1. Two Sample t-test vs. Paired t-test
Layouts and Models
Two Sample t-test vs. Paired t-test

2. The Layout of an Experiment
The Layout for an Experiment is a graphical display indicating elements of the structure of an Experiment and the data which go along with it.
It indicates how many factors there are and any structural relationship among factors (important for more complicated designs).
It also indicates the numbers of observations (data points) and where they are located.

3. Layout for simple comparative experiment

4. Associated Model
The layout suggest two independent samples and so we can use the simple linear model: Yij=µi+εij This suggests an independent sample t-test assuming that: εij ~ Normal(0,σ2)

5. Suppose there is more structure to the data
Suppose that the individual data points have more “structure”.
Suppose that each data point is a Pre or Post Treatment score for subjects/experimental units.
This needs to reflected in the Layout.

6. Corrected Layout

7. Correct Model Model including Subject for Paired data: Yij= µi+Sj+ εij
Note that in the second model we have partitioned the experimental error into two terms, because Subject is now included in the Model. This is a fundamental idea in ANOVA where the variation in the data is partitioned into its components. That is why it is referred to as Analysis of Variance (ANOVA).
The important implication is that σ for the second model (paired data model) is smaller than σ for the first model (independent sample model). This is because our estimate of σ in the second model does not include Subject to subject variation.

8. How they differ
For testing, our estimates of the means for each group are the same, but our estimates of experimental error variation are quite different.
In terms of the signal to noise ratio, our estimate of the signal is the same, but the estimate of “noise” is different, since the subject to subject variation is controlled for.
This does not affect the hypotheses, but it can affect the likelihood of finding significance since the second model has more Power.

9. This can be simplified to a Paired t-test
We can simplify the model by simply computing the differences between Pre and Post values.
Then instead of testing
H0:µ1= µ2
Ha: µ1≠ µ2,
we are testing
H0: µd=0
Ha: µd≠0
where µd= µ1-µ2, with a one sample t-test.