Linear models

How maximise power?

Linear models

How maximize power?

• increase replication
• add covariates (account for conditions)
• block (control conditions)

Hierarchical models

How do we increase power - without more sites (replicates)?

Hierarchical models

Subreplicates - yet these are not independent

Nested design

Hierarchical models

How do we increase power?…

Hierarchical models

Block treatments together - yet these are not independent

Randomized complete design

Hierarchical models

Linear modelling assumptions

• Normality
• Homogeneity of Variance
• Linearity
• Independence

Non-independence

• one response is triggered by another
• temporal/spatial autocorrelation
• nested (hierarchical) design structures

Hierarchical models

• linear model with separate covariance structure per block
• fixed and random factors (effects)

Example

         y        x  block
1 281.1091 18.58561 Block1
2 295.6535 26.04867 Block1
3 328.3234 40.09974 Block1
4 360.1672 63.57455 Block1
5 276.7050 14.11774 Block1
6 348.9709 62.88728 Block1

Example

Example

Simple linear regression

Example

Simple ANCOVA

Example

• Looks good, but for INDEPENDENCE
• Can we deal with that with correlation structure?

\[ \text{Variance-covariance per Block:} \mathbf{V} = \begin{pmatrix} \sigma^2&\rho&\cdots&\rho\\ \rho&\sigma^2&\cdots&\vdots\\ \vdots&\cdots&\sigma^2&\vdots\\ \rho&\cdots&\cdots&\sigma^2\\ \end{pmatrix} \]

Example

Hierarchical model