## Split-plot-in-time

Considering a 2 block experiment with 3 experimental units per block each with a different treatment. Measurements are taken on each unit for each of 3 years, but the treatment is the same in each unit from year to year. Thus there are a total of (2x3x3=) 18 observations.

### Full design

A full design would include these effects (degrees of freedom):

• (Intercept) (1)
• Treatment (2)
• Block (1)
• Year (2)
• Block x Treatment (2)
• Year x Treatment (4)
• Block x Year (2)
• Block x Treatment x Year (4)

This model has no error degrees of freedom left.

### Design based on an experimental unit

Suppose we take averages for each experimental unit across the years. Then a design could be

• Treatment (2)
• Block (1)
• Year (2)
• Block x Treatment (2)

This still has no error degrees of freedom.

### Split-plot-in-time design

Taking the design based on an experimental unit but expanding it to include year produces this design

• (Intercept) (1)
• Treatment (2)
• Block (1)
• Block x Treatment (2) (random effect for repeated measure)
• Year (2)
• Year x Treatment (4)

This design has 6 degrees of freedom which arise by eliminating

• Block x Year (2)
• Block x Treatment x Year (4)

#### Correlation structure

Treating the block x treatment interaction as a standard random effect will result in a compound symmetry structure for observations across years.

In SAS, specify this by using `repeated / type = cs subject = Block x Treatment CS`.

This means that the correlation between every combination of years is the same which may not be reasonable since some of the years are more spread apart in time than others. An alternative is to use an autoregressive structure, e.g. AR(1), which can be specified using `type=ar(1)`.