Comparison of Two Groups

CHAPTER 7: Comparison of Two Groups

7.1 PRELIMINARIES FOR COMPARING GROUPS

• Analysis in social and behavioral sciences often involves comparing two groups.
    - Example 1: Comparing mean income of men and women in similar jobs.
    - Example 2: Comparing proportions of Americans and Canadians in favor of certain gun control laws.

• Two types of comparison:
    - Means for quantitative variables
    - Proportions for categorical variables.

• Response Variable: The variable we measure, e.g., time spent on housework.

• Explanatory Variable: The variable we categorize by, e.g., sex (male vs. female).

Data Overview

• Table 7.1: Cooking and Washing Up Minutes per Day in Britain for Men and Women
    - Men: Sample Size = 1219, Mean = 23 minutes, Standard Deviation = 32 minutes
    - Women: Sample Size = 733, Mean = 37 minutes, Standard Deviation = 16 minutes

Binary and Bivariate Analysis

• Two groups compared lead to a binary variable (dichotomous).

• Bivariate statistical methods are applied, where:
    - Response Variable: Variable measured for comparison.
    - Explanatory Variable: Defines groups.
    - Example: Housework time is the response variable influenced by sex.

Dependent vs. Independent Samples

• Dependent Samples: Same subjects across samples (e.g., longitudinal studies).
    - Example: Examining pre- and post-treatment scores of the same individuals.

• Independent Samples: Subjects in different samples do not overlap.
    - Example: Comparing distinct groups of people from different sex categories.

Accuracy & Variability of Estimates

• Difference of Estimates: Includes calculations on population means.
    - For means $ext{Difference} = \bar{y_2} - \bar{y_1}$

• Estimated Standard Error for Difference:
     - $se = rac{s_1}{ ext{n}_1} + rac{s_2}{ ext{n}_2}$ (for independent samples)
     - Larger standard errors reflect greater variability across studies.

7.2 CATEGORICAL DATA: COMPARING TWO PROPORTIONS

• Understanding how to compare proportions inferentially is key.

• Example: Prayer's effect on coronary surgery outcomes studied among patients.
    - Randomly assigned to prayer and non-prayer groups to observe complications post-surgery.
    - Table 7.2: Structure of results in complications due to prayer
      - Prayer Group: Yes (315 Complications), No (289 Complications)
      - No Prayer Group: Yes (304 Complications), No (293 Complications)

• Sample Proportions:
    - $ilde{p_1} = rac{315}{604} ext{ and } ilde{p_2} = rac{304}{597}$( Prayer and No Prayer)

• Significance Testing for Proportions:
    - Requirements: Large sample sizes help approximate normal distributions, resulting in efficient analysis.

• Confidence Interval Construction for Differences:
    - $( ilde{p_1} - ilde{p_2}) ext{ ± } z(se)$ for four or larger observations in each category.

Prayer Study Example Continued

• Estimated Proportion Differences:
    - $ilde{p_1} - ilde{p_2} ext{ leads to a confidence interval.}$ E.g., for the difference we compute M1-M2 with given standard errors.

7.3 QUANTITATIVE DATA: COMPARING TWO MEANS

• Statistical inferences are focused on mean comparisons: $ext{μ1 and μ2}$.

• Use t-distribution for small sample means for robustness.

• Example: Time spent on housework comparing men vs. women with given data.
     - Required calculations often include total sample sizes, individual means, and standard deviations.

• Confidence Interval for Means:
    - $( ilde{y_2} - ilde{y_1}) ± t(se)$.