ANOVA & Chi-Square (χ2) Analysis Part 4
Introduction
Purpose: To provide a comprehensive understanding of how chi-square tests are applied in business contexts, particularly for analyzing enrollment and retention data, and to demonstrate critical interpretation of statistical results.
Chi-Square Test #1: Relationship Between Two Nominal Variables
Objective: To statistically test for independence between two nominal (categorical) variables, such as 'City' (recruitment location) and 'Status' (enrollment outcome).
Null Hypothesis (H₀): City and Status are independent. This implies that the proportion of students in each status category is the same across all cities; there is no relationship between where a student is recruited and their enrollment outcome.
Alternative Hypothesis (H₁): City and Status are not independent (i.e., they are related). This suggests that the proportion of students in the status categories differs significantly across cities, indicating a relationship between recruitment city and enrollment outcome.
Variables:
City (6 categories): Represents the different geographical locations from which students are recruited, a nominal variable.
Status (3 levels): 'Offered admission but didn't enroll', 'Enrolled but didn't stay enrolled', 'Stayed'. This is also a nominal variable representing the final outcome of the admission process.
Analysis Step: Perform the chi-square test for independence using a statistical software or the provided Excel template. The test calculates a chi-square statistic comparing observed frequencies to expected frequencies under the assumption of independence.
The Chi-Square Statistic (\chi^2): Calculated as \chi^2 = \sum \frac{(Oi - Ei)^2}{Ei}, where Oi are the observed frequencies and E_i are the expected frequencies for each cell in the contingency table.
Sample Size Check: Essential assumptions for the validity of the chi-square test:
Rule 1: No expected count should be under 1. If this rule is violated, it indicates very sparse data in some cells, making the chi-square approximation unreliable.
Rule 2: Less than 20% of cells should have expected counts under 5. Violating this rule also compromises the accuracy of the \chi^2 approximation, suggesting categories might need to be pooled or an exact test used.
Results: The initial analysis indicated that all expected counts were valid, satisfying both rules, which means the chi-square test can be reliably interpreted.
Observed Counts: These are the actual frequencies of occurrences derived directly from the pivot table of the raw data, showing how many students from each city fall into each status category.
Key Observations: A preliminary review of the observed data revealed that Houston had a noticeably different enrollment pattern compared to other cities. This visual or intuitive observation necessitates a formal statistical test to determine if the difference is statistically significant or merely due to random sampling error.
Expected Counts Table: These counts represent the frequencies that would be anticipated in each cell if 'City' and 'Status' were truly independent. They are calculated as \frac{(\text{row total} \times \text{column total})}{\text{grand total}} for each cell. Notably, expected counts are often non-integer values because they are theoretical averages.
P-value: The p-value obtained was 0.0000 (effectively less than 0.0001), which is a statistically significant result. The p-value represents the probability of observing a chi-square statistic as extreme as, or more extreme than, the one calculated, assuming the null hypothesis of independence is true. A small p-value (typically less than a significance level \alpha of 0.05) indicates strong evidence against H₀.
Interpretation: Since p < 0.05, we reject H₀ in favor of H₁. This means we conclude that City and Status are not independent; there is a statistically significant relationship between the city of recruitment and the enrollment status.
Chi-Square Statistic: The calculated value was 111.27 with 10 degrees of freedom (df = (rows-1)(columns-1) = (6-1)(3-1) = 5 \times 2 = 10). A larger chi-square value generally indicates a greater discrepancy between observed and expected frequencies, suggesting a stronger relationship between the variables, especially when compared against the critical value for the specified degrees of freedom and significance level.
Interpretation of Results
Grand Total Comparison Difficulty: Directly comparing raw counts (grand totals) across cities is misleading because the total number of admissions (the base) varies significantly by city. For instance, a city with 100 total admissions cannot be directly compared to one with 1000 total admissions based solely on raw counts.
Column Percentages for Clarity: To make meaningful comparisons, column percentages are crucial. They normalize the data, expressing each status category as a percentage of the city's total admissions. For example, Houston had only 10% of those offered admission enroll and persist, significantly lower compared with 42% in San Francisco and an impressive 81% in Atlanta.
Visual Representations: Visuals like stacked bar charts or mosaic plots can effectively illustrate these percentages and differences. However, caution is advised: presenters must explain the figures thoroughly, especially percentages and base counts, to avoid misinterpretation or confusion among the audience.
Cities Compared by Percentages: The visual analysis using stacked bars often categorize results by color: Blue (did not enroll), Red (did not stay), Green (stayed). This allows for quick, intuitive comparison of enrollment and retention patterns.
Findings: The consistent finding across various analyses highlights that Houston stands out for its significantly lower enrollment and retention rates compared to other recruitment cities.
Chi-Square Test #2: Analysis Without Houston
Objective: To determine if the initial statistically significant relationship was primarily driven by Houston, or if a relationship still exists among the remaining cities.
Sample Size Check: Passed after removing Houston, ensuring the validity of the test for the reduced dataset.
Analysis concluded: Even after removing Houston, the chi-square test still resulted in statistical significance (p < 0.05).
Implication: This indicates that rejecting H₀ and asserting a relationship between city and status remains valid for the other cities as well, not just Houston. There are still significant differences in enrollment patterns among the remaining cities.
Chi-Square Test #3: Removing San Francisco
Variable Reduction: This test involved four cities (excluding Houston and San Francisco) versus three status levels.
Sample Size Check: Failed for the 20% rule. This means that more than 20% of the cells had expected counts under 5, which weakens the reliability of the chi-square approximation and suggests caution in interpreting the p-value.
Statistical Results: The p-value was above 5% (p > 0.05), indicating that with Houston and San Francisco removed, there was no statistically significant relationship between the remaining cities and enrollment status. For these specific cities, independence between city and status is established (or we fail to reject H₀).
Chi-Square Test #4: Proportions from k Independent Samples
Question: Do the enrollment rates (proportions of students who stayed) differ significantly across the k independent samples (cities)? This is essentially a test of homogeneity of proportions.
H₀: p1 = p2 = p3 = \dots = pk. This null hypothesis states that the true population proportions of enrollment (or staying) are the same across all k cities.
H₁: At least two proportions differ. This alternative hypothesis suggests that there is a significant difference in at least one pair of proportions among the cities.
P-value: The test demonstrated statistical significance (p < 0.05), leading to the rejection of H₀.
Observations: This result confirms that at least some cities have different enrollment proportions. Houston's enrollment proportions were distinctly different from others, further reinforcing its status as an outlier; patterns for other cities might still be unclear without further post-hoc analysis.
Chi-Square Test #5: Freshman Retention Rates
Focus: This test specifically compared the proportion of students that stayed after one year across various cities, a crucial metric for institutional success and student support effectiveness.
Sample Size Check: Passed, ensuring the test's reliability.
Results: Houston was again identified as a significant outlier, showing consistently lower retention rates. San Francisco's statistics also suggested potential concerns, indicating retention issues even if its initial enrollment might be better than Houston's.
Chi-Square Test #6: Yearly Comparisons
Analysis: This test aimed to determine if there was a significant difference in enrollment patterns between year one and year two samples.
H₀: Proportions are the same for both years. This hypothesis suggests temporal stability in the enrollment process.
H₁: Proportions differ. This hypothesis suggests a significant change in enrollment patterns between the two years.
Statistical Validation: A p-value of 0.36 was obtained, which is greater than the standard significance level of 0.05. This suggests no statistically significant difference between the enrollment proportions for year one and year two; therefore, the null hypothesis is not rejected.
Implications: The finding of no significant yearly difference is positive for future analysis, as it allows for the pooling of data across these two academic years, creating a larger and potentially more robust dataset for further studies without concerns about temporal shifts in patterns.
Summary and Recommendations
Conclude on the merits of city recruiting strategies based on analysis: The series of chi-square tests provides strong empirical evidence to inform strategic decisions regarding recruitment efforts.
Recommendations: Based on consistently low enrollment and retention rates, consider dropping Houston from future recruitment efforts. There is also a strong basis to consider dropping San Francisco as well, primarily due to its demonstrated low retention rates, even if its initial enrollment might be acceptable.
Reporting Findings & Methods: When presenting these findings, it's crucial to use data from both qualitative comparisons (e.g., anecdotal observations, student feedback if available) and quantitative dichotomous comparisons (using chi-square tests for categorical variables) to provide a holistic view. If applicable, ANOVA could be used for comparing means of continuous outcomes (e.g., GPA of students from different cities) to complement the chi-square analysis.
Final Task for Students: Replicate the analysis using the class dataset and complete homework assignments. This practical application reinforces understanding of findings and interpretations, emphasizing that statistical significance must always be coupled with practical implications for effective business decision-making.