STATISTICAL MODELLING PRACTICAL VI SOLUTIONS

Transcription

1 VI-1. TATITICAL MODELLING PRACTICAL VI OLUTION VI.1 The following daa are from a Lain square experimen designed o invesigae he moisure conen of urnip greens. The experimen involved he measuremen of he percen moisure conen of five leaves of differen sizes from each of five plans. The reamens were ime of measuremen in days since he beginning of he experimen. Plan A B Leaf ize C (A = smalles, D E = larges) E Wha are he feaures of his experimen? 1. Observaional uni a leaf. Response variable Moisure Conen 3. Plan, Leaf ize 4. Time 5. Type of sudy Lain square Wha is he experimenal srucure for his experimen? rucure unrandomized randomized Formula 5 Plan*5 ize 5 Time Wha are he Hasse diagrams of generalized-facor marginaliies, wih degrees of freedom and wih M and Q marices, for his sudy? The sources in he analysis are: Plan*ize = Plan + ize + Plan#ize and Time = Time. The Hasse diagrams, wih degrees of freedom, for his sudy are:

2 VI-. Hasse diagrams for he Turnip moisure experimen Plan 5 P ize Time T Plan ize 5 P# 16 The Hasse diagrams, wih M and Q marices, for his sudy are: Hasse diagrams for he Turnip moisure experimen Plan M P P ize Time T M P M M M T M T Plan ize M P P# M P M P M + Derive he maximal expecaion and variaion models for his sudy? In his example, Plan is likely o be a random facor and ize and Time o be fixed facors. In paricular, ize may well show a sysemaic rend and his would be modelled used expecaion erms.

3 VI-3. Hence, he maximal models are: ψ = E[Y] = ize + Time and var[y] = Plan + Plan ize. Deermine he expeced mean squares for his sudy, using where appropriae he Hasse diagrams of generalized-facor marginaliies. Hasse diagrams for he Turnip moisure experimen Plan 5 P P P + 5P ize q ( ψ) + q ψ P ( ) Plan ize P P# P In addiion he randomized facor Time will conribue qt ( ψ ). Give he analysis of variance able, including he degrees of freedom, sums of squares and expeced mean squares. ource df q E[Mq] Plan 4 YQ PY + 5 ize 4 Plan#ize 16 YQ PY Time 4 T P YQ Y ( ) P + q ψ YQ Y ( ) P + qt ψ P Residual 1 YQ Y P Res P Toal 4 ( ψ ) = 5 ( ) j 4, q ( ) ( ) T ψ = τk τ. q β β. 5 4

4 VI-4. VI. A chemis has four differen conainers of soil. He wans o deermine wheher he moisure conens of hese four soils differs. He randomly selecs 10 samples from each conainer and deermines he moisure conen of each sample. Wha are he feaures of his experimen? 1. Observaional uni a sample. Response variable Moisure conen 3. oil, ample 4. none 5. Type of sudy R Wha is he experimenal srucure for his experimen? rucure unrandomized randomized Formula 4 oils/10 amples Wha are he Hasse diagrams of generalized-facor marginaliies, wih degrees of freedom and wih M and Q marices, for his sudy? The sources in he analysis are: oils/amples = oils + amples[oils]. The Hasse diagram, wih degrees of freedom, for his sudy are: Hasse diagrams for he oil survey oils 4 3 oils amples 40 [] 36

5 VI-5. The Hasse diagram, wih M and Q marices, for his sudy are: Hasse diagrams for he oil survey oils M M oils amples M [] M M Derive he maximal expecaion and variaion models for his sudy? In his survey amples are likely o be random as hese samples are mean o be represenaive of a larger populaion of samples and one envisages using a probabiliy disribuion funcion o model sample variabiliy. I i s no clear wheher soils are fixed or random. Do we have four soils ha difffer in an arbirary way or are hese 4 soils represenaive of a large group of similar soils? Assume hey are fixed. Based on his he maximal models would be: var[y] = oils amples. ψ = E[Y] = oils and

6 VI-6. Deermine he expeced mean squares for his sudy, using where appropriae he Hasse diagrams of generalized-facor marginaliies. Hasse diagrams for he oil survey oils q ( ψ) + ψ q ( ) oils amples [] Give he analysis of variance able, including he degrees of freedom, sums of squares and expeced mean squares. ource df q E[Mq] oils 3 YQ Y ( ) + q ψ amples[oils] 36 YQ Y Toal 39

7 VI-7. VI.3 In an experimen o invesigae he yield ( /hr) of machine producing a chemical, four randomly seleced machines were operaed a five differen emperaures for an hour and he yield measured. The order in which each machine was operaed a he differen emperaures was randomized for each machine. Wha are he feaures of his experimen? 1. Observaional uni a machine a a ime. Response variable Yield 3. Machine, Time 4. Temperaure 5. Type of sudy RCBD Wha is he experimenal srucure for his experimen? rucure unrandomized randomized Formula 4 Machine/ 5 Time 5 Temperaure Wha are he Hasse diagrams of generalized-facor marginaliies, wih degrees of freedom and wih M and Q marices, for his sudy? The sources in he analysis are: Machine/Time = Machine + Time[Machine] and Temperaure = Temperaure. The Hasse diagrams, wih degrees of freedom, for his sudy are: Hasse diagrams for he chemical experimen Machine M Temperaure Machine Time 0 T[M] 16

8 VI-8. The Hasse diagrams, wih M and Q marices, for his sudy are: Hasse diagrams for he chemical experimen Machine M Temperaure M M M M M M Machine Time T[M] M MT M MT M M Derive he maximal expecaion and variaion models for his sudy? In his experimen Machine and Time are likely o be random. I seems reasonable o regard all machines as having he same mean value and ha differen machine vary haphazardly around his mean value. Also, having randomized o he imes wihin a machine, i would appear ha we are no anicipaing a sysemaic difference beween he imes and so again i would be appropriae o model he imes wihin a machine as having he same mean value and he differen imes as varying haphazardly abou he mean value. The facor Temperaure would be fixed. Based on his he maximal models would be: ψ = E[Y] = Temperaure and var[y] = Machine + Machine Time.

9 VI-9. Deermine he expeced mean squares for his sudy, using where appropriae he Hasse diagrams of generalized-facor marginaliies. Hasse diagrams for he chemical experimen Machine 5 M M MT + 5M Temperaure ψ q ( ψ ) q ( ) Machine Time MT T[M] MT Give he analysis of variance able, including he degrees of freedom, sums of squares and expeced mean squares. ource df Q E[Mq] Machine 3 YQ MY + 5 Time[Machine] 16 YQ MTY Temperaure 4 MT M YQY + q ( ψ ) MT Residual 1 YQ Y MT MT Res Toal 19 ( ψ ) = ( ) k q 4 τ τ 4. VI.4 A sudy is o be conduced o compare wo mehods of measuring he concenraion of a cerain componen of a liquid produc. Three facories are seleced from hose ha rouinely deermine he concenraion of he componen. A sample of he produc is obained and divided ino 3 los of 4 porions and each lo is randomly assigned o be sen o one of he facories. A each facory he concenraion of heir 4 porions is deermined using boh mehods. The order in which a porion is esed using a paricular mehod is compleely randomized. Wha are he feaures of his experimen?

10 VI Observaional uni a es a a facory. Response variable Concenraion 3. Facories, Tess 4. Los, Porions, Mehods 5. Type of sudy RCBD wih many randomized facors Wha is he experimenal srucure for his experimen? rucure Formula unrandomized 3 Facories/ 8 Tess randomized (3 Los/4 Porions)* Mehods Wha are he Hasse diagrams of generalized-facor marginaliies, wih degrees of freedom and wih M and Q marices, for his sudy? The sources in he analysis will be: Facories/Tess = Facories + Tess[Facories] (Los/Porions)*Mehods = (Los + Porions[Los])*Mehods = Los + Porions[Los] + Mehods + Los#Mehods + Porions#Mehods[Los] The Hasse diagrams, wih degrees of freedom, for his sudy are: Hasse diagrams for he concenraion experimen Facories 3 F Los L Mehods M 3 1 Facories Tess 4 T[F] Los Porions P[L] Los Mehods L#M 9 6 Los Porions Mehods 4 P#M[L] 9

11 VI-11. The Hasse diagrams, wih M and Q marices, for his sudy are: Hasse diagrams for he concenraion experimen Facories F Los L Mehods M M F M F M L M L M M M M Facories Tess T[F] Los Porions P[L] Los Mehods L#M M FT M FT M LP M LP M L M LM M LM M L M M + Los Porions Mehods M LPM P#M[L] M LPM M LP M LM +M L Derive he maximal expecaion and variaion models for his sudy? In his experimens he random facors would be Facories, Tess, Los and Porions and he fixed facors would be Mehods. Hence he maximal models would be: ψ = E[Y] = Mehods and var[y] = Facories + Facories Tess + Los + Los Porions + Los Mehods + Los Porions Mehods

12 VI-1. Deermine he expeced mean squares for his sudy, using where appropriae he Hasse diagrams of generalized-facor marginaliies. Hasse diagrams for he concenraion experimen Facories 8 F F FT + 8 F Los L LPM + 4LM + LP + 8L Mehods LPM LM q L M ( ψ ) + + q ( ψ ) M LP M Facories Tess FT T[F] FT Los Porions LP P[L] LPM + LP Los Mehods 4 LM L#M LPM + 4LM Los Porions Mehods LPM P#M[L] LPM Give he analysis of variance able, including he degrees of freedom, sums of squares and expeced mean squares. ource df q E[Mq] Facories YQ FY Los YQ LY Tess[Facories] 1 YQ FTY FT F LPM LM LP L Porions[Los] 9 YQ LPY + + Mehods 1 M FT LPM LP YQ Y q ( ψ ) FT LPM LM M Los#Mehods YQ LMY FT LPM LM Porions#Mehods[Los] 9 YQ LPMY + Toal 4 ( ψ ) = ( ) k q 1 τ τ 1 M. FT LPM

13 VI-13. o which raio of mean squares would be used o es for an average difference beween he wo mehods? Take he raio of he Mehods and Los#Mehods mean squares o make he es. Noice ha he degrees of freedom for his es are very low. VI.5 Adap he R expressions for randomizing a randomized complee block design o obain a randomized layou and dummy analysis for he experimen described in exercise VI.4. > b <- 3 > <- 8 > n <- b* > FacFacory.uni <- lis(facories=b, Tess=) > FacFacory.nes <- lis(tess = "Facories") > FacFacory.ran <- fac.gen(lis(los = 3, Porions = 4, Mehods = )) > FacFacory.lay <- fac.layou(unrandomized = FacFacory.uni, + nesed.facors = FacFacory.nes, + randomized = FacFacory.ran, seed = 1015) > FacFacory.lay Unis Permuaion Facories Tess Los Porions Mehods > # add a column y wih random normal daa and analyze > FacFacory.da <- FacFacory.lay > FacFacory.da$y <- rnorm(n) > FacFacory.aov <- aov(y ~ (Los/Porions)*Mehods + Error(Facories/Tess), FacFacory.da) > summary(facfacory.aov) Error: Facories Df um q Mean q Los Error: Facories:Tess Df um q Mean q Mehods Los:Porions Los:Mehods Los:Mehods:Porions This confirms he form of he previously derived analysis.