ANOVA Notes Page 1. Analysis of Variance for a One-Way Classification of Data

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1 ANOVA Notes Page Aalss of Varace for a Oe-Wa Classfcato of Data Cosder a sgle factor or treatmet doe at levels (e, there are,, 3, dfferet varatos o the prescrbed treatmet) Wth a gve treatmet level there are measuremets or scores he subscrpt or dex s betwee ad ad labels the dfferet factor levels or treatmet varatos he j th score the th level s desgated as j, s the the treatmet factor dex; j labels the score the wth the th treatmet: j Note: It s ot requred ad t s tpcall ot the case that ou have a equal umber of measuremets from each treatmet group he ull hpothess asserts that o treatmet populato dffers from a other hus, the populatos of scores of the treatmet levels should have the same mea ad varace For theoretcal coveece, we wll assume the scores are ormall dstrbuted wth a commo varace wth each treatmet level, but the tests whch follow are farl robust for departures from ormalt Null Hpothess: H 0: μ = μ = μ3 = μ hs s equvalet to sag that the dfferet treatmets are reall just dfferet samples all tae from a sgle populato hus, all of the varato see the measured scores s due to sample varatos, e, radomess he total umber of scores s = I R Johso s text, ths s desgated as upper case N, however, to be cosstet wth our earler otato that lower case s assocated wth a sample ad upper case wth a populato, these otes wll use = () = he average of all scores s ofte called the grad mea ad s gve b = ,,,3,,,,3,,,,3, For otatoal coveece, ths s wrtte as a double summato j, = = () If the ull hpothess s true ad all of the treatmets gve the same mea result, the should be a far estmate of ever score Formall, we ca wrte the followg ( ) ( ) = + + (3) j, j, Al Lehe Madso Area echcal College 4/9/00

2 ANOVA Notes Page Here s the mea of treatmet varato j, = (4) I equato (3), ca be looed upo as a correcto term to the grad mea to tae accout of how treatmet group dffers from the grad mea, whle j, s a addtoal correcto term to tae care of varatos wth treatmet group If the ull hpothess s true, both of these terms should be of the same magtude ad both reflect radom devatos from the grad mea From equato (4) wth each treatmet the sum of the devatos about the treatmet mea vashes ( j, ) = 0 (5) j, Furthermore, ( ) = = =, j = 0, = = = = where the zero s a cosequece of equato () ( ) = = 0 (6) = = So, the sum of the devatos of the treatmet meas about the grad mea vashes hs meas that the sample meas have degrees of freedom sce ol of them ca be arbtrarl specfed gve the value of the grad mea Cosder the total varato of, the sum of squared devatos of each score about the grad mea he devato of a score s gve b the followg ( ) ( ) = + (7) j, j, So the sum of squared devatos or total varato s ( ) ( ) ( )( ) ( ) j, = j, + j, + = = (8) Now from equato (5), ( )(, j ) = ( ) (, j ) = 0 = = So the total varato s made up of two cotrbutos SS = (, j ) = (, j ) + ( ) = SSE+ SSr = = (9) Al Lehe Madso Area echcal College 4/9/00

3 ANOVA Notes Page 3 SSr = = () = ( ) (0) SSE = ( ) j, he term SSr s the reatmet sum of squares ad represets that part of the total varato that s due to the dffereces betwee the treatmets (factor levels) hs varato has degrees of freedom he secod term, SSE, s the resdual (left over) varato due to dffereces scores wth the dfferet treatmet groups he th treatmet level has degrees of freedom due to the costrat mposed b equato (5) he total degrees of freedom of the resdual or Error sum of squared devatos s gve b ( ) = = () = = he sum of the degrees of freedom of the treatmet sum of squares ad the error sum of squares s + = hs s the degrees of freedom of the total varato SS hs s sce the grad mea uses up a degree of freedom R Johso s text uses the otato SS for SS ad SS(r) for SS r If the ull hpothess s deed true, the the mea squares of treatmet ad error both estmate the same commo populato varace of each treatmet populato he mea squares are computed as a varato dvded b assocated degrees of freedom ( ) SS he reatmet or Betwee Mea Square: MS r r = = = (3) he Error or Wth Mea Square: ( ) j, SSE = MSE = = (4) o test the ull hpothess that all treatmet levels have the same mea, we compute the observed Fsher F score MS F = r (5) MSE For a gve level of sgfcace,α, ths value s compared agast a crtcal score calculated from a F dstrbuto used to compare two varaces obtaed from samplg varaces from two ormall dstrbuted populatos he umerator degrees of freedom s ad the deomator degrees of freedom s If F > Fα (, ), the ull hpothess s rejected, whle f F < Fα (, ) we fal to reject H 0 o facltate the actual calculatos, we defe the followg termedate varables Al Lehe Madso Area echcal College 4/9/00

4 ANOVA Notes Page 4 =, j (6) A = (8) = B ( ) =, j (7) = (9) = B = B (0) = From equatos () ad (4), = () = () B Also the th sample varace s gve b s = (3) Now, ( ) SSr = + = + = = = = = = Smlarl,, so SSr = A (4) (,, ), j j j j, j j = = = = = = = =, so SSE = + = + = SSE = B A (5) Fall, as a chec, the total varato must be the sum of treatmet ad error varatos SS = (, j ) =, j, j + = B = = = = A + ( B A) = SSr + SSE From equatos (3) ad (4), the mea squares are gve b the followg Al Lehe Madso Area echcal College 4/9/00

5 ANOVA Notes Page 5 he reatmet or Betwee Mea Square: MSr A SS = r = (6) he Error or Wth Mea Square: SSE B A MSE = = (7) So a scheme to calculate a oe-wa classfcato aalss of varace s to la out the data colums as a spreadsheet, wth each colum represetg a dfferet treatmet or factor level Leave room betwee treatmets for a secod colum whch s the square of the frst colum hus, each treatmet s assocated wth a par of colums Sum the colum of scores to obta ad sum the colum of squares of scores to obta B he for each treatmet, calculate the B sample mea,, ad the sample varace s = he sum over the treatmet groups to obta, B, A, ad Note: he table below should ot be terpreted as mplg that the umber of rows each colum s the same I geeral, t s ot true that = = 3 = reatmet reatmet reatmet 3 reatmet,,,, 3, 3,,,,,,, 3, 3,,,,3,3,3,3 3,3 3,3,3,3, 3,,,, 3, 3, B B 3 B 3 B 3 s s For each par of colums:, Summg over the colums: j At ths pot we ca costruct a ANOVA table B 3 = ( ) s =, j A = B = B = = = s = = Source Sum of Squares Degrees of Freedom Mea Square reatmet SS r = A / - MS r = SS r /( ) Error SSE = B - A - MSE = SSE /( ) otal SS = B / - SS s =, Al Lehe Madso Area echcal College 4/9/00

6 ANOVA Notes Page 6 MS Compute F = r ad compare ts value agast Fα ( ν =, ν = ) for a stated level of MSE sgfcace,α If F < Fα (, ), we fal to reject H 0: μ = μ = μ3 = μ If F > Fα (, ), we reject H0 ad coclude that at least two of the populato meas are dfferet o determe whch treatmet meas are dfferet more testg eeds to be doe Some procedures whch are used are Scheffe s test ad ue s test However, the method we wll use ( ) s multple t testg, also ow as the Boferro procedure here are = dstct pars of treatmets whch ca be pced from the dfferet treatmet groups For each such par a observed t - score s calculated If ths observed score s larger absolute value tha a crtcal t score, the the ull hpothess that asserts the equalt of the two populato meas assocated wth the two treatmets s rejected Specfcall, let ad m stad for the treatmet group dces of the two treatmets beg examed he, obs (, ) m t m = (8) sp + m s the b ow famlar observed t - score for comparg two depedet sample meas he absolute value s used sce ths s a two-sded test for a dfferece betwee two populatos Sce we have alread assumed that the treatmet populatos have measuremets whch are ormall dstrbuted wth a commo varace, we use the pooled varace whch best estmates ths commo wth treatmet group varace, amel MSE B A s p = MSE = (9) Hece, our observed t score s computed from obs (, ) m t m = (30) MSE + m Now, oe would expect that ths t obs (, m) would be compared agast a crtcal t score tc = t α /, where α s the level of sgfcace However, we watα to be the maxmum ( ) probablt of a pe I error for the etre sequece of m = = comparsos If each comparso were made at a level of sgfcace of α, the the probablt of a pe I error somewhere the m comparsos made would be ( α ) m hs probablt s larger tha α ad cosderabl larger f m s large o have a actual level of sgfcace of α, we wll therefore use tc = t α /( m ), so that the probablt of a pe I error the m comparsos becomes Al Lehe Madso Area echcal College 4/9/00

7 ANOVA Notes Page 7 m α α m Fall, whe we fd t c, we eed to ow the degrees of freedom Sce we are usg MSE for s p, the degrees of freedom sν = hus, summar we have tc = t α /( m) wth degrees of freedom (3) Cosder the followg example Suppose there are fve dfferet treatmets, e, = 5, ad suppose we wsh to wor at a level of sgfcace of 0%,,e, α = 00 he the umber of 5 54 ( ) par wse comparsos s m = = = 0 Suppose that the total umber of measuremets over the fve treatmets s 6, e, = 6 hus, from equato (3), tc = tα /( m) = t0/0 = t005 wth 6 5 = degrees of freedom Hece, t c = 83 Now to facltate the 0 dfferet comparsos, t helps to summarze the results tabular form Each ope cell the followg table represets oe of the 0 possble comparsos reatmet 3 4 t obs (, ) 3 t obs (,3) t obs (,3) 4 t obs (, 4) t obs (,4) t obs (3,4) 5 t obs (,5) t obs (,5) t obs (3,5) t obs (4,5) I these cells the computed value of t obs for the two treatmets beg compared s dsplaed Wheever ths value of t obs exceeds 83 oe ca coclude at a 0% level of sgfcace that the two populato meas are dfferet Al Lehe Madso Area echcal College 4/9/00