9.1 - Approach 1: Split-plot ANOVA

The Split-plot ANOVA is perhaps the most traditional approach, for which hand calculations are not too unreasonable. It involves modeling the data using the linear model shown below:

Model: \(Y_{ijk} = \mu + \alpha_i + \beta_{j(i)}+ \tau_k + (\alpha\tau)_{ik} + \epsilon_{ijk}\)

Using this linear model we are going to assume that the data for treatment i for dog j at time k is equal to an overall mean μ plus the treatment effect \(\alpha_i\), the effect of the dog within that treatment \(\beta_{j \left( i \right)}\), the effect of time \(τ_k\), the effect of the interaction between time and treatment \(\left(ατ \right)_{ik}\), and the error \(\varepsilon_{ijk}\).

Such that:

  • \(\mu\) = overall mean
  • \(\alpha_i\) = effect of treatment i
  • \(\beta_{j \left( i \right)}\) = random effect of dog j receiving treatment i
  • \(\tau_{k}\)= effect of time k
  • \(\left( \alpha \tau \right)_{ik}\) = treatment by time interaction
  • \(\varepsilon_{ijk}\) = experimental error

Assumptions:

We are going to make the following assumptions about the data:

1. The errors \(\varepsilon_{ijk}\) are independently sampled from a normal distribution with mean 0 and variance \(\sigma^2_{\epsilon}\).

2. The individual dog effects \(\beta_{j \left( i \right)}\) are are also independently sampled from a normal distribution with mean 0 and variance \(\sigma^2_{\beta}\).

3. The effect of time does not depend on the dog; that is, there is no time by dog interaction. Generally,

we need to have this assumption otherwise the results would depend on which animal you were looking at - which would mean that we could not predict much for new animals.

With these assumptions, the random effect of dog and fixed effects for treatment and time, this is called a mixed effects model.

The analysis is carried out in this Analysis of Variance Table shown below:

ANOVA

Source d.f SS MS F
Treatment \(a - 1\) \(SS_{\text {treat}}\) \(\dfrac { \mathrm { SS } _ { \text { treat } } } { a - 1 }\) \(\dfrac { \mathrm { MS } _ { \text { treat } } } { \mathrm { MS } _ { \text { error } ( a ) } }\)
Error (a) \(N - a\) \(SS_{\text {error (a)} }\) \(\dfrac { S S _ { \text { error } ( a ) } } { ( N - a ) }\)  
Time \(t - 1\) \(SS_{\text {time}}\) \(\dfrac { S S _ { \text { time} } } { ( t - 1 ) }\) \(\dfrac { \mathrm { MS } _ { \text { time} } } { \mathrm { MS } _ { \text { error } ( a ) } }\)
Treat x Time \(\left( a - 1 ) ( t - 1 \right)\) \(SS_{\text {treat x time}}\) \(\dfrac { S S _ { \text { treat x times } ( b ) } } { ( a - 1 ) ( t - 1 ) }\) \(\dfrac { \mathrm { MS } _ { \text { treat x time } } } { \mathrm { MS } _ { \text { error } ( b ) } }\)
Error (b) \(\left( N - a ) ( t - 1 \right)\) \(SS_{\text {error (b) }}\) \(\dfrac { S S _ { \text { error } ( b ) } } { ( N - a ) ( t - 1 ) }\)  
Total \(Nt - 1\) \(SS_{\text {total}}\)    

where, 

     a: the number of treatments

     N: the total number of all experimental units

      t: number of time points

The sources of the variation include treatment; Error (a); the effect of Time; the interaction between time and treatment; and Error (b). Error (a) is the effect of subjects within treatments and Error (b) is the individual error in the model.  All these add up to the total.

Sum of Squares Formulas

Here are the formulas that are used to calculate the various Sums of Squares involved:

\(\begin{array}{lll}SS_{total}& =& \sum_{i=1}^{a}\sum_{j=1}^{n_i}\sum_{k=1}^{t}Y^2_{ijk}-Nt\bar{y}^2_{...}\\SS_{treat} &= &t\sum_{i=1}^{a}n_i\bar{y}^2_{i..} - Nt\bar{y}^2_{...}\\SS_{error(a)}& =& t\sum_{i=1}^{a}\sum_{j=1}^{n_i}\bar{y}^2_{ij.} - t\sum_{i=1}^{a}n_i\bar{y}^2_{i..}\\SS_{time}& =& N\sum_{k=1}^{t}\bar{y}^2_{..k}-Nt\bar{y}^2_{...}\\SS_{\text{treat x time}} &=& \sum_{i=1}^{a}\sum_{k=1}^{t}n_i\bar{y}^2_{i.k} - Nt\bar{y}^2_{...}-SS_{treat} -SS_{time}\end{array}\)

Mean Square (MS) is always derived by dividing the Sum of Square term by the corresponding degrees of freedom.

To get the main effects for the treatment we compare the MS treatment to MS error (a)

We will compare these results with the results we get from the MANOVA, the next approach covered in this lesson.