The Model
The normal linear model describes treatment groups with probability distributions which are identically bell-shaped (normal) curves with different means. Thus fitting the models requires only the means of each treatment group and a variance calculation (an average variance within the treatment groups is used). Calculations of the means and the variance are performed as part of the hypothesis test.
The commonly used normal linear models for a completely randomized experiment are:
- (the means model)
or
- (the effects model)
where
- is an index over experimental units
- is an index over treatment groups
- is the number of experimental units in the jth treatment group
- is the total number of experimental units
- are observations
- is the mean of the observations for the jth treatment group
- is the grand mean of the observations
- is the jth treatment effect, a deviation from the grand mean
- , are normally distributed zero-mean random errors.
The index i over the experimental units can be interpreted several ways. In some experiments, the same experimental unit is subject to a range of treatments; i may point to a particular unit. In others, each treatment group has a distinct set of experimental units; i may simply be an index into the list.
Read more about this topic: One-way ANOVA, The Case of Fixed Effects, Fully Randomized Experiment, Unbalanced Data
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