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CHI-SQUARE, CROSS TABULATION, AND NON-PARAMETRIC ASSOCIATION
Chi-Square, Cross Tabulation, and Non-Parametric Association
Karli Bryant
BUSI 820 Quantitative Research Methods
November 26th, 2023
Discussion Board 5
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CHI-SQUARE, CROSS TABULATION, AND NON-PARAMETRIC ASSOCIATION
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? (b)
What does the difference between them tell you?
Understanding the terminology is an important factor in analyzing the information
provided. In Output 7.1, the terms “count” and “expected count” are utilized. The term count
represents the actual number of subjects in that cell whereas the expected count is what one
would expect to find in the cell given the marginal totals if there were no relationship between
variables (Morgan, Barrett, Leech & Gloeckner, 2020). Now with the basic understanding of
what these terms mean, what the difference between them tells the researcher can also be
determined. Essentially, if the expected count and the actual count are similar there is typically
not a significant difference in the data set whereas if there is a large difference it would be
expected to find a significant chi-square (Morgan, Barrett, Leech & Gloeckner, 2020). With this
information, a more well-rounded understanding of the data can be determined.
D.5.7.2 – In Output 7.1: (a) Is the (Pearson) chi-square statistically significant? Explain
what it means.
In order to determine if the Pearson chi-square is statistically significant, a few factors
have to be taken into consideration. In reviewing the P-value of 0.056, it can be determined that
the evidence does not support the assumption that fast-track students are more likely than
expected by chance to have a low or high math grade than regular track students (Morgan,
Barrett, Leech & Gloeckner, 2020). In addition, with the chi-square value of 3.65, it can be
interpreted that there is no significant difference between the two categories and the symmetric
measures table shows the strength of the relationship should be + or - .50 or more to indicate a
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CHI-SQUARE, CROSS TABULATION, AND NON-PARAMETRIC ASSOCIATION
large relationship; however, with the phi value of .22 a weak association and small effect size is
indicated (Morgan, Barrett, Leech & Gloeckner, 2020). Therefore, it can be seen that the fast
track and regular track are not significantly different.
(b) Are the expected values in at least 80% of the cells ≥ 5? How do you know? Why is this
important?
It can be determined that the expected values in at least 80% of the cells are greater than or
equal to 5 in viewing the footnote on the table in output 7.1 as it reflects no cells have an
expected count of less than 5 in the data (Morgan, Barrett, Leech & Gloeckner, 2020). This is
important because it indicates the valid condition for using the chi-square. The chi-square
requires a relatively large sample size because the expected count for at least 80% of cells should
be greater than five and a relatively even split of participants among the levels is often necessary
(Morgan, Barrett, Leech & Gloeckner, 2020). In addition, if the condition is not met, Fisher’s
exact test should be used instead of the chi-square.
D.5.7.3 – In Output D.5.7.2: (a) How is the risk ratio calculated? What does it tell you?
Understanding how the risk ratio is calculated, provides more insight into the calculation
itself. Specifically, when calculating the risk ratio for students with low math grades, the
researcher should divide the percentage of students who did not take Algebra 2 who have a low
math grade by the percentage of students who did take Algebra 2 who have a low math grade
(Morgan, Barrett, Leech & Gloeckner, 2020). Therefore, this would give the calculation of 70%
divided by 45% determining that students who do not take Algebra 2 are approximately 1.5 times
as likely to have a low math grade in comparison to those who take Algebra 2 (Morgan, Barrett,
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CHI-SQUARE, CROSS TABULATION, AND NON-PARAMETRIC ASSOCIATION
Leech & Gloeckner, 2020). With this information, a better understanding of risk is also
determined.
(b) How is the odd ratio calculated and what does that tell you?
Similarly, when understanding how the odd ratio is calculated more information can be
gathered. The odd ratio is calculated by taking the ratio of the risk ratios for students with low
math grades and high math grades which in this case tells the researcher that one student is
approximately three times as likely to get low grades as high grades if they did not take Algebra
2 (Morgan, Barrett, Leech & Gloeckner, 2020). With this interpretation, a more informed
decision about the importance of taking Algebra 2 can be determined for students.
(c) How could information about the odds ratio be useful to people wanting to know the
practical importance of research results?
Information about the odds ratio can be useful to people wanting to know the practical
importance of research results for one primary reason. Essentially, the odds ratio describes the
likelihood that individuals will have a certain outcome given a certain condition which can be
used to judge whether or not the odds justify the treatment (Morgan, Barrett, Leech & Gloeckner,
2020). Being able to justify what type of intervention or treatment is being applied is often
necessary to ensure good decisions are being implemented.
(d) What are some of the limitations of the odds ratio as an effect size measure?
There is one primary limitation of the odds ratio as an effect size measure. Unfortunately,
there is not a specific standard to determine what is a large odds ratio that has been agreed upon
(Morgan, Barrett, Leech & Gloeckner, 2020). Due to the lack of agreement, there is no set
standard to rule by which could cause some discrepancies.
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CHI-SQUARE, CROSS TABULATION, AND NON-PARAMETRIC ASSOCIATION
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 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?
Phi and Cramer’s V provide a test for both statistical significance and information about the
strength of the association between two categorical variables such as being used as measures of
the effect size (Morgan, Barrett, Leech & Gloeckner, 2020). If a 2 x 2 cross-tabulation is being
examined then phi is the appropriate statistic whereas for larger cross-tabulations Cramer’s V
should be utilized (Morgan, Barrett, Leech & Gloeckner, 2020). In the example that father’s and
mother’s education revised are 3-level variables with at least ordinal data Kendall’s tau-b would
be the most appropriate statistic to measure the strength of the relationship. Kendall’s tau-b
should be utilized because both mother’s education and father’s education are ordered variables
and Cramer’s V and phi treat the cross-tabulated variables as if they were nominal even if they
are ordered (Morgan, Barrett, Leech & Gloeckner, 2020). This key difference is why Cramer’s V
or phi would not be a good choice.
D.5.7.5– In Output 7.4: (a) How do you know which is the appropriate value of eta?
When interpreting the appropriate value of eta, it can be seen that the eta with math
courses taken as the dependent variable is the appropriate selection because gender is being used
to predict math courses taken instead of the reverse (Morgan, Barrett, Leech & Gloeckner, 2020).
Reversing the values would lead to illogical results.
(b) Do you think it is high or low? Why?
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CHI-SQUARE, CROSS TABULATION, AND NON-PARAMETRIC ASSOCIATION
Gender does not explain very much variance ,as it is .11, in how many math courses are
taken; however, in terms of being high or low it is average to large using Cohen’s criteria
(Morgan, Barrett, Leech & Gloeckner, 2020).
(c) How would you describe the results?
Depending on what results are being taken into consideration, the results can be described
in a few different ways. However, simply put, the results can be interpreted to indicate that girls
are less likely to take more math courses than boys (Morgan, Barrett, Leech & Gloeckner, 2020).
Essentially, all of the results are in alignment with this interpretation.
References
Morgan, G. A., Barrett, K. C., Leech, N. L., & Gloeckner, G. W. (2020). IBM SPSS for
introductory statistics: Use and interpretation, sixth edition (6th ed.). Routledge.
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