N WAY ANOVA

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N WAY ANOVA - PART 1 OF 2


Purpose

To analyze the effect of N fixed factors on a CT characteristic (see concept Factors and Levels). A fixed effects model is one in which the conclusions apply to the factor levels considered in the analysis, and cannot be extended to those not explicitly considered.


Anatomy

ANOVA

Reference: Design and Analysis of Experiments

Terminology - N Way ANOVA

A. The model where Yijkl is the (ijkl)th observation of the CT characteristic (i = 1, 2, …, a, j = 1, 2, …, b, k = 1, 2, …, c, l = 1, 2, 3, …, n) for "a" levels of factor A, "b" levels for factor B, "c" levels for factor C and n is the number of observations in each of the combination of the factor levels. m represents the overall mean, ti (tau i) the effect of factor A, bj (beta j) the effect of factor B, gj (gamma j) the effect of factor C. (tb)ij represents the interaction effect between A and B, (tbg)ijk represents the full interaction between all the factors, and eijkl (epsilon ijkl) is the error component. Note that this is an example of a three-way two-level ANOVA.

B. Null (H0) and alternative (Ha) hypotheses where the variance (s2) of each effect are compared with 0. For each effect, if H0 is true the levels are similar, but if HA is true variability exists between the levels.


Major Considerations - N Way ANOVA

The example presented is a three-factor, two-level ANOVA. The complexity of the linear model, and the statistical hypotheses, increases rapidly as the number of factors and interaction increases.

Application Cookbook - N Way ANOVA

1. Define problem and state the objective of the study.

2. Identify the factors for study, and the levels associated with these factors. Establish sample size.

3. Measure the CT characteristic.

4. Analyze data with Minitab (part 1 of 2):

5. In order to use the Minitab functions, the data has to be formatted in a certain way. For N-Way ANOVA, N+1 columns are required. N columns contain the identification of the levels for each factor. The (N+1)th column contains the measured data from the CT characteristic. To help create the level codes for the two factors, the function under Stat>DOE>Create Factorial Design can be used.

6. See part 2 for continuation of the application cookbook.

N WAY ANOVA - PART 2 OF 2


Purpose

To summarize, in tabular form, the results of an Analysis of Variance calculation for a multi-factor, two-level Factorial Experiment.

Anatomy - N Way ANOVA


Reference: Design and Analysis of Experiments

Terminology - N Way ANOVA

A. Source – Indicates the different variation sources decomposed in the ANOVA table, the Factors, or Main Effects, and the Interactions. The "Error" is the variation within each treatment combination. The "Total" is the total variation in the CT characteristic (experimental response).

B. DF – The number of degrees of freedom related to each sum-of-squares (SS). They are the denominators of the estimate of variance.

C. Seq SS – The Sequential sums-of-squares measure the variability associated with each source. They are the variance estimate's numerators. The SS for the Main Effects and Interactions is due to the change in factor levels. The formula for Sum-of-Squares is: SS = (Contrast)2/nT where Contrast = (S+Y)-(S-Y), n = number of replications, and T = number of runs of the basic design matrix, before replication.

D. Adj SS – The adjusted Sum of Squares. When the experimental model is orthogonal, the sequential and adjusted Sum of Squares will be the same. Calculations of the MS, F and p factors are all based upon the adjusted sum of squares.

E. MS – Mean Square is the estimate of the variance for the factors, interactions, and error sources. Computed by MS = Adj SS/DF.

F. F – The ratio of the mean square for the "Factor" or interaction and the mean square for the "Error". When the error is zero, the F statistic cannot be calculated.

G. P-Value – This value has to be compared with the alpha level (a) and the following decision rule is used: if P < a, reject H0 and accept Ha with (1-P) 100% confidence; if P ³ a, don't reject H0.

Application Cookbook

1. Analyze the data using Minitab (part 2 of 2).

  • Use the function Stat>ANOVA>General Linear Model.
  • Input the name of the column that contains the measurement of the CT characteristic into the "Response" field, and the names of the columns containing the factors into the "Model" field. The "Pipe" symbol, |, should be introduced between all factors, e.g. A|B|C, to obtain the complete model.
  • In order to verify the assumptions of the model, select the "Graphs" option and highlight all the available Residual Plots (to interpret these plots, refer to the tool "Residual Plot").
  • If the residual plots show any abnormalities, the results of the analysis of variance may be distorted. An investigation should be conducted to determine the causes for such abnormalities.

2. Make a statistical decision from the Session window output from Minitab. Either accept or reject HO for each effect. When HO is rejected, we can conclude that an effect is statistically significant. In other words, the effect has an influence on the CT characteristic.

3. Translate the statistical conclusion into a practical decision about the CT characteristics.

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