Analysis of paired and independent

Question 1

Paired samples design

Answer 1

Each case in one sample corresponds to a particular case in another sample e.g. repeated measures design
Test of depended means (=correlated means)
A set of data where each observation in one group is directly linked to a corresponding observation in another group

E.g. “before and after” or “treatment vs control”

Question 2

Independent samples design

Answer 2

Cases in each sample are completely independent e.g. between-group design
Test of independent means
Samples are selected randomly so that their observations do not depend on the values of other observations

Question 3

Dependent means analysis

Answer 3

Used to analyse data from paired sample design (repeated measures)

Difference score method:
○ Reduce 2 scores to 1 difference score, and use the same procedures as before
○ Calculate the difference between two related measurements taken from the same individual or group, subtracting one score from another, shows change or disparity between them

Question 4

P-value

Answer 4

The probability of obtaining a test statistics as deviates as the one obtained, under the null hypothesis (when it is true

A small p-value (typically below 0.05) is considered statistically significant, meaning you can reject the null hypothesis

If you conduct a t-test comparing the average height of two groups and get a p-value of 0.02, this means there is only a 2% chance of observing such a difference in heights between the groups if there was truly no difference in the population.

Question 5

Natural pairs

Answer 5

Researcher doesn’t randomly allocate participants to one group; it occurs naturally (prior to study)
While the pairs are made up of different participants the scores of each pair are likely to be correlated (dependent means analysis)

Question 6

Matched pairs

Answer 6

Researcher has control over the ways the pairs are formed so they are matched by some variable
While the pairs are made up of different participants the scores of each pair are likely to be correlated (dependent means analysis)

Question 7

3 assumptions of independent means t test

Answer 7

1. Normal distribution of scores in each population
2. Population variances are equal
3. Observations are independent, within and between groups (Measuring on person’s productivity isn’t influenced by another person)
   - When all 3 met, type 1 error rate = nominal rate (alpha)
   - When assumptions violated, actual distribution of t values does not follow theoretical t distribution

Question 8

Robustness

Answer 8

A test is said to be robust against violation of an assumption if the actual type 1 error rate is close to alpha even when that assumption is violated
Robustness in statistics means a test still works well even if some of its assumptions are broken

Question 9

What happens when the assumptions are violated

Answer 9

t test is robust against violation of normality assumption (Central Limit Theorem)
○ t-tests remain robust even if the data is not normally distributed, especially with large sample sizes.
t test is robust against violation of assumption of equal variances provided n1 = n2
○ But when n1 ≠ n2 , outcome depends on which group has larger variance:

When the larger sample is associated with the larger variance, the t-test is conservative (actual Type I error rate < a)
When the larger sample is associated with the smaller variance, the t-test is liberal (actual Type I error rate > a)

The t-test assumes independent observations. If observations are dependent (e.g., students tested in the same classroom influencing each other), the test is not robust, leading to invalid results.

t test is not robust against violation of independence of observations ○ e.g. groups tested together, such as a comparison between 2 classrooms
Here the behaviour of one participant may influence the behaviour of other participants