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D5.1 Interpreting Chi-Square
            In Problem 8.1 chi-square test was used to analyze math grades for male and female students to determine whether the relationship between gender and grades is statistically significant. However, this analysis makes no representation about the strength of that relationship (Morgan, Leech, Gloeckner, & Barrett, 2013). In this exercise, the math grades for females were expected to be lower than the actual results. Figure 8.1a. shows that the expected count for females with lower math scores was 24.1 as compared to the actual count of 20.  Similarly, the expected count for females with higher math scores was 16.9 as compared to the actual count of 21. This data signals a directional hypothesis which also suggests that the one-sided chi-square test with a significance of p = .046 is more applicable for this analysis than the two-sided chi-square value of p = .064.  The one-sided test indicates that the relationship between math grades and gender is statistically significant.
            In addition, the math grades – gender crosstabulation table (Figure 8.1a) illustrates that none of the cells have expected counts less than 5. The lowest expected value in the table is the expected count of 14.1 for male to have mostly A and B math grades. This shows that the condition for chi-square, that no more than 20% of the cells should have expected frequencies less than 5, was violated in this example. This is important because if the expected frequencies are less than 5, “the test of significance is too liberal” (Morgan et al., 2013).  If that were the case, the Fisher’s exact test would have been the more appropriate method to evaluate significance of the relationship between math grades and gender.
Figure 8.1a
D5.2 Measure Strength of the Relationship
  Because ‘father’s education revised’ and ‘mother’s education revised’ are at least ordinal data, which of the statistics used in Problem 8.3 is the most appropriate to measure the strength of the relationship; phi, Cramer’s V, or Kendall’s tau‐b? Explain your choice. Why are the Kendall’s tau‐b and Cramer’s V different?
            According to Morgan et al. (2013), there are several nonparametric measures of association that are good options to measure the strength of association between two variables. In the specific case of exercise 8.3, the author concluded that Kendall’s tau-b is the most appropriate statistic because ordinal data is being assessed. The alternatives like Cramer’s V and phi treat even ordinal variables as nominal, therefore they are not ideal options for this particular exercise. Figure 8.3a shows that the effect size of (tau-b = .572) is large and p is <.001 indicating statistical significance. While Cramer’s V shows a lower effect size (Cramer’s V = .0502) and statistical significance, it is as less desirable statistic because it treats the data values as nominal rather than ordinal.
Figure 8.3b
References
Morgan, G., Leech, N., Gloeckner, G., Barrett, K. (2013). IBM SPSS for Introductory Statistics. New York: Routledge.