Chapter+8+--+Complete

Chapter 8: Comparing More Than Two Proportions

Introduction

• Focuses on inference related to more than two categorical variables.

• Aims to explore interactions between categorical variables in depth.

• This chapter shifts focus towards theoretical approaches for inference.

• Chapter 9 will address Mean Group Differences.

Section 8.1: Comparing Multiple Proportions

• Qualitative or categorical measurements include examples such as:

  • M&M colors (6 possible colors)

  • Airline ticket classes (coach, business, first)

  • Survey responses (strongly disagree to strongly agree)

• Such data can be recorded as counts across categories, representing a multinomial experiment.

• Binomial experiments are limited to two categories.

Exact Multinomial Tests (Simulation)

• For experiments with two categories, can model with a weighted coin flip.

• For k > 2 categories, use weighted dice to simulate data with specific probabilities for each category.

• Use frequency observations to create the sample statistic (n1, n2, …, nk).

• P-value is computed by finding combinations of probabilities that match or are lower than our case.

• Larger samples require more computational power, leading to a preference for theoretical methods.

Example: Ice Cream Preferences

• A local pharmacy's ice cream sales data is analyzed to verify if flavor preferences have changed from five years ago:

  • Previous proportions: Strawberry (25%), Chocolate (40%), Vanilla (20%), Butterscotch (15%).

  • Owner collects customer preference data over one day.

  • Plan to evaluate the evidence of preference change at a 10% significance level.

Using R

• The analysis is a multinomial experiment with one variable.

• Utilize the xmutlti formula from the XNominal package in R to perform inference.

• Example output shows 4960 different tables can be constructed; the observed situation has a probability of 0.002638.

Hypothesis Test — Ice Cream Preferences Solution

1. Assumption: Simple random sample from a multinomial distribution.

2. Hypotheses:

   • H0: Ice cream preference remains the same.

   • Ha: Ice cream preference differs.

3. Test Statistic: 7

4. P-value: Simulated p = 0.5666

5. Conclusions: p > 0.10

   • Fail to reject H0; insufficient evidence for a preference difference.

The Hypergeometric (Simulation) Method (Fisher's Exact Test)

• This test allows simulation for 2x2 tables and is based on a multivariate hypergeometric distribution.

• The test statistic utilizes contingency tables, with the p-value derived by summing probabilities of observed configurations.

• Larger samples often warrant theoretical tests instead due to computational complexity.

Section 8.3: Chi-Square Goodness-of-Fit Test

• Focuses on theoretical applications of inferring single qualitative variables with two or more categories.

• Traditional z-procedures and t-procedures do not fit scenarios with multiple categories; Chi-Square tests are applied.

• The Chi-Square distribution is introduced, leading to the Goodness-of-Fit Test.

The Chi Square Distribution

• Chi-square distribution is right-skewed with degrees of freedom (df).

• Notation: χ2(df),α indicates the critical χ2 value at significance level α.

• Basic properties:

  1. Total area under the curve = 1.

  2. Begins at 0 and extends to the right indefinitely.

  3. Right-skewed curve.

  4. As df increases, the curve resembles a normal distribution.

Exploring the Goodness-of-Fit Test Example

• M&M's are produced in claimed proportions; a sample is analyzed to see if the observed distribution aligns with the expected.

• Null Hypothesis (H0): M&M color distribution is accurate according to the company's claims.

• Alternative Hypothesis (Ha): Distribution differs from the claimed proportions.

Setting up the Problem

• Hypotheses expressed as:

  • H0: π1 = π1,0, π2 = π2,0, ..., πk = πk,0

  • Ha: At least one πi ≠ πi,0

• Aim is to show at least one category has a different proportion.

How to Reject H0

• Compare actual counts (Oi) with expected counts (Ei) under H0.

• Compute expected counts using the formula: E_i = n * π_i,0

Validity Conditions for Goodness-of-Fit Test

• The chi-square test statistic approximates chi-square distribution under specific conditions:

  1. Simple random sample.

  2. Sample size large enough that each expected frequency Ei ≥ 5.

• Alternative conditions can be applied regarding expected frequencies.

Additional Notes

• If expected counts are accurate, differences between observed and expected counts are minor (χ² ≈ 0).

• High differences indicate errors in expected counts resulting in a high χ² value.

• Degrees of freedom calculated as k - 1, where k = number of categories.

Hypothesis Test — M&M Example Continued

• R output used for hypotheses testing:

  1. H0: M&M color distribution matches company claims.

  2. Ha: Distribution differs.

  3. Test Statistic: χ² = 1.2468, 5 df

  4. P-value: p = 0.9403

  5. p > 0.05 ➔ Fail to reject H0; insufficient evidence for claim discrepancy.

Section 8.2: Chi-Square Test for Independence

• Introduces the concept of contingency tables to assess the relationship between two categorical variables.

• Questions focus on whether variables are associated.

• If two variables are independent, one variable provides no information on the other.

Example: Happiness and Income

• The survey investigates the relationship between happiness and family income.

• Assess observed versus expected counts under the null hypothesis to find suggestions of dependence.

Validity Conditions for Chi-Square Test for Independence

• Applies a similar form as the Goodness-of-Fit Test:

  1. Simple random sample.

  2. Sufficiently large sample size.

• Reject H0 if the test statistic is large enough.

Conclusion of Chi-Square Test for Independence Example

• Assess results from GSS to determine if perceived happiness is associated with income and understand significance levels.

• Highlight analysis where the chi-square tests provide mere procedures but require careful interpretation.