Paired t test as a block design
Section of Analysis of variance, part of Statistical inference
Paired t test as a block design
| Type | Section |
| Taught in | Analysis of variance |
| Part | Statistical inference |
Two tyre types, A and B, are fitted one each to five cars, and the tread left after 10 000 km is measured.
| Car | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Type A | 10.6 | 9.8 | 12.3 | 9.7 | 8.8 |
| Type B | 10.2 | 9.4 | 11.8 | 9.1 | 8.3 |
| Difference | 0.4 | 0.4 | 0.5 | 0.6 | 0.5 |
Wear depends heavily on the car and the driver, so this is a randomised block design with 5 blocks and 2 treatments. Under that model the within-car difference D_i=Y_{i1}-Y_{i2}=\tau_1-\tau_2+\varepsilon_{i1}-\varepsilon_{i2} is \operatorname{N}(\tau_1-\tau_2,\ 2\sigma^2), independently across cars, and the car effect has cancelled. Testing \tau_1=\tau_2 is a one-sample t test on the differences. Their mean is 0.48 and standard deviation 0.0837, so
t=\frac{0.48}{0.0837/\sqrt5}=12.8
on 4 degrees of freedom, with two-sided p-value 0.0002. The tyres differ in wear. The block F test gives F=164.6=12.8^2, the same test.
The completely randomised design is the several-group version of the two-sample t test. The block design is the several-treatment version of the paired t test.
A Bonferroni comparison and a worked haemoglobin table are in Analysis of variance examples.
Used in
See also
One-way analysis of variance · ANOVA sum of squares decomposition · ANOVA F distribution · Contrast · Tukey intervals · Randomised block design · Between and within · One-factor analysis of variance