Dependent T-Tests

Independent vs. Dependent T-Tests

Independent Samples T-Test: Involves two different groups for comparison.
- Example: Half of the participants receive a treatment (e.g., Ozempic) while the other half serve as a control group.
- Design: Between-subjects design, comparing two distinct groups.
Dependent Samples T-Test: Involves the same group measured multiple times.
- Example: The same group of participants receives the treatment for six weeks and then is assessed again after a period without treatment.
- Design: Within-subjects design, comparing different measurements from the same individuals.

Definition: Tests the difference between two unrelated groups with respect to a variable.
Design Characteristics:
- Two groups are independent of each other (e.g., men vs. women).
- Requires comparative measures between the two groups.
Example:
- Study on intervention effectiveness:
- Half of third graders receive a reading intervention, while the other half does not.
- Scores are compared to determine intervention effectiveness.

Definition: Compares means from the same group at different times or under different conditions.
Design Characteristics:
- Same subjects are measured at both time points (like before and after an intervention).
- Reduces variability by controlling for individual differences.
Example:
- Scores from the same students tested for reading skills before and after an intervention.

Fewer participants required: Reduces the number of subjects needed for the study as each participant serves as their own control.
Controls for individual differences: Same individuals are compared, reducing the variation that might occur with different groups.

Potential for order effects: Changes in performance could be influenced by the order in which tests are taken (e.g., participants might perform better on the second test due to practice).
Limitations in some studies: Not all studies can use the same participants for both measures, particularly if they involve different treatments.

Null Hypothesis ($H_0$): There is no effect of the intervention on the measurements (e.g., drivers' test scores).
Research Hypothesis ($H_a$): The intervention improves the test scores significantly.

Collect Data:
- Obtain scores for the same participants before and after the intervention.
Calculate the Differences (D):
- Determine $D_i$ as the difference between each pair of scores (post-intervention - pre-intervention).
Summarize Your Data:
- Calculate the sum of differences ($ ext{Sum of } D = ext{Sum}(D_i)$).
- Calculate the sum of squared differences ($ ext{Sum of } D^2 = ext{Sum}(D_i^2)$).
Compute the t-Statistic:
- Use the formula:
  t = rac{ ext{Sum of } D}{ ext{Standard Error}}
Degrees of Freedom:
- For dependent samples: $df = n - 1$ where $n$ = number of participants.
Determine Critical Value:
- Compare calculated t-value with the critical value from t-distribution tables based on chosen alpha level (typically 0.05).

If obtained t-value is greater than critical value: Reject the null hypothesis.
If obtained t-value is less than critical value: Fail to reject the null hypothesis.

Data Collection Example:
- Scores from a driving test before and after a safe driving program:
  - Before: [10, 8, 9, 5, 6]
  - After: [11, 8, 10, 7, 9]
Step 1: Calculate Differences (D):
- D: [1, 0, 1, 2, 3]
Step 2: Compute Summary Statistics:
- Sum of D = 7
- Sum of $D^2$ = 1 + 0 + 1 + 4 + 9 = 15
Step 3: Compute t-value using the formula:
t = rac{7}{ ext{Standard Error}}
Step 4: Degrees of Freedom:
- $n = 5
  ightarrow df = 4$
Find Critical Value using alpha level of 0.05.

After analysis, results should clearly state whether the null hypothesis was rejected or not.
Report the t-value, degrees of freedom, and p-value:
- Example: "$t(4) = 4.47$, $p < 0.05$ -> Reject $H_0$ indicating significant intervention effect on driving test scores.