Page 1 Analysis of Variance and Analysis of Variance and Analysis of Variance and Analysis of Variance and Design Design of Experiments of Experiments- -I I MODULE MODULE – – II II g g p p GENERAL LINEAR HYPOTHESIS GENERAL LINEAR HYPOTHESIS LECTURE LECTURE - -7 7 GENERAL LINEAR HYPOTHESIS GENERAL LINEAR HYPOTHESIS AND ANAL YSIS OF VARIANCE AND ANAL YSIS OF VARIANCE 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 Design of Experiments of Experiments- -I I MODULE MODULE – – II II g g p p GENERAL LINEAR HYPOTHESIS GENERAL LINEAR HYPOTHESIS LECTURE LECTURE - -7 7 GENERAL LINEAR HYPOTHESIS GENERAL LINEAR HYPOTHESIS AND ANAL YSIS OF VARIANCE AND ANAL YSIS OF VARIANCE Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur C 2 Tt f bt f t h d 0 :12 Hk ßß < 2 2 Case 2: Test of asubset of parameters when and are unknown 0 0 : , 1, 2,.., kk Hk rp ßß = = < 12 , ,..., rr p ß ßß ++ 2 s In case 1, the test of hypothesis was developed when all ’s are considered in the sense that we test for each ß In case 1, the test of hypothesis was developed when all s are considered in the sense that we test for each Now consider another situation, in which the interest is to test only a subset of i.e., not all but only a few parameters. This type of test of hypothesis can be used e.g., in the following situation. Suppose five levels of voltage are applied to check the rotations per minute (rpm) of a fan at 160 volts, 180 volts, 200 volts, 220 volts and 240 volts. ß 0 ,1,2,...,. ii ip ßß == 12 , ,..., , p ßßß It can be realized in practice that when the voltage is low, then the rpm at 160, 180 and 200 volts can be observed easily. At 220 and 240 volts, the fan rotates at the full speed and there is not much difference in the rotations per minute at these voltages. So the interest of the experimenter lies in testing the hypothesis related to only first three effects, viz., , for 160 volts, for 180 volts and for 200 volts. 1 ß 2 ß 3 ß 00 0 0 01 1 2 2 3 3 :, , H ß ßß ßß ß == = The null hypothesis in this case can be written as: when and are unknown. 45 , ß ß 2 s Note that under case 1, the null hypothesis will be 00 0 0 0 0 0 1 1 2 23 3 4 45 5 :, , , , . H ß ßß ßß ßß ßß ß == = = = Page 3 Analysis of Variance and Analysis of Variance and Analysis of Variance and Analysis of Variance and Design Design of Experiments of Experiments- -I I MODULE MODULE – – II II g g p p GENERAL LINEAR HYPOTHESIS GENERAL LINEAR HYPOTHESIS LECTURE LECTURE - -7 7 GENERAL LINEAR HYPOTHESIS GENERAL LINEAR HYPOTHESIS AND ANAL YSIS OF VARIANCE AND ANAL YSIS OF VARIANCE Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur C 2 Tt f bt f t h d 0 :12 Hk ßß < 2 2 Case 2: Test of asubset of parameters when and are unknown 0 0 : , 1, 2,.., kk Hk rp ßß = = < 12 , ,..., rr p ß ßß ++ 2 s In case 1, the test of hypothesis was developed when all ’s are considered in the sense that we test for each ß In case 1, the test of hypothesis was developed when all s are considered in the sense that we test for each Now consider another situation, in which the interest is to test only a subset of i.e., not all but only a few parameters. This type of test of hypothesis can be used e.g., in the following situation. Suppose five levels of voltage are applied to check the rotations per minute (rpm) of a fan at 160 volts, 180 volts, 200 volts, 220 volts and 240 volts. ß 0 ,1,2,...,. ii ip ßß == 12 , ,..., , p ßßß It can be realized in practice that when the voltage is low, then the rpm at 160, 180 and 200 volts can be observed easily. At 220 and 240 volts, the fan rotates at the full speed and there is not much difference in the rotations per minute at these voltages. So the interest of the experimenter lies in testing the hypothesis related to only first three effects, viz., , for 160 volts, for 180 volts and for 200 volts. 1 ß 2 ß 3 ß 00 0 0 01 1 2 2 3 3 :, , H ß ßß ßß ß == = The null hypothesis in this case can be written as: when and are unknown. 45 , ß ß 2 s Note that under case 1, the null hypothesis will be 00 0 0 0 0 0 1 1 2 23 3 4 45 5 :, , , , . H ß ßß ßß ßß ßß ß == = = = Let be the p parameters. We can divide them into two parts such that out of and we are interested in testing 12 , ,..., p ß ßß 12 1 , ,..., , ,..., , rr p ß ßßß ß + 3 the hypothesis of a subset of it. Suppose, we want to test the null hypothesis when and are unknown. p 0 0 :,1,2,.., kk Hk rp ßß = =< 12 , ,..., rr p ß ßß ++ 2 s The alternative hypothesis under consideration is In order to develop a test for such a hypothesis, the linear model under the usual assumptions can be rewritten as follows: 0 1 : 1, 2,.., . or at least one f kk Hkrp ßß ? =< YX ß e =+ under the usual assumptions can be rewritten as follows: Partition (1) 12 (2) (), XXX ß ß ß ?? == ?? ?? Th d l b itt where (1) 1 2 (2) 1 2 ( , ,..., ) , ( , ,..., ) rrr p ß ßß ß ß ß ß ß ++ '' == with the orders of matrices as and 12 (1) :, : ( ), : 1 Xn rX n prr ß × ×- × (2) :( ) 1. pr ß - × The model can be rewritten as (1) 12 (2) () YX XX ß e ß e ß =+ ?? =+ ?? ?? (2) 1(1) 2 (2) . XX ß ß ße ?? =+ + Page 4 Analysis of Variance and Analysis of Variance and Analysis of Variance and Analysis of Variance and Design Design of Experiments of Experiments- -I I MODULE MODULE – – II II g g p p GENERAL LINEAR HYPOTHESIS GENERAL LINEAR HYPOTHESIS LECTURE LECTURE - -7 7 GENERAL LINEAR HYPOTHESIS GENERAL LINEAR HYPOTHESIS AND ANAL YSIS OF VARIANCE AND ANAL YSIS OF VARIANCE Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur C 2 Tt f bt f t h d 0 :12 Hk ßß < 2 2 Case 2: Test of asubset of parameters when and are unknown 0 0 : , 1, 2,.., kk Hk rp ßß = = < 12 , ,..., rr p ß ßß ++ 2 s In case 1, the test of hypothesis was developed when all ’s are considered in the sense that we test for each ß In case 1, the test of hypothesis was developed when all s are considered in the sense that we test for each Now consider another situation, in which the interest is to test only a subset of i.e., not all but only a few parameters. This type of test of hypothesis can be used e.g., in the following situation. Suppose five levels of voltage are applied to check the rotations per minute (rpm) of a fan at 160 volts, 180 volts, 200 volts, 220 volts and 240 volts. ß 0 ,1,2,...,. ii ip ßß == 12 , ,..., , p ßßß It can be realized in practice that when the voltage is low, then the rpm at 160, 180 and 200 volts can be observed easily. At 220 and 240 volts, the fan rotates at the full speed and there is not much difference in the rotations per minute at these voltages. So the interest of the experimenter lies in testing the hypothesis related to only first three effects, viz., , for 160 volts, for 180 volts and for 200 volts. 1 ß 2 ß 3 ß 00 0 0 01 1 2 2 3 3 :, , H ß ßß ßß ß == = The null hypothesis in this case can be written as: when and are unknown. 45 , ß ß 2 s Note that under case 1, the null hypothesis will be 00 0 0 0 0 0 1 1 2 23 3 4 45 5 :, , , , . H ß ßß ßß ßß ßß ß == = = = Let be the p parameters. We can divide them into two parts such that out of and we are interested in testing 12 , ,..., p ß ßß 12 1 , ,..., , ,..., , rr p ß ßßß ß + 3 the hypothesis of a subset of it. Suppose, we want to test the null hypothesis when and are unknown. p 0 0 :,1,2,.., kk Hk rp ßß = =< 12 , ,..., rr p ß ßß ++ 2 s The alternative hypothesis under consideration is In order to develop a test for such a hypothesis, the linear model under the usual assumptions can be rewritten as follows: 0 1 : 1, 2,.., . or at least one f kk Hkrp ßß ? =< YX ß e =+ under the usual assumptions can be rewritten as follows: Partition (1) 12 (2) (), XXX ß ß ß ?? == ?? ?? Th d l b itt where (1) 1 2 (2) 1 2 ( , ,..., ) , ( , ,..., ) rrr p ß ßß ß ß ß ß ß ++ '' == with the orders of matrices as and 12 (1) :, : ( ), : 1 Xn rX n prr ß × ×- × (2) :( ) 1. pr ß - × The model can be rewritten as (1) 12 (2) () YX XX ß e ß e ß =+ ?? =+ ?? ?? (2) 1(1) 2 (2) . XX ß ß ße ?? =+ + The null hypothesis of interest is now h d k 000 0 () H ßß ßß ß ß 2 4 where and are unknown. The complete parametric space is 000 0 0(1) (1) 1 2 : ( , ,..., ) r H ßß ßß ß == (2) ß 2 s { } 22 ( , ); , 0, 1,2,..., ßs ß s O= - 8 < < 8 > = i ip and sample space under is 0 H { } 02 2 (1) (2) ( , , ); , 0, 1, 2,..., . i ir r p ?ß ß s ß s =-8<<8>=++ Th i l f lik lih d f ti d i bt i d b b tit ti th i lik lih d The likelihood function is 2 2 22 11 (| , ) exp ( )( ) . 22 n Ly yX yX ßsßß ps s ?? ? ? ' =- - - ?? ? ? ?? ? ? O The maximum value of likelihood function under is obtained by substituting the maximum likelihood estimates of and , i.e., O ß 2 s 1 2 ˆ () 1 ˆˆ ˆ ()() XX Xy yX yX ß sß ß - '' = ' as ()() yX yX n sß ß =- - 2 2 22 11 ˆˆ (| , ) exp ( )( ) ˆˆ 22 n MaxL y y X y X ßsßß ps s O ?? ? ? ' =- - - ?? ? ? ?? ? ? 2 ' exp . ˆˆ 2 2( )( ) n nn yX yX pß ß ?? ?? =- ?? ?? -- ?? ?? Page 5 Analysis of Variance and Analysis of Variance and Analysis of Variance and Analysis of Variance and Design Design of Experiments of Experiments- -I I MODULE MODULE – – II II g g p p GENERAL LINEAR HYPOTHESIS GENERAL LINEAR HYPOTHESIS LECTURE LECTURE - -7 7 GENERAL LINEAR HYPOTHESIS GENERAL LINEAR HYPOTHESIS AND ANAL YSIS OF VARIANCE AND ANAL YSIS OF VARIANCE Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur C 2 Tt f bt f t h d 0 :12 Hk ßß < 2 2 Case 2: Test of asubset of parameters when and are unknown 0 0 : , 1, 2,.., kk Hk rp ßß = = < 12 , ,..., rr p ß ßß ++ 2 s In case 1, the test of hypothesis was developed when all ’s are considered in the sense that we test for each ß In case 1, the test of hypothesis was developed when all s are considered in the sense that we test for each Now consider another situation, in which the interest is to test only a subset of i.e., not all but only a few parameters. This type of test of hypothesis can be used e.g., in the following situation. Suppose five levels of voltage are applied to check the rotations per minute (rpm) of a fan at 160 volts, 180 volts, 200 volts, 220 volts and 240 volts. ß 0 ,1,2,...,. ii ip ßß == 12 , ,..., , p ßßß It can be realized in practice that when the voltage is low, then the rpm at 160, 180 and 200 volts can be observed easily. At 220 and 240 volts, the fan rotates at the full speed and there is not much difference in the rotations per minute at these voltages. So the interest of the experimenter lies in testing the hypothesis related to only first three effects, viz., , for 160 volts, for 180 volts and for 200 volts. 1 ß 2 ß 3 ß 00 0 0 01 1 2 2 3 3 :, , H ß ßß ßß ß == = The null hypothesis in this case can be written as: when and are unknown. 45 , ß ß 2 s Note that under case 1, the null hypothesis will be 00 0 0 0 0 0 1 1 2 23 3 4 45 5 :, , , , . H ß ßß ßß ßß ßß ß == = = = Let be the p parameters. We can divide them into two parts such that out of and we are interested in testing 12 , ,..., p ß ßß 12 1 , ,..., , ,..., , rr p ß ßßß ß + 3 the hypothesis of a subset of it. Suppose, we want to test the null hypothesis when and are unknown. p 0 0 :,1,2,.., kk Hk rp ßß = =< 12 , ,..., rr p ß ßß ++ 2 s The alternative hypothesis under consideration is In order to develop a test for such a hypothesis, the linear model under the usual assumptions can be rewritten as follows: 0 1 : 1, 2,.., . or at least one f kk Hkrp ßß ? =< YX ß e =+ under the usual assumptions can be rewritten as follows: Partition (1) 12 (2) (), XXX ß ß ß ?? == ?? ?? Th d l b itt where (1) 1 2 (2) 1 2 ( , ,..., ) , ( , ,..., ) rrr p ß ßß ß ß ß ß ß ++ '' == with the orders of matrices as and 12 (1) :, : ( ), : 1 Xn rX n prr ß × ×- × (2) :( ) 1. pr ß - × The model can be rewritten as (1) 12 (2) () YX XX ß e ß e ß =+ ?? =+ ?? ?? (2) 1(1) 2 (2) . XX ß ß ße ?? =+ + The null hypothesis of interest is now h d k 000 0 () H ßß ßß ß ß 2 4 where and are unknown. The complete parametric space is 000 0 0(1) (1) 1 2 : ( , ,..., ) r H ßß ßß ß == (2) ß 2 s { } 22 ( , ); , 0, 1,2,..., ßs ß s O= - 8 < < 8 > = i ip and sample space under is 0 H { } 02 2 (1) (2) ( , , ); , 0, 1, 2,..., . i ir r p ?ß ß s ß s =-8<<8>=++ Th i l f lik lih d f ti d i bt i d b b tit ti th i lik lih d The likelihood function is 2 2 22 11 (| , ) exp ( )( ) . 22 n Ly yX yX ßsßß ps s ?? ? ? ' =- - - ?? ? ? ?? ? ? O The maximum value of likelihood function under is obtained by substituting the maximum likelihood estimates of and , i.e., O ß 2 s 1 2 ˆ () 1 ˆˆ ˆ ()() XX Xy yX yX ß sß ß - '' = ' as ()() yX yX n sß ß =- - 2 2 22 11 ˆˆ (| , ) exp ( )( ) ˆˆ 22 n MaxL y y X y X ßsßß ps s O ?? ? ? ' =- - - ?? ? ? ?? ? ? 2 ' exp . ˆˆ 2 2( )( ) n nn yX yX pß ß ?? ?? =- ?? ?? -- ?? ?? Now we find the maximum value of likelihood function under . The model under becomes Th lik lih d f ti d i 0 H 0 H 0 YX X ßß 5 . The likelihood function under is 0 1(1) 2 2 YX X ßß e =+ + 0 H 2 20 0 1(1) 2 (2) 1(1) 2 (2) 22 11 (| , ) exp ( )( ) 22 11 n n Ly y X X y X X ßs ßß ßß ps s ?? ? ? ' = - -- -- ?? ? ? ?? ? ? ?? ? ? where Note that and are the unknown parameters. This likelihood function looks like as if it is 2 s 2 2(2) 2(2) 22 11 exp ( * ) ( * ) 22 yX yX ßß ps s ?? ? ? ' =- - - ?? ?? ?? ? ? (0) 1(1) *. yyX ß =- (2) ß written for This helps is writing the maximum likelihood estimators of and directly as 2 2(2) *~ ( , ). ß s yNX 2 s (2) ß '1' (2) 22 2 ˆ () * XX X y ß - = () 2 2 (2) 2 (2) 1 ˆˆ ˆ (* )( * ). yX yX n sß ß ' =- - Note that is a principal minor of Since is a positive definite matrix, so is also positive definite. Thus exists and is unique. ' 22 XX . XX ' XX ' ' 22 XX '1 22 () XX - definite. Thus exists and is unique. Thus the maximum value of likelihood function under is obtained as 22 () XX 0 H 2 2 2(2) 2(2) 22 11 ˆˆˆ (*| , ) exp ( * )( * ) ˆˆ 22 n MaxL y y X y X ? ßs ß ß pss ?? ? ? ' =- - - ?? ? ? ?? ? ? 2 2(2) 2(2) exp. ˆˆ 2 2(*)'(*) n nn yX yX pß ß ?? ? ? ?? ?? =- ?? ?? ?? -- ?? ??Read More

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