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Journal ArticleDOI

Large-sample tests for comparing Likert-type scale data

TL;DR: Asymptotic tests for identical distribution of responses in two independent sets of Likert-type scale data using latent variable models were developed in this article, where the proposed tests were compared with regard to the distribution of response scores.
Abstract: Asymptotic tests for identical distribution of responses in two independent sets of Likert-type scale data using latent variable models are developed. The proposed tests are compared with regard to...
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MonographDOI
TL;DR: In this article, the authors present a generalized linear model for categorical data, which is based on the Logit model, and use it to fit Logistic Regression models.
Abstract: Preface. 1. Introduction: Distributions and Inference for Categorical Data. 1.1 Categorical Response Data. 1.2 Distributions for Categorical Data. 1.3 Statistical Inference for Categorical Data. 1.4 Statistical Inference for Binomial Parameters. 1.5 Statistical Inference for Multinomial Parameters. Notes. Problems. 2. Describing Contingency Tables. 2.1 Probability Structure for Contingency Tables. 2.2 Comparing Two Proportions. 2.3 Partial Association in Stratified 2 x 2 Tables. 2.4 Extensions for I x J Tables. Notes. Problems. 3. Inference for Contingency Tables. 3.1 Confidence Intervals for Association Parameters. 3.2 Testing Independence in Two Way Contingency Tables. 3.3 Following Up Chi Squared Tests. 3.4 Two Way Tables with Ordered Classifications. 3.5 Small Sample Tests of Independence. 3.6 Small Sample Confidence Intervals for 2 x 2 Tables . 3.7 Extensions for Multiway Tables and Nontabulated Responses. Notes. Problems. 4. Introduction to Generalized Linear Models. 4.1 Generalized Linear Model. 4.2 Generalized Linear Models for Binary Data. 4.3 Generalized Linear Models for Counts. 4.4 Moments and Likelihood for Generalized Linear Models . 4.5 Inference for Generalized Linear Models. 4.6 Fitting Generalized Linear Models. 4.7 Quasi likelihood and Generalized Linear Models . 4.8 Generalized Additive Models . Notes. Problems. 5. Logistic Regression. 5.1 Interpreting Parameters in Logistic Regression. 5.2 Inference for Logistic Regression. 5.3 Logit Models with Categorical Predictors. 5.4 Multiple Logistic Regression. 5.5 Fitting Logistic Regression Models. Notes. Problems. 6. Building and Applying Logistic Regression Models. 6.1 Strategies in Model Selection. 6.2 Logistic Regression Diagnostics. 6.3 Inference About Conditional Associations in 2 x 2 x K Tables. 6.4 Using Models to Improve Inferential Power. 6.5 Sample Size and Power Considerations . 6.6 Probit and Complementary Log Log Models . 6.7 Conditional Logistic Regression and Exact Distributions . Notes. Problems. 7. Logit Models for Multinomial Responses. 7.1 Nominal Responses: Baseline Category Logit Models. 7.2 Ordinal Responses: Cumulative Logit Models. 7.3 Ordinal Responses: Cumulative Link Models. 7.4 Alternative Models for Ordinal Responses . 7.5 Testing Conditional Independence in I x J x K Tables . 7.6 Discrete Choice Multinomial Logit Models . Notes. Problems. 8. Loglinear Models for Contingency Tables. 8.1 Loglinear Models for Two Way Tables. 8.2 Loglinear Models for Independence and Interaction in Three Way Tables. 8.3 Inference for Loglinear Models. 8.4 Loglinear Models for Higher Dimensions. 8.5 The Loglinear Logit Model Connection. 8.6 Loglinear Model Fitting: Likelihood Equations and Asymptotic Distributions . 8.7 Loglinear Model Fitting: Iterative Methods and their Application . Notes. Problems. 9. Building and Extending Loglinear/Logit Models. 9.1 Association Graphs and Collapsibility. 9.2 Model Selection and Comparison. 9.3 Diagnostics for Checking Models. 9.4 Modeling Ordinal Associations. 9.5 Association Models . 9.6 Association Models, Correlation Models, and Correspondence Analysis . 9.7 Poisson Regression for Rates. 9.8 Empty Cells and Sparseness in Modeling Contingency Tables. Notes. Problems. 10. Models for Matched Pairs. 10.1 Comparing Dependent Proportions. 10.2 Conditional Logistic Regression for Binary Matched Pairs. 10.3 Marginal Models for Square Contingency Tables. 10.4 Symmetry, Quasi symmetry, and Quasiindependence. 10.5 Measuring Agreement Between Observers. 10.6 Bradley Terry Model for Paired Preferences. 10.7 Marginal Models and Quasi symmetry Models for Matched Sets . Notes. Problems. 11. Analyzing Repeated Categorical Response Data. 11.1 Comparing Marginal Distributions: Multiple Responses. 11.2 Marginal Modeling: Maximum Likelihood Approach. 11.3 Marginal Modeling: Generalized Estimating Equations Approach. 11.4 Quasi likelihood and Its GEE Multivariate Extension: Details . 11.5 Markov Chains: Transitional Modeling. Notes. Problems. 12. Random Effects: Generalized Linear Mixed Models for Categorical Responses. 12.1 Random Effects Modeling of Clustered Categorical Data. 12.2 Binary Responses: Logistic Normal Model. 12.3 Examples of Random Effects Models for Binary Data. 12.4 Random Effects Models for Multinomial Data. 12.5 Multivariate Random Effects Models for Binary Data. 12.6 GLMM Fitting, Inference, and Prediction. Notes. Problems. 13. Other Mixture Models for Categorical Data . 13.1 Latent Class Models. 13.2 Nonparametric Random Effects Models. 13.3 Beta Binomial Models. 13.4 Negative Binomial Regression. 13.5 Poisson Regression with Random Effects. Notes. Problems. 14. Asymptotic Theory for Parametric Models. 14.1 Delta Method. 14.2 Asymptotic Distributions of Estimators of Model Parameters and Cell Probabilities. 14.3 Asymptotic Distributions of Residuals and Goodnessof Fit Statistics. 14.4 Asymptotic Distributions for Logit/Loglinear Models. Notes. Problems. 15. Alternative Estimation Theory for Parametric Models. 15.1 Weighted Least Squares for Categorical Data. 15.2 Bayesian Inference for Categorical Data. 15.3 Other Methods of Estimation. Notes. Problems. 16. Historical Tour of Categorical Data Analysis . 16.1 Pearson Yule Association Controversy. 16.2 R. A. Fisher s Contributions. 16.3 Logistic Regression. 16.4 Multiway Contingency Tables and Loglinear Models. 16.5 Recent and Future? Developments. Appendix A. Using Computer Software to Analyze Categorical Data. A.1 Software for Categorical Data Analysis. A.2 Examples of SAS Code by Chapter. Appendix B. Chi Squared Distribution Values. References. Examples Index. Author Index. Subject Index. Sections marked with an asterisk are less important for an overview.

4,650 citations

Book
01 Dec 1971
TL;DR: Theoretical Bases for Calculating the ARE Examples of the Calculations of Efficacy and ARE Analysis of Count Data.
Abstract: Introduction and Fundamentals Introduction Fundamental Statistical Concepts Order Statistics, Quantiles, and Coverages Introduction Quantile Function Empirical Distribution Function Statistical Properties of Order Statistics Probability-Integral Transformation Joint Distribution of Order Statistics Distributions of the Median and Range Exact Moments of Order Statistics Large-Sample Approximations to the Moments of Order Statistics Asymptotic Distribution of Order Statistics Tolerance Limits for Distributions and Coverages Tests of Randomness Introduction Tests Based on the Total Number of Runs Tests Based on the Length of the Longest Run Runs Up and Down A Test Based on Ranks Tests of Goodness of Fit Introduction The Chi-Square Goodness-of-Fit Test The Kolmogorov-Smirnov One-Sample Statistic Applications of the Kolmogorov-Smirnov One-Sample Statistics Lilliefors's Test for Normality Lilliefors's Test for the Exponential Distribution Anderson-Darling Test Visual Analysis of Goodness of Fit One-Sample and Paired-Sample Procedures Introduction Confidence Interval for a Population Quantile Hypothesis Testing for a Population Quantile The Sign Test and Confidence Interval for the Median Rank-Order Statistics Treatment of Ties in Rank Tests The Wilcoxon Signed-Rank Test and Confidence Interval The General Two-Sample Problem Introduction The Wald-Wolfowitz Runs Test The Kolmogorov-Smirnov Two-Sample Test The Median Test The Control Median Test The Mann-Whitney U Test and Confidence Interval Linear Rank Statistics and the General Two-Sample Problem Introduction Definition of Linear Rank Statistics Distribution Properties of Linear Rank Statistics Usefulness in Inference Linear Rank Tests for the Location Problem Introduction The Wilcoxon Rank-Sum Test and Confidence Interval Other Location Tests Linear Rank Tests for the Scale Problem Introduction The Mood Test The Freund-Ansari-Bradley-David-Barton Tests The Siegel-Tukey Test The Klotz Normal-Scores Test The Percentile Modified Rank Tests for Scale The Sukhatme Test Confidence-Interval Procedures Other Tests for the Scale Problem Applications Tests of the Equality of k Independent Samples Introduction Extension of the Median Test Extension of the Control Median Test The Kruskal-Wallis One-Way ANOVA Test and Multiple Comparisons Other Rank-Test Statistics Tests against Ordered Alternatives Comparisons with a Control Measures of Association for Bivariate Samples Introduction: Definition of Measures of Association in a Bivariate Population Kendall's Tau Coefficient Spearman's Coefficient of Rank Correlation The Relations between R and T E(R), tau, and rho Another Measure of Association Applications Measures of Association in Multiple Classifications Introduction Friedman's Two-Way Analysis of Variance by Ranks in a k x n Table and Multiple Comparisons Page's Test for Ordered Alternatives The Coefficient of Concordance for k Sets of Rankings of n Objects The Coefficient of Concordance for k Sets of Incomplete Rankings Kendall's Tau Coefficient for Partial Correlation Asymptotic Relative Efficiency Introduction Theoretical Bases for Calculating the ARE Examples of the Calculations of Efficacy and ARE Analysis of Count Data Introduction Contingency Tables Some Special Results for k x 2 Contingency Tables Fisher's Exact Test McNemar's Test Analysis of Multinomial Data Summary Appendix of Tables Answers to Problems References Index A Summary and Problems appear at the end of each chapter.

2,988 citations


"Large-sample tests for comparing Li..." refers background in this paper

  • ...Gibbons and Chakraborti (2011) further give conditions under which ARE of tests based on asymptotically normal statistics are comparable....

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  • ...…K12K 122 K012 q ¼ ffiffiffiffiffiffiffiffiffiffi K11:2 p > 0 is 1ffiffiffimp times the square root of the efficacy (see Gibbons and Chakraborti 2011) of the test based on Tð1Þ: For the test under “a priori” model recall that Tð2Þm, n ¼ ĥ ð2Þ m, n…...

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  • ...1 dEðTð2Þm, nÞ=dh h i h¼0ffiffiffiffi m p VðTð2Þm, nÞjh¼0 ¼ kffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1þ kÞPiPj sijq > 0 is 1ffiffiffimp times the square root of the efficacy (see Gibbons and Chakraborti 2011) of the test based on Tð2Þ:...

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01 Jan 2005
TL;DR: This classic textbook, now available from Springer, summarizes developments in the field of hypotheses testing and indicates that optimality considerations continue to provide the organizing principle, but they are now tempered by a much stronger emphasis on the robustness properties of the resulting procedures.
Abstract: This classic textbook, now available from Springer, summarizes developments in the field of hypotheses testing Optimality considerations continue to provide the organizing principle However, they are now tempered by a much stronger emphasis on the robustness properties of the resulting procedures This book is an essential reference for any graduate student in statistics

1,960 citations


"Large-sample tests for comparing Li..." refers background in this paper

  • ...…hrFðhÞ , 1 : Hence, APFFðhÞ 1 U sa hrFðhÞ ¼ U sa hrFðhÞ For testing H0 : h 0 against H1 : h > 0 it can be noted that the approximate expression for APFFðhÞ exceeds a for h > 0 and lies below a for h 0: Hence the test is asymptotically unbiased at significance level a (see Lehmann and Romano 2008)....

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Journal Article
TL;DR: The differences between Lkert-type and Likert scale data are discussed and recommendations for descriptive statistics to be used during the analysis are provided and once a researcher understands the difference, the decision on appropriate statistical procedures will be apparent.
Abstract: This article provides information for Extension professionals on the correct analysis of Likert data. The analyses of Likert-type and Likert scale data require unique data analysis procedures, and as a result, misuses and/or mistakes often occur. This article discusses the differences between Likert-type and Likert scale data and provides recommendations for descriptive statistics to be used during the analysis. Once a researcher understands the difference between Likert-type and Likert scale data, the decision on appropriate statistical procedures will be apparent.

1,119 citations

Book
01 Jan 1996
TL;DR: Basic probability, large sample theory, and efficient Estimation and Testing: a meta-thesis on large sample estimation and testing.
Abstract: A Course in Large Sample Theory is presented in four parts. The first treats basic probabilistic notions, the second features the basic statistical tools for expanding the theory, the third contains special topics as applications of the general theory, and the fourth covers more standard statistical topics. Nearly all topics are covered in their multivariate setting.The book is intended as a first year graduate course in large sample theory for statisticians. It has been used by graduate students in statistics, biostatistics, mathematics, and related fields. Throughout the book there are many examples and exercises with solutions. It is an ideal text for self study.

1,029 citations


Additional excerpts

  • ...An iterative procedure is needed to solve these equations, with each iteration further consisting of two steps: successively updating estimates of h and x ¼ ðx1, x2, :::, xkÞ: The estimate of h for given x1, x2, :::, xk for the first part of each step of iteration, as well as, for the second part, those of the xj’s for given h can be obtained by the scoring method: @2lðhÞ @h2 ¼ Xkþ1 j¼1 gj 1 dj @ 2dj @h2 1 d2j @dj @h 2" # , so i11ðh, xÞ ¼ E @ 2lðhÞ @h2 ¼ Xkþ1 j¼1 mdj 1 dj @ 2dj @h2 1 d2j @dj @h 2" # ¼ Xkþ1 j¼1 m dj @dj @h 2 : while the matrix I22ðh, xÞ has entries iij, 22ðh, xÞ ¼ nf 2ðxiÞ 1pi þ 1 piþ1 þmf 2ðxi hÞ 1di þ 1 diþ1 , j ¼ i nf ðxiÞf ðxi 1Þ pi mf ðxi hÞf ðxi 1 hÞ di , j ¼ i 1 nf ðxiÞf ðxiþ1Þ piþ1 mf ðxi hÞf ðxiþ1 hÞ diþ1 , j ¼ iþ 1 0 otherwise 8>>>>< >>>>: Also, the entries of the vector i12ðh, xÞ are given by E @ 2lðh, xÞ @h@xj " # ¼ mf ðxj hÞ f ðxjþ1 hÞ f ðxj hÞ djþ1 f ðxj hÞ f ðxj 1 hÞ dj " # for j ¼ 1, 2, :::, k: Now, we can conclude from Serfling (1980) or Ferguson (1996) that the regularity conditions imply, for the solution of the likelihood equations, the following: ðĥM , x̂1, x̂2, :::, x̂kÞ a Nkþ1ððh, x1, x2, :::, xkÞ,I 1m, nÞ: Hence, ðĥMm, nÞ a N 0, ið11Þððh, xÞÞ as m, n !...

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  • ...…multinomial pmfs, is given by lððh, xÞÞ ¼ c1 þ Xkþ1 j¼1 fj log pj þ c2 þ Xkþ1 j¼1 gj log dj where c1, c2 > 0: Standard regularity conditions are given, for example, in Serfling (1980) or Ferguson (1996), guaranteeing efficient and consistent asymptotic normality of the maximum likelihood estimator....

    [...]

  • ...…1 0 otherwise 8>>>>< >>>>: Also, the entries of the vector i12ðh, xÞ are given by E @ 2lðh, xÞ @h@xj " # ¼ mf ðxj hÞ f ðxjþ1 hÞ f ðxj hÞ djþ1 f ðxj hÞ f ðxj 1 hÞ dj " # for j ¼ 1, 2, :::, k: Now, we can conclude from Serfling (1980) or Ferguson (1996) that the regularity conditions imply, for…...

    [...]

  • ...The loglikelihood, arising from the product of two independent multinomial pmfs, is given by lððh, xÞÞ ¼ c1 þ Xkþ1 j¼1 fj log pj þ c2 þ Xkþ1 j¼1 gj log dj where c1, c2 > 0: Standard regularity conditions are given, for example, in Serfling (1980) or Ferguson (1996), guaranteeing efficient and consistent asymptotic normality of the maximum likelihood estimator....

    [...]

Trending Questions (2)
What is the correlation between VARK scores and Likert scale scores?

The provided paper does not mention anything about the correlation between VARK scores and Likert scale scores.

How do Likert scales compare to other types of scales used in questionnaires?

The paper compares asymptotic tests for Likert-type scale data with other tests for comparing distributions of responses in questionnaires.