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Two Way ANOVA is used to analyze the effect of two random factors on a Critical-To-Characteristic (see concept Factors and Levels). A factor is said to be random when levels are randomly chosen from a population of possible levels and we wish to draw conclusions about the entire population of levels, not just those used in the study. For example, Two Way ANOVA type of analysis is generally used in Gage R&R studies.


Two Way Anova Model

Reference: Juran's Quality Control Handbook

Terminology - Two Way ANOVA

A. The model where yijk is the (ijk)th observation of the CT characteristic (I = 1, 2, …, a, j = 1, 2, …, b, k = 1, 2, …, n) for a levels of factor A, b levels for factor B and n the number of observa-tions 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, (tb)ij the interaction effect between A and B and eijk (epsilon ijk) the error component. For example, with a Gage R&R study factor, A can be the Operator, factor B the Parts and the interaction is the interaction between the Operators and the Parts. In this case, 'a' will be the number of Operators, 'b' the number of parts and 'n' the number of repeated measurements (see tool Measurement-Variable Gage R&R Study).

B. The variance (V) of any observation (yijk) where st2,sb2, stb2 and s2 are called variance components.

C. 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 - Two Way ANOVA

For a random effect model, we assume that the levels of the two factors are chosen at random from a population of levels. The data should come from independent random samples taken from normally distributed populations. The adequacy of this model has to be verified using residual analysis (see tool Residual Plots)

Application Cookbook - Two Way ANOVA

1. Define problem and state the objective of the study

2. Identify the two factors for study, and the levels associated with these factors. E.g. in a Gage R&R study, the factors are Parts and Operators and generally the number of levels for the parts are 10 and the number of levels for the operators are 3 (see tool – Measurement-Variable Gage R&R Study).

3. Establish sample size.

4. Measure the CT characteristic

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

  • In order to use the Minitab functions, the data has to be formatted in a certain way. Three columns are required. Two columns contain the identification of the levels for each factor. The third column contains the measured data from the CT characteristic. To help create the level codes for the two factors, the function under Calc > Make Patterned Data > Simple Set of Numbers can be used.
  • E.g. if we have one factor 'A' with 2 levels and the second factor 'B' with 4 levels, the following screens would be used in Minitab to create the two columns identifying the factor levels.

See part 2 for continuation of the application cookbook.



To summarize the results of an analysis of variance calculation in a table.


Two Way Anova Table

Reference: Design and Analysis of Experiments by D. C. Montgomery

Terminology - Two Way ANOVA

A. Source – Indicates the different sources of variation that are decomposed in the ANOVA table. These sources are: factor 1, factor 2, the interaction between factors 1 & 2, the error of the repeated observations within each combination of the factor levels, and the total variation.

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

C. SS – The sums of squares measure the variability associated with each source. They are the numerators of the estimate of variance.

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

E. F – F statistics that permit hypothesis testing for factors 1, 2 and the interaction effect.

F. 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 ³a, don't reject H0.

G. The estimators of variance components.

Major Considerations - Two Way ANOVA

A drawback of the analysis of variance method to estimate the variance components is that it may provide negative estimates. This indicates that the model being fit is inappropriate for the data. One should assume that the negative estimate means that the variance component is really zero and fit a reduce model that does not include the corresponding effect.

Application Cookbook

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

  • Use the function under Stat > ANOVA > Balanced ANOVA.
  • Input the name of the column which contains the measurement of the CT characteristic into the 'Response' field, and the names of the columns that contains the factors into the 'Model' field. The symbol (|) should be introduced between the two factors e.g. Parts | Operators to obtain the complete model.
  • In order to obtain a random effect model for the two way Anova, you have to input the name of the two factors into the option 'Random Factors'. To obtain the estimators of the variance component go to 'Options' and select the 'Display expected mean squares'.
  • In order to verify the assumption of the model, select the 'Graphs' option and highlight all the available residual Plots (to interpret these plots see tool 'Residual Plots').
  • 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.
  • For a gage R&R study, use the function under Stat > Quality Tools > Gage R&R Study and ensure that the method of analysis is ANOVA.

2. Make a statistical decision from the session window output of Minitab. Either accept or reject H0 for each effect. When H0 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 statistical conclusion into practical decision about the CT characteristic.

From Two Way ANOVA to Free Six Sigma Tools.

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