17 Nested and Higher Order Designs
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1 54 17 Nested and Hgher Order Desgns 17.1 Two-Way Analyss of Varance Consder an experment n whch the treatments are combnatons of two or more nfluences on the response. The ndvdual nfluences wll be called factors, and the possble values for the factors wll be called the levels of the factor. The mportant concepts can be best llustrated n the case of two factors n a fxed effects model: In an experment wth two factors A and B, a specfc treatment combnaton conssts of factor A at level and factor B at level. Assume that there are a levels of factor A and b levels of factor B under nvestgaton. Thus the experment conssts of a b treatment combnatons. Assume the lnear model Y k = µ + ε k, wth =1,...,a; =1,...,b; k =1,...,n ; and ε k N(0,σ 2 ) ndependent Theorem: The least-squares estmates are ˆµ = Ȳ = n k=1 Y k/n Note: Ths model s often called the cell means model. Ths model has ab 1 degrees of freedom and the error degrees of freedom are (n 1). However, such an analyss s not of much nterest snce we have not ncluded anystructure on the way nwhch the factors nfluence the response. One useful structure postulates the exstence of an addtve structure. Wrte µ = µ +( µ µ )+( µ µ )+(µ µ µ + µ ) =µ + α + β + γ, µ = µ s the overall mean, α = µ µ s the effect of the th level of factor A, β = µ µ s the effect of the th level of factor B, γ = µ µ µ + µ s the effect of an nteracton between levels and of factors A and B Note: We use the dentfablty constrants a α =0, =1 b β =0, =1 a b γ = γ =0. =1 = Note: Hypothess of nterest are: H AB : γ =0, ; H A : α =0 ; H B : β = Note: No nteracton means that the dfference between two groups and defned by one factor s ndependent of the level of the other factor: µ µ = µ µ, µ µ = µ µ,, µ µ = µ µ,,
2 Note: H A s equvalent to µ µ =0,. e. averaged across levels of factor B, the average mean s constant across levels of factor A Note: If H AB : γ =0s true, then H A s equvalent to µ µ =0,,,. e. the mean s constant across levels of factor A wthn each level of factor B. Thus, hypotheses on man effects make most sense f there s no nteracton Theorem: In a balanced desgn (. e. n = r, ), the least-squares estmates are ˆµ = Ȳ, ˆα = Ȳ Ȳ, ˆβ = Ȳ Ȳ, ˆγ = Ȳ Ȳ Ȳ + Ȳ. The ANOVA decomposton s gven by SS TOTAL = SS A + SS B + SS AB + SS E SS TOTAL = SS A = SS B = SS AB = SS E = (Y k Ȳ ) 2, (Ȳ Ȳ ) 2 = rb (Ȳ Ȳ ) 2 = rb ˆα 2 = RSS HA RSS, (Ȳ Ȳ ) 2 = ra (Ȳ Ȳ ) 2 = ra ˆβ 2 = RSS HB RSS, (Ȳ Ȳ Ȳ + Ȳ ) 2 = r ˆγ 2 = RSS HAB RSS, (Y k Ȳ ) 2 = RSS Theorem: For the fxed effects model we have: (a) E[MS A ]=E[SS A /(a 1)] = σ 2 + rb (b) E[MS B ]=E[SS B /(b 1)] = σ 2 + ra b 1 α2. β 2. (c) E[MS AB ]=E[SS AB /((a 1)(b 1))] = σ 2 + (d) E[MS E ]=E [SS E /(ab(r 1))] = σ 2. r ()(b 1) γ Theorem: For the fxed effects model we have: (a) MS A /MS E F,ab(r 1) (b) MS B /MS E F b 1,ab(r 1) f H A s true. f H B s true. (c) MS AB /MS E F ()(b 1),ab(r 1) f H AB s true.
3 Note: The above s usually summarzed n an ANOVA table: source sum of squares df mean squares test statstc Factor A SS A a 1 MS A = SS A /(a 1) MS A /MS E Factor B SS B b 1 MS B = SS B /(b 1) MS B /MS E Interacton SS AB (a 1)(b 1) MS AB = SS AB /((a 1)(b 1)) MS AB /MS E Error SS E ab(r 1) MS E = SS E /(ab(r 1)) Total SS TOTAL abr Note: The above ANOVA table s for a fxed effects model. To calculate the correct F statstcs for the hypothess tests n 2-way ANOVA random effects and mxed effects models, we have to consder the expected mean squares for the dfferent factors. The expected mean squares for fxed, random, and mxed (A fxed and B random) effects ANOVAs are shown n the table below: source fxed effects random effects mxed effects Factor A Factor B σ 2 + rb σ 2 + ra b 1 Interacton σ 2 + r ()(b 1) α2 σ 2 + rσ 2 AB + rbσ2 A σ 2 + rσ 2 AB + rb β 2 σ 2 + rσab 2 + raσ2 B σ 2 + raσb 2 γ2 σ 2 + rσab 2 σ 2 + rσab 2 Error σ 2 σ 2 σ 2 α Nested Desgns In the two-way classfcaton the factors were crossed,. e. each level of factor A occurred wth each level of factor B. In contrast, B s called nested wthn A f the levels of B are sampled wthn each of the levels of A, and so dfferent levels of A have dfferent levels of B Example: Assume factor A represents dfferent laboratores. Wthn each lab, 3 techncans are sampled from all the techncans n that lab, and each techncan performs some experment. Laboratory 1 2 Techncan (1, 1) (1, 2) (1, 3) (2, 1) (2, 2) (2, 3) µ 11 µ 12 µ 13 µ 21 µ 22 µ 23 If nstead the same 3 techncans performed the experment n all the labs, the factors would be crossed: Techncan Laboratory µ 11 µ 12 µ 13 2 µ 21 µ 22 µ 23
4 Note: The model for the nested desgn s µ = µ + α + β, whle the model for the crossed desgn s µ = µ + α + β + γ. The nested desgn can be seen as an ncomplete 2-way layout: Techncan Laboratory (1,1) (1,2) (1,3) (2,1) (2,2) (2,3) 1 µ 11 µ 12 µ 13 2 µ 21 µ 22 µ Example: Assume we have a hosptals, b techncans per hosptal, and each techncan performs r replcatons of the experment. The ANOVA model s Y k = µ + ε k wth =1,...,a; =1,...,b; k =1,...,r; and ε k N(0,σ 2 ) ndependent. We wrte µ = µ +( µ µ )+(µ µ ) =µ + α + β, α s the Hosptal effect, and β s the effect of th techncan n th hosptal. In a random effects model, we assume α N(0,σ 2 A ) and β N(0,σ 2 B A ) Note: Hypothess of nterest are: H A : σ 2 A =0 and H B : σ 2 B A = Note: H B mples that µ = µ, µ µ =0,,,. e. there are no dfferences between levels of B wthn each level of A Theorem: The ANOVA decomposton n a balanced desgn s gven by SS TOTAL = SS A + SS B A + SS E SS TOTAL = SS A = SS B A = SS E = (Y k Ȳ ) 2, (Ȳ Ȳ ) 2 = rb (Ȳ Ȳ ) 2 = r (Y k Ȳ ) 2. (Ȳ Ȳ ) 2, (Ȳ Ȳ ) 2,
5 Theorem: For the random effects model we have: (a) E[MS A ]=E[SS A /(a 1)] = σ 2 + rσb A 2 + rbσ2 A. (b) E[MS B A ]=E[SS AB /(a(b 1))] = σ 2 + rσb A 2. (c) E[MS E ]=E[SS E /(ab(r 1))] = σ Theorem: For the random effects model we have: (a) MS A /MS B A F,a(b 1) f H A s true. (b) MS B A /MS E F a(b 1),ab(r 1) f H B A s true Note: The above s usually summarzed n an ANOVA table: source sum of squares df mean squares test statstc Factor A SS A a 1 MS A = SS A /(a 1) MS A /MS B A Factor B A SS B A a(b 1) MS B A = SS B A /(a(b 1)) MS B A /MS E Error SS E ab(r 1) MS E = SS E /(ab(r 1)) Total SS TOTAL abr Note: The above ANOVA table s for a random effects model. In general, by the defnton of a nested model, the nested factor s usually consdered random. However, the expected mean squares can be calculated for a fxed effects model too. In the table below we summarze the expected mean squares for random, mxed (A fxed and B random), and fxed effects nested ANOVA models. source random effects mxed effects fxed effects Factor A σ 2 + rσb A 2 + rbσ2 A σ2 + rσb A 2 + rb α2 σ 2 + rb Factor B A σ 2 + rσ 2 B A σ 2 + rσ 2 B A σ 2 + r a(b 1) Error σ 2 σ 2 σ 2 α2 β2
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