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ANALYSIS OF VARIANCE (ANOVA) test of equality of population means for 3 and more samples H0 : 1 = 2 = 3 = … = k H1: AT LEAST between two population means is a difference 1 ANOVA We want to evaluate the infuence of fertilisation on height growth of seedlings H0: fertilisation has no influence without strong middle fertilisation fertilisation fertilisation H1: fertilisation has significant influence 2 3 ANOVA FERTILISATION strong middle none x x x3 2 1 3 2 1 ANOVA x3 x x 1 2 without middle strong fertilisation fertilisation fertilisation H0 : 1 = 2 = 3 are random and in population statistically insignificant 4 ANOVA x3 x 2 x 1 without middle strong fertilisation fertilisation fertilisation H1: 1 2 3 are significant and in population statistically significant 5 ANOVA – parts of variance TOTAL VARIANCE (VARIANCE OF ALL EXPERIMENT) KNOWN SOURCES OF VARIABILITY (part of total variability explanable by known factor (effect) and is expressed by the difference among sample means UNKNOWN SOURCES OF VARIABILITY (part of total variability explanable by random variations of values within samples and source of variation is unknown) 6 ANALÝZA ROZPTYLU (ANOVA) 1 2 3 Variability within samples is SMALL in comparison with variability between samples GOOD DIFFERENTIATION OF MEANS Variability within samples is LARGE in comparison with variability between samples BAD DIFFERENTIATION OF 7 12 3 MEANS ANOVA 2 SAMPLES 3 AND MORE SAMPLES t – test ANOVA paired t – test ANOVA with repeteated measurements Mann-Whitney Kruskal-Wallis test 8 ANOVA - conditions all samples are independent all samples come from populations with normal distributions all samples come from populations with equal variances 9 ANOVA – checking of conditions independence sample B sample B sample A sample A samples A and B are dependent samples A and B are independent normality – test of normality equality of variances – Cochran test – for equal sample size, 10 Barttlet test – for unequal sample size ANOVA –model Model of 1-F ANOVA: yij = + i + ij EFFECT ERROR MEASURED MEAN change of sample VALUE mean caused by factor 11 ANOVA 12 1-F ANOVA - result table Degrees Mean square Test Source of variability Sum of squares of (variance) criterion freedom between samples within samples (residual) Total 13 F > F,k-1,N-k H1 1-F ANOVA – next step? H0 accepted STOP DATA ANOVA multiply H0 rejected comparison tests 14 ANOVA – multiply comparisons H0: A = B , (A B) H1: A B Comparisons are made for possible combinations of samples. 15 ANOVA – multiply comparisons Fisher Tuckey Scheffe x1 x2 x3 and many others … 16 TuckeyHSD test H0: A = B , (A B) H1: A B, xA xB q SE MR MR 1 1 SE SE 2 n n n A B For q > q; N-k; k; (quantil of studentized range) we reject H0 and difference of means of samples A and B is significant 17 Scheffe test H0: A = B , (A B) H1: A B, xA xB 1 1 S n n SE M R SE A B If S > S k 1 F ;k 1; N k then there is significant difference between population means A and B. 18 2-F ANOVA - model y ij = μ + αi + β j + αiβ j + εij yij measured value influenced by i-th level of factor A a j-th level of factor B average common value of yij (common for all samples) i effect of level Ai j effect of level Bi α i β j interaction between factors (optional , there are models with interaction or without interaction) ij random error N(0,2) 19 2-F ANOVA - interaction Study is dealing with the impact of various doses of nitrogen (N) and phospforus (P) on yield of crop. Both elements have two levels – N (40, 60), P(10,20). Results of first three experiments: Exper. N P Yield T1 60 10 145 T2 40 10 125 20 T3 40 20 160 2-F ANOVA - interaction expectation What yield we obtain when for N = measured 60 we increase dose of P to 20? Pokus N P Výnos T1 60 10 145 T2 40 10 125 T3 40 20 160 21 T4 60 20 ? we expect 180 2-F ANOVA - interaction Pokus N P Výnos T1 60 10 145 T2 40 10 125 T3 40 20 160 Result of 4th experiment: T4 60 20 130 expected skutečnost measured 22 2-F ANOVA - interaction Paralel lines – effects of factors are additive (factors are independent) Crossed lines– – effects of factors are not additive - there is interaction between factors Interaction is present if effect of one factor is not the same when levels of the second factors are changed. So factors are not independent but response on one factor 23 depends on levels of other factors.
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