One-way analysis of variance

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1 Analysis of variance From R.R. Sokal and F.J. Rohlf, Biometry, 2nd Edition (1981): A knowledge of analysis of variance is indispensable to any modern biologist and, after you have mastered it, you will undoubtedly use it numerous times to test scientific hypotheses. However, the analysis of variance is more than a technique for statistical analysis. Once understood, analysis of variance provides an insight into the nature of variation of natural events, into Nature in short, which is possibly of even greater value than the knowledge of the method as such. However, like other models in science, analysis of variance may create constructions of nature in the mind of the scientist that give rise to misleading or unproductive conclusions.

2 One-way analysis of variance We have measured a response variable Y in g groups. Example: Yield under a control and two treatments 1 Box-plots of yield Yield ctrl trt1 trt2 Question: Is yield different under control and treatments? 1 data from Dobson, A. J. (1983) An Introduction to Statistical Modelling in the standard R library

3 Formulae in analysis of variance Y ik N(µ i, σ 2 ), H 0 : µ i µ, σ 2 > 0. H 1 : differences among µ i, σ 2 > 0. i = 1,..., g (groups) Note; if g = 2 test on equality of means (t-test). Method: decomposition of variance (Y ik Ȳ )2 = n i (Ȳi Ȳ )2 + (Y ik Ȳi) 2. i,k i i k total variation variation variation among groups within groups Ȳ i : mean in group i, Ȳ general mean.

4 Hypothesis testing in analysis of variance Under H 0 (all µ i equal) 1 σ 2 SS among = 1 σ 2 n i (Ȳ i Ȳ ) 2 χ 2 (g 1) 1 σ 2 SS within = 1 σ 2 i i (Y ik Ȳi) 2 χ 2 ((n 1)g) and they are independent. [Formulae true with equal size n of all groups] Hence the distribution of k SSamong/(g 1) SS within /((n 1)g) is known and is called F g 1,(n 1)g.

5 Analysis of variance: computations Recipe for hypothesis testing: Compute SSamong/(g 1) SS within /((n 1)g) ; if it is below the (1 α)-quantile of F g 1,(n 1)g, accept H 0 at level α; otherwise, reject it. Example: Analysis of Variance Table Response: weight Df Sum Sq Mean Sq F value Pr( > F) group * Residuals

6 Analysis of variance: R 2 A related quantity R 2 = SS among SS total = variance explained by model. total variance Necessarily 0 R 2 1. HIgh R 2 and rejection of H 0 are different (though related) facts. R-squared in the example:

7 Paired comparisons Rejecting H 0, we find that not all group means are equal. But which are different from which ones? We can compare the means of any given pair of groups by a t-test for equality of means. By default R compares each group to one taken as reference 2. Results in the example: Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) < 2e 16 *** grouptrt grouptrt grouptrt1 is the difference between the mean of trt1 and that of the reference (ctrl). No treatment appears different from the control at the 5% level. 2 The first one in alphabetical order

8 Paired comparisons But which is the correct level of significance? We may be comparing only groups whose means appear particularly different. If there are 6 groups, there would be 15 possible comparisons. The probability that at least one of them would be beyond the nominal 5% level is much higher... One solution is called Tukey s HSD (Honest Significant Difference) that looks at the distribution of maximum difference: diff lwr upr p adj trt1-ctrl trt2-ctrl trt2-trt We can conclude that the yield with treatment 2 is significantly different (higher) than with treatment 1.

9 2-way analysis of variance We may be considering the effect of two factors (A and B) on our response variable, for instance fertilizer and variety on maize yield; or therapy and sex on cholesterol level. The most general model would be Y ijk N(µ + α i + β j + (αβ) ij, σ 2 ), k = 1,..., n ij µ is the general mean, α i the effect of level i of factor A, β j the effect of level j of factor B; (αβ) ij their interaction. n ij number of observations at combination (i, j) of the factors [much easier if n ij are all the same]. What does the interaction mean? Without that term, the effect of the two factors would be additive; on the other hand, it may well be that variety 1 grows better with fertilizer 1 than with 2, while variety 2 has the opposite trend. Such features are captured by the interaction term.

10 Illustrazione grafica di analisi della varianza a 2 vie con effetti additivi da M.C. Whitlock, Dolph Schluter, ANALISI STATISTICA DEI DATI BIOLOGICI, Zanichelli editore S.p.A. Copyright

11 Illustrazione grafica di analisi della varianza a 2 vie con interazione da M.C. Whitlock, Dolph Schluter, ANALISI STATISTICA DEI DATI BIOLOGICI, Zanichelli editore S.p.A. Copyright

12 2-way analysis of variance We may be considering the effect of two factors (A and B) on our response variable, for instance fertilizer and variety on maize yield; or therapy and sex on cholesterol level. The most general model would be Y ijk N(µ + α i + β j + (αβ) ij, σ 2 ), k = 1,..., n ij µ is the general mean, α i the effect of level i of factor A, β j the effect of level j of factor B; (αβ) ij their interaction. n ij number of observations at combination (i, j) of the factors [much easier if n ij are all the same]. What does the interaction mean? Without that term, the effect of the two factors would be additive; on the other hand, it may well be that variety 1 grows better with fertilizer 1 than with 2, while variety 2 has the opposite trend. Such features are captured by the interaction term.

13 Variance decomposition SS total = SS A + SS B + SS A B + SS within. First, test of interaction: SS A B /(a 1)(b 1) SS within /(ab(n 1)) a = number of levels of factor A; b = number of levels of factor B; n = number of replicates of each combination of the two factors. If the interaction is significant, it is difficult to talk of the effect of factor A or factor B... If the interaction is not significant, then one proceeds with an additive model with factors A and B.

14 An example I consider a dataset (found somewhere on the Web) that presents the survival time of 48 rats after having received a poison (first factor) of 3 different types (I, II or III), and one of 4 different treatments (second factor) named A to D. See the data in the figure: survival time A.I B.I C.I D.I A.II B.II C.II D.II A.III B.III C.III D.III

15 Interaction plot An interaction plot is a visual guidance to see interaction: y-axis: mean response (survival time); x-axis: first factor (treatment); lines: second factor (poison). If additive model, parallel lines. Lines in opposite directions: relevant interaction. survival time A B C D treatment In the example, lines not far from parallel: weak interaction. poison II I III

16 Test for interaction Note: if n = 1 (no replicates) impossible testing for interactions (# of parameters = # of interactions); analysis possible only using additive model, after an interaction plot. Results of complete 2-way anova: Response: time Df Sum Sq Mean Sq F value Pr(> F ) poison e-07 *** treat e-06 *** poison:treat Residuals poison:treat are the interaction terms. Relevant F test is ratio between Mean Sq of interaction and of Residuals (within). The result shows that interaction is not significant.

17 Cautions Further steps that ought to be done: Check assumptions of the model (equal variance, and normality). [They are clearly not... ] Look for paired comparisons. Assume you are analysing the effects of treatments without knowing different poisons were used. Would the same results be obtained?

18 How to check assumptions (visually) Check for normality: QQ-plot of residuals Quantiles of residuals vs. normal ones Check for equal variances: Box-plots of residuals Residuals at each poison+treat combination If from a normal distribution, points should lie on a line. If equal variances, boxes should have approximately the same width.

19 Transformations of data to correct the problems A classical transformation is the logarithm. Repeat the analysis of variance on the logarithm of survival, and plot the residuals of the regression: Q-Q plot of residuals of ANOVA of logarithms Box-plots of residuals of ANOVA of logarithms Slightly better...

20 Transformations. II Use the inverse: 1/survival mortality Q-Q plot of residuals of ANOVA of inverses Box-plots of residuals of ANOVA of logarithms This seems the best one so far.... present the analysis on this.

2-way analysis of variance

2-way analysis of variance We may be considering the effect of two factors (A and B) on our response variable, for instance fertilizer and variety on maize yield; or therapy and sex on cholesterol level.