Chi Square Analysis: Few insights and glances of some of its applications in the industry in Marketing; Some key learnings from the statistic : Gargi Ghosh (Roll: 12135)

Insights on Chi Square Statistic:
 The chi-square test is a non parametric statistical test used to examine differences with categorical variables. Chi-square statistics use nominal (categorical) or ordinal level data, thus instead of using means and variances, this test uses frequencies.
In SPSS, it can be easily found out by the sequence: Analyse --> Descriptive statistics --> Crosstabs. 

The value of the chi-square statistic is given by: c² _{= S (f observed - f expected)}² _{/ f expected,} where
_{f observed} is the observed frequency and _{f expected} is the expected frequency.
The expected frequency is calculated by: _{f expected = (row total * column total) / total frequency}

_{and, the degree of freedom is determined as: c}² _{df = (no. of rows – 1)*(no. of columns – 1)}

Generally the chi-squared statistic summarizes the discrepancies between the expected number of times each outcome occurs (assuming that the model is true) and the observed number of times each outcome occurs, by summing the squares of the discrepancies, normalized by the expected numbers, over all the categories.

Data used in a chi-square analysis has to satisfy the following conditions:

1. Randomly drawn from the population,
2. Reported in raw counts of frequency,
3. Measured variables must be independent,
4. Observed frequencies cannot be too small, and
5. Values of independent and dependent variables must be mutually exclusive.

There are two types of chi-square test for two similar but distinct circumstances:

·         The Chi-square test for goodness of fit which compares the expected and observed values to determine how well an experimenter's predictions fit the data.

·         The Chi-square test for independence which compares two sets of categories to determine whether the two groups are distributed differently among the categories.

Goodness of fit means how well a statistical model fits a set of observations. A measure of goodness of fit typically summarizes the discrepancy between observed values and the values expected under the model in question. Such measures can be used in statistical hypothesis testing, e.g., to test for normality of residuals, to test whether two samples are drawn from identical distributions.

The chi-square test for independence is used to determine the relationship between two variables of a sample. In this context, independence means that the two factors are not related.

Applications in industry:
Let us consider the following decision situations for a Marketer:

 1) Are all package designs equally preferred?

2) Are all brands equally preferred?

3) Is there any association between income level and brand preference?

4) Is there 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 requires the help of Chi-Square (c²) analysis.

The first two questions can be solved using Chi-Square test of goodness of fit for a single variable.

The solutions to questions 3, 4 and 5 need Chi-Square test of independence in contingency table.

Some Learnings:
 
Some disadvantages of chi-squared tests are that its value depends upon the degree of freedom of the crosstabs and the sample size. And it cannot detect the direction of an association. So one has to be careful while interpreting its value when comparing two different crosstabs with different degrees of freedom or with very different sample sizes.

 Chi-square distribution is a skewed distribution particularly with smaller degree of freedom (d.f). As the sample size, and therefore the d.f. increases and becomes large, the Chi Square  distribution approaches normality.

Chi Square Analysis is widely applied in a number of service industries, like, Retail, banking, Airlines, Healthcare, Education where a huge amount of data are dealt with and in most of the cases, the data are highly inter related with each other.

P.S.: Few of the above insights are adopted from: http://cnx.org/content/m13487/latest/