Natasha Fort
BUSI 820 – Quantitative Analysis June 16, 2026 Discussion Forum D5.7
D.5.7.1 Count and Expected CountD.5.7.1.a Count and Expected Count
In a chi-square cross-tabulation, the count refers to the actual observed frequency of cases within a cell. The expected count represents the frequency that would be expected if there were no relationship between the variables being analyzed (Barrett et al., 2026).
D.5.7.1.b Difference Between Count and Expected Count
The difference between the observed count and expected count indicates the extent to which the observed data differ from what would be expected by chance alone. Larger differences contribute more heavily to the chi-square statistic and suggest a stronger relationship between variables (Barrett et al., 2026).
D.5.7.2 Pearson Chi-SquareD.5.7.2.a Statistical Significance
A statistically significant Pearson chi-square indicates that the observed relationship between the variables is unlikely to have occurred by chance. When the significance value (p) is less than .05, researchers reject the null hypothesis of independence and conclude that a relationship exists between the variables (Barrett et al., 2026).
D.5.7.2.b Expected Cell Values
The assumption for chi-square analysis requires that at least 80% of the expected cell frequencies be 5 or greater. SPSS reports this information in the chi-square output table. This requirement is important because small, expected frequencies can distort the chi-square statistic and reduce the validity of the results (Barrett et al., 2026).
D.5.7.3 Risk Ratio and Odds RatioD.5.7.3.a Risk Ratio
The risk ratio is calculated by dividing the probability of an outcome occurring in one group by the probability of the outcome occurring in another group. It indicates how much more likely one group is to experience a particular outcome compared to another group.
D.5.7.3.b Odds Ratio
The odds ratio compares the odds of an event occurring in one group to the odds of the same event occurring in another group. An odds ratio greater than 1 indicates increased odds, whereas a value less than 1 indicates reduced odds.
D.5.7.3.c Practical Importance
The odds ratio helps researchers evaluate practical significance by showing the magnitude of a relationship. Even when results are statistically significant, the odds ratio helps determine whether the effect is meaningful in real-world settings.
D.5.7.3.d Limitations of Odds Ratios
One limitation of odds ratios is that they can overestimate effect sizes when outcomes are common. Additionally, odds ratios may be more difficult for non-statistical audiences to interpret than probabilities or percentages.
D.5.7.4 Strength of RelationshipBecause father’s education revised and mother’s education revised are ordinal variables with three levels, Kendall’s tau-b is the most appropriate measure of association. Unlike Phi and Cramer’s V, which are designed primarily for nominal data, Kendall’s tau-b incorporates the ordering of categories and measures both the strength and direction of association (Barrett et al., 2026).
Tau-b and Cramer’s V often produce different values because tau-b considers ordinal ranking information while Cramer’s V treats categories as nominal without considering their natural order. Therefore, tau-b provides a more informative measure for ordinal variables.
D.5.7.5 EtaD.5.7.5.a Appropriate Value of Eta
When interpreting eta, researchers typically focus on the eta value for the dependent variable, as it indicates the strength of the relationship between the independent and dependent variables.
D.5.7.5.b Magnitude of Eta
Whether eta is considered high or low depends on its numerical value. Values closer to 0 indicate weaker relationships, whereas values approaching 1 indicate stronger relationships. Researchers often compare eta values to established effect size guidelines when interpreting results.
D.5.7.5.c Description of Results
Eta measures the strength of association between a categorical independent variable and a scale-dependent variable. A larger eta suggests that group membership explains a greater proportion of variability in the dependent variable. Therefore, eta provides valuable information regarding the practical significance of group differences beyond statistical significance alone.
ReferencesBarrett, K. C., Leech, N. L., Gloeckner, G. W., & Morgan, G. A. (2026). IBM SPSS for introductory statistics (7th ed.). Taylor & Francis.
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