So far in ANOVA we have treated all categorical vars as the same

There are actually two types of categorical variables

• fixed effects
• random effects

We need to deal with random effects differently, but it can be tricky at first to tell them apart

ANOVA model: \[Y_{ij}=\mu + A_i + \epsilon_{ij}\]

Fixed effects: (\(A_i\)) affect the mean of groups in a meaningful way

Mixed Model: \[Y_{ij}=\mu + A_i + U_i + \epsilon_{ij}\]

Random effects: (\(U_i\)) structure variance, but not in an additive way like fixed effects. Think of these as additional, structured error terms

Fixed Effects:

• meaningful factor labels (e.g., chimp vs. gorilla)
• small possible range of values….you have data for all of them
• affect only mean of \(y\) variable
• effect predictable in different studies (e.g., gorillas predictably larger than chimps)

Random Effects:

• uninformative factor labels (e.g., site A)
• very large range of possible values…you have data for a random sample of them
• affect only the variance of \(y\) variable
• effect unpredictable or meaningless across studies

Almost always, you choose LMM because you have one or more fixed effects, but you also have pseudo-replication

Recall from our discussion of ANOVA that independent observations in a treatment group are called replicates

Pseudoreplication occurs when something masquerading as a replicate of the treatment actually isn’t independent

This situation is common, and is easy to identify with a bit of practice.

There are several major kinds of pseudo-replication, most important for us are:

• temporal pseudoreplication - typically involves repeated measures on same individual
• spatial pseudoreplication - typically involves observations that may be similar due to spatial association

Challenge: give me some examples of instances where data may be pseudoreplicated

Crossed effects is the more common situation. Crossed effects occur when all levels of one effect appear within all the levels of a second effect. For example Parity and Social Group are crossed.

Nested effects occur when levels of one effect only occur within a single level of a second effect.

In our primate dataset, Individual is nested within SocialGroup, e.g. Louise and Pat only occur within the GoldStars, and they may have similarities due to being in the same group that we want to account for.

Models are specified much like normal linear models, using a formula

Random effects are specified as: (1|randomEffect)

The 1 stands in for the intercept. Effectively, you are saying: “allow each level of randomEffect to have its own independent intercept”

Fixed effects are specified just like you always have in standard linear models

Question: with our primate data, do individuals of parity status differ in aggression rates?

Note: this is fake data, the primatedata object doesn’t exist. I just want to show you the model formula.

libray(lme4)
lmer(AggEncounters ~ Parity + (1|SocialGroup/Individual),  data=primatedata)

Notice: Parity is a fixed effect. The nested random effects are given after the | symbol, starting from the most general, to most specific.

Another option is to fit a model allowing both the slope and the intercept to vary for each subject:

randSlopeInt <- lmer(Reaction ~ Days + (Days | Subject), sleepstudy)
summary(randIntercept)

## Linear mixed model fit by REML ['lmerMod']
## Formula: Reaction ~ Days + (1 | Subject)
##    Data: sleepstudy
## 
## REML criterion at convergence: 1786.5
## 
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -3.2257 -0.5529  0.0109  0.5188  4.2506 
## 
## Random effects:
##  Groups   Name        Variance Std.Dev.
##  Subject  (Intercept) 1378.2   37.12   
##  Residual              960.5   30.99   
## Number of obs: 180, groups:  Subject, 18
## 
## Fixed effects:
##             Estimate Std. Error t value
## (Intercept) 251.4051     9.7467   25.79
## Days         10.4673     0.8042   13.02
## 
## Correlation of Fixed Effects:
##      (Intr)
## Days -0.371

One tool is the Akaike Information Criterion (AIC)

Quantifies goodness of fit, while penalizing for model complexity.

AIC(randIntercept)

## [1] 1794.465

AIC(randSlopeInt)

## [1] 1755.628

Simplest model just uses body mass

mod1 <- lm(log(ECV..cc.) ~ log(Body.mass..g.), data=brains)

Slightly more complicated model includes wild versus captive

mod2 <- lm(log(ECV..cc.) ~ log(Body.mass..g.) + Wild.captive, data=brains)

Which model is better? In this simple case, we can look at significance of individual terms, and \(R^2\), but in LMM we don’t have this option.

Recall that anova() doesn’t do analysis of variance, it creates an ANOVA table from a model

You can also compare two models with anova() to test the hypothesis that they are significantly different

The models are compared with a likelihood-ratio test

Like any other linear models, a basic assumption of general linear mixed-models is a normal error term

However, the structure of data may make this assumption invalid (e.g. binary data or count data)

Two very common types of non-normal error structures are:

• poisson - for count data
• binomial - for binary (e.g. presence absence data) or proportion data