2 Way Analysis of Variance (ANOVA)

1. 2 Way Analysis of Variance (ANOVA) Peter Shaw RU

2. ANOVA - a recapitulation. • This is a parametric test, examining whether the means differ between 2 or more populations. • It generates a test statistic F, which can be thought of as a signal:noise ratio. Thus large Values of F indicate a high degree of pattern within the data and imply rejection of H0. • It is thus similar to the t test - in fact ANOVA on 2 groups is equivalent to a t test [F = t2 ]

3. How to do an ANOVA: • table 1: Calculate total Sum of Squares for the data Sstot = Σi(xi - μ)2 • =Σi(xi2)– CF • where CF = Correction factor = (Σixi * Σixi)/N • 2: calculate Treatment Sum of Squares SStrt = Σt(Xt.*Xt.)/r - CF • where Xt. = sum of all values within treatment t • 3: Draw up ANOVA table

4. ANOVA tables • Exact layout varies somewhat - I dislike SPSS’s version! • Learn as parrots: • Source DF SS MS F Source df SS MS F Treatment (T-1) SStrt SStrt / (T-1) MStrt / MSerr Error Sserr by subtraction = SSerr / DFerr Total N-1 Sstot Variance

5. One way ANOVA’s limitations • This technique is only applicable when there is one treatment used. • Note that the one treatment can be at 3, 4,… many levels. Thus fertiliser trials with 10 concentrations of fertiliser could be analysed this way, but a trial of BOTH fertiliser and insecticide could not.

6. Linear models.. • Although rather worrying-looking, these equations formally define the ANOVA model being used. (By understanding these equations you can readily derive all of ANOVA from scratch) • The formal model underlying 1-Way ANOVA with Treatment A and r replicates: • Xir = μ + Ai + Errir • Xiris the rth replicate of Treatment A applied at level i • Aiis the effect of treatment i (= difference between μ and mean of all data in treatment i. • Errtr is the unexplained error in Observation Xtr • Note that ΣAi = Σerrir = 0

7. Basic model: Data are deviations from the global mean: • Xir = μ + Errir • Sum of vertical deviations squared = SStot μ Trt 1 Trt 2 • 1 way model: Data are deviations from treatment means: • Xir = μ + Ai + Errir • Sum of vertical deviations squared = SSerr A2 A1 Trt 1 Trt 2

8. No model • Xir just is! • H0 model: • Xir = μ + Errir μ A1 1 way anova model: Xir = μ + Ai + Errir μ

9. Two-way ANOVA • Allows two different treatments to be examined simultaneously. • In its simplest form it is all but identical to 1 way, except that you calculate 2 different treatment sums of squares: • Calculate total Sum of Squares • Sstot= Σi(xi2)– CF • Calculate Sum of Squares for treatment A • SSA = ΣA(XA.*XA.)/r - CF • Calculate Sum of Squares for treatment B • SSB = ΣB(XB.*XB.)/r - CF

10. 2 Way ANOVA table Source df SS MS F Treatment A (NA-1) SSA SSA / (NA-1) MSA / MSerr Treatment B (NB-1) SSB SSB / (NB-1) MSB / MSerr Error By By Subtraction Subtraction SSerr / DFerr Total N-1 SStot Variance

11. The 2 way Linear model • The formal model underlying 2-Way ANOVA, with 2 treatments A and B • Xikr = μ + Ai + Bk + errikr • Xikris the rth replicate of Treatment A level i and treatment B level k • Aiis the effect of the ith level of treatment A (= difference between μ and mean of all data in this treatment. • Bkis the effect of the kth level of treatment B (= difference between μ and mean of all data in this treatment. • Errijr is the unexplained error in Observation Xijr • Note that ΣAi = ΣBk = Σerrikr = 0

12. To take a worked example (Steel & Torrie p. 343). Effect of 2 treatments on blood phospholipids in lambs. 1 was a handling treatment, one the time of day. A1B1 A1B2 A2B1 A2B2 8.53 17.53 39.14 32.00 20.53 21.07 26.20 23.80 12.53 20.80 31.33 28.87 14.00 17.33 45.80 25.06 10.80 20.07 40.20 29.33 totals: 66.39 96.80 182.67 139.06

13. 2 Way ANOVA on these data: Start by a preliminary eyeballing of the data: They are continuous, plausibly normally distributed. There are 2 handling treatments and 2 time treatments, which are combined in a factorial design so that each of the 4 combinations is replicated 5 times. Get the basics: n = 20 Σx = 484.92 Σx^2 = 13676.7 CF = 484.92^2 / 20 = 11757.37 SS = 13676.7 - cf = 1919.33

14. Now get totals for treatments A and B A1 A2 Σ B1 66.39 182.67 249.06 B2 96.80 139.06 235.86 Σ 163.19 321.73484.92 Hence the sums of squares for A and B can be calculated: SSA = 163.19^2/10 + 321.73^2 / 10 - CF = 1256.75 SSB = 249.06^2/10 + 235.86^2/10 - CF = 8.712

15. A aloneSource Df SS MS F A 1 1256.75 1256.75 34.14** error 18 662.58 36.81 total 19 1919.33 B alone Source Df SS MS F B 1 8.71 8.71 0.08 NS error 18 1910.62 106.15 total 19 1919.33 Pooled (the correct format) Source Df SS MS F A 1 1256.75 1256.75 32.67** B 1 8.71 8.71 0.24NS error 17 653.87 38.86 total 19 1919.33

16. Note that we have reduced error variance and DF by incorporating 2 treatments into one table. This is not just good practice but technically required - by including only one treatment in the table you are implicitly calling the effects of the other treatment random noise, which is incorrect. ANOVA tables can have many different treatments included. The skill in ANOVA is not working out the sums of squares, it is the interpretation of ANOVA tables. The clues to look for are always in the DF column. A treatment with N levels has N-1 DF - this always applies and allows you to infer the model a researcher was using to analyse data.

17. A B 1 1 18 1 1 22 1 2 25 1 2 35 1 3 47 1 3 53 2 1 29 2 1 31 2 2 38 2 2 42 2 3 45 2 3 51 3 1 38 3 1 42 3 2 46 3 2 44 3 3 35 3 3 45 Your turn! These data come from a factorial experiment with 2 treatments applied at 3 levels each, with 2 replicates of each treatment. Hence the design contains 3 (A)*3 (B)*2(reps) = 18 data points. They are specially contrived to make the calculations easy for ANOVA Remember the sequence: Get: n, Σx, Σx^2 Calculate CF then SStot Get the totals for each treatment: A1, A2, A3, B1, B2 and B3 hence get SSA and SSB

18. These model data: • N = 18 • Σx = 686.00 • Σx^2 = 27822.00 • CF = 26144.22 • SStot = 27822.00 -26144.22 = 1677.78

19. Totals for each treatment: • A1 A2 A3 Σ • B1 40 60 80 180 • B2 60 80 90 230 • B3 100 96 80 276 • Σ 200 236 250 686

20. Sums of squares: • SSa = 200^2/6 + 236^2/6 + 250^2/6 - CF = 221.78 • SSb = 180^2/6 + 230^2/6 + 276^2/6 - CF = 768.44 • Source Df SS MS F • A 2 221.78 110.89 2.1 NS • B 2 768.44 384.22 7.26** • error 13 687.56 52.89 • Σ 17 1677.78

21. Interaction terms • We now meet a unique, powerful feature of ANOVA. It can examine data for interactions between treatments - synergism or antagonism. • No other test allows this, while in ANOVA it is a standard feature of any 2 way table. • Note that this interaction analysis is only valid if the design is perfectly balanced. Unequal replication or missing data points make this invalid (unlike 1 way, which is robust to imbalance).

22. Synergism and antagonism • Some treatments intensify each others’ effects: • The classic examples come from pharmacology. • Alcohol alone is lethal at the 20-40 unit range. Barbiturates are lethal. Together they are a vastly more lethal combination, as the 2 drugs synergise. (In fact most sedatives and depressants show similar dangerous synergism). • In ecology, SO2 + NO2 is more damaging than the additive effects of each gas alone - a synergism.

23. Antagonism. • is the opposite - 2 treatments nullifying each other. • Drought antagonises effects of air pollution on plants, as drought leads to closed stomata excluding the noxious gas.

24. No interaction Response 2 I Treatment B I I I 1 I I I 1 2 3 Treatment A Synergism Response I Antagonism I I I I I I I I I I 1 2 3 Treatment A 1 2 3

25. How to do this? • Easy! We work out a sum of squares caused by ALL treatments at ALL levels. Thus for a 3*3 design there are really 9 treatments, etc. Call this SStrt • Now we can partition this Sum of squares: SStrt = SSA + SSB + SSInteraction • We know SSA, we know SSB, so we get SSinteraction by subtraction. • To get SStrt we just add up all data in each treatment, square this total, divide by replicates, add up and remove CF.

26. For the lamb blood data: • We have 4 separate treatments: A1B1, A1B2, A2B1, A2B2 • The data within these 4 groups add to: 66.39, 182.67, 96.80, 139.06. There are 5 replicates • SStrt = 66.39^2/5 + 182.67^2/5 + 96.8^2/5 + 139.06^2/5 - CF = 1539.407

27. 2 Way anova table with interaction • Source Df SS MS F • All trts 3 1539.07 *********** • A 1 1256.75 1256.75 52.93* • B 1 8.71 8.71 0.37NS • A*B 1 273.95 273.95 11.54** • error 16 379.92 23.75 • Σ 19 1919.33

28. Interpreting the interaction term The hardest part of 2 way anova is trying to explain what a significant interaction term means, in terms that make sense to most people! Formally it is easy; you are testing H0: Ms for interaction term is same population as MS for error. In English let’s try “It means that you can’t reliably predict the effect of Treatment A at level m with B at level n, knowing only the effect of Am and Bn on their own.”

29. Treatment A – big effect (A2>A1) Treatment B – mean (B1) is v close to mean (B2) so no effect Interaction: When A=1, B1<B2 but when A =2, B1> B2 A1B1 A1B2 A2B1 A2B2