Herve Ngate Discussion Week 5: Chi-Square Cross Tabulation

Herve Ngatediscussion Week 5chi Square Cross Tabulation And Non Pa

Herve Ngate Discussion Week 5: Chi-Square, Cross Tabulation, and Non-parametric Association D.5.7.1 In Output 7.1: (a) What do the terms “count” and “expected count” mean? In Output 7.1, the term “count” refers to the number of cases measured or observed during data collection, while the term “expected count” indicates the results after computing data collected. (b) What does the difference between them tell you? According to Morgan et al. (2020), the difference between the “expected count” and the “count” demonstrates that the Chi-Square Tests must clarify if the discrepancies are systematic and find out if the relationship between variables is statistically significant.

D.5.7.2 In Output 7.1: (a) Is the (Pearson) chi-square statistically significant? Explain what it means. The asymptotic significance of the Pearson Chi-Square for two-sided shows the value of .056, indicating it is not statistically significant. Morgan et al. (2020) interpreted it as the fact that there is no certainty that the difference between fast and regular track is systematic in the student’s grades. (b) Are the expected values in at least 80% of the cells ≥ 5? How do you know? Why is this important? The Crosstabulation table shows that the minimum expected count is 14.1, indicating that more than 80% of the cells have an expected frequency ≥ 5. It is crucial because the number of expected counts will determine the use of the Pearson Chi-Square or Fisher’s exact test. When the expected count is at least 5, the Pearson Chi-Square is used, but Fisher's exact test is used if it is less than five (Morgan et al., 2020).

D.5.7.3 in Output 7.2 : (a) how is the risk ratio calculated? What does it tell you? The risk ratio is calculated by dividing the percentage of students who did not take algebra 2 by the percentage of students who did (Morgan et al., 2020). For example, the first risk ratio of 1.531 =70%/45.7% indicates that students with low math grades were 1.5 times more likely or not to take algebra 2. The second risk ratio of .533=30%/54.3% demonstrates that the students with high math grades were ½ times more likely or not to take algebra 2. (b) how is the odds ratio calculated, and what does that tell you? The odds ratio is calculated by dividing the ratio with the highest value by the one with the lowest. For example, the OR of 2.77 = 1.531/.553. This number shows that “the odds of failing to take algebra 2 are 2.77 times higher for those with low math grades than for those with high math grades" (Morgan et al., 2020, p.144). (c) how could information about the odds ratio be useful to people wanting to know the practical importance of research results? The odds ratio provides important information on the relationship's strengths between two nominal variables. People who want to know the practical importance of the research results could use the odds ratio to decide which variable to focus on and which one to disregard to measure the effectiveness of their studies. (d) what are some of the limitations of the odds ratio as an effect size measure? The odds ratio as an effect size measure seems to apply only to a limited number of fields like healthcare and prevention science. Therefore, a more significant number of industries might not be interested in using it. In addition, the odds ratio poses the challenge of deciding what represents a big ratio (Morgan et al., 2020), considering that cases could be complex to find or too famous for some studies.

D.5.7.4 Because father’s and mother’s education revised are 3-level variables with at least ordinal data, which of the statistics used in Problem 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? Morgan et al. (2020) posit that Kendall’s tau-b is most appropriate to measure the strength of the relationship between father’s and mother’s education because both variables are ordered. The Symmetric measures show that tau’s value is less than .001, indicating a statistically positive association between father’s and mother’s education. This could be interpreted as highly educated parents marrying each other and less educated parents marrying among themselves. Tau-b considers variables ordered to measure the relationship's strengths among variables, whereas Cramer’s V will be effective only if variables are nominals with three or more levels.

D.5.7.5 In Output 7.4: (a) How do you know which is the appropriate value of eta? The Directional Measures show two values of eta: One for math courses taken with a value of .328 and one for academic track with a value of .419. The appropriate value of eta would be .328 because math courses taken as a dependent variable are the ones being viewed (Morgan et al., 2020). (b) Do you think it is high or low? Why? SPSS computes eta from zero to 1 (Morgan et al., 2020). Therefore, the eta value of .328 is relatively low even if it confirms an association between math courses taken and academic track. (c) How would you describe the results? The results could be described as “those in the fast track were more likely to take several or all the math courses than those in the regular track” (Morgan et al., 2020, p.149).

Paper For Above instruction

In the analysis of categorical data, statistical tools such as the chi-square test, cross-tabulation, risk ratios, odds ratios, and measures of association like phi, Cramér’s V, and Kendall’s tau-b are essential for understanding relationships between variables. This essay explores these concepts in detail, drawing on Morgan et al. (2020) to interpret their application in research contexts.

Firstly, understanding the fundamental terms of "count" and "expected count" in chi-square tests is crucial. "Count" refers to the actual observed frequency of cases within a category during data collection, while "expected count" is the frequency anticipated under the null hypothesis—assuming no association between variables. For example, if 70% of students are on a fast track, the expected count of students in a particular category would be based on this percentage applied to the total sample size. The difference between observed and expected counts indicates the extent of deviation from the null hypothesis and helps determine whether any association exists between variables.

The chi-square test assesses whether those deviations are statistically significant. In the specific output discussed, the Pearson chi-square value had a p-value of .056, exceeding the typical threshold of .05, thus indicating that the result is not statistically significant. This implies that there is no sufficient evidence to assert a systematic difference between groups, such as students in fast versus regular tracks concerning grades. Also, the appropriateness of the chi-square test depends on the expected cell counts; more than 80% of cells should have expected counts of at least 5 for the test's assumptions to hold. The analysis confirmed this criterion, with the minimum expected count being 14.1, validating the use of the chi-square test in this context (Morgan et al., 2020).

Moving to risk and odds ratios, these measures assess the strength of association between categorical variables. The risk ratio compares the probability of an event occurring between two groups. For instance, students in the low math grade group had a risk ratio of 1.531, suggesting they were 1.5 times more likely not to take algebra 2 compared to students in higher grades. Conversely, the risk ratio of 0.533 for high-grade students indicates they were about half as likely to avoid algebra 2. The odds ratio further quantifies this relationship by comparing the odds of an event across groups. Here, an odds ratio of 2.77 demonstrates that students with low math grades are approximately 2.77 times more likely to fail to enroll in algebra 2 than their high-grade counterparts (Morgan et al., 2020).

Beyond these ratios, measures of association such as phi, Cramer’s V, and Kendall’s tau-b help quantify relationships between categorical variables with different levels of measurement. Since parents’ education levels are ordinal, Kendall’s tau-b is most appropriate, as it accounts for the order of categories. The calculated tau-b value of less than .001 indicates a very weak but statistically significant positive association, implying that highly educated parents tend to marry each other, and similarly for less educated parents. Cramer’s V, suitable for nominal data, provides a different perspective and often yields different values because it does not consider order. These differences highlight the importance of selecting the appropriate statistic based on the nature of the variables involved.

Finally, eta coefficients evaluate the strength of association between independent and dependent variables. In the case of students’ course selections related to academic tracks, eta values of .328 for math courses and .419 for academic track suggest a moderate association. However, from a practical standpoint, an eta of .328 representing only roughly 11% of variance explained indicates a relatively low practical significance, despite a statistically significant association. The findings suggest that students in fast tracks are more likely to take comprehensive math courses, reinforcing the relationship between academic pathways and subject choices (Morgan et al., 2020).

In conclusion, these statistical measures provide valuable insights into the nature and strength of relationships within categorical data. Correct interpretation of "count" versus "expected count," significance levels of chi-square tests, and measures like risk ratios, odds ratios, and association statistics are fundamental for rigorous research analysis. Selecting appropriate measures based on data levels and understanding their limitations allows researchers to draw meaningful and accurate conclusions from their data, fostering deeper understanding across varied research fields.

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