...How to calculate an ANOVA table Calculations by Hand We look at the following example: Let us say we measure the height of some plants under the effect of 3 different fertilizers. Treatment X Y Z Overall mean STEP 0: The model: Yij = µ + Ai + i ij Measures Mean 1 2 2 ... 5 6 5 ... 2 1 ... // . . . ˆ Ai ... ... ... (0.1) (0.2) n i Ai = 0 Interpretation: An observation yij is given by: the average height of the plants (µ), plus the effect of the fertilizer (Ai ). and an ”error” term ( ij ), i.e. every seed is different and therefore any plant will be different. All these values (µ, Ai , ij ) are UNKNOWN! Our GOAL is to test if the hypothesis A1 = A2 = A3 = 0 is plausible1 . Remark 1 If we have a control group (for example treatment ”X” is ”without any fertilizer”, then we assume that the values of X are in some way the best approximation for µ, therefore we can choose A1 = 0 is spite of condition (0.2). STEP 1: complete the first table. For the treatment means it is enough to calculate the mean of the values M eanX M eanY M eanZ 1 = = = 1+2+2 = 1.667 3 5+6+5 = 5.333 3 1+2 = 1.5 2 We DO NOT find ”the correct value” for the Ai We WILL NOT find which factor (treatment) has an effect, we just look if in general treatments has effect on the results. 1 The (estimated) overall mean (ˆ, which is an estimation of the exact, unµ known overall mean µ) is calculated as follows2 : µ = ˆ 1+2+2+5+6+5+2+1 =3 8 ˆ The estimated effects Ai are the difference between the ”estimated...
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