Chapter 8 -- Complete

Chapter 8: Comparing More Than Two Proportions

Introduction

This chapter focuses on performing statistical inference specifically aimed at analyzing relationships among multiple categorical variables. It emphasizes examining interactions among these variables, thereby building a comprehensive understanding of dynamics that may not be visible when only comparing one or two proportions. By exploring interactions among multiple proportions, researchers can draw more nuanced conclusions about population characteristics and behaviors.

Going Beyond 2-Samples / Variables

Previously, studies predominantly covered inference based on 1-sample or 2-sample means and proportions, or regression methods tailored for simpler comparisons. This chapter expands the scope to include comparisons across multiple categorical variables and highlights the significance of interactions within these datasets. Additionally, the chapter adopts a theoretical approach, contrasting with the simulation methods elaborated in Chapter 5. This foundational understanding sets the stage for Chapter 9, which will dive deeper into mean group differences and their implications within various contexts.

Section 8.1: Comparing Multiple Proportions

Experiments in this domain often yield qualitative data, which can arise from various contexts, such as:

• Colors of M&Ms: Analyzing the distribution of 6 varieties of M&Ms.

• Airline Ticket Classes: Investigating consumer preferences among different ticket classes (coach, business, first).

• Survey Responses: Evaluating varying levels of agreement on survey statements from a target population. Data collection can be summarized by counting the number of measurements within each category, thereby facilitating analysis of multinomial experiments. In simpler cases with only two categories, a binomial model is employed, as covered in prior discussions.

Exact Multinomial Tests (Simulation)

Modeling one-sample proportions previously involved the analogy of a weighted coin flip. For scenarios where k > 2 categories are present, a related concept employs weighted dice. Data simulation can therefore be achieved by rolling n dice and counting the number of observations in each category (represented as n = n1, n2, ..., nk).The computation of p-values involves summing probabilities for numerous combinations of sample outcomes, specifically those less than or equal to the observed values. It is important to note that while exact p-value computations are feasible for small sample sizes, larger samples necessitate theoretical methodologies due to computational complexities.

Example: Ice Cream Preference at Pharmacy

A store owner seeks to understand ice cream flavor preferences among customers, considering options such as strawberry, chocolate, vanilla, and butterscotch. Historical data suggests proportions of 25% for strawberry, 40% for chocolate, 20% for vanilla, and 15% for butterscotch. Collecting data over a single day at a 10% significance level enables the evaluation of whether customer preferences have shifted.Using R for Analysis: To analyze this experiment as a multinomial one, utilize the xmultinomial function from the XNomial package. The function generates extensive outputs, detailing 4960 possible tables, culminating in the calculation of the necessary p-value based on simulated data.

Hypothesis Test (Solution)

Assuming that the sample is randomly gathered from a multinomial distribution, we establish the following hypotheses:

• Null Hypothesis (H0): Ice cream preference remains unchanged compared to prior years.

• Alternative Hypothesis (Ha): There exists a significant difference in preference. The test statistic is derived from observed data collections, leading to a simulated p-value of 0.5666. Since the p-value exceeds the 0.10 threshold, we fail to reject H0, indicating no significant evidence to support a difference in preferences at the 10% significance level.

The Hypergeometric Method (Fisher’s Exact Test)

Fisher’s Exact Test, explored earlier in Chapter 5 concerning 2x2 contingency tables, utilizes margins derived from the contingency analysis framework and applies hypergeometric distribution properties.This method computes p-values through combinations of corresponding frequencies across the contingency table, making it especially suitable for small sample sizes, where direct statistical inference may be tenuous. As sample sizes increase, the emphasis on relying on theoretical tests becomes apparent, enhancing the reliability of the analysis.

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

In this section, we focus on single-variable inferences to introduce the Chi-Squared distribution and the Goodness-of-Fit Test. Traditional methods such as z-procedures or t-procedures may not be adequately equipped to handle multiple categories' proportions; thus, this chapter emphasizes developing inference procedures rooted in the Chi-Square distribution.

Properties of the Chi-Square Distribution

The Chi-Square distribution possesses several distinct properties:

• The total area under the curve equals 1.

• It initiates from 0 and extends towards the right, nearing the horizontal axis without ever contacting it, illustrating a right-skewed form.

• As the degrees of freedom increase, the curve's shape shifts towards a normal distribution.

• Notation for critical Chi-Square values shifts according to degrees of freedom and significance levels.

Exploring the Goodness-of-Fit Test

A thorough example involving M&M color distribution versus the expected claims from the manufacturer will be analyzed. This involves setting hypotheses aimed at either confirming or refuting the observed color distribution data:

• Null Hypothesis (H0): The observed color distribution is consistent with manufacturer claims.

• Alternative Hypothesis (Ha): The observed color distribution differs significantly. For hypothesis formulation, proportions are established for k categories, providing clarity to the research question.

How to Reject H0

One must compare the observed frequencies (Oi) against the expected frequencies (Ei). The expected counts are derived based on hypothesized proportions rooted in understanding the population from which the sample originates.

Validity Conditions for Goodness-of-Fit Test

To apply the Goodness-of-Fit Test effectively, certain conditions must be adhered to:

• The data must constitute a simple random sample from a multinomial distribution.

• Ideally, the expected counts for all categories should be at least 5 to maintain the test's reliability.While these validity conditions are essential, they can sometimes be relaxed based on the size of the sample, providing flexibility in application.

Additional Notes

Testing through Goodness-of-Fit provides critical insights by comparing observed frequencies against expected frequencies, hence helping ascertain the reliability of the test and understanding its distribution characteristics. Degrees of freedom for the Chi-Square tests are generally calculated as (k - 1), where k represents the number of categories in question.

Example Analysis Using R Output

Revisit the practical application involving the M&M example, leveraging R output to identify the test statistic and derive corresponding p-value for substantive analysis.

Chi-Square Test for Independence

Engagement in analyzing two categorical variables' associations is executed via contingency tables. A relevant observational example comprises assessing the relationship between self-reported happiness levels against income brackets via data collected from the General Social Survey (GSS).

Conclusion

It is vital to exercise caution against misinterpreting statistical associations and underlying significance. Readers must recognize the impact of statistical validity and the lurking potential for Simpson’s Paradox, where trends appear in different groups of data but disappear or reverse when these groups are combined, thus reiterating the importance of thorough analysis and careful interpretation of results.