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

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

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?

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 3 238.67 79.556 17.46 0.002 pH 2 1896.00 948.000 208.10 0.000 Error 6 27.33 4.556 Total 11 2162.00

2-Way ANOVA Without Replication Step 3: Determine the probability(ies) there are two, one for each factor… 0.002 and 0.000 (<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…

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

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 3 725.44 241.81 197.85 0.000 pH 2 5756.22 2878.11 2354.82 0.000 Interaction 6 82.89 13.81 11.30 0.000 Error 24 29.33 1.22 Total 35 6593.89

2-Way ANOVA With Replication Step 3: Determine the probability(ies) there are three, one for each factor, plus interaction… all 0.000 (<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’

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

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

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

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

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

2-Way ANOVA: Randomised Block Minitab… MTB > anova c3 = c1 c2 ANOVA: b.p. versus subject, drug Factor Type Levels Values subject fixed 10 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 drug fixed 3 1, 2, 3 Analysis of Variance for b.p. Source DF SS MS F P subject 9 5530.80 614.53 21.71 0.000 drug 2 370.40 185.20 6.54 0.007 Error 18 509.60 28.31 Total 29 6410.80 Drug effect is sig (<0.0005); interesting! explore further (post-hoc testing) Subject effect sig; not interesting - people vary!

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