Analysis of Variance and Design of Experiments-II

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1 Analysis of Variance and Design of Experiments-II MODULE I LECTURE - 8 INCOMPLETE BLOCK DESIGNS Dr Shalabh Department of Mathematics & Statistics Indian Institute of Technology Kanpur

2 Generally, we are not interested merely in the interblock analysis of variance but we want to utilize the infmation from interblock analysis along with the intrablock infmation to improve upon the statistical inferences After obtaining the interblock estimate of treatment effects, the next question that arises is how to use this infmation f an improved estimation of treatment effects and use it further f the testing of significance of treatment effects Such an estimate will be based on the use of me infmation, so it is expected to provide better statistical inferences We now have two different estimates of the treatment effect as based on intrablock analysis ˆ CQ and GE based on interblock analysis ( N' N) N' B v bk Let us consider the estimation of linear contrast of treatment effects L l' Since the intrablock and interblock estimates of are based on Gauss-Markov model and least squares principle, so the best estimate of L based on intrablock estimation is L ˆ l' lcq ' and the best estimate of L based on interblock estimation is L l' GE l' ( N' N) N' B bk v l NN NB le '( ' ) ' ( since ' v 0 being contrast )

3 3 The variances of L and L are Var L σ l C l ( ) ' and Var L l N N l ( ) σ f '( ' ), respectively The covariance between Q (from intrablock) and B (from interblock) is (, ) ( ' *, ) Cov Q B Cov V N K B B (, ) ( ' *, ) Cov V B Cov N K B B N' σ N' K Kσ 0 f f Note that B* denotes the block total based on intrablock analysis and B denotes the block totals based on interblock analysis We are using two notations B and B* just to indicate that the two block totals are different The reader should not misunderstand that it follows from the result of Cov( Q, B ) 0 in case of intrablock analysis Thus Cov( L, L ) 0 irrespective of the values of l The question now arises that given the two estimats ˆ and of, how to combine them and obtain a minimum variance unbiased estimat of It is illustrated with following example:

4 Example Let ˆ ϕ and ˆ ϕ be any two unbiased estimats of a parameter ϕ with Var( ϕ ) σ and Var( ϕ ) σ Consider a linear combination ˆ ϕ ϕˆ + ϕˆ with weights and In der that ˆϕ is an unbiased estimat of, we need E( ˆ ϕ) ϕ E( ˆ ϕ ) + E( ˆ ϕ ) ϕ ϕ+ ϕ ϕ + So modify ˆϕ as ϕˆ + ϕˆ which is the weighted mean of + Further, if ˆ ϕ and ˆ ϕ are independent, then Var( ˆ ϕ) σ + σ Now we find and such that Var( ˆ ϕ) is minimum such that + Var( ˆ ϕ) 0 σ ( ) σ 0 σ σ 0 σ σ weight variance ˆ ϕ and ˆ ϕ ˆ ˆ 4 Alternatively, the Lagrangian function approach can be used to obtain such result as follows The Lagrangian function with as Lagrangian multiplier is given by Solving φ φ φ 0, and 0 λ* φ Var( ˆ ϕ) λ* ( + ) also gives the same result that σ σ * λ

5 We note that a pooled estimat of in the fm of weighted arithmetic mean of uncrelated L and L is the minimum variance unbiased estimat of when the weights and of L and L, respectively are chosen such that Var( L), Var( L ) 5 ie, the chosen weights are reciprocal to the variance of respective estimats, irrespective of the values of l So consider the weighted average of L and L with weights and, respectively as with L + L * + l '( ˆ + ) + lc ' lσ l'( N' N) lσ f The linear contrast of * is L* l' * and its variance is Var( L) + Var( L) Var( L*) l ' l (since Cov( L, L) 0) ( + ) ll ' ( + ) because the weights of estimats are chosen to be inversely proptional to the variance of the respective estimats

6 * We note that can be obtained provided and are known But and are known only when σ and σ are * known So can be obtained if σ and σ β are known In case, if σ and σ β are unknown, then their estimates can be used A question arises how to obtain such estimats? 6 β One such approach to obtain the estimates of σ and σ β is based on utilizing the results from intrablock and interblock analysis both and is as follows: From intrablock analysis, we have so an unbiased estimat of An unbiased estimat of σ σ β is E n b v σ ( Err () t ) ( + ), () ˆ Err t σ n b v + is obtained by using the following results based on the intrablock analysis: Treat( unadj) v V j j G j n, Block ( unadj) b B G k n i i i, Q ˆ, Treat( adj) j j j v Total b v G yij n i j,

7 7 where Hence + + Total Treat( adj) Block ( unadj) Err () t + + Treat( unadj) Block ( adj) Err () t + Block ( adj) Treat( adj) Block ( unadj) Treat( unadj) Under the interblock analysis model E[ ] E[ ] + E[ ] E[ ] Block ( adj) Treat( adj) Block ( unadj) Treat( unadj) which is obtained as follows: E[ ] ( b ) σ + ( n v) σ Block ( adj) β b E Block ( adj) Err () t n v n b v+ ( ) σ β Thus an unbiased estimat of σ β is b n v n b v+ ˆ σ β Block ( adj) Err () t

8 8 Now the estimates of weights and can be obtained by replacing σ and σ by ˆ σ and, respectively Then the estimate of * can be obtained by replacing and by their estimates and can be used in place of * It may be are replaced by ˆ σ and σ, respectively in * Some approximate results are possible which we will present while dealing with the balanced incomplete block design An increase in the precision using interblock analysis as compared to intrablock analysis is measured by noted that the exact distribution of associated sum of squares due to treatments is difficult to find when σ and σ β ˆβ / variance of pooled estimate / variance of intrablock estimate β σˆβ In the interblock analysis, the block effects are treated as random variable which is appropriate if the blocks can be regarded as a random sample from a large population of blocks The best estimate of the treatment effect from the intrablock analysis is further improved by utilizing the infmation on block totals Since the treatments in different blocks are not all the same, so the difference between block totals is expected to provide some infmation about the differences between the treatments So the interblock estimates are obtained and pooled with intrablock estimates to obtain the combined estimate of called the recovery of interblock infmation The procedure of obtaining the interblock estimates and then obtaining the pooled estimates is