INTRODUCTION TO MATHEMATICAL MODELS IN MARKET AND OPINION RESEARCH

INTRODUCTION TO MATHEMATICAL MODELS IN MARKET AND OPINION RESEARCH

THEODOR HARDER INTRODUCTION TO MATHEMATICAL MODELS IN MARKET AND OPINION RESEARCH WITH PRACTICAL APPLICATIONS, COMPUTING PROCEDURES, AND ESTIMATES OF COMPUTING REQUIREMENTS D. REIDEL PUBLISHING COMPANY / DORDRECHT-HOLLAND

ELEMENT ARE MATHEMATISCHE MODELLE IN DER MARKT- UND MEINUNGSFORSCHUNG First published in 1966 by R. Oldenbourg, Munich-Vienna Translated/rom the German by Peter H.Friedlander and Eva H. Friedlander ISBD-13: 978-94-010-3398-5 e-isbn-13: 978-94-010-3396-1 DOl: 10.1007/978-94-010-3396-1 1969. D. Reidel Publishing Company, Dordrecht, Holland Softcover reprint of the hardcover 15t edition 1969 No part of this book may be reproduced in any form, by print, photoprint, microfilm, or any other means, without written permission from the publisher

PREFACE In the introduction to his book Dr. Harder has very clearly described its purpose and organization. I only want to add for the English-speaking reader a few words on the place the present text is likely to have in the current literature. At first Dr. Harder's undertaking might come as a surprise. Only a few years ago, Zeisel's Say it with Figures gave the market research practitioner some ideas of how simple figures and tables could be successfully employed; Langhoff's publication for the American Marketing Association presented some pertinent mathematical models in the most elementary form; why should a German author believe he can already introduce us to serious mathematical procedures for use in product management and advertising? After reading the book, incredulity turns into pleasure because of the skill with which the author has pursued his task. As a matter of fact, the book can serve two audiences who at first glance might appear to have quite opposing interests. For the mathematically trained market researcher, the book has the marked advantage of combining a variety of approaches not ordinarily mixed in one volume. If the market researcher began as an economist he is already familiar with difference equations and time series analysis; if he moved in from psychology, he is already acquainted with factor analysis. But as he reads this book, he finds the two worlds well integrated. The interpenetration of econometric and psychometric thoughtways is a dominant characteristic of Dr. Harder's text. The mathematically naive reader might at first claim that the book is not for him. For some of the sections, he is correct. But Dr. Harder's presentation of certain techniques, such as scaling, could be excellent introductions for the untutored reader. The widespread fear of mathematical symbols can frequently be overcome by meeting them in a familiar substantive context. The unsophisticated mathematician should be intrigued v

PREFACE by discovering how common but vague topics take on a new shape if an effort is made to translate them into mathematical language. Even if the mathematical analysis cannot be followed to its end, the early parts of most of the sections and chapters will be of interest to quite a large audience. There is an old saying that a city that introduces buses later usually has a much better transportation system than the city that instituted them earlier. Seeing Dr. Harder's book from the American viewpoint, I feel that an analogy is appropriate. We began early with market research and developed a large group of practitioners; only tentatively do we dare to introduce them to new methods. Here is a European who speaks to a more recently formed audience. He can be bolder in offering them more advanced techniques. I am hopeful that this English translation will have a salutary feedback effect on the American marketing profession. New York. N.Y. October 1969 PAUL F. LAZARSFELD VI

TABLE OF CONTENTS PREFACE BY PAUL F. LAZARSFELD INTRODUCTION v 1 CHAPTER 1/ ADVERTISING MEDIA CALCULATIONS 1. Introduction 2. Media Selection based on Reach 3. Cost Analysis 4. Utilization of the Selected Media 5. Selection and Utilization of Media in a Single Step 6. Multiple Utilization of a Medium 7. Cumulation of Reach using Several Media 8. Forecasting Cumulation 6 6 7 16 23 38 40 52 56 CHAPTER II / ELEMENTARY FORECASTING CALCULATIONS 1. Introduction 2. Linear, Static Forecasts, with an Example 3. Computation of Regression Coefficients 4. Computing Requirements 5. Forecasts, using Time Functions 6. Difference Equations (applied as an Example to Election Forecasting) 7. Markov Chains 8. Application of Markov Chains to the Case of Competition between Brand Name Products 8.1. Markov Chains for more than two Products 8.2. Forecast of Market Share 60 60 60 69 74 77 84 88 93 94 98 VII

TABLE OF CONTENTS 9. Non-Stationary Markov Chains 99 10. Concluding Remarks on Forecasting 101 CHAPTER III / FACTOR ANALYSIS 105 1. Introduction 105 1.1. Fundamentals 105 1.2. Purpose of Factor Analysis 107 2. Metric Problems 108 2.1. Non-Response Problems of Scaling Techniques 109 2.2. Reference to the Metric Problem 113 2.3. Dispensing with the Use of Individual Scales 113 3. Characteristics of Results of Factor Analysis and Factor Forecasting 114 3.1. Computation of Correlation from Raw Data 114 3.2. Correlation Matrix 115 3.3. A Fundamental Task: Partitioning the Data Matrix 115 3.4. Factor Forecasting 121 3.4.1. Problem of Feed-Back 124 3.4.2. Analysis of an Individual Forecast 125 3.4.3. External Factor Forecasts 127 3.4.4. Concluding Remarks on Factor Forecasting 130 4. Computational Procedures of Factor Analysis 131 4.1. Calculation of Factor Loadings (Centroid Method) 131 4.2. Rotation 136 4.2.1. Interpretation of Factors 138 4.2.2. Rotation Calculations (Example) 141 4.3. Calculation of Factor Loadings (Component Analysis) 145 5. Calculation Requirements 147 5.1. Brief Discussion of the Optimization of Computational Requirements 150 5.2. Computational Requirements of Rotations 152 5.3. Total Computing Requirements 154 6. Method of Computation and Computing Requirements of Semantic Distance 155 CHAPTER IV / MARKET MODELS 158 1. Present Practices and Problems in Data Collection 158 VIII

T ABLE OF CONTENTS 2. Essential Nature of Market Models (Autonomy ofindicators or Model Dependency) 168 3. Image Aspects 171 4. A Psychological-Economic Market Model 173 5. Introduction to the Economic Aspects of Models 177 6. The Psychological Market Model as a Special Variant of an Economic Market Model 181 7. Problems Inherent in a Market Model 184 BIBLIOGRAPHY 193 IX