Summary

• Two different models for generating tables lead to the same analysis of two-way count data. In the first model, a SRS of size n is drawn from a population, and samples are classified according to two categorical variables having r and c possible values. In the second model, independent SRS of size are drawn from each of c populations, and each sample is classified according to a categorical variable with r possible values.
• The null hypothesis is that there is no relationship between the column variable and row variable. In the first model, this means that the two variables are independent. In the second, it means that the distributions of the row categorical variable are the same for all c populations.
• Expected counts are computed using the formula

  

  where is the ith row total and is the jth column total.

• The null hypothesis is tested by the chi-square statistic

  

• Under the null hypothesis, has approximately the distribution. The p-value for the test is where is a random variable having the distribution.

EXAMPLE: Each of 250 job applicants at a large firm was classified in two ways: (1) whether or not they got a job offer; and (2) their ethnic group.

Do the data indicate that receiving a job offer is independent of the ethnicity of the applicant? We obtain the following output from Stataquest:

           | col                                    
        row|       1       2       3 |  Total
-----------+-------------------------+-------
         1 |      24      13      18 |     55
         2 |     124      39      32 |    195
-----------+-------------------------+-------
      Total|     148      52      50 |    250
                                                        
          Pearson chi2(2) =   8.8688   Pr = 0.012

• Receiving a job offer is independent of ethnicity
• There is a relationship between receiving a job offer and ethnicity
• Test Statistic: = 8.869
• p-value: p-value = 0.012
• The is rejected at 0.05 level.