3: ANOVA Models Part I

Overview Section

In Lesson 2, we learned that ANOVA is based on testing the effect of the treatment relative to the amount of random error. In statistics, we call this the partitioning of variability (some due to the treatment and some due to random measurement variability). This partitioning of the deviations can be written mathematically as:

\(\underbrace{Y_{ij}-\bar{Y}_{..}}_{\enclose{circle}{\color{black}{1}}} = \underbrace{\bar{Y}_{i.}-\bar{Y}_{..}}_{\enclose{circle}{\color{black}{2}}} + \underbrace{Y_{ij}-\bar{Y}_{i.}}_{\enclose{circle}{\color{black}{3}}}\)

Thus, the total deviation \(Y_{ij}-\bar{Y}_{..}\) in \(\enclose{circle}{\color{black}{1}}\) can be viewed as the sum of two components:

  1. \(\enclose{circle}{\color{black}{2}}\) Deviation of estimated factor level mean around overall mean, and
  2. \(\enclose{circle}{\color{black}{3}}\) Deviation of the jth response of the ith factor around the estimated factor level mean.

These two deviations are also called variability between groups, a reflection of differences between treatment levels, and the variability within groups, which serves as a proxy for the error variability among individual observations. A practitioner would be more interested in the variability between groups, as it is the indicator of treatment level differences, and may have little interest in the within-group variability, expecting it to be small. However, both these variability measures will play an important role in the ANOVA statistical procedures.

There are several mathematically equivalent forms of ANOVA models describing the relationship between the response and the treatment. In this lesson, we will focus on the effects model and in the next lesson, the alternative models will be introduced.   

This lesson will cover the model assumptions needed to employ the ANOVA. Model diagnostics, which deal with verifying the validity of model assumptions, are discussed along with power analysis techniques to assess the power associated with a statistical study. Software methods using the statistical techniques discussed will also be presented.

Objectives

Upon completion of this lesson, you should be able to:

  1. Apply the Effects Model for a one-way ANOVA and interpret the results.
  2. Examine the assumptions for ANOVA and associated diagnostics.
  3. Use statistical software to conduct an ANOVA (in SAS, R, and Minitab).
  4. Conduct a power analysis and recognize the role of power analysis in a statistical hypothesis test.