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1. BIOL 582 Lecture Set 10 Nested Designs Random Effects

2. BIOL 582 Background • Thus far, we have concerned ourselves with factors that have 2 or more levels • A linear model with such factors is said to have “fixed” effects • A good way to think of a fixed effect is as something that cannot be changed by switching to another individual. • E.g., being female or male will not change by looking at another female or male, respectively • E.g., examining other subjects within an experimental treatment does not change the treatment level. • However, it is often convenient or essential to sample (sometimes more than once) a portion of available subjects (nested) within a fixed effect. The variance of responses among subjects within “subgroup” and/or among subgroups within groups might influence inferences made about group (fixed) effects.

3. BIOL 582 Background • Research designs (usually experimental) that incorporate subordinate classification within a higher level are called nestedor hierarchicaldesigns, and ANOVA performed on models that properly deal with this kind of design are called nested-ANOVA. • Before getting too far, we need to understand a few basics and introduce some new nomenclature. • We have used the linear equation • to indicated that a response can be modeled by factor A (whose effects are called alpha). We can also state • and • which means all fixed effects sum to zero, and that the residuals are normally distributed with mean 0 and a variance equal to

4. BIOL 582 Background • From now on, we will just write: , , • Now let’s assume that factor A is not a fixed effect, but rather a random effect. This is the case when the levels of factor A are not standard (like sex, treatment, population) but are sample from a hypothetical continuous distribution of possibilities Assuming equal sample sizes

5. BIOL 582 Background • From now on, we will just write: , , • Now let’s assume that factor A is not a fixed effect, but rather a random effect. This is the case when the levels of factor A are not standard (like sex, treatment, population) but are sample from a hypothetical continuous distribution of possibilities • For example, a breeding design, which means randomly selecting females from a colony of individuals and mating them with selected males. The researcher might want to measure a response in offspring but realizes that – because of heritability of traits – the arbitrary selection of females might introduce variation in the response • We can write the linear equation similarly, but we will use Arabic letters for random effects Assuming equal sample sizes Note the key difference

6. BIOL 582 Background • Expanding from before, let’s assume that females are randomly chosen from colonies that have been raised in different experimental treatments • Which translates roughly as follows. The sum of treatment effects is equal to zero. Random effects of females nested within treatment effects are normally distributed with a mean of zero and an expected variance. Residuals are also normally distributed with a mean of zero and an expected variance. • Another important note: • When a model contain only fixed effects  fixed effects model (Model I ANOVA performed) • When a model contains only random effects  random effects model (Model II ANOVA performed) • When a model contains both fixed and random effects  mixed model (or mixed effects model)

7. BIOL 582 Background • IMPORTANT DISCLAIMER: When random effects are imbalanced (unequal sample sizes), then the mean effects might not be 0. How to estimate model parameters and analyze variance is not trivial. But it is largely beyond the scope of this class. • THE ONE IMPORTANT POINT is that F statistics – if one wants to use an F distribution in ANOVA – are not calculated the same for each model effect. • Random effects are calculated with respect to other random effects or model error. Fixed effects are calculated with respect to certain random effects!

8. BIOL 582 Mixed Model hypotheses • A mixed linear model with A fixed and B random effects, where B is nested within A (i.e., levels of B are sampled within levels of A).

9. BIOL 582 Random Model hypotheses (Pure Model II) • A mixed linear model with A and B random effects, where B is nested within A (i.e., levels of B are sampled within levels of A).

10. BIOL 582 Random/Mixed Model Evaluation (ANOVA) • Do not worry about SS type right now • Assume equal sample sizes, for simplicity of illustration • a and b are the number of levels for factors A and B, respectively; n is the number of measurements within levels of B We will not explore it at this point, but when one uses random effects, differences among subjects or subgroups that make up the random effects is usually not important. However, one might analyze variance components – relative portions of variance explained by different random components – to understand how meaningful any particular random variable is. E.g., Error: Groups: 50%, Subgroups within Groups 32%, Within Subgroups 18%

11. BIOL 582 Comment • We are not going to go into much detail about mixed models • There are enough nuances to warrant an entire course on the subject • The important thing to realize is that if one has random effects, on must make proper statistical inference. • Using the breeding design example, the differences among treatments are not evaluated with respect to variation among subjects born to different mothers (within-subgroup) or error; rather, the differences among treatments are evaluated with respect to variation among mothers (among-subgroup). • Evaluating fixed effects with respect to model error is a common mistake made by clicking buttons in canned ANOVA software! • Don’t make that mistake! • Also, much more complicated nested designs can be analyzed… • Multiple comparison tests are problematic – there are special tests for nested designs • Consider this coverage of random effects, nested effects, mixed models, presented here to be only a springboard.