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Constrained Dual Scaling for Detecting Response Styles in Categorical Data.

Schoonees PC, van de Velden M, Groenen PJ - Psychometrika (2015)

Bottom Line: 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.

View Article: PubMed Central - PubMed

Affiliation: Econometric Institute, Erasmus University Rotterdam, Rotterdam, The Netherlands. schoonees@gmail.com.

ABSTRACT
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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The three I-spline basis functions for quadratic monotone splines with a single interior knot .
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Fig2: The three I-spline basis functions for quadratic monotone splines with a single interior knot .

Mentions: The basis functions and in (7), as depicted in Figure 2, are piecewise quadratic, with only two of them nonconstant in each of the intervals and . This is convenient because it means the second derivative of , and hence the curvature, depends only on two parameters in each interval. Rescaling without loss of generality so that and , the curvature of (not necessarily defined at ) is given by9\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{d^{2}}{dx^{2}} f(x) = {\left\{ \begin{array}{ll} - 8 \alpha _{1} + 4\alpha _{2}, &{}\text {if}\quad 0 \le x < 1/2;\\ - 4\alpha _{2} + 8\alpha _{3}, &{}\text {if}\quad 1/2 < x \le 1;\\ \end{array}\right. } \end{aligned}$$\end{document}d2dx2f(x)=-8α1+4α2,if0≤x<1/2;-4α2+8α3,if1/2<x≤1;Fig. 2


Constrained Dual Scaling for Detecting Response Styles in Categorical Data.

Schoonees PC, van de Velden M, Groenen PJ - Psychometrika (2015)

The three I-spline basis functions for quadratic monotone splines with a single interior knot .
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC4644217&req=5

Fig2: The three I-spline basis functions for quadratic monotone splines with a single interior knot .
Mentions: The basis functions and in (7), as depicted in Figure 2, are piecewise quadratic, with only two of them nonconstant in each of the intervals and . This is convenient because it means the second derivative of , and hence the curvature, depends only on two parameters in each interval. Rescaling without loss of generality so that and , the curvature of (not necessarily defined at ) is given by9\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{d^{2}}{dx^{2}} f(x) = {\left\{ \begin{array}{ll} - 8 \alpha _{1} + 4\alpha _{2}, &{}\text {if}\quad 0 \le x < 1/2;\\ - 4\alpha _{2} + 8\alpha _{3}, &{}\text {if}\quad 1/2 < x \le 1;\\ \end{array}\right. } \end{aligned}$$\end{document}d2dx2f(x)=-8α1+4α2,if0≤x<1/2;-4α2+8α3,if1/2<x≤1;Fig. 2

Bottom Line: 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.

View Article: PubMed Central - PubMed

Affiliation: Econometric Institute, Erasmus University Rotterdam, Rotterdam, The Netherlands. schoonees@gmail.com.

ABSTRACT
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.

Show MeSH
Related in: MedlinePlus