ANOVA - ONE WAY

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ANOVA: ONE WAY - PART 1 OF 2


Purpose

To compare the means of two or more populations on a continuous CT characteristic. Since we don't know the population means, an analysis of data samples is required. ANOVA (Analysis of Variance) is usually used to determine if there is a statistically significant change in the mean of a CT characteristic under two or more conditions introduced by one factor (see concept Factor and Levels).


Anatomy

ANOVA

Reference: Juran's Quality Control Handbook

Terminology

A. Null (H0) and alternative (Ha) hypotheses where the means (m1, m2, …, mg) of the g levels of the factor are compared. There is only one alternative hypotheses: which is at least the means of two levels are significantly different. (Note: these hypotheses correspond to a fixed effects model, for more information see reference above.)

B. Model where yij is the (ij)th observation of the CT characteristic (i = 1, 2, …, g and j = 1, 2, …, n, for g levels of size n), m is the overall mean, ti is the ith level effect and eij is an error component.

C. Minitab session window output.

D. Descriptive Statistics – Sample sizes, means, standard deviations (StDev), and the Pooled Standard Deviation, which is a combination of the standard deviations of the g groups.

E. Confidence Intervals around the mean for each individual level.

F. ANOVA Table (see part 2).


Major Considerations

The assumptions for using this tool is that the data comes from independent random samples taken from normally distributed populations, with the same variance. When using ANOVA, we are using a model where its adequacy has to be verified using residual analysis (see tool Residual Plots).

Application Cookbook

Define problem and state the objective of the study.

State Null and Alternative Hypothesis.

Establish sample size (see tool Sample Size – Continuous Data – One Way ANOVA).

Select random samples.

Measure the CT characteristic.

Analyze data with Minitab (part 1 of 2):

In order to fully use the Minitab functions associated with ANOVA it is recommended that the data be stacked into one column and a second column to contain the group codes. Using the function under Manip > Stack/Unstack > Stack Columns.

See Part 2 for continuation of the application cookbook.

ANOVA: ONE WAY - PART 2 OF 2 (ANOVA TABLE)


Purpose

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

Anatomy

ANOVA Table

Terminology

A. Source – Indicates the different variation sources decomposed in the the table. "Factor" represents the variation introduced between the factor levels. The "Error" is the variation within each of the factor levels. The "Total" is the total variation in the CT characteristic.

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 variance estimate's numerators. "Factor" SS is due to the change in factor levels. The larger the difference between the means of a factor levels, the larger the factor sum of squares will be. The "Error" SS is due to the variation within each factor level. The "Total" SS is the sum of the Factor and Error sum of squares (see tool Sum of Squares).

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

E. F – The ratio of the mean square for the "Factor" and the mean square for the "Error".

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 a,, don't reject H0.


Major Considerations

The assumptions for using this tool is that the data comes from independent random samples taken from normally distributed populations, with the same variance. When using it we are using a model where its adequacy has to be verified using residual analysis (see tool Residual Plots).

Application Cookbook

1. Analyze the data with Minitab (part 2 of 2):
Verify the assumption of equality of variance for all the levels with the function under Stat > ANOVA > Homogeneity of Variance (to interpret this analysis, see tool Homogeneity of Variance Tests).

  • Use the function under Stat > ANOVA > One Way.
  • Input the name of the column which contains the measurement of the CT characteristic into the 'Response' field, and the name of the column that contains the level codes into the 'Factor' field.
  • In order to verify the assumption of the model, select the options 'Store Residuals' and 'Store Fits'. Select the graphs option and highlight all the available residual Plots (to interpret these plots see tool Residual Plots).
  • In the event of non-compliance with either of these assumptions, the results of the analysis of variance may be distorted. In this case, the use of graphical tools such as a Box Plot can be used depict the location of the means and the variation associated with each factor level. In the case of outliers, these should be investigated.

2. Make a Statistical decision from the session window output of Minitab. Either accept or reject H0. When H0 is rejected we can conclude that there is a significant difference between the means of the levels.

3. Translate statistical conclusion into practical decision about the CT characteristic.

From ANOVA to Free Six Sigma Tools.

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