Page 1 Analysis of Variance and Analysis of Variance and Analysis of Variance and Analysis of Variance and Design of Experiment Design of Experiment- -I I MODULE MODULE â€“ â€“IX IX gp gp ANALYSIS OF ANALYSIS OF LECTURE LECTURE - - 36 36 ANALYSIS OF ANALYSIS OF NONORTHOGONAL NONORTHOGONAL DATA DATA Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur Page 2 Analysis of Variance and Analysis of Variance and Analysis of Variance and Analysis of Variance and Design of Experiment Design of Experiment- -I I MODULE MODULE â€“ â€“IX IX gp gp ANALYSIS OF ANALYSIS OF LECTURE LECTURE - - 36 36 ANALYSIS OF ANALYSIS OF NONORTHOGONAL NONORTHOGONAL DATA DATA Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur Orthogonal data 2 g The concept of orthogonality of data is associated with two or higher way classified data. Consider the set up of two-way classified data. Let A and B be two factors at p and q levels respectively. Let n ij : number of observations in (i, j) th cell. Let y ijk = k th observation in cell, 1, 2,..., ; 1, 2,..., ; 1, 2,..., , ij ipj qk n = == : marginal total correspondence to level of th i ijk jk Ay i A = ?? ` : : th where cell total marginal total correspondence to level of ij i ij ijk k j ijk T Ty By j B = = = ? ? ?? . : Let Gra jj ik ij j i j T ABG = == ?? ? , nd total ?? j ,, Ë† :, i marginal mean of j ij io ij oj io oj ji j j i i ij nn nn n n n A tA n == = = = ?? ?? ? ? ? Ë† :. marginal mean of j j j ij j B bB n = ? Page 3 Analysis of Variance and Analysis of Variance and Analysis of Variance and Analysis of Variance and Design of Experiment Design of Experiment- -I I MODULE MODULE â€“ â€“IX IX gp gp ANALYSIS OF ANALYSIS OF LECTURE LECTURE - - 36 36 ANALYSIS OF ANALYSIS OF NONORTHOGONAL NONORTHOGONAL DATA DATA Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur Orthogonal data 2 g The concept of orthogonality of data is associated with two or higher way classified data. Consider the set up of two-way classified data. Let A and B be two factors at p and q levels respectively. Let n ij : number of observations in (i, j) th cell. Let y ijk = k th observation in cell, 1, 2,..., ; 1, 2,..., ; 1, 2,..., , ij ipj qk n = == : marginal total correspondence to level of th i ijk jk Ay i A = ?? ` : : th where cell total marginal total correspondence to level of ij i ij ijk k j ijk T Ty By j B = = = ? ? ?? . : Let Gra jj ik ij j i j T ABG = == ?? ? , nd total ?? j ,, Ë† :, i marginal mean of j ij io ij oj io oj ji j j i i ij nn nn n n n A tA n == = = = ?? ?? ? ? ? Ë† :. marginal mean of j j j ij j B bB n = ? 3 If any contrast of marginal means of A is orthogonal to any contrast of the other marginal means, then the data are called orthogonal other ise non orthogonal Ë† ii i lt ? Ë† j j j mb ? data are called orthogonal, otherwise non-orthogonal. If each cell has constant number of observations then the data are orthogonal as shown below. Note that in this case , . qq io ijj nn nnqnnp == = = ?? The definition extends to higher classification if we treat the marginal means of every pair of factors of classification. jj jj 1 1 Ë† (... ). ii i ii i i iq ii lA Llt lT T qn qn == = ++ ? ?? Similarly 1 1 Ë† (... ). jj j jjjjpj jj n mB M mb m T T Bpn == = ++ ? ?? The sum of products of the coefficients of identical observations in the two contrasts is 2 11 0. ij i j ij i j lmn l m pqn pqn ?? ?? == ?? ?? ?? ?? ?? ? ? So the data are orthogonal as the two contrasts are orthogonal. When cell frequencies are proportional, i.e., ( , ' 1, 2,..., ; 1,..., ) ij j nC jjri p nC == = then also the data are orthogonal. The same is true for higher order classification if the number of observations in the ultimate cell is constant. '' ij j nC Page 4 Analysis of Variance and Analysis of Variance and Analysis of Variance and Analysis of Variance and Design of Experiment Design of Experiment- -I I MODULE MODULE â€“ â€“IX IX gp gp ANALYSIS OF ANALYSIS OF LECTURE LECTURE - - 36 36 ANALYSIS OF ANALYSIS OF NONORTHOGONAL NONORTHOGONAL DATA DATA Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur Orthogonal data 2 g The concept of orthogonality of data is associated with two or higher way classified data. Consider the set up of two-way classified data. Let A and B be two factors at p and q levels respectively. Let n ij : number of observations in (i, j) th cell. Let y ijk = k th observation in cell, 1, 2,..., ; 1, 2,..., ; 1, 2,..., , ij ipj qk n = == : marginal total correspondence to level of th i ijk jk Ay i A = ?? ` : : th where cell total marginal total correspondence to level of ij i ij ijk k j ijk T Ty By j B = = = ? ? ?? . : Let Gra jj ik ij j i j T ABG = == ?? ? , nd total ?? j ,, Ë† :, i marginal mean of j ij io ij oj io oj ji j j i i ij nn nn n n n A tA n == = = = ?? ?? ? ? ? Ë† :. marginal mean of j j j ij j B bB n = ? 3 If any contrast of marginal means of A is orthogonal to any contrast of the other marginal means, then the data are called orthogonal other ise non orthogonal Ë† ii i lt ? Ë† j j j mb ? data are called orthogonal, otherwise non-orthogonal. If each cell has constant number of observations then the data are orthogonal as shown below. Note that in this case , . qq io ijj nn nnqnnp == = = ?? The definition extends to higher classification if we treat the marginal means of every pair of factors of classification. jj jj 1 1 Ë† (... ). ii i ii i i iq ii lA Llt lT T qn qn == = ++ ? ?? Similarly 1 1 Ë† (... ). jj j jjjjpj jj n mB M mb m T T Bpn == = ++ ? ?? The sum of products of the coefficients of identical observations in the two contrasts is 2 11 0. ij i j ij i j lmn l m pqn pqn ?? ?? == ?? ?? ?? ?? ?? ? ? So the data are orthogonal as the two contrasts are orthogonal. When cell frequencies are proportional, i.e., ( , ' 1, 2,..., ; 1,..., ) ij j nC jjri p nC == = then also the data are orthogonal. The same is true for higher order classification if the number of observations in the ultimate cell is constant. '' ij j nC Analysis of non-orthogonal two way data 4 When the data are non-orthogonal, then the analysis is no longer simple as the straight solution of normal equations is not available. Consider the following usual two way model: 1,...,; 1,....,; 1,..., . iik i j ijk ij yiIjJkn µ aß e =+ + + = = = 2 '(0) Assume that are identically and independently distribution as N 2 '(0, ). ijk Assume that are identically and independently distribution as s N e s Using the least squares principal, we minimize the sum of squares due to error as 22 () Ë† Ë† Ë†0(1) ijk ijk i j ij k i j k oo io i oj j ijk i j i j k Ey E nn n yG eµaß µa ß µ == --- ? =? + + = = ? ??? ??? ?? ??? 1 Ë† Ë† Ë† 0 ( 1, 2,..., ) (2) Ë† Ë† 0 ( 1, 2,..., ). (3) jj io io i ij j ijk i jjk oj i oj j ijk j k j E nn n y Ai I E nn y B j J µa ß a aß ß ? =? + + = = = ? ? =? + = = = ? ??? i ?? k j ß ? (3), Ë† Ë† Ë†(). From we obtain we use instead of to avoid confusion j i joj jmm m oj oj n nm i nn ßµ ßa =- - ? Page 5 Analysis of Variance and Analysis of Variance and Analysis of Variance and Analysis of Variance and Design of Experiment Design of Experiment- -I I MODULE MODULE â€“ â€“IX IX gp gp ANALYSIS OF ANALYSIS OF LECTURE LECTURE - - 36 36 ANALYSIS OF ANALYSIS OF NONORTHOGONAL NONORTHOGONAL DATA DATA Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur Orthogonal data 2 g The concept of orthogonality of data is associated with two or higher way classified data. Consider the set up of two-way classified data. Let A and B be two factors at p and q levels respectively. Let n ij : number of observations in (i, j) th cell. Let y ijk = k th observation in cell, 1, 2,..., ; 1, 2,..., ; 1, 2,..., , ij ipj qk n = == : marginal total correspondence to level of th i ijk jk Ay i A = ?? ` : : th where cell total marginal total correspondence to level of ij i ij ijk k j ijk T Ty By j B = = = ? ? ?? . : Let Gra jj ik ij j i j T ABG = == ?? ? , nd total ?? j ,, Ë† :, i marginal mean of j ij io ij oj io oj ji j j i i ij nn nn n n n A tA n == = = = ?? ?? ? ? ? Ë† :. marginal mean of j j j ij j B bB n = ? 3 If any contrast of marginal means of A is orthogonal to any contrast of the other marginal means, then the data are called orthogonal other ise non orthogonal Ë† ii i lt ? Ë† j j j mb ? data are called orthogonal, otherwise non-orthogonal. If each cell has constant number of observations then the data are orthogonal as shown below. Note that in this case , . qq io ijj nn nnqnnp == = = ?? The definition extends to higher classification if we treat the marginal means of every pair of factors of classification. jj jj 1 1 Ë† (... ). ii i ii i i iq ii lA Llt lT T qn qn == = ++ ? ?? Similarly 1 1 Ë† (... ). jj j jjjjpj jj n mB M mb m T T Bpn == = ++ ? ?? The sum of products of the coefficients of identical observations in the two contrasts is 2 11 0. ij i j ij i j lmn l m pqn pqn ?? ?? == ?? ?? ?? ?? ?? ? ? So the data are orthogonal as the two contrasts are orthogonal. When cell frequencies are proportional, i.e., ( , ' 1, 2,..., ; 1,..., ) ij j nC jjri p nC == = then also the data are orthogonal. The same is true for higher order classification if the number of observations in the ultimate cell is constant. '' ij j nC Analysis of non-orthogonal two way data 4 When the data are non-orthogonal, then the analysis is no longer simple as the straight solution of normal equations is not available. Consider the following usual two way model: 1,...,; 1,....,; 1,..., . iik i j ijk ij yiIjJkn µ aß e =+ + + = = = 2 '(0) Assume that are identically and independently distribution as N 2 '(0, ). ijk Assume that are identically and independently distribution as s N e s Using the least squares principal, we minimize the sum of squares due to error as 22 () Ë† Ë† Ë†0(1) ijk ijk i j ij k i j k oo io i oj j ijk i j i j k Ey E nn n yG eµaß µa ß µ == --- ? =? + + = = ? ??? ??? ?? ??? 1 Ë† Ë† Ë† 0 ( 1, 2,..., ) (2) Ë† Ë† 0 ( 1, 2,..., ). (3) jj io io i ij j ijk i jjk oj i oj j ijk j k j E nn n y Ai I E nn y B j J µa ß a aß ß ? =? + + = = = ? ? =? + = = = ? ??? i ?? k j ß ? (3), Ë† Ë† Ë†(). From we obtain we use instead of to avoid confusion j i joj jmm m oj oj n nm i nn ßµ ßa =- - ? 5 1 Ë† Ë†Ë† Ë†Ë† (2), Put in j ji ii ij j i B nn n n A ßµa µ a ?? ++ -- = ?? ?? (2), 1 Ë†Ë† Ë†Ë† Put in or jio ioiij mj m i jm oj oj j iij io ioi ij ij mjm jjj oj oj nn n n A nn B An n n n n n nn ßµa µ a µa µ a ++ ?? ?? ?? ?? ? ? -=+ - - ?? ? ? ?? ? ? ?? ? ? ?? ???? 2 1 Ë†Ë† 1 Ë†Ë† . jj io i ij mj m jm oj ij ii ij mj j jj nn n n n nnn nn aa aa ?? ? ? =- ?? =- - ?? ?? ?? ?? ? 2 Ë† ij mi jm j j n n a ?? - ?? ?? ?? ?? ? or j oj oj nn ?? jm j oj n ?? 2 Ë†Ë† j ij ij mj iij iio m jjmij oj oj oj B nnn An n nn n aa ? ??? ? ? ? -= - - ??? ? ? ? ??? ? ? ? ??? ? ? ? ???? iii im QC C ?? ? or ? These are referred to as the reduced normal equations. Ë†Ë† ( 1, 2,..., ). (4) iiii imm mi QC C i I a a ? =+ = ? is called the adjusted treatment total of i th level of A. i QRead More

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