ANALYSIS OF VARIANCE (ANOVA) In spite of its name it serves for comparing means, not for comparing variances. ANOVA1

Remember: Two-sample t test Two independent samples Assuming the equality of variance for the two populations: ANOVA1

One-way ANOVA Several independent samples of size ANOVA1

The sample means are different, even if there is no difference between groups ANOVA1

The sample means are more different. Is the difference significant? group ANOVA1

order of experiments is important Example 51 (Box-Hunter-Hunter: Statistics for Experimenters, J. Wiley, 1978, p. 165) Blood coagulation times (seconds) with four different diets order of experiments is important ANOVA1

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Open Data Table: blood.xls Analyze>Fit Y by X Y: CTIME X: DIET click on the red triangle, choose Display options error bar ANOVA1

The variance within the i-th group (deviations from the group mean) The pooled within-groups variance (if constant across groups): ANOVA1

Variance of the group mean, if there is no real difference (H0) but sampling fluctuation repetitions as it is estimated from the group means (if p=const) more generally (if pconst) ANOVA1

if there exists real difference (H1) (between) (within) if there exists real difference (H1) ANOVA1

The ANOVA Table Effect of factor A is significant (H0 hypothesis is rejected) if ANOVA1

Balanced design: p1=p2=...=pr=p ANOVA1

Click on the red triangle, choose Means/Anova Example (cont.) Click on the red triangle, choose Means/Anova F0 between p within decision? ANOVA1

Minitab>Stat>ANOVA>One-Way between within decision? ANOVA1

pi different ANOVA1

Statistics>Advanced Linear/Nonlinear Models> >General Linear Models>One-way ANOVA ANOVA1

Summary tab: Descriptive cell statistics Summary tab: Test all effects between within ANOVA1

Assumptions expected value of the ij experimental errors is zero the ij experimental errors are independent both within groups (across j) and between groups (across i) the variance of experimental errors is constant the ij experimental errors follow normal distribution to be checked! In order to avoid misunderstanding for error variance will be used ANOVA1

experimental error (not just measurement error) i-th level of the factor (i-th diet) j-th repetition in the i-th group Model experimental error (not just measurement error) measured value true value expected value ANOVA1

i effect of the i-th level (i-th diet) means model i effect of the i-th level (i-th diet)  is a common value; r+1 parameters i=1,…,r sum to zero set to zero effects model ANOVA1

effect, only r-1 independent Estimates grand mean effect, only r-1 independent group mean ANOVA1

Estimation =0 grand mean ANOVA1

effect, only r-1 of them are independentfüggetlen mean of the i-th group ANOVA1

Fisher-Cochran-theorem all ANOVA1

Summary tab: Coefficients sum to zero sigma-restricted set to zero ANOVA1

Summary tab: Coefficients sigma-restricted set to zero ANOVA1

Confidence interval for the expected value of group means Point estimator: Interval estimator : degrees of freedom: Confidence interval for thee expected value of the i-th group: ANOVA1

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All of them are different? rejected All of them are different? Comparisons: planned, post hoc ANOVA1

df would be only n2+n3-2=6+6-2=10 and pooling df for is LSD test (Least Significant Difference) ANOVA1

cik contrast coefficients Generalisation: (kth null hypothesis) cik contrast coefficients contrast c11=0, c21=1, c31=-1, c41=0 ANOVA1

orthogonal contrasts if kl independent comparisons ANOVA1

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for a comparison α* (e.g. 0.05) (individual error rate) comparisons (1-2, 1-3, 1-4, 2-3, 2-4, 3-4) for a comparison α* (e.g. 0.05) (individual error rate) not committing type I error: 1- α* not committing type I error at any of r independent comparisons: committing type I error at some comparison: (family error rate) e.g. ANOVA1

In case of non-independent comparisons Bonferroni inequality e.g. for 6 non-independent comparisons 60.05=0.3 ANOVA1

Post hoc comparisons ANOVA1

B-A C-A D-A C-B D-B D-C ANOVA1

B-A C-A D-A C-B D-B D-C Minitab12

Post-hoc tab: Bonferroni Post hoc comparisons Post-hoc tab: LSD Post-hoc tab: Bonferroni 0.003803·6=0.022815 ANOVA1

Planned comparisons Planned comps tab: Specify contrasts ANOVA1

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Calculate the effect estimates: ANOVA1

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is rejected None of them are equal? further questions: ANOVA1

Click on the red triangle next to Oneway analysis, choose Compare Means, Each Pair, Student’s t threshold in t (LSD) significant: C-A, C-D,... C and B are connected by A ANOVA1

Click on the red triangle next to Oneway analysis, choose Compare Means, All Pairs, Tukey HSD more conservative (less easily states significance) ANOVA1

Power: probability of detecting an existing difference Click on the red triangle next to Oneway analysis, choose Power If =2.5, with 20 experiments (5 for each diet) we will be able to detect Delta as large as 3 with 98.6% probability ANOVA1

E.g. if 1=-3, 2=3, 3= 4=0 ANOVA1

The size of detectable difference Statistics>Power Analysis>Several Means, ANOVA 1-Way ANOVA1

E.g. if 1=-3, 2=3, 3= 4=0 ANOVA1

The size of detectable difference Minitab>Power and Sample Size>One-Way ANOVA ANOVA1

Required sample size for detecting difference of 5 units ANOVA1

Checking the assumption on homogeneity of variances click on the red triangle, choose Unequal Variances p ANOVA1

Contrast analysis click LSMeans Contrast ANOVA1

A+, B+, C-, D- ANOVA1

Homogeneity of (within-group) variance Minitab>Stat>ANOVA>Homogeneity of variance sensitive to the normaality assumption ANOVA1

Homogeneity of (within-group) variance Minitab>Stat>ANOVA>Homogeneity of variance sensitive to the normality assumption ANOVA ANOVA

Checking residuals (Graphs) ANOVA ANOVA

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Homogeneity of variance ? More results>Assumptions tab: Homogeneity of variances ... Bartlett test sensitive to normality Levene test ANOVA1

Checking assumptions by examining the residuals Residuals 1 tab Normality Pred & resids Predicted results (histogram) ANOVA1

Residuals 2 tab X: Order Y: Resids ANOVA1

Checking residuals Graphs>Four in One ANOVA1

Two-way ANOVA All levels of the first factor are combined with all levels of the second factor equal number of repetitions in each cell (balanced design). The structure of the design ensures the orthogonality. ANOVA1

Survival time of animals Example 52 (Box-Hunter-Hunter: Statistics for Experimenters, J. Wiley, 1978, p. 228) Survival time of animals ANOVA1

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poison (i) treatment (j) i=1,…,r; j=1,…,q, k=1,…,p Model repetition (k) poison (i) treatment (j) means model (all are equal) ANOVA1

effect of the i-th poison j-th treatment interaction Model effect of the i-th poison j-th treatment interaction effects model ANOVA1

The ANOVA Table 34(4-1)=36 ANOVA1

Open Data Table: poison.xls, (poison, treatment Nominal) Analyze>Fit Model Y: survival Add: poison, treatment (Macros, Full factorial) Emphasis: Minimal Report N! ANOVA1

Click on the red triangle next to Response SURVIVAL, choose Factor Profiling, Profiler and Interaction Plots ANOVA1

34(4-1)=36 ANOVA1

(r-1)(q-1)=(3-1)(4-1)=6 independent r-1=3-1=2 independent q-1=4-1=3 independent (r-1)(q-1)=(3-1)(4-1)=6 independent ANOVA1

Minitab>Stat>ANOVA>Two-Way poison.mtw Minitab>Stat>ANOVA>Two-Way ANOVA1

Minitab>Stat>ANOVA>Main Effects Plot

Minitab>Stat>ANOVA>Interactions Plot

Advanced Linear/Nonlinear Models> Statistics> Advanced Linear/Nonlinear Models> >General Linear Models>Factorial ANOVA> Means tab: Observed, unweighted, Plot Summary tab: All effects ANOVA1

Checking the assumptions by plotting residuals Click on the red triangle next to Response SURVIVAL, choose Row Diagnostics, Plot Residual by Predicted is not justified ANOVA1

Box-Cox transformation Click on the red triangle next to Response SURVIVAL, choose Factor Profiling, Box-Cox Y transformation variance stabilising transformation ANOVA1

Homogeneity of variance ? More results>Assumptions tab: Homogeneity of variances sensitive to normality ANOVA1

Checking the assumptions by plotting residuals Residuals1 tab: Pred. & resid. is not justified ANOVA1

Checking the assumptions by plotting residuals satisfied? ANOVA1

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Box-Cox transformation File>Open: (Program Files>StatSoft>Statistica8>Examples>Macros> >Analysis Examples>BoxCox) variance stabilising transformation ANOVA1

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straight line is fitted ANOVA1

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New column: recsurv=1/survival Repeat the analysis not transformed ANOVA1

Click on the red triangle next to Response SURVIVAL, choose Save Columns, Residuals Analyze: Distributions, Normal Quantile Plot random fluctuation around the line: Normal ANOVA1

Minitab>Stat>Basic Statistics>Store Descriptive Statistics Mean, Standard Deviation ANOVA1

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Minitab>Stat>Regression ANOVA1

Box-Cox transformation Minitab>Control Charts>Box-Cox transformation ANOVA1

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The effects are more convincing (F values are larger), p for interaction is 0.112 → 0.387 ANOVA1

The residuals ANOVA1

Minitab>Stat>ANOVA>Interactions Plot y to 1/y Minitab>Calc Minitab>Stat>ANOVA>Interactions Plot ANOVA1

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Minitab>Stat>ANOVA>Two-Way

Checking residuals Graphs>Four in One ANOVA1

Comparisons Do Poisons 1 and 2 differ? Planned comparisons fülön Compute ANOVA1

estimated effect ANOVA1