• What problem do mixed effect models solve?
• Random versus fixed effects
• Parameters
• Equations for dragons

• Mixed models are also known as mixed effects models or multilevel models
• used when the data have some sort of hierarchical form such as in longitudinal or panel data, repeated measures, time series and blocked experiments
• which can have both fixed and random coefficients together with multiple error terms.
• Greatest benefit is they save degrees of freedom

• Super easy to code in R (end of next lecture)
• Difficult to interpret (end of next lecture)
• Somewhere in between to understand how they work (today)

• Always categorical predictor variable
• Not so much the variables themselves, but rather what you are interested in
• Are you measuring a few specific instances of interest in themselves (=fixed) or a few randomly chosen instances interesting only as representatives of a population (=random).
• fixed reporting n site estimates (s1, s2,…,sn). If it is random it is reporting a single \(\sigma^2\) value with \(\sigma^2\) =variance(s1, s2,…,sn). Which one do you want?
• Long discussion https://dynamicecology.wordpress.com/2015/11/04/is-it-a-fixed-or-random-effect/

• degrees of freedom = sample size (n) - number of parameters (p)
• the more parameters you have, the more data you need
• rule of thumb you need 10 times more data than parameters you are trying to estimate
• degrees of freedom eat up power (next slide)

• Degrees of freedom (n-p)
  • Garden: 2 levels, 1 parameter, therefore 2-1
  • Error: 20 samples, 2 parameters (look at the equation). 20-2
  • Total: Add up the other two
• Mean squares (Mean squared deviation - lecture 2) = SS/df
• F = Mean squares (treatment) / Mean squares (error) = 20/1.333 [Think signal over noise]

• Taken from https://gkhajduk.github.io/2017-03-09-mixed-models/
• Next lecture we will analyse this data
• Today lets thinks about the model

Imagine that we decided to train dragons and so we went out into the mountains and collected data on dragon intelligence (testScore) as a prerequisite. We sampled individuals over a range of body lengths and across three sites in eight different mountain ranges.

\[ \text{Model 1: } Y_i = \alpha + \beta bodylengths_i + \epsilon_i \text{ and } \epsilon_i \sim N(0,\sigma^2) \]

• Going back to the \(\alpha\) as intercept nomenclature
• This model has three unknown parameters (one intercept, one slope and the error term)
• Assumes that testscore/body length relationship is the same in each mountain

\[\text{Model 2: } Y_{ij} = \alpha_j + \beta_j bodylengths_{ij} + \epsilon_{ij} \text{ and } \epsilon_{ij} \sim N(0,\sigma^2) \]

• \(j\) is 1, … ,8 and \(i\) is 1, … , number of samples per mountain
• This is an ANCOVA using mountain as a factor, body length as a continous explanatory variable and an interaction term (mountain:bodylength).
• Here the regression lines are allowed to have different intercepts and different slopes for each mountain. That is , the model does not assume that testscore/body length relationship is the same in each mountain
• But it has 17 parameters (8 (mountains) x 2 (slope and intercept) + 1 (error term))
• If we included site, that would increase by a factor of 3 (49 = (8 x 3 x 2) + 1)

\[\text{Model 3: } Y_{ij} = \alpha_j + \beta bodylengths_{ij} + \epsilon_{ij} \text{ and } \epsilon_{ij} \sim N(0,\sigma^2) \]

• Different intercepts but same slope for each mountain
• 10 parameters (8 (mountains) x 1 (intercept) + 1 (slope) + 1 (error term))

\[\text{Model 4: } Y_{ij} = \alpha + \beta_j bodylengths_{ij} + \epsilon_{ij} \text{ and } \epsilon_{ij} \sim N(0,\sigma^2) \]

• the intercepts are kept same and the slopes are allowed to differ.
• 10 parameters (8 (mountains) x 1 (slope) + 1 (intercept) + 1 (error term))

• The price of 14 (17-3) extra regression parameters can be rather large, namely in the loss of precious degrees of freedom.
• To avoid this, mixed modelling can be used.
• But there is another motivation for using mixed modelling with these data:
  • If mountain is used as a fixed term, we can only make a statement of testscore/bodylength relationships for these particular mountains
  • whereas if we use it as a random component, we can predict the testscore/bodylength relationship for all similar mountains.

\[\text{Model 3: } Y_{ij} = \alpha_j + \beta bodylengths_{ij} + \epsilon_{ij} \text{ and } \epsilon_{ij} \sim N(0,\sigma^2) \]

\[\text{Model 5: } Y_{ij} = \alpha + \beta bodylengths_{ij} + a_j + \epsilon_{ij} \text{ where } \] \[ a_j \sim N(0,\sigma_a^2) \text{ and } \epsilon_{ij} \sim N(0,\sigma^2) \]

• We assume there is only one regression line with a single intercept and a single slope. The single intercept (\(\alpha\)) and the single slope (\(\beta\)) are called the fixed parameters
• Additionally there is a random intercept \(a_j\), which adds a certain amount of random variation to the intercept at each mountain
• So the unknown parameters are \(\alpha\), \(\beta\), the variance of the noise \(\sigma^2\) and the variance of the random intercept \(\sigma_a^2\). That is only 4 parameters. This is the magic of the mixed effect model.

• Slopes and intercepts can change \[\text{Model 6: } Y_ij = \alpha + a_j + \beta bodylengths_{ij} + b_j bodylengths_{ij} + \epsilon_{ij} \text{ where } \]

\[ a_j \sim N(0,\sigma_a^2) \text{ and } b_j \sim N(0,\sigma_b^2) \text{ and }\epsilon_{ij} \sim N(0,\sigma^2) \]

• \(b_j\) random variation of the slope at each mountain
• 5 parameters as oppose to 17!!!!!!!!

• We will code this example as a linear mixed effect model
• You can extend mixed effect models to be generalized (generalized mixed effect models), but that’s getting quite advanced (one for your own enjoyment)