## Linear Statistical Models

### Between-Subjects Designs

In behavioral and educational research, subjects may be randomly sampled from some
population and randomly assigned to one of two or more groups (control, treatment level 1,
treatment level 2, etc.). A comparison of the groups tells us about the effects of
the treatments. These treatment level effects represent differences between subjects.

In a between-subjects design, responses from a given subject appear in only one group.

The variability of scores within each group reflects individual differences and is accounted for
by the chance elements in sampling. This sampling variability is thus due to chance.

The variability of means between groups reflects both individual (chance) differences
and differences due to the treatment.

**Consider the following 4-group experiment:**

Grp1 | Grp2 | Grp3 | Grp4 | |

y11 y21 y31 ... yn1 |
y12 y22 y32 ... yn2 |
y13 y23 y33 ... yn3 |
y14 y24 y34 ... yn4 |

y.1 | y.2 | y.3 | y.4 | y.. |

where,

**y.1** -- mean of group 1

**y.2** -- mean of group 2

**y.3** -- mean of group 3

**y.4** -- mean of group 4

**y..** -- grand mean

**Sources of Variability**

Variability within group one is due to sampling variability -- chance.
Variability within group two is due to sampling variability -- chance.
Variability within group three is due to sampling variability -- chance.
Variability within group four is due to sampling variability -- chance.
Variability between groups is due to a combination of sampling variability and treatment
effect -- chance + treatment.
**The F-ratio**

The F-ratio for a between subjects design reflects:

Linear Statistical Models Course

Phil Ender, 12Feb98