1. Comparing Multiple Factors:
Scientific Practice
Comparing Multiple Factors:
Analysis of Variance
ANOVA (2-way)
@dave_lush
@UWE_JT9

2. Where We Are/Where We Are Going
1-way ANOVA allows many groups to be compared in terms of one factor (treatment)
each group is exposed to one factor
but what if we want to look at two different factors?
and what if we wanted to know if factors interact?
eg the effects of temperature and pH on cell growth
does temp have an effect? Does pH? Do they affect each other?
This needs a 2-way ANOVA
we normally think of having to carry out experiments over and over again, but 2-way ANOVA can be carried out without replication
though interaction effects cannot be analysed

3. 2-Way ANOVA Without Replication
Here are some results looking at growth rate of cells at 3 pH values and 4 temps…
Every combination present (factorial design)
but note only one trial at each combination
Can we discover if…
temp has an effect
pH has an effect
whether the two factors interact in some way?

4. 2-Way ANOVA Without Replication
Step 1: The Null Hypothesis
there are two…
temp has no effect on cell growth
pH has no effect on cell growth
temp and pH do not interact to affect cell growth
Step 2: Generate the Test Statistic, F (x2)
cell growth in one col; two factors in separate cols
using Minitab gives…
Two-way ANOVA: Growth versus Temp, pH
Source DF SS MS F P
Temp pH
Error
Total

5. 2-Way ANOVA Without Replication
Step 3: Determine the probability(ies)
there are two, one for each factor…
0.002 and (<0.0005)
Step 4: Interpret the Probabilities
can reject the Null Hypo in both cases, so…
pH and temp both significantly affect cell growth
NB, the small Error MS (4.556) suggests that most of the variation in data caused by separate effects of temp and pH
ie no interaction
but to really look at interaction, we need replicates…

6. 2-Way ANOVA With Replication
Say we did our experiment at 3 pH values and 4 temps three times…

7. 2-Way ANOVA With Replication
Step 1: The Null Hypothesis
there are three…
temp has no effect on cell growth
pH has no effect on cell growth
temp and pH do not interact to affect cell growth
Step 2: Generate the Test Statistic, F (x3)
using Minitab gives…
Two-way ANOVA: Growth versus Temp, pH
Source DF SS MS F P
Temp pH
Interaction
Error
Total

8. 2-Way ANOVA With Replication
Step 3: Determine the probability(ies)
there are three, one for each factor, plus interaction…
all (<0.0005)
Step 4: Interpret the Probabilities
can reject the Null Hypo in all cases, so…
pH and temp both significantly affect cell growth
furthermore, the two factors interact significantly to influence growth
This analysis does not report means (or SDs) of the groups
means can be determined separately and plotted as a ‘main effects’ plot and ‘interaction plot’

9. 2-Way ANOVA: Main Effects Plot
Graph shows means of the main effects…
rise in growth with temp (except highest temp)
rise in growth with pH

10. 2-Way ANOVA: Interaction Plot
Graph shows…
rise in growth with pH
rise in growth with temp (except highest temp)
Interaction visualised by how parallel lines are
parallel lines suggest no interaction
non-parallel lines show interaction
eg high pH changed effect of higher temp

11. 2-Way ANOVA: ‘Paired’ Approach
The ‘paired’ approach to experimental design and analysis is very powerful…
Q: but how can it be used in ANOVA?
A: we need to consider the individual as a factor
Eg, looking at effect of 3 compounds on BP
‘unpaired’ approach (with 30 subjects)
randomly assign each individual to take one drug
analyse with 1-way ANOVA (drugs are single factor)
‘paired’ approach (only need 10 subjects)
give each person each drug (in a random sequence)
2-way ANOVA (factors are drugs, subjects)
called randomised block design

12. 2-Way ANOVA: Randomised Block
A block is a factor that we deliberately control to organise our subject allocation
not our primary interest, such as the drugs
but it helps control for variation
not a block as in road-block, rather a block of flats
each person is an individual ‘testing unit’ in which different things can be tested
Eg, looking at effect of 3 compounds on BP
give each person each drug
drugs are the experimental factor
subject is the blocking factor
importantly, order of drugs randomised per subject

13. 2-Way ANOVA: Randomised Block
Eg, effect of 3 drugs on BP in 10 subjects

14. 2-Way ANOVA: Randomised Block
Minitab…
MTB > anova c3 = c1 c2
ANOVA: b.p. versus subject, drug
Factor Type Levels Values
subject fixed , 2, 3, 4, 5, 6, 7, 8, 9, 10
drug fixed , 2, 3
Analysis of Variance for b.p.
Source DF SS MS F P
subject
drug
Error
Total
Drug effect is sig (<0.0005); interesting!
explore further (post-hoc testing)
Subject effect sig; not interesting - people vary!

15. Summary
2-way ANOVA allows many groups to be compared in terms of >1 factors (treatments)
Allows effects of different factors to be teased out
Usual form includes replication
permits interaction between factors to be shown
increased complexity of interpretation required
One factor can be a ‘blocking’ factor
a way to organise the application of treatments
if that is the individual, then it permits a ‘paired’ approach