ANOVA & Chi-Square Analysis (Part3)

Applied Business Statistics: ANOVA & Chi-Square (χ2) Analysis Part 3

Introduction

• Course Title: Applied Business Statistics ANOVA & Chi-Square (χ2) Analysis Part 3

• Instructor: Tamar Kugler, Associate Professor of Management and Organizations

• Institution: University of Arizona Eller MBA

Chi-Square (χ2) Test

Overview of Variables in Analysis

• Context: Smithe College case focuses on analyzing ordinal or nominal variables, with only GPA being quantitative.

• Objective: Determine analysis methods available for non-quantitative data.

Chi-Square Distribution

• Characteristics:

  • The chi-square distribution is always positive and positively skewed.

  • It is naturally right-tailed which simplifies the analysis as it eliminates the need for distinguishing between one-tailed or two-tailed tests.

Purpose and Type of Data for Chi-Square Tests

• Data Requirements:

  • Involves two variables typically nominal or ordinal with few discrete levels.

• Types of Chi-Square Tests Covered:

  1. χ2 Independence Test:

  • Purpose: To verify if there is a relationship between two non-quantitative variables.

  1. χ2 K Proportions Test:

  • Purpose: To determine if two or more proportions differ from each other.

Equations for Chi-Square Tests

• Formula:
  $<br>ext{χ}^2 = rac{(O - E)^2}{E} <br>$
  Where:

  • O = observed frequency count

  • E = expected frequency count under the null hypothesis of independence

• Null Hypotheses:

  • χ2 Independence Test: $H_0$: the two variables are independent/unrelated

  • χ2 K Proportions Test: $H0$: $p1 = p2 = p3 = … = p_t$

Examples of Chi-Square Tests

• χ2 Independence Test Example: Analyzing the relationship between income level and educational level (both ordinal).

• χ2 K Proportions Test Example: Comparing the proportion of women across five different professions.

Considerations for Appropriate Data

• Each person/object in the analysis must be counted only once.

• At least one expected frequency in each cell must be present, with no more than 20% of expected counts being less than 5.

• The proportions test is a specialized case of the independence test when one of the variables is dichotomous.

Chi-Square Independence Test

Purpose

• To investigate whether observed patterns in frequency counts are systematic and arise from true relationships rather than random variation.

Example Case of Cities and Status

• Qualitative Status Variable: Includes three levels - did not enroll, enrolled but did not stay, enrolled and stayed.

• Observation: Notable differing patterns observed, especially for Houston where enrollment numbers deviate significantly from other cities.

General Procedures for Performing a Chi-Square Test

1. Create a Pivot Table: Calculate frequency counts for two variables. These comprise the observed counts.

2. Matrix Size Requirements: Ensure at least a 2 x 2 table; more rows/columns are acceptable as long as sufficient data points are available for each cell.

3. Expected Counts: Calculate what frequency counts would be if the two variables were unrelated, preserving row/column sums.

   • Formula:
     $<br>E = rac{(row ext{ sum}) imes (column ext{ sum})}{n} <br>$

4. Comparison:

   • Small difference in counts indicates no significant relationship (p ≥ α = 0.05).

   • Large difference indicates significant relationship (p < α = 0.05).

Theoretical Framework Behind the Chi-Square Test

Setting Up Hypotheses

• Null Hypothesis (H0): The two variables are independent (not related).

• Alternative Hypothesis (H1): The variables are dependent (related).

Calculating Expected Frequencies

• Utilize the given null hypothesis to calculate expected counts.

• Comparative Framework: Expected counts should reflect assumed independence, and must be compared against observed counts to validate this assumption.

  • Expected Cell Frequency Formula:
    $<br>E<em>{ij} = n</em>{ij} p<em>{i} p</em>{j} = rac{(r<em>i)(c</em>j)}{n} <br>$

Performing the Chi-Square Calculation

• Use the following formula to derive the chi-square statistic:
  $<br>ext{χ}^2 = ext{Sum}\bigg( rac{(observation - expectation)^2}{expectation}\bigg) <br>$

• Degrees of Freedom: Calculated as $ext{d.f.} = (r-1)(k-1)$ Where:

  • r = number of row categories

  • k = number of column categories

Interpreting Chi-Square Results

• Increasing χ2 value indicates growing differences between observed and expected counts, suggesting lower likelihood of independence.

• If p-value < α: Null hypothesis rejected, indicating a significant relationship.

Essential Assumptions

Rules of Thumb for Expected Frequency Counts

• Each category must have an expected frequency of at least 1 (no empty cells).

• Not more than 20% of the categories should have expected values below 5.

Corrective Actions

• Strategies if rules aren’t met:

  • Increase sample size.

  • Combine or eliminate categories to ensure validity.

Assumption of Mutually Exclusive Categories

• Each observation is counted in only one category; no overlaps allowed in the analysis.

Chi-Square Analysis Templates in Excel

Utilizing Templates

• A designated Excel template is available containing multiple chi-square analysis formats.

• Procedure:

  1. Input observed counts into a highlighted red box in the template, which will auto-populate expected counts and results.

  2. Analyze output for chi-square value, p-value, and counts to gauge analysis validity.

Conclusion

• Templates facilitate efficient analysis for both independence tests and multiple proportions, reducing potential for errors and improving functionality of statistical examination.