA

UNDERSTANDING THE TWO-WAY ANOVA UNDERSTANDING THE e have seen how the one-way ANOVA can be used to compare two or more sample means in studies involving a single independent variable. This can be extended to two independent variables

More information

MISSING DATA TECHNIQUES WITH SAS. IDRE Statistical Consulting Group

MISSING DATA TECHNIQUES WITH SAS. IDRE Statistical Consulting Group MISSING DATA TECHNIQUES WITH SAS IDRE Statistical Consulting Group ROAD MAP FOR TODAY To discuss: 1. Commonly used techniques for handling missing data, focusing on multiple imputation 2. Issues that could

More information

A Primer on Mathematical Statistics and Univariate Distributions; The Normal Distribution; The GLM with the Normal Distribution

A Primer on Mathematical Statistics and Univariate Distributions; The Normal Distribution; The GLM with the Normal Distribution A Primer on Mathematical Statistics and Univariate Distributions; The Normal Distribution; The GLM with the Normal Distribution PSYC 943 (930): Fundamentals of Multivariate Modeling Lecture 4: September

More information

Handling missing data in Stata a whirlwind tour

Handling missing data in Stata a whirlwind tour Handling missing data in Stata a whirlwind tour 2012 Italian Stata Users Group Meeting Jonathan Bartlett www.missingdata.org.uk 20th September 2012 1/55 Outline The problem of missing data and a principled

More information

Lecture 7: Binomial Test, Chisquare

Lecture 7: Binomial Test, Chisquare Lecture 7: Binomial Test, Chisquare Test, and ANOVA May, 01 GENOME 560, Spring 01 Goals ANOVA Binomial test Chi square test Fisher s exact test Su In Lee, CSE & GS suinlee@uw.edu 1 Whirlwind Tour of One/Two

More information

Regression in SPSS. Workshop offered by the Mississippi Center for Supercomputing Research and the UM Office of Information Technology

Regression in SPSS. Workshop offered by the Mississippi Center for Supercomputing Research and the UM Office of Information Technology Regression in SPSS Workshop offered by the Mississippi Center for Supercomputing Research and the UM Office of Information Technology John P. Bentley Department of Pharmacy Administration University of

More information

LOGISTIC REGRESSION. Nitin R Patel. where the dependent variable, y, is binary (for convenience we often code these values as

LOGISTIC REGRESSION. Nitin R Patel. where the dependent variable, y, is binary (for convenience we often code these values as LOGISTIC REGRESSION Nitin R Patel Logistic regression extends the ideas of multiple linear regression to the situation where the dependent variable, y, is binary (for convenience we often code these values

More information

Principal Components Analysis (PCA)

Principal Components Analysis (PCA) Principal Components Analysis (PCA) Janette Walde janette.walde@uibk.ac.at Department of Statistics University of Innsbruck Outline I Introduction Idea of PCA Principle of the Method Decomposing an Association

More information

, then the form of the model is given by: which comprises a deterministic component involving the three regression coefficients (

, then the form of the model is given by: which comprises a deterministic component involving the three regression coefficients ( Multiple regression Introduction Multiple regression is a logical extension of the principles of simple linear regression to situations in which there are several predictor variables. For instance if we

More information

CHAPTER 11 CHI-SQUARE: NON-PARAMETRIC COMPARISONS OF FREQUENCY

CHAPTER 11 CHI-SQUARE: NON-PARAMETRIC COMPARISONS OF FREQUENCY CHAPTER 11 CHI-SQUARE: NON-PARAMETRIC COMPARISONS OF FREQUENCY The hypothesis testing statistics detailed thus far in this text have all been designed to allow comparison of the means of two or more samples

More information

Linda K. Muthén Bengt Muthén. Copyright 2008 Muthén & Muthén www.statmodel.com. Table Of Contents

Linda K. Muthén Bengt Muthén. Copyright 2008 Muthén & Muthén www.statmodel.com. Table Of Contents Mplus Short Courses Topic 2 Regression Analysis, Eploratory Factor Analysis, Confirmatory Factor Analysis, And Structural Equation Modeling For Categorical, Censored, And Count Outcomes Linda K. Muthén

More information

Describe what is meant by a placebo Contrast the double-blind procedure with the single-blind procedure Review the structure for organizing a memo

Describe what is meant by a placebo Contrast the double-blind procedure with the single-blind procedure Review the structure for organizing a memo Readings: Ha and Ha Textbook - Chapters 1 8 Appendix D & E (online) Plous - Chapters 10, 11, 12 and 14 Chapter 10: The Representativeness Heuristic Chapter 11: The Availability Heuristic Chapter 12: Probability

More information

Chapter 11: Linear Regression - Inference in Regression Analysis - Part 2

Chapter 11: Linear Regression - Inference in Regression Analysis - Part 2 Chapter 11: Linear Regression - Inference in Regression Analysis - Part 2 Note: Whether we calculate confidence intervals or perform hypothesis tests we need the distribution of the statistic we will use.

More information

Confidence Intervals for the Area Under an ROC Curve

Confidence Intervals for the Area Under an ROC Curve Chapter 261 Confidence Intervals for the Area Under an ROC Curve Introduction Receiver operating characteristic (ROC) curves are used to assess the accuracy of a diagnostic test. The technique is used

More information

1. χ 2 minimization 2. Fits in case of of systematic errors

1. χ 2 minimization 2. Fits in case of of systematic errors Data fitting Volker Blobel University of Hamburg March 2005 1. χ 2 minimization 2. Fits in case of of systematic errors Keys during display: enter = next page; = next page; = previous page; home = first

More information

Probability Calculator

Probability Calculator Chapter 95 Introduction Most statisticians have a set of probability tables that they refer to in doing their statistical wor. This procedure provides you with a set of electronic statistical tables that

More information

An example ANOVA situation. 1-Way ANOVA. Some notation for ANOVA. Are these differences significant? Example (Treating Blisters)

An example ANOVA situation. 1-Way ANOVA. Some notation for ANOVA. Are these differences significant? Example (Treating Blisters) An example ANOVA situation Example (Treating Blisters) 1-Way ANOVA MATH 143 Department of Mathematics and Statistics Calvin College Subjects: 25 patients with blisters Treatments: Treatment A, Treatment

More information

What Does the Correlation Coefficient Really Tell Us About the Individual?

What Does the Correlation Coefficient Really Tell Us About the Individual? What Does the Correlation Coefficient Really Tell Us About the Individual? R. C. Gardner and R. W. J. Neufeld Department of Psychology University of Western Ontario ABSTRACT The Pearson product moment

More information

The Big 50 Revision Guidelines for S1

The Big 50 Revision Guidelines for S1 The Big 50 Revision Guidelines for S1 If you can understand all of these you ll do very well 1. Know what is meant by a statistical model and the Modelling cycle of continuous refinement 2. Understand

More information

LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING

LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING In this lab you will explore the concept of a confidence interval and hypothesis testing through a simulation problem in engineering setting.

More information

Introduction to Confirmatory Factor Analysis and Structural Equation Modeling

Introduction to Confirmatory Factor Analysis and Structural Equation Modeling Introduction to Confirmatory Factor Analysis and Structural Equation Modeling Lecture 12 August 7, 2011 Advanced Multivariate Statistical Methods ICPSR Summer Session #2 Today s Class An Introduction to:

More information

MAT140: Applied Statistical Methods Summary of Calculating Confidence Intervals and Sample Sizes for Estimating Parameters

MAT140: Applied Statistical Methods Summary of Calculating Confidence Intervals and Sample Sizes for Estimating Parameters MAT140: Applied Statistical Methods Summary of Calculating Confidence Intervals and Sample Sizes for Estimating Parameters Inferences about a population parameter can be made using sample statistics for

More information

The Exponential Family

The Exponential Family The Exponential Family David M. Blei Columbia University November 3, 2015 Definition A probability density in the exponential family has this form where p.x j / D h.x/ expf > t.x/ a./g; (1) is the natural

More information

II. DISTRIBUTIONS distribution normal distribution. standard scores

II. DISTRIBUTIONS distribution normal distribution. standard scores Appendix D Basic Measurement And Statistics The following information was developed by Steven Rothke, PhD, Department of Psychology, Rehabilitation Institute of Chicago (RIC) and expanded by Mary F. Schmidt,

More information

Lecture 2: Descriptive Statistics and Exploratory Data Analysis

Lecture 2: Descriptive Statistics and Exploratory Data Analysis Lecture 2: Descriptive Statistics and Exploratory Data Analysis Further Thoughts on Experimental Design 16 Individuals (8 each from two populations) with replicates Pop 1 Pop 2 Randomly sample 4 individuals

More information

CS395T Computational Statistics with Application to Bioinformatics

CS395T Computational Statistics with Application to Bioinformatics CS395T Computational Statistics with Application to Bioinformatics Prof. William H. Press Spring Term, 2010 The University of Texas at Austin Unit 6: Multivariate Normal Distributions and Chi Square The

More information

Multiple regression - Matrices

Multiple regression - Matrices Multiple regression - Matrices This handout will present various matrices which are substantively interesting and/or provide useful means of summarizing the data for analytical purposes. As we will see,

More information

When to Use Which Statistical Test

When to Use Which Statistical Test When to Use Which Statistical Test Rachel Lovell, Ph.D., Senior Research Associate Begun Center for Violence Prevention Research and Education Jack, Joseph, and Morton Mandel School of Applied Social Sciences

More information

Assessing the Relative Fit of Alternative Item Response Theory Models to the Data

Assessing the Relative Fit of Alternative Item Response Theory Models to the Data Research Paper Assessing the Relative Fit of Alternative Item Response Theory Models to the Data by John Richard Bergan, Ph.D. 6700 E. Speedway Boulevard Tucson, Arizona 85710 Phone: 520.323.9033 Fax:

More information

How To Use A Monte Carlo Study To Decide On Sample Size and Determine Power

How To Use A Monte Carlo Study To Decide On Sample Size and Determine Power How To Use A Monte Carlo Study To Decide On Sample Size and Determine Power Linda K. Muthén Muthén & Muthén 11965 Venice Blvd., Suite 407 Los Angeles, CA 90066 Telephone: (310) 391-9971 Fax: (310) 391-8971

More information

Crosstabulation & Chi Square

Crosstabulation & Chi Square Crosstabulation & Chi Square Robert S Michael Chi-square as an Index of Association After examining the distribution of each of the variables, the researcher s next task is to look for relationships among

More information

Latent Class Regression Part II

Latent Class Regression Part II This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License. Your use of this material constitutes acceptance of that license and the conditions of use of materials on this

More information

4. Introduction to Statistics

4. Introduction to Statistics Statistics for Engineers 4-1 4. Introduction to Statistics Descriptive Statistics Types of data A variate or random variable is a quantity or attribute whose value may vary from one unit of investigation

More information

" Y. Notation and Equations for Regression Lecture 11/4. Notation:

 Y. Notation and Equations for Regression Lecture 11/4. Notation: Notation: Notation and Equations for Regression Lecture 11/4 m: The number of predictor variables in a regression Xi: One of multiple predictor variables. The subscript i represents any number from 1 through

More information

Multivariate Analysis of Ecological Data

Multivariate Analysis of Ecological Data Multivariate Analysis of Ecological Data MICHAEL GREENACRE Professor of Statistics at the Pompeu Fabra University in Barcelona, Spain RAUL PRIMICERIO Associate Professor of Ecology, Evolutionary Biology

More information

Introduction to General and Generalized Linear Models

Introduction to General and Generalized Linear Models Introduction to General and Generalized Linear Models General Linear Models - part I Henrik Madsen Poul Thyregod Informatics and Mathematical Modelling Technical University of Denmark DK-2800 Kgs. Lyngby

More information

Module 9: Nonparametric Tests. The Applied Research Center

Module 9: Nonparametric Tests. The Applied Research Center Module 9: Nonparametric Tests The Applied Research Center Module 9 Overview } Nonparametric Tests } Parametric vs. Nonparametric Tests } Restrictions of Nonparametric Tests } One-Sample Chi-Square Test

More information

Analysing Tables Part V Interpreting Chi-Square

Analysing Tables Part V Interpreting Chi-Square Analysing Tables Part V Interpreting Chi-Square 8.0 Interpreting Chi Square Output 8.1 the Meaning of a Significant Chi-Square Sometimes the purpose of Crosstabulation is wrongly seen as being merely about

More information

Outline. Topic 4 - Analysis of Variance Approach to Regression. Partitioning Sums of Squares. Total Sum of Squares. Partitioning sums of squares

Outline. Topic 4 - Analysis of Variance Approach to Regression. Partitioning Sums of Squares. Total Sum of Squares. Partitioning sums of squares Topic 4 - Analysis of Variance Approach to Regression Outline Partitioning sums of squares Degrees of freedom Expected mean squares General linear test - Fall 2013 R 2 and the coefficient of correlation

More information