Discussion Thread: Chi-Square, Cross Tabulation, and Non-parametric Association

The reply must summarize the students findings and indicate areas of agreement, disagreement, and improvement. It must be supported with scholarly citations in the latest APA format and a corresponding list of references.

Below Is the post that you will respond to

Discussion Board 5

DB5 Chi-Square, Cross Tabulation, and Non-Parametric Association

D.5.7.1(a) What do the terms count and expected count mean?

            In cross tabulation, the term count is the actual number of observations in a sample that belong to a category.  Expected count is the projected frequency that would be expected in a cell, if the variables are independent.  In Output 7.1, 24 fast track students and 20 regular track students had low math grades.  The expected count for fast track and regular track were 19.9 and 24.1 respectively.

D.5.7.1(b) What does the difference between them tell you?  

            The differences between the observed counts and the expected counts tells the researcher which variables have the largest differences, which could indicate dependence.  The researcher can also compare the standardized remainders to see which variables have the largest difference between expected counts and the actual counts. 

D.5.7.2(a) Is the (Pearson) chi-square statistically significant? Explain what it means.

            If the probability is less than the preset alpha level, the results are statistically significant (Morgan et. al., 2020).  The Pearson chi-square in Output 7.1 is p = .056 and is not statistically significant.  The X2 value of 3.65 and degrees of freedom of 1 are associated with a p value of greater than 0.50.  This means that a chi-square value this large (or differences between expected and observed numbers this great or greater) would occur by chance between 50% and 75% of the time. 

D.5.7.2(b) Are the expected values in at least 80% of the cells 5? How do you know? Why is this important? 

           Yes, at least 80% of the cells are greater than 5.  As shown in the chi-square test, 0 cells have expected count less than 5 and the minimum expected count is 14.05.  If there are more than 20% of the cells with frequencies less than 5, then the condition for using chi-square is violated. 

D.5.7.3(a) How is the risk ratio calculated? What does it tell you?

            The risk ratio for Output 7.2 is calculated by dividing the percentage of students who did not take algebra 2 by the percentage who did take algebra 2.  The risk ratio for students with low math grades is 1.53 which tells us that students who had low math grades were about 1.5 times as likely to not take algebra 2 as they were to take algebra 2. The risk ratio for students with high math grades is .553 which tells us that students who had high math grades were about .55 times as likely to not take algebra 2 as to take it. 

D.5.7.3(b) How is the odds ratio calculated and what does that tell you?

            The odds ratio for Output 7.2 is calculated by dividing the risk ratio of students with low math grades not taking algebra 2 (1.53) by the students with high math grades not taking algebra 2 (.553).  The odds ratio is 2.77 which tells us the odds of students failing to take algebra 2 are 2.77 times higher for those who had low math grades as for those who had high math grades. 

D.5.7.3(c) How could information about the odds ratio be useful to people wanting to know the practical importance of research results?

            The odds ratio can be used to determine whether a particular exposure is a risk factor for a particular outcome, and to compare the magnitude of various risk factors for that outcome (Szumilas, 2010).  The 95% confidence interval (CI) is used to estimate the precision of the odds ratio.  The upper and lower bounds are either greater than 1.0 or less than 1.0.  If the odds ratio =1 then exposure does not affect odds of outcome.  If the odds ratio is >1, then the exposure is associated with higher odds of outcome.  If the odds ratio is <1 then exposure is associated with lower odds of outcome (Szumilas, 2010).

D.5.7.3(d) What are some of the limitations of the odds ratio as an effect size measure?

            There are no set standards for what represents a large ratio because the ratio may approach infinity if the outcome is very rare or common.  In addition, it would be inappropriate to interpret an odds ratio with 95% CI that spans the null value as indicating evidence for lack of association between the exposure and the outcome (Szumilas, 2010). 

D.5.7.4(a) Because fathers and mothers 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, Cramers V, or Kendalls tau-b?

            Kendall tau-b is most appropriate to measure the strength of the relationship between the ordinal variables fathers education and mothers education.

D.5.7.4(b) Interpret the results. Why are tau-b and Cramers V different?

            Kendalls tau-b is a nonparametric measure of the strength and direction of association that exists between two variables measured on at least an ordinal scale (Morgan et. al., 2020).  The tau-b in Problem D.5.7.3 is .572.  Since p is < .001 is clearly significant, and the effect size is large (tau-b = .572), there is a statistically positive association between fathers education and mothers education.  The Cramers V is different because it treats the variables as nominal despite the fact that theyre ordered. 

D.5.7.5(a) How do you know which is the appropriate value of eta?

            Eta is appropriate for a dependent variable measured on an interval scale.  It quantifies the percentage of variance in the dependent variable (math courses taken) that is explained by one or more independent variables (academic track).  It is also noted that eta is always a positive number (Morgan et. al, 2020). 

D.5.7.5(b) Do you think it is high or low? Why?

            The eta (.328) is medium to large.  The value for eta squared ranges from 0 to 1, where values closer to 1 indicate a higher proportion of variance that can be explained by a given variable.  For eta squared values, .01 = small effect size, .06 = medium effect size, and .14 and higher = large effect size (Emerson, 2019).  The eta squared in Output 7.4 is .11 and is therefore considered to have a medium to large effect size. 

D.5.7.5(c) How would you describe the results? 

            Students who are in fast track were more likely to take several or all the math courses than those in the regular track.