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Randomized Block Design Randomized Block Design If a large number of treatments are to be compared then a large number of experimental units are required This will increase the variation among the responses and CRD may not be appropriate to use In such a case when the experimental material is not homogeneous and there are v treatments to be compared then it may be possible to group the experimental material into blocks of sizes v units 2 Randomized Block Design Blocks are constructed such that the experimental units within a block are relatively homogeneous and resemble to each other more closely than the units in the different blocks If there are b such blocks we say that the blocks are at b levels are at v levels Similarly if there are v treatments we say that the treatments The responses from the b levels of blocks and v levels of layout The treatments can be arranged in a two way observed data set is arranged as follows 3 Arrangement of Data in RBD Treatments Factor B 2 j y12 y22 yi2 yb2 T2 y1j y2j yij ybj Tj 1 y11 y21 yi1 yb1 T1 v y1v y2v yiv ybv Tv A r o t c a F s k c o B l 1 2 i b Treatment totals Block totals B1 B2 Bi Bb Grand total G 4 Randomized Block Design Layout A two way layout is called a randomized block design RBD or a randomized complete block design RCB if within each block the v treatments are randomly assigned to v experimental units such that each of the v ways of assigning the treatments to the units has the same experiment and the statistically independent probability assignment the in blocks are being adopted in different of The RBD utilizes the principles of design replication and local control in the following way randomization 5 Randomized Block Design Randomization 1 Randomization Number the v treatments 1 2 v Randomly the allocate experimental units in each block Treatments Factor B j 2 v treatments to v 1 2 i b 1 y11 y21 yi1 yb1 T1 A r o t c a F s k c o B l Treatment totals y12 y22 yi2 yb2 T2 y1j y2j yij ybj Tj v y1v y2v yiv ybv Tv Block totals B1 B2 Bi Bb 6 Grand total G Randomized Block Design Replication 2 Replication Since each treatment is appearing in each block so every treatment will appear in all the blocks So each treatment can be considered as if replicated the number of times as the number of blocks Thus in RBD the number of blocks and the number of replications are same 7 Randomized Block Design Local Control 3 Local control Local control is adopted in RBD in the following way First form the homogeneous blocks of the experimental units Then allocate each treatment randomly in each block The error variance now will be smaller because of homogeneous blocks and some variance will be parted away from the error variance due to the difference among the blocks 8 Randomized Block Design Example denoted there are 7 treatment Suppose corresponding to 7 levels of a factor to be included in 4 blocks So one possible layout of the assignment of 7 treatments to 4 different blocks in a RBD is as follows as T1 T2 T7 Block 1 Block 2 Block 3 Block 4 T2 T1 T7 T4 T7 T6 T5 T1 T3 T7 T1 T5 T5 T4 T6 T6 T1 T5 T4 T2 T4 T3 T2 T7 T6 T2 T3 T3 9 Randomized Block Design Analysis Let yij Individual measurements of block jth treatment in ith i 1 2 b j 1 2 v yij s are independently distributed following N i j 2 where overall mean effect ith block effect jth treatment effect i j such that i 0 j b i 1 0 v j 10 Randomized Block Design Hypothesis There are two null hypotheses to be tested related to the block effects H0 B 1 2 b 0 related to the treatment effects H0T 1 2 v 0 11 Randomized Block Design The linear model in this case is a two way as model i j ij i 1 2 b j 1 2 v yij are identically and independently distributed random where ij errors following a normal distribution with mean 0 and variance 2 The tests of hypothesis can be derived using the likelihood ratio test or the principle of least squares The use of likelihood ratio test has already been demonstrated earlier so we now use the principle of least squares 12 Randomized Block Design Minimizing S v b i 1 j 1 1 2 ij and solving the normal equation y i 1 j b v j i 2 ij S 0 2 b S 0 S 0 for all i 1 j 1 2 v j i the least squares estimators are obtained as yoo i j yoj yoo y y io oo 13 Randomized Block Design Using the fitted model we can write yio yoo yoj yoo yij yio yoj yoo yij yoo Squaring both sides and summing over i and j gives b v ij y i 1 j 1 y 2 b oo i 1 v y TSS or y io oo 2 v b y oj j 1 y oo 2 b v y y ij io i 1 j 1 y y oj oo 2 SSBl SSTr SSE with degrees of freedom partitioned as bv 1 b 1 v 1 b 1 v 1 The reason for the number of degrees of freedom for different sums of squares is the same as in the case of CRD 14 Randomized Block Design 2 Total SS TSS yij oo b v y i 1 j 1 v ni G yij Grand total SS due to blocks SSBl i 1 j 1 n v yio v iB j 1 th ijy i block total i 1 b v i 1 j 1 y 2 ij G2 bv Correction factor G2 bv 2 oo y G2 b i 1 B2 i bv v oo T 2 G2 bv b 2 j v j 1 SS due to treatments SSTr b y y jT treatment total ijy j j 1 oj th b i 1 b v SS due to error SSE y y y i 1 j 1 ij io oj y 2 oo 16 Randomized Block Design Moreover b 1 SSBl 2 b 1 v 1 SSTr 2 v 1 2 2 b 1 v 1 SSE 2 b 1 v 1 2 Under H 0 B 1 2 b 0 E MSBl E MSE F MSBl F b 1 b 1 v 1 bl and SSBl and SSE are independent so MSE 16 Randomized Block Design Similarly under H0T 1 2 v 0 E MSTr E MSE Also SSTr and SSE are independent so TrF MSTr F v 1 b 1 v 1 MSE Decision Reject H0 B Reject H0T FTr if Fbe F b 1 b 1 v 1 F v 1 b 1 v 1 if 17 is accepted …

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VIT MAT 2001 - Randomized Block Design

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