7.
Product and Process Comparisons
7.4.
Comparisons based on data from more than two processes
7.4.3.
Are the means equal?
7.4.3.1.
|
1-Way ANOVA overview
|
|
| Overview
and principles |
This section gives
an overview of the one-way ANOVA. First we explain the principles involved
in the 1-way ANOVA. |
| Partition
response into components |
In an analysis
of variance the variation in the response measurements is partitoned into
components that correspond to different sources of variation.
The goal in this procedure is to split the total variation in the data
into a portion due to random error and portions due to changes in the values
of the independent variable(s). |
| Variance
of n measurements |
The variance of
n measurements is given by
![s**2 = SUM[i=1 to n][(Y(i) - Ybar)**2/(n-1)](eqns/variance.gif)
where  is
the mean of the n measurements. |
| Sums of
squares and degrees of freedom |
The numerator part
is called the sum of squares of deviations from the mean, and the
denominator is called the degrees of freedom.
The variance, after some algebra, can be rewritten as:
![s**2 = {SUM[i=1 to n][y(i)**2] - (SUM[i=1 to n][y(i)])**2/n}/(n-1](eqns/anova3.gif)
The first term in the numerator is called the "raw sum of squares"
and the second term is called the "correction term for the mean".
Another name for the numerator is the "corrected sum of squares",
and this is usually abbreviated by Total SS or SS(Total).
The SS in a 1-way ANOVA can be split into two components, called the
"sum of squares of treatments" and "sum of squares of error",
abbreviated as SST and SSE, respectively. |
| The guiding
principle behind ANOVA is the decomposition of the sums of squares, or
Total SS |
Algebraically,
this is expressed by
![Total SS = SST + SSE; SUM[i=1 to k]SUM[j=1 to n(i)][(Y(ij) - ybar(..))**2 = SUM[i=1 to k][n(i)*(ybar(i.) - ybar(..))**2 + SUM[i=1 to k][j=1 to n(i)][(y(ij) - ybar(i.)**2]](eqns/anova4.gif)
where k is the number of treatments and the bar over the
y.. denotes the "grand" or "overall" mean. Each
ni
is the number of observations for treatment
i. The total number
of observations is N (the sum of the ni). |
| Note on subscripting |
Don't be alarmed by the double subscripting.
The total SS can be written single or double subscripted. The double subscript
stems from the way the data are arranged in the data table. The table is
usually a rectangular array with k columns and each column consists
of ni rows (however, the lengths of the rows, or the
ni, may be unequal). |
| Definition
of "Treatment" |
We introduced the
concept of treatment. The definition is: A treatment is a specific
combination of factor levels whose effect is to be compared with other
treatments. |
