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.

Appendices

Appendix 1

1.1 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?"

Fig. 24

Observed counts, 5 point scale to 11 point scale

Fig. 25

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

Fig. 26

Observed counts, 5 point scale to 5 point scale

Fig. 27

Observed counts, 11 point scale to 5 point scale

Fig. 28

Observed counts, 11 point scale to 11 point scale

Appendix 2

1.1 Inferential aspects of the MCE algorithm

The MCE algorithm can be considered as the function Q = Ψ(P, P^'), where P = (P₁₊,  … , 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

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

(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 ℙ_i = ℙ'_j then i = R and j = C.

In that case the asymptotics is simple, since if ℙ_i < ℙ'_j then ℚ_ij = ℙ_i, and the CI of ℙ_i is the CI of ℚ_ij as well.

As for Q_ij 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 ℚ_{0, j} ≡ ℚ_{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 ℙ₀ = 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 − αconfidence 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

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