Module 16 – Dependent Samples T-Tests

Dependent Samples T-Tests

Overview

• Dependent samples t-tests go by several names:

  • Dependent samples t-tests

  • Paired samples t-tests

  • Matched samples t-tests

  • Repeated measures t-tests

  • t-test for dependent means

• These are all different names for the same statistical test.

• The most common names are dependent samples and paired samples.

• Dependent samples is preferred because it contrasts well with the independent samples t-test (discussed later).

Comparison to One-Sample T-Tests

• One-sample t-tests require knowing the population mean.

  • This is less restrictive than z-tests, which require both population mean and standard deviation.

  • However, it's still limited because population means are often unknown.

• The module addresses what to do when the population mean is unknown.

Determining the Appropriate Statistical Test

• The choice of statistical test depends on:

  • Types of variables: continuous or categorical

  • Comparison distribution

• Each test is designed for a specific study type.

• Flowchart for test selection:

  • IV and DV both continuous? If yes, correlation.

  • DV categorical? If yes, Chi-square test.

  • IV and DV not both continuous, DV not categorical:

    • More than 2 IVs? If yes, Factorial ANOVA.

    • Not more than 2 IVs:

      • 3+ groups? If yes, One-way ANOVA.

      • Not 3+ groups:

        • 1 group? Dependent samples t-test.

        • 2 groups? Independent samples t-test.

Requirements for Dependent Samples T-Test

• To use a dependent samples t-test:

  • The IV is categorical.

  • The DV is continuous.

  • There is only 1 IV.

  • There is only 1 group.

Dependent Samples T-Test vs. One-Sample T-Test

• One-sample t-test: 1 group with one set of scores compared to a known population mean.

• Dependent samples t-test: 1 group with two sets of scores, comparing one set of scores to the other.

• This involves a specific research design.

Research Design for Dependent Samples T-Test

• Typical design: obtain initial scores, expose subjects to an independent variable (intervention, drug, etc.), and then obtain scores again.

• This yields two scores from each participant, one score under each level of the IV.

Example: Memory Drug

• Scenario: Testing a new memory drug.

• Procedure:

  • Administer a memory test to establish a baseline score.

  • Have participants take the drug for a period.

  • Administer the memory test again (or a similar one).

  • Compare the results of the first test to the second test.

Calculation

• The formula is similar to the one-sample t-test, but uses difference scores.

• Difference scores: the difference between the scores collected in each level of the IV.

• Formula: $t = \frac{\overline{D} - 0}{s<em>{\overline{D}}} = \frac{\overline{D} - 0}{\frac{s</em>D}{\sqrt{n}}}$

  • $\overline{D}$ = mean of the difference scores

  • $s_{\overline{D}}$ = standard error of the difference scores

  • $s_D$ = standard deviation of the difference scores

  • $n$ = sample size

Example Data

• Memory drug example: words recalled before and after taking the drug.

• Calculate difference scores by subtracting scores for each subject consistently (e.g., After Drug – Before Drug).

• A positive difference score indicates more words recalled after taking the drug.

Using Difference Scores for One-Sample T-Test

• The difference scores can be used to perform a one-sample t-test.

• The mean of the 5 difference scores is 1 (denoted as $\overline{D}$ instead of $\overline{X}$).

• $s_{\overline{D}}$ is calculated as the standard deviation of the difference scores (SD) divided by $\sqrt{n}$.

• $s_{\overline{D}} = \frac{1.22}{\sqrt{5}} = 0.55$

The Zero in the Equation

• The 0 in the equation represents the null hypothesis that there is no difference between the means (H0: µD = µ1 - µ2 = 0).

• If there is no difference, the mean of the difference scores will be 0.

Steps for Conducting a Dependent Samples T-Test

• Calculate the difference scores.

• Perform a one-sample t-test using the difference scores.

Example: Revisiting the Moon Illusion

• Holway and Boring (1940) proposed that the moon illusion is caused by the difference in eye elevation when viewing the moon on the horizon versus at its zenith.

• Kaufman and Rock (1962) tested this by forcing subjects to view the zenith moon with eyes level and then with eyes raised.

• The goal is to see if eye elevation affects the moon illusion.

Hypotheses

• Research hypothesis: Ratings will differ between the eyes level and eyes elevated conditions.

• Null hypothesis: The means will be equal under the two viewing conditions.

Characteristics of the Comparison Distribution

• For a dependent samples t-test, the mean of the comparison distribution is 0.

• Given the data set (m16 examples):

  • $\overline{D}$ = 0.019

  • $s_D$ = 0.137

  • n = 10

• Standard error: $s_{\overline{D}} = \frac{0.137}{\sqrt{10}} = 0.043$

• Degrees of freedom: df = n – 1 = 9

Cutoff Sample Score

• Two-tailed test, alpha = 0.05, df = 9.

• Cutoff t-values: ±2.262

Sample Score on the Comparison Distribution

• Plug the numbers into the dependent samples t formula:

  • $\overline{D}$ = 0.019

  • $s_{\overline{D}}$ = 0.043

  • $t = \frac{(0.019 - 0)}{0.043} = 0.44$

Decision Regarding the Null Hypothesis

• Cutoff values: ±2.262

• Sample t-score: 0.44

• Fail to reject the null hypothesis because the sample’s t-score is not more extreme than the cutoff.

• Conclusion: No significant effect of viewing angle on size ratings, t(9) = 0.44, p > 0.05.

Summary

• Dependent samples t-tests are similar to one-sample t-tests, but use difference scores to calculate t.

T-Tests in Excel

• Can be performed using functions in the XLMiner Toolpak.

• Steps:

  • Go to the Home ribbon.

  • Click on the add-ins button.

  • Search for xlminer.

  • Add the xlminer analysis toolpak.

  • Choose ‘t-Test: Paired Two Sample for Means’.

Filling Out the Excel Tool Box

• Select the cells with the data for both samples.

• Hypothesized mean difference is always 0 (testing difference scores).

• If column headers are selected, check the 'Labels' box; otherwise, leave unchecked.

• Set alpha to 0.05.

• Specify where to output the results.

Excel Output

• The output provides:

  • Degrees of freedom (df)

  • t-score

  • Critical cutoff for t

  • p-value

• Excel provides critical cutoff and p-value for one-tail and two-tailed tests; use the appropriate one.

Example: Sleeping Pill

• Testing if a new sleeping pill affects sleep time.

Hypotheses

• Research hypothesis: The sleeping pill will affect people's average sleep times.

• Null hypothesis: The sleeping pill has no effect on the amount of time a person sleeps.

Using Excel for the Sleeping Pill Example

• With Excel, many steps can be skipped.

• Excel provides all necessary values directly.

• No need to manually find difference scores.

Decision Using Excel Output

• Use the t-Test: Paired Two Sample for Means tool in Excel.

• Input the data range, including column headers (and check the 'Labels' box if headers are included).

• Excel output provides the t-score and p-value.

• Determine whether to reject the null hypothesis based on the p-value.

Interpreting Results

• If p < 0.05, reject the null hypothesis.

• If p > 0.05, fail to reject the null hypothesis.

• Example: t(9) = -1.5, p = 0.168; since p > 0.05, the sleeping pill has no significant effect on sleep.

Significant Results

• If the t-stat is significant, examine the means for each variable to determine which is higher.

• This indicates whether there was a significant increase or decrease.

Additional Practice

• A third data set is provided in the example file for more practice.