BUSI820D5

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School of Business, Liberty University Faizan Malik Week 5 Discussion Assignment Author Note: Faizan Malik I have no known conflict of interest to disclose. Correspondence concerning this article should be addressed to Faizan Malik: Fmalik@Liberty.edu D.5.7.1 In Output 7.1: (a) What do the terms “count” and “expected count” mean? (b) What does the difference between them tell you? D.5.7.1.a. In the context of statistics, "count" refers to the number of occurrences or observations of a specific category or value within a dataset and represents the frequency of that particular category or value in the data, whereas “expected count” refers to the theoretical value that indicates the number of occurrences we would expect to see in a specific category or cell of a contingency table if the data followed a certain distribution or if there were no association between the variables being analyzed (Morgan et al., 2020). D.5.7.1.b. The differences in the count and expected count can provide researchers insight into potential discrepancies or expected results for a particular hypothesis, indicating whether a particular value is more or less than would be expected by chance alone. D.5.7.2 In Output 7.1: (a) Is the (Pearson) chi-square statistically significant? Explain what it means. (b) Are the expected values in at least 80% of the cells ≥ 5? How do you know? Why is this important? D.5.7.2.a. To determine if a chi-square test is statistically significant, the 'Sig.' (significance) value under the 'Asymp. Sig. (2-sided)' column in the 'Chi-Square Tests' section should be less than the chosen alpha level (Sharpe, 2015). In Output 7.1, the footnote indicates that the chi-square test is statistically significant at the 0.05 alpha level, which means there is a significant relationship. Additionally, the absence of any cell with an expected count less than 5 is important for the validity of the chi-square test results. D.5.7.2.b. To determine if the expected values in at least 80% of the cells are greater than or equal to 5, we need to examine the "Expected" values in each cell of the contingency table. If at least 80% of the cells have expected values of 5 or more, then the condition is met. The importance of having expected values of 5 or more in at least 80% of the cells is

related to the validity of the chi-square test results, when the expected values are too low, the chi-square test may not be reliable, and the test's assumptions may be violated. D.5.7.3 In output D.5.7.2: (a) how is the risk ratio calculated? What does it tell you? (b) how is the odds ratio calculated and what does that tell you? (c) how could information about the odds ratio be useful to people wanting to know the practical importance of research results? (d) what are some of the limitations of the odds ratio as an effect size measure? D.5.7.3.a. The risk ratio is calculated by dividing the proportion of events (successes) in one group by the proportion of events in another group, indicating the relative risk or likelihood of an event occurring in one group compared to another (Morgan et al., 2020). A risk ratio value greater than 1 indicates that the event is more likely to occur in the first group, while a value less than 1 suggests that the event is less likely to occur in the first group compared to the second group. D.5.7.3.b. The odds ratio is calculated by dividing the odds of an event occurring in one group by the odds of the event occurring in another group, indicating the relative odds of an event occurring in one group compared to another (Morgan et al., 2020). Similar to the risk ratio, an odds ratio value greater than 1 indicates that the event is more likely to occur in the first group, while a value less than 1 suggests that the event is less likely to occur in the first group compared to the second group. D.5.7.3.c. The odds ratio provides a measure of the strength of the association between two variables. By comparing the odds of an event in different groups, researchers can assess the magnitude of the effect and its practical significance. Szumilas (2010) explains that odds ratios are often used in medical research and allow for the comparison of risks, logistical regression, and an understanding of the measure of association (Szumilas, 2010). D.5.7.3.d. Odds ratios do carry some limitations in that can often exaggerate the size of the effect compared to the relative risk or can even be mistaken for relative risk ratios and can be misleading when event rates are high (Deeks, 1998). D.5.7.4 Because the father’s and mother’s education revised are 3-level variables with at least ordinal data, which of the statistics used in Problem D.5.7.3 is the most appropriate to measure the strength of the relationship: phi, Cramer’s V, or Kendall’s tau-b? Interpret the results. Why are tau-b and Cramer’s V different? D.5.7.4.a. In Problem, D.5.7.3, where the father's and mother's education are 3-level variables with at least ordinal data, Cramer's V is indeed an appropriate statistic to measure the strength of the relationship between these variables. Cramer's V is an extension of the Phi coefficient, and it quantifies the strength of association between two categorical variables with ranges from 0 to 1, where 0 indicates no association, and 1 indicates a perfect association between the variables. Kendall's tau-b, however, is a non- parametric measure specifically designed to assess the strength of association between ordinal variables. It considers the rankings and the direction of the relationship between the variables, making it more suitable for ordinal data with multiple levels. D.5.7.5 In Output 7.4: (a) How do you know which is the appropriate value of eta? (b) Do you think it is high or low? Why? (c) How would you describe the results?

D.5.7.5.a. Morgan et al. (2020) explain that the Eta is utilized in instances where one variable is nominal and the other either normal or scale, with Eta being a measure of effect size used to quantify the strength of association between two categorical variables in a chi-square analysis (Morgan et al., 2020). In Output 7.4, the appropriate Eta value is the one where math courses are taken to serve as the dependent variable, as the focus is on examining how gender predicts the number of math courses taken, rather than the reverse. D.5.7.5.b. Generally, a larger value of Eta (closer to 1) indicates a stronger association between the variables, suggesting that they are more related or dependent on each other. Conversely, a smaller value of Eta (closer to 0) indicates a weaker association, suggesting that the variables are less related or independent of each other. Output 7.4 has an Eta of 0.328, indicating a low or weak association between variables. D.5.7.5.c. The results in Output 7.4 indicate that gender and the number of math courses taken have a weak association, meaning there is little dependence between variables. References Deeks, J. (1998). When can odds ratios mislead?: Odds ratios should be used only in case-control studies and logistic regression analyses. BMJ: British Medical Journal , 317 (7166), 1155. Morgan, G., Leech, N., Gloeckner, G., Barrett, K. (2020). IBM SPSS for Introductory Statistics (5th Ed.). New York, NY Sharpe, D. (2015). Chi-square test is statistically significant: Now what?. Practical Assessment, Research, and Evaluation , 20 (1), 8. Szumilas, M. (2010). Explaining odds ratios. Journal of the Canadian academy of child and adolescent psychiatry , 19 (3), 227. Hey Roger, great post! In your post, you stated, “Because odds ratios are calculated on dichotomous variables, this value can be used by researchers to give an idea of whether a specific treatment is more or less likely to provide an impact” and cited an example that involved additional variables that could impact math grades. This is accurate, as one key limitation of odds ratios is their sensitivity to rare events or imbalanced data. As you mentioned, when the outcome of interest is infrequent, odds ratios tend to overestimate the effect size, leading to inflated interpretations. In situations where the outcome is not well-balanced across groups, the odds ratio may not accurately reflect the true association between the variables, potentially leading to misleading conclusions. Deeks (1998) also explains that odds ratios do carry some limitations of often exaggerating the size of the effect compared to the relative risk or can even be mistaken for relative risk ratios and can be misleading when event rates are high (Deeks, 1998). Essentially, when event rates are high, the odds ratio may not accurately reflect the magnitude of the effect. In such scenarios, the odds ratio can become inflated, leading to an overestimation of the association between the variables. Schmidt and Kohlmann (2008) also explain that, while odds ratios provide information about the odds of an event occurring in one group compared to another, they do not directly convey the absolute risk or the probability of an event happening, which may be more meaningful in some contexts (Schmidt & Kohlmann, 2008). Absolute risk provides a clearer understanding of the actual likelihood of an event happening in each group, which is especially important in studies around medical decision-making and public health interventions.

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References Deeks, J. (1998). When can odds ratios mislead?: Odds ratios should be used only in case-control studies and logistic regression analyses. BMJ: British Medical Journal , 317 (7166), 1155. Morgan, G., Leech, N., Gloeckner, G., Barrett, K. (2020). IBM SPSS for Introductory Statistics (5th Ed.). New York, NY Schmidt, C. O., & Kohlmann, T. (2008). When to use the odds ratio or the relative risk?. International journal of public health , 53 (3), 165.

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