Modelling Assumed Metric Paired Comparison Data - Application to Learning Related Emotions

Alexandra Grand; Regina Dittrich

doi:10.17713/ajs.v44i1.25

Authors

Alexandra Grand WU University of Economics and Business
Regina Dittrich WU University of Economics and Business

DOI:

https://doi.org/10.17713/ajs.v44i1.25

Abstract

In this article we suggest a beta regression model that accounts for the degree of preference in paired comparisons measured on a bounded metric paired comparison scale. The beta distribution for bounded continuous random variables assumes values in the open unit interval (0,1). However, in practice we will observe paired comparison responses that lie within a fixed or arbitrary fixed interval [-a,a] with known value of a. We therefore transform the observed responses into the interval (0,1) and assume that these transformed responses are each a realization of a random variable which follows a beta distribution. We propose a simple paired comparison regression model for beta distributed variables which allows us to model the mean of the transformed response using a linear predictor and a logit link function -- where the linear predictor is defined by the parameters of the logit-linear Bradley-Terry model. For illustration we applied the presented model to a data set obtained from a student survey of learning related emotions in mathematics.

References

Agresti A (1992). Analysis of Ordinal Paired Comparison Data." Applied Statistics, 41(2),

{297.

Aitken R (1969). Measurement of Feelings Using Visual Analogue Scales." Proceedings of

the Royal Society of Medicine, 62, 989{993.

Bockenholt U, Dillon W (1997). Modeling within-subject dependencies in ordinal paired

comparison data." Psychometrika, 62(3), 411{434.

Bradley R, Terry M (1952). Rank Analysis of Incomplete Block Designs: I. The Method of

Paired Comparisons." Biometrika, 39(3/4), 324{345.

Cribari-Neto F, Zeileis A (2010). Beta Regression in R." Journal of Statistical Software,

(2), 1{24.

De Ruiz K (1990). A Mathematical Model for a Paired Comparison Experiment on a Contin-

uum of Response. PhD dissertation, University of California-Riverside.

Dittrich R, Francis B, Hatzinger R, Katzenbeisser W (2007). A paired comparison approach

for the analysis of sets of Likert-scale responses." Statistical Modelling, 7(1), 3{28.

Dittrich R, Hatzinger R, Katzenbeisser W (1998). Modelling the eect of subject-specic

covariates in paired comparison studies with an application to university rankings." Applied

Statistics, 47(4), 511{525.

Dittrich R, Hatzinger R, KatzenbeisserW(2004). A log-linear approach for modelling ordinal

paired comparison data on motives to start a PhD programme." Statistical Modelling, 4,

{193.

Ferrando P (2002). Theoretical and Empirical Comparisons between Two Models for Continuous

Item Response." Multivariate Behavioral Research, 37(4), 521{542.

Ferrari S, Cribari-Neto F (2004). "Beta Regression for Modelling Rates and Proportions." Journal of Applied Statistics, 31(7), 799-815.

Francis B, Dittrich R, Hatzinger R, Penn R (2002). "Analysing partial ranks by using smoothed paired comparison methods: an investigation of value orientation in Europe." Applied Statistics, 51(3), 319-336.

Frenzel A, Pekrun R, Goetz T (2007). "Girls and mathematics. A hopeless" issue? A control-value approach to gender dierences in emotions towards mathematics." European Journal of Psychology of Education, 22(4), 497-514.

Goetz T, Pekrun R, Zirngibl A, Jullien S, Kleine M, vom Hofe R, Blum W (2004). "Leistung und emotionales Erleben im Fach Mathematik." Zeitschrift für Pädagogische Psychologie, 18(3/4), 201-212.

Grün B, Kosmidis I, Zeileis A (2012). "Extended Beta Regression in R: Shaken, Stirred, Mixed, and Partitioned." Journal of Statistical Software, 48(11), 1-25.

Götz T, Keller M, Martiny S (2012). "Emotionales Erleben in den MINT-Fächern: Ursachen, Geschlechtsunterschiede und Interventionsmöglichkeiten." In H Stöger, A Ziegler, M Heilemann (eds.), Mädchen und Frauen in MINT., pp. 135-161. Berlin: Lit Verlag.

Hatzinger R (2012). "prefmod: Utilities to Fit Paired Comparison Models for Preferences." R package Version 0.8-31. URL http://cran.R-project.org/web/packages/prefmod/index.html.

Kieschnick R, McCullough B (2003). "Regression analysis of variates observed on (0,1): percentages, proportions and fractions." Statistical Modelling, 3, 193-213.

Kousgaard N (1976). "Models for Paired Comparisons with Ties." Scandinavian Journal of Statistics, 3(1), 1-14.

Paolino P (2001). "Maximum Likelihood Estimation of Models with Beta-Distributed Dependent Variables." Political Analysis, 9, 325-346.

Pekrun R, Goetz T, Frenzel A, Barchfeld P, Perry R (2011). "Measuring Emotions in students' learning and performance: The Achievement Emotions Questionnaire (AEQ)." Contempo-

rary Educational Psychology, 36, 36-48.

R Development Core Team (2013). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0. URL http://www.R-project.org.

Rao P, Kupper L (1967). "Ties in Paired-Comparison Experiments: A Generalization of the Bradley-Terry Model." Journal of the American Statistical Association, 62(317), 194-204.

Simas A, Barreto-Souza W, Rocha A (2010). "Improved estimators for a general class of beta regression models." Computational Statistics and Data Analysis, 54, 348-366.

Smithson M, Verkuilen J (2006). "A Better Lemon Squeezer? Maximum-Likelihood Regression with Beta-Distributed Dependent Variables." Psychological Methods, 11(1), 54-71.

Stern S (2011). "Moderated paired comparisons: a generalized Bradley-Terry model for continuous data using a discontinuous penalized likelihood function." Applied Statistics, 60(3), 397-415.

Zeileis A, Cribari-Neto F, Grün B, Kosmidis I, Simas A, Rocha A (2013). "betareg." R package Version 3.0-2. URL http://CRAN.R-project.org/package=betareg.