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A study into mechanisms of attitudinal scale conversion: A randomized stochastic ordering approach

Abstract

This paper considers the methodological challenge of how to convert categorical attitudinal scores (like satisfaction) measured on one scale to a categorical attitudinal score measured on another scale with a different range. This is becoming a growing issue in marketing consulting and the common available solutions seem too few and too superficial. A new methodology for scale conversion is proposed, and tested in a comprehensive study. This methodology is shown to be both relevant and optimal in fundamental aspects. The new methodology is based on a novel algorithm named minimum conditional entropy, that uses the marginal distributions of the responses on each of the two scales to produce a unique joint bivariate distribution. In this joint distribution, the conditional distributions follow a stochastic order that is monotone in the categories and has the relevant optimal property of maximizing the correlation between the two underlying marginal scales. We show how such a joint distribution can be used to build a mechanism for scale conversion. We use both a frequentist and a Bayesian approach to derive mixture models for conversion mechanisms, and discuss some inferential aspects associated with the underlying models. These models can incorporate background variables of the respondents. A unique observational experiment is conducted that empirically validates the proposed modeling approach. Strong evidence of validation is obtained.

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Correspondence to Zvi Gilula.

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Appendices

Appendix 1

Data tables

Two way data tables for the question "How satisfied or dissatisfied are you with the quality of life in your city of residence?"

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Observed counts, 5 point scale to 11 point scale

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5 point scale to 11 point scale, counts obtained from the MCE algorithm.

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Observed counts, 5 point scale to 5 point scale

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Observed counts, 11 point scale to 5 point scale

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Observed counts, 11 point scale to 11 point scale

Appendix 2

Inferential aspects of the MCE algorithm

The MCE algorithm can be considered as the function Q = Ψ(P, P'), where P = (P1+,  … , PR+), P' = (P'+1,  … , P'+C) are probability vectors and Qij, i = 1, … , R, j = 1, … , C is a contingency vector with marginals P and P′. In practice it is used with sampled value, \( \widehat{Q}=\varPsi \left(\widehat{P},{\widehat{P}}^{\prime}\right) \), where \( \widehat{P} \) and \( {\widehat{P}}^{\hbox{'}} \) are estimators based on a sample.

We want to construct confidence intervals for the estimators \( \widehat{Q} \).

Let \( {\mathbb{P}}_i={\sum}_{k=1}^i{P}_{k+} \) the cumulative distribution function corresponding to P. Define similarly other cdf’s (e.g., ' and ), and \( {\mathbb{Q}}_{ij}={\sum}_{k=1}^i{\sum}_{l=1}^j{Q}_{kl} \). The CME algorithm is simply

$$ {\mathbb{Q}}_{ij}=\min \left\{{\mathbb{P}}_i,\mathbb{P}{\hbox{'}}_j\right\}. $$
(2)

We consider a standard fix point asymptotics. We assume.

  • A1. \( \widehat{P} \), \( {\widehat{P}}^{\hbox{'}} \) are independent and based on a multinomial samples with sizes n,n’ respectively.

  • A2. For all i = 1, …, R and j = 1, …, C, if i = 'j then i = R and j = C.

In that case the asymptotics is simple, since if i < 'j then ij = i, and the CI of i is the CI of ij as well.

As for Qij itself,

$$ {Q}_{ij}={\mathbb{Q}}_{ij}+{\mathbb{Q}}_{i-1,j-1}-{\mathbb{Q}}_{i-1,j}-{\mathbb{Q}}_{i,j-1},i=1,\dots, R,j=1,\dots, C, $$
(3)

where 0, j ≡ i, 0 ≡ 0. Hence

$$ {Q}_{ij}=\min \left\{{\mathbb{P}}_i,\mathbb{P}{\hbox{'}}_j\right\}+\min \left\{{\mathbb{P}}_{i-1},\mathbb{P}{\hbox{'}}_{j-1}\right\}-\min \left\{{\mathbb{P}}_{i-1},\mathbb{P}{\hbox{'}}_j\right\}-\min \left\{{\mathbb{P}}_i,\mathbb{P}{\hbox{'}}_{j-1}\right\}, $$

where 0 = '0 = 0. Without loss of generality, there are three possibilities:

  1. i.

    Suppose i < 'j − 1. Then clearly i − 1 < i < 'j − 1 < 'j and hence Qij = 0. Thus \( {P}_r\left({\widehat{Q}}_{ij}={Q}_{ij}\right)\overset{\mathrm{p}}{\to }1 \).

  2. ii.

    Suppose i − 1 < 'j − 1 < 'j < i. In this case Qij = 'j − 'j − 1 = '+j, and an asymptotic 1 − αconfidence interval for Qij is given by \( {\widehat{Q}}_{ij}\pm {z}_{\alpha /2}{n}^{\prime -1/2}\widehat{P}{\prime}_{+j}\left(1-\widehat{P}{\prime}_{+j}\right) \).

  3. iii.

    Suppose i − 1 < 'j − 1 < i < 'j. In this case Qij = i − 'j − 1, and an asymptotic 1 − α confidence interval for Qij is given by

    $$ {\widehat{Q}}_{ij}\pm {z}_{\alpha /2}{\left(\frac{{\widehat{P}}_{i+}\left(1-{\widehat{P}}_{i+}\right)}{n}+\frac{\widehat{P}{\hbox{'}}_{j-1,+}\left(1-\widehat{P}{\hbox{'}}_{j-1,+}\right)}{n^{\hbox{'}}}\right)}^{1/2}. $$

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Gilula, Z., McCulloch, R.E., Ritov, Y. et al. A study into mechanisms of attitudinal scale conversion: A randomized stochastic ordering approach. Quant Mark Econ 17, 325–357 (2019). https://doi.org/10.1007/s11129-019-09209-3

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Keywords

  • Categorical conversion
  • Conditional entropy
  • Mixture models
  • Ordinal attitudinal scales
  • Stochastic ordering