ST 524 NCSU - Fall 2008 Completely Randomized Design with Subsampling. Completely Randomized Design with subsamples

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1 ST 5 NCSU - Fall 008 Completely Randomzed Desgn wth subsamples Example (ST&D, p 59) Experment to analyze the effect of hours of daylght and nght temperature on the stem growth of mnt plants: 6 treatments (combnatons of temperature and daylght) were randomly assgned to 3 pots each; pots are nested under each level of treatment, and plants were measured from each pot, subsamples are nested wthn each level of pot and treatment. Plants randomly assgned to pots: per pot Treatment randomly assgned to pots: 3 pots per treatment Expermental unt: pot Subsample: plants wthn each pot. Snce every level of the nested factor does not appear wth every level of the treatment factor, no nteracton between these two factors. Sources of Varaton Varaton among the subsamples (plants) from the same expermental unt (pot) and treatment Samplng Error Varaton among expermental unts (pots) treated alke: pots wthn treatments Expermental Error Data: One-week stem growth Low Temp Hgh Temp 8 hs hs 6 hs 8 hs hs 6 hs T T T3 T T5 T6 Pot number Pot number Pot number Pot number Pot number Pot number Plant N o Pot total = Y j Treatment total = Y Treatment mean = Y Addtve Lnear Model = Y μ τ jk j jk Treatment s a Fxed Effect t = τ = 0 Pot s a Random effect, nested on treatments j ~ dn ( 0, ) Plant s a Random effect, nested on pots jk ~ dn ( 0, ), j and jk are ndependent random effects Analyss of Varance - Expected Mean Squares Sum of Squares Decomposton Tuesday September 9, 008 CRD Analyss of varance wth subsamplng

2 ST 5 NCSU - Fall 008 t r s t t r s ( Yjk Y... ) = rs( Y.. Y... ) + s( Yj. Y.. ) + ( Yjk Yj.) = j= k= = = j= j k= SS due to Treatment Effects SS due to Samplng Error Varaton among samples wthn pots SS due to Expermental Error Varaton among pots wthn treatments Analyss of Varance Table Sources df Sum of Squares Mean Square E(MS) τ Treatment 6- = e + ( 3) ( 6 ) Expermental Error Pots(Treatment) Samplng Error Plants(Pots) 6 ( 3 ) = ( ) = e Corrected total 6 3 = Dependent Varable: growth Sum of Source DF Squares Mean Square F Value Pr > F Model <.000 Error Corrected Total R-Square Coeff Var Root MSE growth Mean Source DF Type I SS Mean Square F Value Pr > F treatment <.000 pot(treatment) Note that each mean square s on a per-observaton (plant) bass. Means should be as well on a per-observaton bass. proc GLM data=a; model growth= treatment pot (treatment); random pot (treatment)/test; run; Tuesday September 9, 008 CRD Analyss of varance wth subsamplng

3 ST 5 NCSU - Fall 008 Test of Hypotheses Dependent Varable: growth Tests of Hypotheses for Mxed Model Analyss of Varance Source DF Type III SS Mean Square F Value Pr > F treatment <.000 Error Error: MS(pot(treatment)) Source DF Type III SS Mean Square F Value Pr > F pot(treatment) Error: MS(Error) Treatments : τ = τ = τ = τ = τ = τ = H : At least one τ 0, =,,, F calc = = 6.69 p-value <0.000 Reject.58 Expermental error = 0 H.58 F calc = = p-value = Reject Varance Components Estmaton Var(among subsamples unts wthn pot and treatment) = Var(among expermental unts): = = Varance among plants wthn pots Varance among pots wthn treatments Proc VARCOMP Type Estmates proc varcomp Method= Type; model growth= treatment pot(treatment)/fxed=; run; Varance Component Estmate Var(pot(treatment)) Var(Error) Proc MIXED Mxed Model: Treatments are Fxed effects, Pots(Treatment) are random effects, Plants(pots and treatment) are random effects. proc mxed data=a; model growth = treatment; random pot (treatment); lsmeans treatment; run; Tuesday September 9, 008 CRD Analyss of varance wth subsamplng 3

4 ST 5 NCSU - Fall 008 The Mxed Procedure Covarance Parameter Estmates Plants(pot and treatment) Cov Parm Estmate pot(treatment) Resdual Ft Statstcs - Res Log Lkelhood 07.7 AIC (smaller s better).7 AICC (smaller s better).9 BIC (smaller s better) 3.5 Type 3 Tests of Fxed Effects Num Den Effect DF DF F Value Pr > F treatment <.000 An Observaton, Yjk = μ + τ + j + jk Varance of an observaton Pot mean ( ) Var Y jk Varance of a pot mean Var Y Treatment mean = + estmated by =.387 = μ+ τ + + = μ+ τ + j + s s s j jk k k Y j. (. ) (. ) j Var j j = = +, estmated by μ τ = = rs rs r rs j jk jk jk Y.. μ τ μ τ Varance of a treatment mean Var Y (.. )... = Var μ+ τ + + = + r rs r sr s... j., estmated by Standard error of a treatment mean = Var ( Y... ) = 0.79 = 0.36 Expermental Error MS = Expermental Error MS 3 = 0.79 Tuesday September 9, 008 CRD Analyss of varance wth subsamplng

5 ST 5 NCSU - Fall 008 Least Squares Means Standard Effect treatment Estmate Error DF t Value Pr > t treatment T <.000 treatment T <.000 treatment T <.000 treatment T <.000 treatment T <.000 treatment T <.000 Note: Treatment LSMEANS from PROC MIXED. Testng null hypothess for Expermental Error - Covtest n PROC MIXED The Mxed Procedure Model Informaton Data Set WORK.A Dependent Varable growth Covarance Structure Varance Components Subject Effect pot(treatment) Estmaton Method REML Resdual Varance Method Profle Fxed Effects SE Method Model-Based Degrees of Freedom Method Contanment Class Level Informaton Class Levels Values treatment 6 T T T3 T T5 T6 pot 3 3 proc mxed data=a covtest; model growth = treatment; random ntercept/ subject= pot (treatment); lsmeans treatment; run; Dmensons and Covarance Parameters Columns n X 7 6 treatments + ntercept Columns n Z Per Subject Subjects 8 3 pots*6 treatments = 8 Max Obs Per Subject Number of Observatons Number of Observatons Read 7 Number of Observatons Used 7 Number of Observatons Not Used 0 Convergence crtera met. = 0 Covarance Parameter Estmates H Standard Z Cov Parm Subject Estmate Error Value Pr Z Intercept pot(treatment) Resdual <.000 Ft Statstcs - Res Log Lkelhood 07.7 AIC (smaller s better).7 AICC (smaller s better).9 BIC (smaller s better) 3.5 = 0 H Type 3 Tests of Fxed Effects Num Den Effect DF DF F Value Pr > F treatment <.000 Tuesday September 9, 008 CRD Analyss of varance wth subsamplng 5