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

Transcription

1 Analysis of Variance and Design of Exeriment-I MODULE II LECTURE -4 GENERAL LINEAR HPOTHESIS AND ANALSIS OF VARIANCE Dr. Shalabh Deartment of Mathematics and Statistics Indian Institute of Technology Kanur

2 Regression model for the general linear hyothesis Let 1,,..., n be a sequence of n indeendent random variables associated with resonses. Then we can write it as E( ) = β x, i = 1,,..., n, j = 1,,..., Var i j ij j = 1 ( i ) = σ. This is the linear model in the exectation form where β1, β,..., β are the unknown arameters and x ij ' s are the known values of indeendent covariates X1, X,..., X. Alternatively, the linear model can be exressed as = β x +, i = 1,,..., n; j = 1,,..., i j ij i j= 1 where s are identically and indeendently distributed random error comonent with mean 0 and variance σ, i.e., i E( ) 0, Var( ) Cov(, ) 0( i j). ( i) = 0 ( i) = σ and ( i j) = 0( ) In matrix notations, the linear model can be exressed as where = Xβ + = ( 1,,..., n )' is n 1 vector of observations on resonse variable, the matrix X X11 X1... X1 X1 X... X = Xn 1 Xn... X n is n matrix of n observations on indeendent covariates X 1, X,..., X,

3 3 β = ( β1, β,..., β ) is a 1 vector of unknown regression arameters (or regression coefficients) β1, β,..., β associated with X1, X,..., X, resectively and = (,,..., ) is a n 1 vector of random errors or disturbances. 1 n We assume that E( ) = 0, covariance matrix V E I rank X ( ) = ( ') = σ, ( ) =. In the context of analysis of variance and design of exeriments, the matrix X is termed as design matrix; unknown β1, β,..., β are termed as effects; the covariates X1, X,..., X are counter variables or indicator variables where counts the number of times the effect β occurs in the i th observation x. β j x ij mostly takes the values 1 or 0 but not always. x i The value x ij = 1 indicates the resence of effect β j in x and x i ij = 0 indicates the absence of effect β j in X i. x ij Note that in the linear regression model, the covariates are usually continuous variables. When some of the covariates are counter variables and rest are continuous variables, then the model is called as mixed model and is used in the analysis of covariance.

4 4 Relationshi between the regression model and analysis of variance model The same linear model is used in the linear regression analysis as well as in the analysis of variance. So it is imortant to understand the role of linear model in the context of linear regression analysis and analysis of variance. Consider the multile linear model = β0 + X1β1 + X β X β +. In the case of analysis of variance model, the one-way classification considers only one covariate, two way-classification model considers two covariates, three-way classification model considers three covariates and so on. If β, γ and δ denote the effects associated with the covariates X, Z and W which are counter variables, then in One-way model: = α + Xβ + Two-way model: = α + Xβ + Zγ + Three-way model : = α + Xβ + Zγ + Wδ + and so on. Consider an examle of agricultural yield. The study variable denotes the yield which deends on various covariates X1, X,..., X. In case of regression analysis, the covariates X1, X,..., X are the different variables like temerature, quantity of fertilizer, amount of irrigation, etc.

5 5 Now consider the case of one way model and try to understand its interretation in terms of multile regression model. The covariate X is now measured at different levels, e.g., if X is the quantity of fertilizer then suose there are ossible values, say 1 Kg., Kg.,,..., Kg. then X1, X,..., X denotes these values in the following way. The linear model now can be exressed as = β + β X + β X + + β X + o by defining X 1 1 if effect of 1 Kg. fertilizer is resent = 0 if effect of 1 Kg. fertilizer is absent 1 if effect of Kg. fertilizer is resent X = 0 if effect of Kg. fertilizer is absent 1 if effect of Kg. fertilizer is resent X = 0 if effect of Kg. fertilizer is absent. If effect of 1 Kg. of fertilizer is resent, then other effects will obviously be absent and the linear model is exressible as = β + β ( X = 1) + β ( X = 0) β ( X = 0) = β + β If effect of Kg. of fertilizer is resent then = β + β ( X = 0) + β ( X = 1) β ( X = 0) = β + β

6 6 If effect of Kg. of fertilizer is resent then = β + β ( X = 0) + β ( X = 0) β ( X = 1) = β + β + 0 and so on. If the exeriment with 1 Kg. of fertilizer is reeated n 1 number of times then n 1 observation on resonse variables are recorded which can be reresented as = β + β.1 + β β = β + β.1 + β β = β + β.1 + β β n n1 If X = 1 is reeated n times, then on the same lines n number of times then n 1 observation on resonse variables are recorded which can be reresented as = β + β.0 + β β = β + β.0 + β β = β + β.0 + β β.0 +. n 0 1 n

7 7 The exeriment is continued and if X = 1 is reeated n times, then on the same lines = β + β.0 + β β P1 = β + β.0 + β β P = β + β.0 + β β.1 +. n 0 1 n All these n1, n,.., n observations can be reresented as y y y 1n n1 y y β β = 1 + yn n β y y y n n or = Xβ +.

8 8 In the two way analysis of variance model, there are two covariates and the linear model is exressible as = β + β X + β X β X + γ Z + γ Z γ Z q q where X1, X,..., X denotes, e.g., the levels of quantity of fertilizer, say 1 Kg., Kg.,..., Kg. and Z 1, Z,..., Zq denotes, e.g., the q levels of level of irrigation, say 10 Cms., 0 Cms.,,10q Cms. etc. The levels X1, X,..., X, Z1, Z,..., Zq are defined as counter variable indicating the resence or absence of the effect as in the earlier case. If the effect of X 1 and Z 1 are resent, i.e., 1Kg of ffertilizer and d10c Cms. of irrigation i is used then the linear model is written as = β0 + β1.1 + β β.0 + γ1.1 + γ γ.0 + = β + β + γ If X = 1 and Z = 1 is used, then the model is = β0+ β+ γ +. The design matrix can be written accordingly as in the one way analysis of variance case. In the three way analysis of variance model = α + β X β X + γ Z γ Z + δw δ W q q 1 1 r r

9 9 The regression arameters β 's can be fixed or random. β 's If all are unknown constants, they are called as arameters of the model and the model is called as a fixed-effects model or model I. The objective in this case is to make inferences about the arameters and the error variance σ. i n β β If for some j, x ij = 1 for all i = 1 1,,..., n then j is termed as additive constant. In this case, occurs with every observation and so it is also called as general mean effect. β j If all β 's are observable random variables excet the additive constant, then the linear model is termed as random-effects model, model II or variance comonents model. The objective in this case is to make inferences about the variances of ' s, i.e., σ, σ,..., σ and error variance σ and/or certain functions of them. β 1 β β β If some arameters are fixed and some are random variables, then the model is called as mixed-effects model β or model III. In mixed effect model, at least one j is constant and at least one j is random variable. The objective is to make inference about the fixed effect arameters, variance of random effects and error variance σ. β