Statistics for the Behavioral Sciences

STATISTICS FOR THE BEHAVIORAL SCIENCES

Chapter 10: Testing Means - The Related-Samples t Test

Author: Gregory J. Privitera
Reference: Statistics for the Behavioral Sciences, 3rd Edition, SAGE Publishing, 2018.

Chapter Outline

Related and Independent Samples
Repeated-Measures Design
Matched-Pairs Design
Introduction to the Related-Samples t Test
The Test Statistic
Measuring Effect Size for the Related-Samples t Test
Advantages for Selecting Related Samples
APA in Focus: Reporting the t Statistic and Effect Size for Related Samples

Related and Independent Samples

Independent Sample:
- Participants in each group or sample are unrelated, being observed once in only one sample.
- To collect independent samples:
- Participants can be selected from two or more different populations.
- Participants can be selected from a single population and randomly assigned to different groups.
Related or Dependent Sample:
- Participants in each group or sample are related.
- Participants can be related in two ways:
- Observed in more than one group (repeated-measures design).
- Matched experimentally or naturally based on shared characteristics or traits (matched-pairs design).

Repeated-Measures Design

A research design in which the same participants are observed in each sample.
This is the most common related-samples design.

Types of Repeated-Measures Designs

Pre–Post Design:
- Type of repeated-measures design where researchers measure a dependent variable for participants before (pre) and after (post) a treatment.
- Limited to observing participants at two times.
- Example: Measuring athletic performance in athletes before and after a training camp.
Within-Subjects Design:
- Researchers observe the same participants across many treatments but not necessarily before and after a treatment.
- Example: In studying the effect of exercise on memory, participants take a memory test after aerobic exercise and again after anaerobic exercise.

Matched-Pairs Design

A research design in which participants are matched based on common traits or characteristics before being separated into two groups.
Limited to observing two groups where pairs of participants are matched.
Different but matched participants are observed in each treatment, and scores from each matched pair are compared.

Matching Types

Experimental Manipulation:
- Matching participants through manipulation involves measuring a trait or characteristic before matching.
- Example: Measuring intelligence and matching participants based on their scores.
Natural Occurrence:
- In quasi-experiments, participants are matched based on pre-existing traits or characteristics.
- Example: Matching participants based on genetics, such as biological twins.

Introduction to the Related-Samples t Test

Related-Samples t Test:
- A statistical procedure used to test hypotheses concerning two related samples selected from populations where the variance is unknown.
- Involves comparing mean differences between pairs of scores in a population to observed scores in a sample.
- Difference Score:
- The score obtained by subtracting two scores. In a related-samples t test, computed before calculating the test statistic, reducing error from different participants.

Understanding Errors

Between-Persons Error:
- Occurs due to differences between participants. Example: Different increases in scores among participants.
Within-Groups Error:
- Differences that arise within each group, even though the same treatment is given.
Between-Groups Effect:
- This is the effect we want to test—mean differences resulting from the treatments being applied.

The Test Statistic

The formula for the t statistic in a related-samples t test is as follows: $t = \frac{M_D - \mu_D}{s_{MD}}$
- Where:
- $M_D$ is the sample mean difference.
- $\mu_D$ is the population mean difference.
A larger t value suggests that the observed sample mean difference is less likely to have occurred under the null hypothesis.
The estimated standard error for difference scores is computed and used in the denominator.

Degrees of Freedom

The degrees of freedom for the related-samples t test is computed as: $df = n - 1$
- Assumptions for the Related-Samples t Test:
1. Normality:
- Assumes that the data in the population of difference scores are normally distributed.
1. Independence within groups:
- Assumes that difference scores arise from different individuals within each group or treatment.

Example 10.1: Related-Samples t Test (Repeated-Measures Design)

A study tests whether teacher supervision influences the reading time of elementary school children in two 6-minute reading sessions (one with a teacher present, the other without).
Hypotheses:
- H0: $\mu = 0$ —No mean difference in reading times between teacher present and absent.
- H1: $\mu \neq 0$ —There is a mean difference in reading times.

Step 2: Setting Criteria

Level of significance: .05
Degrees of freedom: 8 participants → $df = 8 - 1 = 7$
Critical values from t table: ±2.365

Step 3: Compute the Test Statistic

Compute mean, variance, and standard deviation for difference scores.

Mean, Variance and Standard Deviation calculations need to be shown in detail based on the specific data collected.

Compute the estimated standard error for difference scores.
Compute the test statistic using the formulas.

Step 4: Decision-Making

Compare the obtained t-value to the critical value:
If $t_{obt}$ exceeds the critical value, reject the null hypothesis.
Example Outcome: $t(7) = 2.804$ , which exceeds 2.365, indicating a significant effect of teacher presence on reading time (p < .05).

Example 10.2: Related-Samples t Test (Matched-Pairs Design)

Psychologists tested differences in introversion scores between older and younger twins using a scale from 12 to 60.
Hypotheses:
- H0: $\mu = 0$ —No mean difference in introversion scores.
- H1: $\mu \neq 0$ —There is a mean difference in introversion scores.

Step 2: Setting Criteria

Level of significance: .05
Degrees of freedom: 7 pairs → $df = 7 - 1 = 6$
Critical values from t table: ±2.447

Step 3: Compute the Test Statistic

Compute mean, variance, and SD for difference scores.
- Calculation formulas and results should be illustrated here.

Step 4: Decision-Making

Compare obtained t-value with critical value.
If $t_{obt}$ does not exceed critical value, retain the null hypothesis.
Example Outcome: $t(6) = 0.282$ , indicating no significant differences in introversion scores (p > .05).

Measuring Effect Size for the Related-Samples t Test

Three measures of effect size for the related-samples t test:
1. Cohen’s d:
- Common measure of effect size that assesses how many standard deviations the mean difference stands above or below the population mean difference stated in the null hypothesis.
- Estimated Cohen’s d can be calculated using appropriate formulas.
1. Eta-Squared:
2. Omega-Squared:

Proportion of Variance

An estimate of the proportion of variance in a dependent variable explained by treatment.
For example, $\omega^2$ can be calculated as follows: $\omega^2 = \frac{(t^2) - 1}{(t^2) + (df)}$
- Practical interpretation: 46% of variability in reading times can be explained by teacher presence.

Advantages for Selecting Related Samples

Selecting related samples can be more practical, allowing behaviors to be observed in the same participants before and after treatments (e.g., repeated measures) or matched participants in similar abilities (e.g., matched pairs).
Minimizes standard error by eliminating the between-persons source of error, thereby reducing estimates of standard error.
Increases statistical power by reducing error estimates, enhancing the value of the test statistic.

APA in Focus: Reporting the t Statistic and Effect Size for Related Samples

To summarize, report the test statistic, degrees of freedom, and p-value.
Means and standard error or standard deviations should be summarized either in a figure/table or text.
It is typically unnecessary to specify the type of t test used in results reporting.