## 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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## 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?"

### Appendix 2

### Inferential aspects of the MCE algorithm

The MCE algorithm can be considered as the function *Q* = *Ψ*(*P*, *P*^{'}), where *P* = (*P*_{1+}, … , *P*_{R+}), *P*^{'} = (*P*'_{+1}, … , *P*'_{+C}) are probability vectors and *Q*_{ij}, *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., *ℙ*^{'} and ), and \( {\mathbb{Q}}_{ij}={\sum}_{k=1}^i{\sum}_{l=1}^j{Q}_{kl} \). The CME algorithm is simply

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*ℙ*_{i}=*ℙ*'_{j}then*i*=*R*and*j*=*C*.

In that case the asymptotics is simple, since if

ℙ_{i}<ℙ'_{j}thenℚ_{ij}=ℙ_{i}, and the CI ofℙ_{i}is the CI ofℚ_{ij}as well.

As for *Q*_{ij} itself,

where *ℚ*_{0, j} ≡ *ℚ*_{i, 0} ≡ 0. Hence

where *ℙ*_{0} = *ℙ*'_{0} = 0. Without loss of generality, there are three possibilities:

- i.
Suppose

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

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

*ℙ*_{i − 1}<*ℙ*'_{j − 1}<*ℙ*_{i}<*ℙ*'_{j}. In this case*Q*_{ij}=*ℙ*_{i}−*ℙ*'_{j − 1}, and an asymptotic 1 −*α*confidence interval for*Q*_{ij}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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DOI: https://doi.org/10.1007/s11129-019-09209-3

### Keywords

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