Glimpses into Application of Chi-Square Tests in Marketing

By P.K. Viswanathan

Chennai-India

preamble:

In this article, an attempt is made to bring into sharp focus the use of  c²  in marketing function. By no means, the coverage is exhaustive. The aim is to make the reader appreciate the conceptual  framework of Chi-Square analysis through problem illustrations in marketing. The ideas presented in this article certainly can be extended to many decision situations in marketing that can fruitfully employ chi-square tests.

 Contents: 2. Chi-Square Test-Goodness of Fit 3. Chi-Square Test of Independence

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1. Chi-Square (c²) Analysis- Introduction

Consider the following decision situations:

1) Are all package designs equally preferred? 2) Are all brands equally preferred? 3) Is their any association between income level and brand preference? 4) Is their any association between family size and size of washing machine bought? 5) Are the attributes educational background and type of job chosen independent? The answer to these questions require the help of Chi-Square (c²) analysis. The first two questions can be unfolded using Chi-Square test of goodness of fit for a single variable while solution to questions 3, 4, and 5 need the help of Chi-Square test of independence in a contingency table. Please note that the variables involved in Chi-Square analysis are nominally scaled. Nominal data are also known by two names-categorical data and attribute data.

The symbol c² used here is to denote the chi-square distribution whose value depends upon the number of degrees of freedom (d.f.). As we know, chi-square distribution is a skewed distribution particularly with smaller  d.f. As the sample size and therefore the d.f. increases and becomes large, the c² distribution approaches normality.

c² tests are nonparametric or distribution-free in nature. This means that no assumption needs to be made about the form of the original population distribution from which the samples are drawn. Please note that all parametric tests make the assumption that the samples are drawn from a specified or assumed distribution such as the normal  distribution.

For a meaningful appreciation of the conditions/assumptions involved in using chi-square analysis, please go through the contents of hyperstat on chi-square test meticulously.

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2. Chi-Square Test-Goodness of Fit

A number of marketing problems involve decision situations in which it is important for a marketing manager to know whether the pattern of frequencies that are observed fit well with the expected ones.  The appropriate test is the c² test of goodness of fit. The illustration given below will clarify the role of c² in which only one categorical variable is involved.

Problem: In consumer marketing, a common problem that any marketing manager faces is the selection of appropriate colors for package design.  Assume that a marketing manager wishes to compare five different colors of package design. He is interested in knowing which of the five is the most preferred one so that it can be introduced in the market. A random sample of 400 consumers reveals the following:

 Package Color preference by  Consumers Red 70 Blue 106 Green 80 Pink 70 Orange 74 Total 400

Do the consumer preferences for package colors show any significant difference?

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Solution: If you look at the data, you may be tempted to infer that Blue is the most preferred color. Statistically, you have to find out whether this preference could have arisen due to chance. The appropriate test statistic is the c² test of goodness of fit.

Null Hypothesis: All colors are equally preferred.

alternative Hypothesis: They are not equally preferred

 Package Color Observed Frequencies (O) Expected Frequencies (E) Red 70 80 100 1.250 Blue 106 80 676 8.450 Green 80 80 0 0.000 Pink 70 80 100 1.250 Orange 74 80 36 0.450 Total 400 400 11.400

Please note that under the null hypothesis of equal preference for all colors being true, the expected frequencies for all the colors will be equal to 80. Applying the formula

,

we get the computed value of chi-square ( c²) = 11.400

The critical value of   cat 5% level of significance for 4 degrees of freedom is 9.488. So, the null hypothesis is rejected. The inference is that all colors are not equally preferred by the consumers. In particular, Blue is the most preferred one. The marketing manager can introduce blue color package in the market.

3. Chi-Square Test of Independence

The goodness-of-fit test discussed above is appropriate for situations that involve one categorical variable. If there are two categorical variables, and our interest is to examine whether these two variables are associated with each other, the chi-square( c² )  test of independence is the correct tool to use. This test is very popular in analyzing cross-tabulations in which an investigator is keen to find out whether the two attributes of interest have any relationship with each other.

The cross-tabulation is popularly called by the term “contingency table”. It contains frequency data that correspond to the categorical variables in the row and column. The marginal totals of the rows and columns are used to calculate the expected frequencies that will be part of the computation of the c² statistic. For calculations on expected frequencies, refer hyperstat on c² test.

Problem: A marketing firm producing detergents is interested in studying the consumer behavior in the context of purchase decision of detergents in a specific market. This company is a major player in the detergent market that is characterized by intense competition.  It would like to know in particular whether the income level of the consumers influence their choice of the brand. Currently there are four brands in the market. Brand 1 and Brand 2 are the premium brands while Brand 3 and Brand 4 are the economy brands.

A representative stratified random sampling procedure was adopted covering the entire market using income as the basis of selection. The categories that were used in classifying income level are: Lower, Middle, Upper Middle and High. A sample of 600 consumers participated in this study. The following data emerged from the study.

Cross Tabulation of Income versus Brand chosen (Figures in the cells represent number of consumers)

 Brands Brand1 Brand2 Brand3 Brand4 Total Income Lower 25 15 55 65 160 Middle 30 25 35 30 120 Upper Middle 50 55 20 22 147 Upper 60 80 15 18 173 Total 165 175 125 135 600

Analyze the cross-tabulation data above using chi-square test of independence and draw your conclusions.

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Solution:

Null Hypothesis:  There is no association between the brand preference and income level (These two attributes are independent).

alternative Hypothesis: There is association between brand preference and income level  (These two attributes are dependent).

Let us take a level of significance of 5%.

In order to calculate the c²  value, you need to work out the expected frequency in each cell in the contingency table. In our example, there are 4 rows and 4 columns amounting to 16 elements. There will be 16 expected frequencies. For calculating expected frequencies, please go through hyperstat. Relevant data tables are given below:

## Observed Frequencies (These are actual frequencies observed in the survey)

 Brands Brand1 Brand2 Brand3 Brand4 Total Income Lower 25 15 55 65 160 Middle 30 25 35 30 120 Upper Middle 50 55 20 22 147 Upper 60 80 15 18 173 Total 165 175 125 135 600

Expected Frequencies (These are calculated on the assumption of the null hypothesis being true: That is, income level and brand preference are independent)

 Brands Brand1 Brand2 Brand3 Brand4 Total Income Lower 44.000 46.667 33.333 36.000 160.000 Middle 33.000 35.000 25.000 27.000 120.000 Upper Middle 40.425 42.875 30.625 33.075 147.000 Upper 47.575 50.458 36.042 38.925 173.000 Total 165.000 175.000 125.000 135.000 600.000

Note: The fractional expected frequencies are retained for the purpose of accuracy. Do not round them.

 Next-Calculation of Chi-Square previous

Calculation:

Compute

.

There are 16 observed frequencies (O) and 16 expected frequencies (E). As in the case of the goodness of fit, calculate this c²  value. In our case, the computed c²  =131.76 as shown below: Each cell in the table below shows (O-E)²/(E)

 Brand1 Brand2 Brand3 Brand4 Income Lower 8.20 21.49 14.08 23.36 Middle 0.27 2.86 4.00 0.33 Upper Middle 2.27 3.43 3.69 3.71 Upper 3.24 17.30 12.28 11.25

and there are 16 such cells. Adding all these 16 values, we get c²  =131.76

The critical value of c²  depends on the degrees of freedom. The degrees of freedom = (the number of rows-1) multiplied by (the number of colums-1) in any contingency table. In our case, there are 4 rows and 4 columns. So the degrees of freedom =(4-1). (4-1) =9. At  5% level of significance, critical c²  for 9 d.f = 16.92. Therefore reject the null hypothesis and accept the alternative hypothesis.

The inference is that brand preference is highly associated with income level. Thus, the choice of the brand depends on the income strata. Consumers in different income strata prefer different brands.  Specifically, consumers in upper middle and upper income group prefer premium brands while consumers in lower income and middle-income category prefer economy brands. The company should develop suitable strategies to position its detergent products. In the marketplace, it should position economy brands to lower and middle-income category and premium brands to upper middle and upper income category.