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Analysis of Variance
An important technique for analyzing the effect of categorical factors on a response is to perform an Analysis of Variance. An ANOVA decomposes the variability in the response variable amongst the different factors. Depending upon the type of analysis, it may be important to determine: (a) which factors have a significant effect on the response, and/or (b) how much of the variability in the response variable is attributable to each factor.
STATGRAPHICS Centurion provides several procedures for performing an analysis of variance:
1. One-Way ANOVA - used when there is only a single categorical factor. This is equivalent to comparing multiple groups of data.
2. Multifactor ANOVA - used when there is more than one categorical factor, arranged in a crossed pattern. When factors are crossed, the levels of one factor appear at more than one level of the other factors.
3. Variance Components Analysis - used when there are multiple factors, arranged in a hierarchical manner. In such a design, each factor is nested in the factor above it.
4. General Linear Models - used whenever there are both crossed and nested factors, when some factors are fixed and some are random, and when both categorical and quantitative factors are present.
A one-way analysis of variance is used when the data are divided into groups according to only one factor. The questions of interest are usually: (a) Is there a significant difference between the groups?, and (b) If so, which groups are significantly different from which others? Statistical tests are provided to compare group means, group medians, and group standard deviations. When comparing means, multiple range tests are used, the most popular of which is Tukey's HSD procedure. For equal size samples, significant group differences can be determined by examining the means plot and identifying those intervals that do not overlap.
When more than one factor is present and the factors are crossed, a multifactor ANOVA is appropriate. Both main effects and interactions between the factors may be estimated. The output includes an ANOVA table and a new graphical ANOVA from the latest edition of Statistics for Experimenters by Box, Hunter and Hunter (Wiley, 2005). In a graphical ANOVA, the points are scaled so that any levels that differ by more than exhibited in the distribution of the residuals are significantly different.
A Variance Components Analysis is most commonly used to determine the level at which variability is being introduced into a product. A typical experiment might select several batches, several samples from each batch, and then run replicates tests on each sample. The goal is to determine the relative percentages of the overall process variability that is being introduced at each level.
The General Linear Models procedure is used whenever the above procedures are not appropriate. It can be used for models with both crossed and nested factors, models in which one or more of the variables is random rather than fixed, and when quantitative factors are to be combined with categorical ones. Designs that can be analyzed with the GLM procedure include partially nested designs, repeated measures experiments, split plots, and many others. For example, pages 536-540 of the book Design and Analysis of Experiments (sixth edition) by Douglas Montgomery (Wiley, 2005) contains an example of an experimental design with both crossed and nested factors. For that data, the GLM procedure produces several important tables, including estimates of the variance components for the random factors.
Analysis of Variance for Assembly Time
Source
Sum of Squares
Df
Mean Square
F-Ratio
P-Value
Model
243.7
23
10.59
4.54
0.0002
Residual
56.0
24
2.333
Total (Corr.)
299.7
47
Type III Sums of Squares
Source
Sum of Squares
Df
Mean Square
F-Ratio
P-Value
Layout
4.083
1
4.083
0.34
0.5807
Operator(Layout)
71.92
6
11.99
2.18
0.1174
Fixture
82.79
2
41.4
7.55
0.0076
Layout*Fixture
19.04
2
9.521
1.74
0.2178
Fixture*Operator(Layout)
65.83
12
5.486
2.35
0.0360
Residual
56.0
24
2.333
Total (corrected)
299.7
47
Expected Mean Squares
Source
EMS
Layout
(6)+2.0(5)+6.0(2)+Q1
Operator(Layout)
(6)+2.0(5)+6.0(2)
Fixture
(6)+2.0(5)+Q2
Layout*Fixture
(6)+2.0(5)+Q3
Fixture*Operator(Layout)
(6)+2.0(5)
Residual
(6)
Variance Components
Source
Estimate
Operator(Layout)
1.083
Fixture*Operator(Layout)
1.576
Residual
2.333
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