Study that involves multiple measures of the same variable taken on the same or matched subjects either under different conditions or over two or more time periods.
Example: repeated measurements are collected in a longitudinal study in which change over time is assessed.

Repeated Measures Designs

In a dependent-samples $$t$$-test, the initial analysis step involves reducing the two paired scores recorded for each participant into a single value known as a difference score. This is achieved by subtracting one score from the other. It does not matter whether the first set of scores is subtracted from the second or vice versa, provided the order of subtraction is applied consistently across all participants. Often, researchers choose the direction of subtraction so that positive or negative difference scores intuitively align with the expected outcome (e.g., subtracting pretest scores from posttest scores so that a positive difference indicates an increase). Once these difference scores are calculated, the dependent-samples $$t$$-test functions exactly as a one-sample $$t$$-test performed directly on these new values.

Difference Score

To illustrate a dependent-samples $$t$$-test, consider a pretest-posttest study evaluating a training program designed to improve calorie estimates. A sample of $$10$$ participants estimates calories in a cookie both before and after the training. Subtracting the pretest estimates from the posttest estimates yields a set of $$10$$ difference scores with a mean ($$M$$) of $$8.50$$ and a standard deviation ($$SD$$) of $$24.27$$. Using the one-sample $$t$$-test formula on these difference scores against a hypothetical population mean of $$0$$, the computed $$t$$ score is $$1.11$$. For a one-tailed test with $$9$$ degrees of freedom ($$10 - 1$$), the critical value is $$1.833$$. Because the calculated $$t$$ score ($$1.11$$) is less extreme than this critical value, the researcher fails to reject the null hypothesis, concluding that the training program does not significantly increase calorie estimates. The corresponding $$p$$-value is $$.148$$, which is greater than the standard $$.05$$ alpha level.

Example of a Dependent-Samples t-Test

When is it most appropriate to use a dependent-samples t-test in a research study?

The dependent-samples $$t$$-test, sometimes called the paired-samples $$t$$-test, is used to compare two means obtained from the same sample tested at two different times or under two distinct conditions. This statistical comparison is particularly appropriate for analyzing data from pretest-posttest designs or within-subjects experiments. Conceptually, it functions as a special case of the one-sample $$t$$-test.

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The $$t$$-test is the most frequently used null hypothesis test for evaluating the statistical relationship between two means. There are three primary variations of this procedure—the one-sample $$t$$-test, the dependent-samples $$t$$-test, and the independent-samples $$t$$-test—each designed to accommodate slightly different research designs.

t-Test

The one-sample $$t$$-test is a statistical procedure used to compare a single sample mean ($$M$$) against a specific, hypothetical population mean ($$\mu_0$$) that serves as an interesting standard of comparison. By defining a null hypothesis ($$\mu = \mu_0$$) and an alternative hypothesis ($$\mu \neq \mu_0$$), researchers use this test to determine if the sample data provide enough statistical evidence to conclude that the true population mean differs from the hypothesized value.

One-Sample t-Test

James, G., Witten, D., Hastie, T., & Tibshirani, R. (2013). An introduction to statistical learning (Vol. 112, pp. 3-7). New York: springer.

An Introduction to Statistical Learning with Applications in R

https://kpu.pressbooks.pub/psychmethods4e/

KPU Research Methods in Psychology - 4th American Edition

Dependent-Samples t-Test

The independent-samples $$t$$-test is a statistical procedure utilized to determine if there is a significant difference between the means of two distinct groups ($$M_1$$ and $$M_2$$). This analytical method is appropriate when the two samples are tested under varying conditions within a between-subjects experimental design, or when researchers are comparing pre-existing groups (such as extraverts and introverts) in a cross-sectional research design.

Independent-Samples t-Test

When presenting the outcomes of a $$t$$-test in a research manuscript, specific APA style formatting conventions must be followed. The results are typically written in the format: $$t(df) = \text{t-score}, p = \text{p-value}$$. For instance, a result might be reported as $$t(9) = -3.07, p = .01$$. In this structure, the letters $$t$$ and $$p$$ must be italicized. The degrees of freedom ($$df$$) are placed in parentheses immediately following the $$t$$ symbol, and must be presented as a whole number without any decimal remainder. Finally, both the calculated $$t$$ score and the $$p$$-value are rounded to two decimal places.

Reporting t-Test Results in APA Style

The analysis of variance (ANOVA) is the most common null hypothesis test used when researchers need to compare the means of more than two groups or conditions. It serves as a necessary extension of the $$t$$-test, which is strictly limited to evaluating differences between exactly two means (such as a single sample mean against a population mean, or the means of two distinct groups).

Analysis of Variance

The $$t$$ distribution illustrates the spread of possible $$t$$ scores that researchers would expect to observe if the null hypothesis were entirely true. This distribution forms a unimodal and symmetrical curve centered precisely around a mean of $$0$$. By utilizing this known distribution curve, researchers can evaluate any computed $$t$$ score to find its exact $$p$$-value, which represents the proportion of expected scores that are at least as extreme as the observed sample result.

Distribution of t Scores

What is the primary statistical function of a t-test in psychological research?

Statement that says there is no relationship between the two measured items

Null Hypothesis

Statement that says something is HAPPENING! New theory is true instead of an old one

Alternative Hypothesis

1. The dependent variable must be continuous (interval/ratio).
2. The observations are independent of one another.
3. The dependent variable should be approximately normally distributed.
4. The dependent variable should not contain any outliers.

Assumptions

To conduct a one-sample $$t$$-test, researchers compute the $$t$$ statistic using the exact formula: $$t = \frac{M - \mu_0}{\frac{SD}{\sqrt{N}}}$$. In this mathematical equation, $$M$$ represents the sample mean, $$\mu_0$$ is the hypothetical population mean serving as the standard of comparison, $$SD$$ is the standard deviation of the sample, and $$N$$ indicates the total sample size.

Formula for the One-Sample t-Test

In a one-sample $$t$$-test, the null hypothesis posits that the true population mean ($$\mu$$) is identical to the hypothetical population mean ($$\mu_0$$), expressed formally as $$\mu = \mu_0$$. Conversely, the alternative hypothesis argues that the actual population mean significantly deviates from this expected standard of comparison, formalized as $$\mu 
eq \mu_0$$.

Null and Alternative Hypotheses for the One-Sample t-Test

When conducting a one-sample $$t$$-test, the degrees of freedom ($$df$$) are calculated by subtracting one from the total sample size ($$N - 1$$). This specific calculation dictates the exact shape of the $$t$$ distribution, enabling researchers to accurately identify the $$p$$ value associated with their test statistic.

Degrees of Freedom (One-Sample t-Test)

To demonstrate a one-sample $$t$$-test, imagine a health psychologist investigating whether university students accurately estimate the $$250$$ calories in a chocolate chip cookie. The null hypothesis ($$\mu = 250$$) assumes accurate estimation. The researcher asks a sample of $$10$$ students, resulting in a sample mean ($$M$$) of $$212.00$$ calories and a standard deviation ($$SD$$) of $$39.17$$. Plugging these statistics into the one-sample $$t$$-test formula yields a $$t$$ score of $$-3.07$$. For a two-tailed test with $$9$$ degrees of freedom, the critical value is $$\pm 2.262$$. Because the calculated $$t$$ score is more extreme than this critical value, the researcher rejects the null hypothesis and concludes that students significantly underestimate the cookie's calories. If the researcher had a strong theoretical reason to expect an underestimation, a one-tailed test with a less severe critical value of $$-1.833$$ could have been used, making it easier to reject the null hypothesis if the results trended in the expected direction.

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