Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur

Transcription

1 Analyss of Varance and Desgn of Experment-I MODULE VIII LECTURE - 34 ANALYSIS OF VARIANCE IN RANDOM-EFFECTS MODEL AND MIXED-EFFECTS EFFECTS MODEL Dr Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur

2 Fxed-effectseffects model All the factors n a fxed-effects model for an experment have a predetermned set of levels, e, they are fxed The statstcal nferences are drawn only for those levels of the factors whch are actually used n the experment Random-effects model The levels of factors used n experment are randomly drawn from a populaton of possble levels n case of a randomeffects model for an experment The statstcal nferences are drawn from the data for all levels of the factors n the populaton from whch the levels were selected and not only the levels used n the experment For example, n case of qualty control experment about the daly producton of fve machnes from an assembly lne, we have the followng set ups of fxed and random effect models: Fxed-effects: The daly producton of fve partcular machnes from an assembly lne Random-effects: The daly producton of fve machnes, chosen at random, that represent the machnes as a class

3 3 Many studes nvolve factors havng a predetermned set of levels and factors n whch the levels used n the study are randomly selected from a populaton of levels For example, the blocks n a randomzed complete block desgn may represent a random sample of b plots of land taken from a populaton of plots n an agrcultural research land Then the effects due to the blocks are consdered to be random effects Suppose the treatments are four new varetes of wheat that have been developed to be resstant to a specfc bactera The levels of the treatment are fxed because there are only varetes of nterest to the researchers, whereas the levels of the plots of land are random because the researchers are not nterested t n only those plots of land but are nterested t n the effects of these treatments on a wde range of plots of land When some of the factors to be used n the experment have levels randomly selected from a populaton of possble levels and other factors have predetermned levels, the model used to relate the response varable to the levels of the factors s referred to as a mxed effects model

4 4 Mxed effects model In a mxed-effects model for an experment, the levels of some of the factors used n the experment are randomly selected from a populaton of levels, whereas the levels of the other factors n the experment are predetermned The nferences from the data n the experment concernng factors wth fxed levels are only for the levels of the factors used n the experment, whereas the nferences concernng factors wth randomly selected levels are for all levels of the factors n the populaton from whch the levels were selected

5 5 Analyss of varance n one way random-effects model The model wth random effects s of the same structure as the model wth fxed effects gven as μ y = μ + + ε ( = 1,,, s; h= 1,,, n ) j j Here s the general mean effect as usual and s are the usual random error component The meanng of the parameter however has now changed The s are now the random effects of the treatment ( machne) Hence, ' s are the random varables whose dstrbutons we have to specfy We assume E( ) =, Var( ) = and E ( ε j ) =, E( ) = ( j) j ε j ' th th Then yj ~( μ, + ) holds

6 6 In the model wth fxed effects, the treatment effect A was represented by the parameter estmates, or ˆ μ = ˆ μ + ˆ, ˆ respectvely In the model wth random effects, a treatment effect can be expressed by the varance components The varance s estmated as a component of the entre varance The absolute or relatve sze of ths component then makes conclusons about the treatment effect possble The estmaton of the varances and requres no assumptons about the dstrbuton For the test of hypothess and the computaton of confdence ntervals, however, we assume the normal dstrbuton, e, ε j ~ N(, ), ε 's are assumed to be ndependent of each other, j ~ N(, ), 's are assumed to be ndependent of each other and, hence, y N j ~ ( μ, + ) th Unlke, the model wth effects, the response values y of a level of the treatment (e, of the sample) are no longer uncorrected E( y μ)( y μ) = E( + ε )( + ε ), j j' j j ' j j ' = E = ( ) j

7 7 On the other hand, the response values of dfferent samples are stll uncorrelated ( ', for any j, j'): E( y μ)( y μ) = E( ) + E( ε ε ) + E( ε ) = j ' j' ' j' ' j' ' j In the case of a normal dstrbuton, uncorrelated can be replaced by ndependence Test of the null hypothess H : = aganst H : 1 > The hypothess : no treatment effect for the two models s: - fxed effects: H : = for all - random effects: H : = If =, then the random effects are dentcally In ths case, each ˆ estmate ( = 1,,, s) should be close to relatve to the E If >, then the random effects are not dentcally In ths case, the varablty of the estmate ( = 1,,, s) should be large relatve to the E ˆ Testng hypotheses about the equalty of means s meanngless n the random effects case Therefore, we do not perform a multple comparson procedure to compare the means

8 8 The ANOVA table for a random factor s the same as the ANOVA table for a fxed factor wth SS = SS + SS Total Treatment Error To see ths we need to look at the expected mean squares for the random effects model We can partly adopt some of the results of fxed effect model, we have for random effect model; E ( Error ) = e, ˆ = s an unbased estmate of Error We compute ( ) E Tr as follows: n s Tr = ( o oo), = 1 j= 1 SS y y y = μ+ + ε, y o o oo = μ+ + ε = n / n, oo ( y y ) = ( ) + ( ε η ), o oo o oo

9 9 Then E y y E E ( o oo) = ( ) + ( εo εoo), E E E E ( ) = ( ) + ( ) ( ) n n = 1+, n n E E E E ( εo εoo) = ( εo + ( εoo) ( εoεoo) = + n n n 1 1 = n n

10 1 Hence n j= 1 E( y y ) = ne( y y ) o oo o oo n n n n = n n n n n and s = 1 n n E( yo yoo) = n + ( s 1) n We have now In the unbalanced case, e, all sample szes s are not the same, we have wth 1 E( Tr ) = E( SSA) = + k s k = n n s 1 n In the balanced case, we have 1 1 s 1 rs ; k = rs sr = r E ( Tr ) = + r, n ( n = r for all, n= rs)

11 11 Ths yelds the unbased estmate () In the unbalanced case k Tr Error ˆ = ˆ of as follows: () n the balanced case r Tr Error ˆ = In the case of an assumed normal dstrbuton we have Error ~ χ n s and Tr ~( + k ) χs 1 The two dstrbutons are ndependent, hence the rato Tr Error + k has a central F- dstrbuton under the assumpton of equal varances, e, under H : =

12 1 Under H : = Tr Error we thus have ~ F s 1, n s Hence H = s tested wth the same test statstc as H : = : the analyss of varance remans unchanged It s gven as follows: (all ) n the model wth fxed effects The table of Source Sum of Degrees of E() squares freedom Effects Fxed Random Treatments SS Tr s 1 n + + k s 1 Error SS Error n s

13 13 Note: The estmate of can be negatve also ( ˆ < ) But we know that a varance component cannot be negatve The followng are 3 possble ways to handle ths stuaton: 1 Assume = and the negatve estmate occurred due to random samplng The problem s that usng zero nstead of a negatve number can affect the other estmates Estmate usng the restrcted maxmum lkelhood method because t always yelds a nonnegatve estmate Ths method wll adjust other varance component estmates 3 Assume the model s ncorrect, and examne the problem n another way For example, add or remove an effect from the model, and then analyze the new model