# Experimental Designs

## What is an experimental design?

Is the process of planning a study to meet specified objectives.  An experiment that SHOULD be designed to match a specific research question.

## Steps to designing an experiment

1. Define the EXPERIMENTAL UNIT
What is the difference between an EXPERIMENTAL UNIT and a SAMPLING UNIT?
2. Identify the types of variables
3. Define the treatment structure
4. Design the design structure

## Experimental Unit  vs. Sampling Unit

Experimental unit is the unit to which the treatment is applied to.

Sampling unit is a fraction of the experimental unit.

Examples of potential experimental units:

• An animal
• A cage with 5 birds inside
• A plot in a field
• A box of fruit
• A tree
• A pot of plants
• A growth chamber
• A fish tank
• A tray of seedlings
• A taste panelist
• A sample of a new food product
• A bench in a greenhouse

Examples of potential sampling units:

• 1 bird in a cage
• A quadrant in a plot of a field
• 5 apples from a box
• A branch or leaf of a tree
• 1 plant from a pot of plants
• A tray or shelf placed in a growth chamber
• An individual fish from a fish tank
• One pod of seedlings from a tray
• A plot on a bench in a greenhouse

## Experimental Error

Measure of the variation that exists among observations taken on the experimental units that are treated alike.

### Sources of Experimental Error

1. Natural variation among experimental units
2. Variability of the measurements taken (response)
3. Inability to reproduce the treatment conditions exactly from one unit to another
4. Interaction between the treatments and the experimental units
5. Any other extraneous factors that may influence the response

With any statistical analyses, what we are looking for is an estimate of the variation of the experimental error.  So, the variation between our experimental units – We need this  to test treatment differences.

Variation of observations within an experimental unit will not give us treatment differences!

## Completely Randomized Design (CRD)

Treatments that are randomly assigned to experimental units.

Experimental unit is the individual plot/square in the design.  The statistical model is represented by:

Where:

Yij = Observation on the jth experimental unit on the ith treatment
μ = overall mean
τi = the effect of the ith treatment
εij = experimental error or residual

The experimental error is variation among experimental units on the same treatment. The unexplained variation – the residual – what’s left.

## Randomized Complete Block Design (RCBD)

In any experiment we conduct, we have experimental error.  Our goal is to take control over our experimental error so we can study the effects of our treatments.  Blocking is one way to take control of our experimental error.

Blocking occurs when we group experimental units in a way where the variation of the experimental units within the blocks is less than the variation among all the units before blocking.

Each block highlighted as the different colours or the columns in the above table.  Within each block all the treatments will appear an equal amount of time.  The statistical model would be:

What happens though when we have more than one experimental unit/treatment in each block?  If you look at the current design – you have one measurement per treatment in each block – so there is not enough measures to see whether the treatments are doing something different across the blocks.  But when we have more than one experimental unit per treatments in a block, then you have variation to examine.  So your model would now be:

## Split Plot Design

A design where you have 2-3 factors or treatments of interest, yet the experimental units of each treatment are different sizes.

What are the 2 sources of experimental error?

Variation between the Blocks where A was assigned.  Two blocks have the A1 treatment and two blocks have the A2 treatment.  The main plot is the A treatment.

The second source of experimental error is the variation among the experimental units.  The subplot is the B treatment.  The statistical model is:

## Split Block or Strip-plot

Two treatments that are applied as a strip as an example.  Here is one block

If we are interested in looking at the effect of Treatment A – what is the correct error term?  Start by asking yourself what is the experimental unit for treatment A?  Then think about the definition of experimental error – variation between experimental units that were treated the same….