Parametric Tests of Difference: Unrelated & Related T-tests Revision Notes for AQA A-Level Psychology

Parametric Tests of Difference: Unrelated & Related T-tests

What are parametric tests?

Parametric tests are statistical tests that are more powerful than their non-parametric counterparts, but they require specific criteria to be met. These criteria include:

Interval data - the data must be measured on a scale with equal intervals
Homogeneity of variance - the spread of scores should be similar across groups
Normal distribution - the data should be approximately normally distributed

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When these conditions are met, researchers can use parametric tests like t-tests instead of less powerful non-parametric alternatives such as the Mann-Whitney U test or Wilcoxon test. The increased power means you're more likely to detect genuine effects when they exist.

Understanding t-tests

T-tests are parametric tests of difference used to determine whether there is a statistically significant difference between means. The choice between unrelated and related t-tests depends entirely on your experimental design.

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The key decision factor is whether you're comparing different groups of participants (unrelated) or the same participants under different conditions (related). This design choice determines which formula and procedure you must use.

Unrelated t-test

When to use an unrelated t-test

An unrelated t-test is used when you want to compare the means of two separate, independent groups of participants. This test is appropriate when:

You have interval level data
You are using an independent measures design (different participants in each condition)
The data meets parametric assumptions

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Worked Example: Gender differences in jigsaw puzzle completion

Aim: To investigate whether there are gender differences in visuo-spatial ability by comparing the time taken to complete a jigsaw puzzle between boys and girls.

Hypotheses:

Alternative hypothesis: There is a difference in the time taken by males and females to complete a jigsaw puzzle (non-directional, two-tailed)
Null hypothesis: There is no difference in the time taken by males and females to complete a jigsaw puzzle

Procedure: The study involved 20 participants (10 boys, 10 girls) who each completed the same jigsaw puzzle. The time taken was recorded in seconds for each participant.

Calculating the unrelated t-test:

Prepare the data: Calculate the sum of scores (ΣX) and sum of squared scores (ΣX²) for each group
Apply the formula: $t = \frac{\bar{X_1} - \bar{X_2}}{\sqrt{\frac{(S_1 + S_2) \times (N_1 + N_2)}{N_1 \times N_2}}}$

Where $S_1$ and $S_2$ represent the sum of squares calculations for each group
Calculate degrees of freedom: $df = N_1 + N_2 - 2$

Result interpretation: Using the example data, the calculated t-value was -0.116. With df = 18 and using a two-tailed test at p < 0.05, the critical value is 2.101. Since the calculated value (ignoring the negative sign) is less than the critical value, the result is not significant, so we accept the null hypothesis.

Related t-test

When to use a related t-test

A related t-test is used when you want to compare means from the same participants tested under different conditions. This test is appropriate when:

You have interval level data
You are using a repeated measures design (same participants in both conditions)
The data meets parametric assumptions

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Worked Example: Effects of CBT on gambling behaviour

Aim: To investigate the effects of cognitive behavioural therapy (CBT) on the physiological arousal of gamblers by measuring heart rate before and after treatment.

Hypotheses:

Alternative hypothesis: There is a reduction in heart rate activity when comparing heart rate before and after cognitive behavioural therapy (directional, one-tailed)
Null hypothesis: There is no difference in heart rate activity comparing heart rate before and after cognitive behavioural therapy

Procedure: Ten participants categorised as 'persistent gamblers' were given a six-week course of CBT. Their heart rate was measured before treatment and after treatment while playing on a fruit machine for 20 minutes.

Calculating the related t-test:

Calculate differences: For each participant, calculate the difference (d) between their two scores
Apply the formula: $t = \frac{\Sigma d}{\sqrt{\frac{N\Sigma d^2 - (\Sigma d)^2}{N-1}}}$
Calculate degrees of freedom: $df = N - 1$

Result interpretation: Using the example data, the calculated t-value was 2.237. With df = 9 and using a one-tailed test at p < 0.05, the critical value is 1.833. Since the calculated value is greater than the critical value, the result is significant, so we reject the null hypothesis and conclude there is a reduction in heart rate activity following CBT.

Interpreting results

Critical values

Both t-tests require comparison of your calculated t-value with critical values from statistical tables. The critical value depends on:

Degrees of freedom (df)
Level of significance (typically p < 0.05)
Type of hypothesis (one-tailed or two-tailed)

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Decision Rule for Statistical Significance:

If calculated value > critical value: Result is significant, reject null hypothesis
If calculated value < critical value: Result is not significant, accept null hypothesis

Remember to ignore the sign (+ or -) when comparing your calculated value with the critical value - only the magnitude matters for determining significance.

Making conclusions

The interpretation of your results should directly relate back to your original research question and hypothesis. A significant result means you have found evidence for a genuine difference, while a non-significant result suggests any observed difference could be due to chance.

Key differences between the tests

Test	Design	Participants	Formula complexity
Unrelated t-test	Independent measures	Different groups	More complex (accounts for two separate groups)
Related t-test	Repeated measures	Same participants	Simpler (uses difference scores)

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The related t-test is generally more sensitive because it controls for individual differences between participants, as the same people are tested in both conditions. This increased sensitivity makes it more likely to detect genuine effects when they exist.

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Key Points to Remember:

Parametric tests require interval data, normal distribution, and homogeneity of variance
Unrelated t-tests compare different groups of participants using independent measures designs
Related t-tests compare the same participants using repeated measures designs
Always compare your calculated t-value with the appropriate critical value to determine significance
The choice between tests depends entirely on your experimental design, not the type of data
Related t-tests are generally more powerful due to controlling for individual differences

Parametric Tests of Difference: Unrelated & Related T-tests (AQA A-Level Psychology): Revision Notes