6: Random Effects and Introduction to Mixed Models

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Overview

So far, in our discussion of treatment designs, we have made the (unstated) assumption that the treatment levels were chosen intentionally by the researcher as dictated by his/her specific interests. The scope of inference in this situation is limited to the specific (or fixed) levels used in the study. However, this is not always the case. Sometimes, treatment levels may be a (random) sample of possible levels, and the scope of inference is to a larger population of all possible levels.

If it is clear that the researcher is interested in comparing specific, chosen levels of treatment, that treatment is called a fixed effect. On the other hand, if the levels of the treatment are a sample of a larger population of possible levels, then the treatment is called a random effect.

Learning Objectives

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

Extend the treatment design to include random effects.
Understand the basic concepts of random-effects models.
Calculate and interpret the intraclass correlation coefficient.
Combining fixed and random effects in the mixed model.
Work with mixed models that include both fixed and random effects.

6.1: Random Effects
Introduction to modeling single factor random effects, including variance components and Expected Means Squares.
6.2: Battery Life Example
Comparing the effects of battery brand as a fixed vs. a random effect, depending on the study design. Includes a worked example for using R to model a single random effect for the battery data.
6.3: Random Effects in Factorial and Nested Designs
EMS formulas and F-tests for factorial vs nested designs, in two-factor studies. Includes a worked example in R to analyze greenhouse data for two random effects in isolation.
6.4: Special Case - Fully Nested Random Effects Design
Modeling a case with fully and hierarchically nested random effects.
6.5: Quality Control Example
Detailed example for modeling fully nested random effects, using SAS. Subpages model the same example in Minitab and R.
- 6.5.1: Using Minitab
- 6.5.2: Using R
6.6: Introduction to Mixed Models
ANOVA models for two-factor mixed models.
6.7: Mixed Model Example
Example of a mixed model, in which a random effect is nested within factorial fixed treatment combinations.
6.8: Complexity Happens
EMS expressions for additional combinations of fixed and random factors, in both crossed and nested two-factor designs.
6.9: Try It!
Practice problems: data analysis and interpretation of ANOVA tables.
6.10: Chapter 6 Summary