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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           12   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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posted:5/11/2013
language:English
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