9. Paired Samples t-Test

MODULE 9: PAIRED-SAMPLES T-TEST

Overview

The paired-samples t-test is a statistical method used to determine whether there is a significant difference between the means of two related groups. This test is particularly relevant in within-group designs where the same participants provide two separate scores, resulting in paired data.

Checking In

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Outline of Topics to Cover

  • Within-subjects designs: Focus on designs where the same subjects are measured multiple times.

  • Significance testing steps: Detailed protocols for conducting significance tests.

  • Study questions: Formulate guiding questions to foster critical thinking.

  • Analyses using jamovi and RStudio: Practical engagement with statistical software for analysis.

  • Standardized mean difference effect size: Understanding effect sizes to gauge the magnitude of findings.

  • Quick review: Summarizing key concepts for clarity.

Within-Group Research Designs

Definition

Within-group designs generate two or more scores from the same participant, allowing researchers to study changes over time or under different conditions.

Example

A common example is measuring pretest and posttest scores of a participant to observe effects of an intervention.

Dyadic Pairs

This method can also analyze scores from pairs of individuals, such as siblings or friends, which can reveal insights about relational influences in observations.

Randomized Trial Application

Procedure

In randomized trials, researchers collect baseline scores from participants before an intervention and follow-up scores afterward.

Scores Obtained

This results in two scores for the same individuals: one prior to the intervention and one after, allowing researchers to measure the impact of that intervention.

Experimental Application

Procedure

In experimental designs, subjects undergo two distinct experimental conditions, which might involve different treatments or stimuli.

Scores Obtained

Researchers analyze the scores of the same group across different conditions, aiming to understand how variations in stimuli affect outcomes.

Developmental Application

Longitudinal Study Design

This approach evaluates the same group over a period to understand developmental changes.

Goal

The primary goal is to examine how variables such as behavior or cognitive function evolve over time, providing insights into growth patterns and developmental milestones.

Dyadic Application

Definition

Dyadic applications involve scores from naturally occurring dyads, such as siblings or partners.

Example

Scores from Partner 1 and Partner 2 can be linked analytically to explore relational influences in the scoring outcomes.

Paired-Samples T-Test

Appropriate for:

  • Within-group designs with two related measurements from the same subjects.

Research Interest:

Investigations focus on assessing the differences between means from the same subjects to draw conclusions about effects of treatments or experiences.

Comparative Questions

Types of Research Questions

Research often involves questions that require comparison of groups or conditions to determine effects.

Designs

  • Independent-samples t-test: Utilized for comparing two different groups.

  • Paired-samples t-test: Used for comparing the same subjects under different conditions.

  • One-factor ANOVA: This analyzes differences across multiple groups based on one factor.

  • Repeated measures ANOVA: Used to compare three or more related groups.

Quick Review: Sampling Error

True Mean (μ)

The true mean is unknown and can only be accurately calculated from the entire population under study.

Null Hypothesis

The null hypothesis specifies a particular value of the true population statistic, which serves as the baseline for analysis.

Sampling Error

Different samples can yield varying means that may deviate from the true mean, affecting conclusions drawn from research.

Quick Review: Sampling Distribution

Definition

A sampling distribution represents the distribution of means from multiple hypothetical samples drawn from a population.

Central Limit

When the sample size is sufficiently large, the distribution of sample means tends to center around the null hypothesis mean, reflecting the properties of the central limit theorem.

Quick Review: Standard Error

Standard Error

This measures the average distance between sample means and the null mean, offering insights into sample variability.

Expectation

The standard error provides an estimate of variability expected across samples due to sampling error.

Quick Review: Normal Curve Rule

Rule of Thumb

Approximately 95% of sample means will fall within ±1.96 standard errors of the true mean, providing a guideline for evaluating unusual results.

Outliers

Around 5% of samples may be located outside this range, indicating potential anomalies or significant findings that warrant further exploration.

Steps in Significance Testing

  1. Design the study and collect relevant data.

  2. Specify hypotheses concerning the population.

  3. Define the standard of evidence required for significance.

  4. Compare observed data to the null hypothesis.

  5. Evaluate results to draw conclusions about the hypotheses.

Skin Color Satisfaction and Binge Eating Study

Study Objective

The study investigates the impact of skin color satisfaction on binge eating behaviors among Black girls, focusing on the critical period of adolescence.

Key Variables

  • Age: The longitudinal study collects data from participants aged 10 to 18.

  • Body Satisfaction: Assessed as a component of self-concept in relation to body image and weight concerns.

Research Question

The primary question is whether Black girls experience changes in body satisfaction from ages 10 to 18, which is central to understanding the development of their self-image.

Study Design

This employs a within-subjects design, implementing repeated measures on the same individuals over several years to track changes in their perceptions.

Mean Difference Statistic

Population Means

The analysis focuses on the changes observed from the mean body satisfaction at age 10 (μ10) compared to age 18 (μ18).

Hypothesis

Changes in body satisfaction are quantified through the mean difference statistic reflecting these comparisons.

Two-Tailed Hypotheses

Null Hypothesis (H0)

The null hypothesis states there is no significant change in body satisfaction over the evaluated ages.

Alternative Hypothesis (H1)

Contrarily, the alternative hypothesis posits that there exists a significant change (either increase or decrease) in body satisfaction levels.

Standard of Evidence

The study's data are rigorously compared against the null hypothesis to ascertain its plausibility. A significant mean difference suggests that the null hypothesis may be unlikely given the observed outcomes.

Significance Criteria

5% Significance Level

Evidence against the null hypothesis is obtained from rare sample means where p < .05 indicates a statistically significant finding. Sample means with p > .05 support the notion of the null hypothesis being plausible.

Analysis Summary

Participants

The study comprises 882 Black girls, tracked over time to gather meaningful data.

Sample Mean Difference

Findings reveal a sample mean difference of -3.05 with corresponding standard error, indicating statistically significant changes in body satisfaction levels throughout adolescence.

Jamovi Output Summary

Statistical results include detailed analyses of means, t-statistics, effect size calculated using Cohen's d, and confidence intervals, contributing to a robust understanding of the data.

Cohen's Effect Size Guidelines

  • Negligible: < |.20|

  • Small: |.20 - .50|

  • Moderate: |.50 - .80|

  • Large: > |.80|

Conclusion

Significant changes were identified in body satisfaction levels between ages 10 and 18, with a sample mean difference of approximately 3 points demonstrating statistical significance with a p-value < .001, indicating meaningful findings in the context of psychological and sociocultural factors.

Key Takeaways from RStudio Output

Outputs from RStudio provide the essential statistics necessary for conducting, interpreting, and understanding the results from paired-samples t-tests in the study of body satisfaction, empowering researchers with the tools to analyze their data effectively.