# Experimental Design and Analysis of Variance

## Aims and objectives for module 2

After completing this Module you will be familiar with taking an idea from a research problem thinking it through to treatment choices, and the statistical hypotheses and, through to randomisation of the experiment, (using R). Then transferring this design to the plot layout for the field. You will also have the skills for preparing a spreadsheet for data collection, collecting data in the field, and then following through to the appropriate analysis using ANOVA designs and reporting your results. Simple examples are used as demonstrations. Check through the code and then try yourself.

## Contents

- Developing your Research Question
- Introduction to statistical modelling
- A note on R Packages
- Assumptions for ANOVA with a demonstration
- Simple ANOVA or CRD (one way ANOVA) with a demonstration
- Randomised Complete Block (RCBD) ANOVA
- Factorial ANOVA with a demonstration
- Mean comparison
- Split Plot ANOVA
- Unbalanced Designs
- Incomplete Block ANOVA
- Sources of Variance in experiments
- Example of an alpha design

## Developing your Research Question

This short video may help you to think about your research question. This should be thought about carefully as it will help in planning, carrying out the experiment and the analysis. Your research question helps you summarise what (How and why) you are trying to find out in a couple of sentences that most people can understand.

## Introduction to statistical modelling

The following terms are all useful and relevant for understanding ANOVA, so please make sure you are familiar with them, by checking out the page under the Technical Section in our menu under ‘Support’. Also check out our definitions section to help in understanding and remembering the terms (jargon) we use.

##### Factor, treatment, level, blocks, the model, partitioning of variance, residual mean square, experimental error, F distribution, F tests, mean separation, interaction, main effects.

This Module is focusing on the use of Models typically called Analysis of Variance (ANOVA). Their usefulness is our interest in comparing means from more than two groups, and often we are interested in more than one factor.

It will be focused on experimental design using a few different ANOVAs. Design and layout, and analysis, mean separation and display of results. It involves the steps of understanding the experimental terminology. Here are the building blocks.

## A note on R Packages

In this Module we will use a few R packages that we find quite useful. We access these by using the term library(nameofpackage) in our R programs. This will become clear in this Module. In later modules we will introduce some other useful packages.

In this Module we will use functions called, lm, aov, and some others. These functions are available from the base R.

## Assumption for ANOVA

We have assumptions of Independence for the data collected from each experimental unit, this is achieved by applying an appropriate design.

This demonstration goes through the assumption and shows some R code.

## Simple ANOVA (one way ANOVA)

Thinking about your experimental unit and which treatments you are interested in comparing.

Work through the following presentation on One-way ANOVA.

A completely randomised design (CRD) is when the treatments are randomised to any plot in the field. There are no blocks, and no restriction on randomisation.

R Code and output showing the randomisation for a CRD Design

Thinking about your land and site. This means you need to visit your site (such as a field or greenhouse). You need to examine the site to see whether you think there may be an environmental gradient. For example in a greenhouse we may have fans one side and a cooling panel on the other side, resulting in a gradient from cooler to warmer. The blocks are designed so the blocks are perpendicular (at right angles) to the gradient. Within a block we wish to have less variance. Blocking helps reduce extraneous (nuisance) variability.

We can use Excel to randomise simple treatments. We can use also use R software code to plan designs and to make the field book, and layout. We use the package agricolae to use R code.

Here is a demonstration of a one way ANOVA ANOVA

## Randomised Complete Block Design (RCBD) ANOVA

Data capture, and analysis (ANOVA, mean separation, plotting)

We can use R software code to plan designs and to make the field book, and layout

R code for the randomisation of a Randomised Complete Block Design

After the design, we layout the field according to the plan and run the experiment. The data is collected, and needs to be checked carefully, so that mistakes are minimised. The data is often collected in a Excel, saved, backed up and then saved as a csv file. We can then take the csv file and read into R, and then we have our data in a very powerful package for analysis, summary, plotting and so on.

## Factorial ANOVA

In a factorial we may have two Factors A and B , each at a number of levels. the advantage is that we get internal replication that assist in the number of treatment comparisons. For example in the following trial there are five blocks , and a two by 2 factorial. The Factor A and B are each at low and high and so it is called a 2 by 2 factorial and has four treatment combinations. Another advantage of a factorial designs are the ability to examine interactions.

We can use R software code to plan designs and to make the field book, and layout

Data capture, and analysis (ANOVA, mean separation, plotting)

Download this file (a PDF) to run through a demonstration of a factorial experiment, from data entry to analysis and checking of assumptions and a multiple comparison of means.

## Split Plot

This figure demonstrates the concept of the split plot with two sizes of blocks. The four colours show there are 4 blocks and these are the main-plots with A1 and A2 as the two levels of Factor A

The random allocation of Factor A is shown in the right hand side. The experimental plots is shown on the left and there are five levels of Factor B randomised within each Main-plot.

We can use R software code to plan designs and to make the field book, and layout Randomise Split Plot

Data capture, and analysis (ANOVA, mean separation, plotting)

## Unbalanced Designs

We can use R software code to plan designs and to make the field book, and layout.

Data capture, and analysis (ANOVA, mean separation, plotting)

## Balanced Incomplete Block (BIB) Designs

We can use R software code to plan designs and to make the field book, and layout

Data capture, and analysis (ANOVA, mean separation, plotting)

## Sample Size Considerations

Review of sample size and it’s importance

An example of variation within plots, and plant material and how to plan your sampling. An example in Module 7 includes a Random part to the model.

## CASE STUDY – of an alpha design

Look through this presentation and think about alpha designs They are often used when testing a large number of genotypes for some aspect such as disease susceptibility.

An example of alpha design

Please try these out. If you have some data that is more complex, we would be happy to try and help. Contact us. we are developing Module 7 for mixed models and some examples of GLM

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