Chi-Square (χ2) test of Independence

Exhibit 32.22 Computation of expected score (Example — shampoo data, from Exhibit 1.8, Chapter Brand Sensing).

The chi-squared test is used to determine whether there is a significant difference between the expected frequencies and the observed frequencies in one or more categories.

Take for instance the data on the image rating (top2box score) of shampoo brands in Exhibit 32.22. To understand how her brand is perceived, the brand manager wants to know the attributes for which Organics is rated higher, compared to other brands, and those for which it is rated lower.

To begin the brand manager should compute the expected rating or score:

$$Expected Score=\frac{AVG(Brand Score)×AVG(Attribute Score)}{AVG(All Score)}$$ $$ = \frac{AVG(Column)×AVG(Row)}{AVG(All)}$$

Note: Total scores may be used, instead of average score.

You may recall from Chapter Brand Sensing that profile (Profile = Actual Score – Expected Score) provides an understanding of the features that distinguish one brand from another. While a big brand like Organics is rated high on all attributes, the aim of image profiling is to know which of these attributes distinguish it from other brands.

Chi-Square (χ2) test of independence is a universal metric that standardizes the data so that it becomes comparable across data sets of different magnitude.

$$Chi˗square \,statistic:\,χ^2 = \sum\frac{(Observed-Expected)^2}{Expected}=\sum\frac{Profile^2}{Expected}$$

For Organics on the attribute nourishes roots:

$$Chi˗square \,value \,of \,the \,cell=\frac{(Observed-Expected)^2}{Expected}=\frac{(35-24)^2}{24}=4.9$$

For the entire data set, sum of these quantities over all of the cells is the Chi˗square test statistic:

$$Chi˗square\, statistic \,χ2 = \sum\frac{(Observed-Expected)^2}{Expected}=230.4$$

Under the null hypothesis (there is no difference in the proportions across brands), this has approximately a chi-squared distribution whose number of degrees of freedom are:

$$df =degrees \,of \,freedom=(attributes-1)(brands-1)=(11 - 1)(9 -1)=80$$

If the test statistic is improbably large according to that chi-squared distribution, then one rejects the null hypothesis of independence.

In our shampoo example the p-value for χ2 = 230.4 and df = 80 is extremely small (p-value << 0.05). The null hypothesis is rejected. There are significant differences between the ratings across the brands.

The Chi-square test is often used in research studies to test the relationship between a variable pertaining to behaviour or attitude, with a variable pertaining to classification. For instance, the relationship between the consumption of a product with income level, location or age. The variables are cross tabulated, and then tested. The test will reveal whether or not a relationship exists between the two variables.

Note: Chi-square function in Excel is CHISQ.TEST. In SPSS the analysis falls under ‘Descriptive statistics’ -> ‘Crosstab’. Check the Chi-square box within the statistics pop-up page.

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