We recall that the full and reduced model are, respectively:
y_{ij} &= \mu_i + \epsilon_{ij}\\
y_{ij} &= \mu + \epsilon_{ij}.
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The least squares estimators of and are:
\hat\mu_i &= \bar{y}_{i\cdot}\\
\hat\mu &= \bar{y}_{\cdot\cdot}.
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The special notation is commonly used:
y_{i\cdot}&=\sum_{j=1}^{r_i}y_{ij}\\
\bar{y}_{i\cdot} &=\frac{1}{r_i}y_{i\cdot}
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Generally, replacing any index with a dot means summation, and a bar means dividing by the number of terms in a sum.
Thus,
y_{\cdot\cdot} &= \sum_{i=1}^{t}\sum_{j=1}^{r_i}y_{ij}\\
\bar{y}_{\cdot\cdot} &= \frac{1}{N} y_{\cdot\cdot}
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where is the number of samples (observational units).
The sum of squares (SSE or for "full model") of the experimental error is:
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The unbiased estimate of the variance of a group is:
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We will also consider the pooled estimator of the variance of the reference population, assuming that is the same for each group:
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We have the obvious relation:
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Similarly, for the reduced model the total sum of squares is:
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The corresponding estimator of variance is:
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The following decomposition identity is fundamental to the development of the analysis of variance (ANOVA):
where the term SST is the sum of squares of treatment and is defined as follows:
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The proof of (1) follows from the Pythagorean theorem in finite dimensional Hilbert spaces.
We also define the mean sums of squares:
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With these definitions, we can define the Fisher statistic:
It can be shown that this statistic has Fisher distribution with degrees of freedom of and . The statistic is a ratio of two independent -distributed variables with and degrees of freedom, respectively. The standard assumption must be valid, that the distribution of the reference population is normal. The assumption of equal variances of treatment groups must also hold.
The Fisher test is defined in a straightforward fashion. The null hypothesis (all means equal) is rejected when:
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where is the value of the cumulative distribution function (cdf) of the Fisher distribution with significance level and degrees of freedom and .
The power of the Fisher test is determined from the fact that the Fisher statistic given by (2) has a non-central Fisher distribution with non-centrality parameter , where:
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Thus, one decides what level of the means is deemed significant, finds and determines the probability of accepting given (type II error) using the values of the non-central Fisher distribution. The power is 1 minus that. The optimal number of replications may be determined by studying the power curves for the Fisher test, which may be available either in graph form (see [Kuehl]) or obtained by using computer software.
