Constrained Dual Scaling for Detecting Response Styles in Categorical Data.
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Response styles occur when respondents use rating scales differently for reasons not related to the questions, often biasing results.A spline-based constrained version of DS is devised which can detect the presence of four prominent types of response styles, and is extended to allow for multiple response styles.An alternating nonnegative least squares algorithm is devised for estimating the parameters.
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Affiliation: Econometric Institute, Erasmus University Rotterdam, Rotterdam, The Netherlands. schoonees@gmail.com.
ABSTRACT
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Dual scaling (DS) is a multivariate exploratory method equivalent to correspondence analysis when analysing contingency tables. However, for the analysis of rating data, different proposals appear in the DS and correspondence analysis literature. It is shown here that a peculiarity of the DS method can be exploited to detect differences in response styles. Response styles occur when respondents use rating scales differently for reasons not related to the questions, often biasing results. A spline-based constrained version of DS is devised which can detect the presence of four prominent types of response styles, and is extended to allow for multiple response styles. An alternating nonnegative least squares algorithm is devised for estimating the parameters. The new method is appraised both by simulation studies and an empirical application. |
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Mentions: A related question concerns the performance of the method in the presence of a nontrivial correlation structure. To impose such a structure whilst retaining truncated normal marginal distributions for the objects, a copula is used (note that the truncated multivariate normal distribution does not guarantee truncated normal marginals). A copula is a multivariate distribution function with uniform marginals (Hofert & Mächler, 2011). According to Sklar’s theorem (Sklar, 1959; Hofert & Mächler, 2011) a multivariate distribution function with marginals can be constructed as15\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} F(x_1,x_2,\ldots ,x_m) = C(F_1(x_1),F_2(x_2),\ldots ,F_m(x_m)). \end{aligned}$$\end{document}F(x1,x2,…,xm)=C(F1(x1),F2(x2),…,Fm(xm)).The marginal truncated normal distributions can be achieved by the inverse probability integral transform. The dependence structure between the variables is solely determined by the copula. Here two independent Clayton copula (Clayton, 1978) functions will be used to impose a correlation structure in terms of Kendall’s , a well-known measure of rank correlation (see Kendall, 1938; Hofert & Mächler, 2011). The structure induced here for is as follows: the first ten objects are correlated with , independent of the other ten objects which are correlated with . These values amount to Pearson correlations of approximately and , respectively (an approximate relationship is —see Kendall and Gibbons (1990)). It is also possible to introduce negative correlations by using instead of in the inverse probability integral transform. In the application here these reversals are made randomly with differing probability . The theoretical, observed and cleaned correlations given by Kendall’s for one such copula is illustrated in Figure 6, with and .Fig. 6 |
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Affiliation: Econometric Institute, Erasmus University Rotterdam, Rotterdam, The Netherlands. schoonees@gmail.com.