Some Approaches to the Analysis of a Group of Repeated Measurements Experiment on Mahogany Tree with Heteroscedustic Model

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1 J Agric Rural Dev 4(&), 55-59, June 006 ISSN JARD Journal of Agriculture & Rural Develoment Some Aroache to the Analyi of a Grou of Reeated Meaurement Exeriment on Mahogany Tree with Heterocedutic Model MD. NASIRUL ISLAM, ABU SYEED MD. RION ROUF AND M. MAZIBAR RAHMAN Oen School, Bangladeh Oen Univerity, Gaziur, Bangladeh Deartment of Statitic, Jahangirnagar Univerity, Savar, Dhaka, Bangladeh Received 4 March 004; received in revied form 6 November 005; acceted June 006 ABSTRACT Combined analyi of a grou of reeated meaurement exeriment could lay an imortant role in both the caue either the variance might be known or unknown. Ordinarily the variance vary from exeriment to exeriment deending on lace and environmental condition due to great irreconcilable inherient caue. On the other hand the variance are obviouly related a a functional form of the reective error variance in different lace. Relying on the functional form ome tet criterion are rooed for removing any reulting bia. Key word: Reeated meaurement exeriment, divergency and convergency, ercentage rotatability. INTRODUCTION The analyi of reeated meaurement exeriment i a owerful exerimental tet rocedure in the field of agricultural, biological and clinical reearch ( Rahman, 989; Maden, 977; Lana & Lubin, 963). However, in all the cae they ued in a ingle reeated meaurement exeriment. Rahman and Miah (99) ued combined analyi in more than one exerimental deign. They conidered the variance-covariance tructure among the different exeriment are equal. Neverthele, when the exeriment are conducted in different lace the variance and covariance matrix are not equal. In that cae the analytical rocedure i very much comlicated. In thi aer, an analytical rocedure to overcome the roblem through an examle on Mahogany tree ha been rooed. MATERIALS AND METHODS A reeated meaurement exeriment i that which ha treatment tructure, at leat one treatment factor i not randomly aigned having more than one different ize of exerimental unit. The required reeated meaurement model can be exreed a- Xigt=µαgβ(αβ)gε igτt(ατ)gt(βτ)t(αβτ)gtεigt; i =,,... ng g =,,... G =,,... t =,,... T Correonding author: Aociate rofeor, Oen School, BOU. ilamn@bou.edu.bd 006, School of Agriculture and Rural Develoment, Bangladeh Oen Univerity. All right reerved.

2 Md. Nairul Ilam et al. Where, Xigt = the reone of the ith individual in the gth grou on th lace at the tth occaion. µ = the general mean αg = the effect of the gth grou β = the effect of the th lace (αβ)g = the interaction effect between the gth grou at the th lace ε ig = the effect aociated with the ith ubject in the gth grou on the th lace τt = the effect of the tth occaion (ατ)gt = the interaction effect between the gth grou at the tth occaion (βτ)t = the interaction effect between the th lace at the tth occaion. (αβτ)gt = the interaction effect among the gth grou in the th lace at the tth occaion εigt = the error aociated with the ith ubject in the gth grou on the th lace at the tth occaion. Aumtion Under the following aumtion the analyi of variance i given in Table.. ε ig N (0, σ ε ). εigt N (0, e ) 3. both are indeendent 4. all the variance in the different lace are equal i.e. ( ) ( t ) ( ) ( ) σ = σ =... = σ = σ ; t =,,... Table. ANOVA Source of Degree of variation freedom Grou G- lace - Grou lace (G-) (-) e T (G ) Error-I (N-G) σ ε σ ε Occaion Grou Occaion lace Occaion Grou lace Occaion (T-) (G-)(T-) (-)(T-) (G-)(-)(T-) Sn g (a g a.) EMS GT e S(ß ß.) ( ) N e SS{(aß) g (a ß). - (aß) g. (G -)(T ) e N S(t (T ) t t.) (aß)..} σ ε n g {( ατ ) gt ( ατ ). t ( ατ ). ( ατ )..} ( G )( T ) t G σ ε { βτ) t ( βτ).t ( βτ). ( βτ)..} ( )(T ) e Error-II GT(N-G) e ( )(T - )(G SSS(aßt) gt -) 56

3 Analyi of a grou of reeated meaurement exeriment on mahogany tree The following hyothei are to be teted:. H0(grou) : α = α =... = αg i.e. the grou effect are equal. H0(lace) : β = β =... = β i.e. the lace effect are equal 3. H0(grou lace) : The grou lace interaction effect are nil 4. H0(occaion) : τ = τ =... = τt i.e. the occaion effect are equal 5. H0(grou occaion) : The grou occaion interaction effect are nil 6. H0(lace occaion) : The lace occaion interaction effect are nil 7. H0(grou lace occaion) : The grou lace occaion interaction effect are nil The above hyothee about the main effect and the interaction effect can be teted by uing F- tatitic with reective degree of freedom. reviouly the analyi conidered the variance among different lace are equal, which i a reaonable aumtion in any cae. One method for analying data when variance are unequal imly to ignore the fact that they are unequal and calculate the ame tatitic. Surriingly thee tet are quite good, in articular if the variance in different lace are all equal or almot equal. If the variance in different lace are not equal the uual tet tatitic i not valid. Moreover, the variance tructure would be a function with the reective error variance in different lace, then the functional form are a follow:. (variance tructure) = ϕ( ) = a 0 a ; linear functional form. = = Combined variance co-variance matrix = 0 quadratic =. (variance tructure) = ϕ(s ) a a S S Sa S S ; functional form 3. (variance tructure) = ϕ ) = c 4. ϕ( ) ( ; linear contrat = ϕ( ) limt S ds = limt S ; which imlie either divergency or convergency. = Some tet criterion are Suggeted for teting ( ) ( ) ( ) H0 : σ = σ =... = σ ; =,,... ε ε Tet rocedure. If the correonding mean um of quare i ubtantially large than the mean quare error in different lace then the linear equation doe not rereent the true reone urface (llam, 99) i.e. the hyothee i not acceted.. Ilam (99) howed that if the ercentage rotatability i equal to or about to equal ϕn(d)=00 0 = 00 ; then the tet i not ignificant i.e. the null hyothee i acceted. = 57

4 Md. Nairul Ilam et al. 3. Conider the null hyothee Ho(lace)= c = = 0 = Ha(lace)= c 0 Then = = c c =, = c n where n number of amle in different lace Then the tet tatitic Z = = c = c n = c N(0,) the Z tatitic follow tandardized normal ditribution. 4. If the number of lace N>5, then the tet tatitic would be followed: limt ϕ( ) d = = limt ϕ( S ); indicate null hyothei i acceted. = Numerical examle The data were derived from a grou of exeriment on Mahogany (witania macrohylla) tree conducted in different tation of Jahangirnagar Univerity, Savar, Dhaka. The different H value of the oil in which the tree grown are 5.4, 5.6 and 5.8. The land wa reared well and lanted the Mahogany tree at the deth of 5 to 6 cm. with a lant to lant acing 70cm. Five tree were randomly aigned to each fertilizer in each grou of different lace. The time eriod of the exeriment were 0 th November 993 to 9 th July 994. The height of the tree were meaured in every two month interval and were recorded in cm. Thu G=3, T=3, =5, O=5. [G=grou, T=Fertilizer, =lace, O=Occaion]. The objective of the exeriment wa to tudy the effect of a level of lantation in combination with 3 level of Nitrogen on the growth of Mahogany. The analyi technique would be followed by Reeated Meaurement exerimental Deign. The error variance of the three exeriment were oberved a , and in different lace. Thee error variance were heterogeneou a wa oberved. But uggeted tet tatitic (I, II, III, IV) indicated that the null hyothee were acceted. 58

5 Analyi of a grou of reeated meaurement exeriment on mahogany tree Thu, the uual analyi of variance i valid. The analyi of variance Table i given below. Table. ANOVA S.V. D.f. M.S. F.ratio ower lace Fertilizer lace & Fertilizer Error Occaion lace & Occaion Fertilizer & Occaion lace & Fertilizer & Occaion Error Table howed that the lace and occaion effect are highly ignificant. But the fertilizer effect, lace and fertilizer interaction effect, lace and occaion interaction effect, lace and fertilizer and occaion interaction effect are inignificant. The fertilizer and occaion interaction effect i inignificant. From the analytical reult, the lace and occaion effect are highly ignificant i.e. ignificant variation of growth of the lant over the time and the lace are reent in the exeriment. CONCLUSION Deigned field exeriment on the effect of inorganic fertilizer were erformed on the lanted eedling of Mahogany, the mot widely ued lantation foretry ecie in Bangladeh. Therefore, it i very imortant to outline an exerimental rocedure to accomodate all of the lantation at a time. Thi exeriment i very helful for thi uroe. LITERATURE CITED Ilam, M. N. 99. Analyi of reeated meaurement exeriment. Unublihed [Mhil Diertation], Jahangirnagar Univerity, Savar, Dhaka. Lana, R. E. and Lubin, A The effect of correlation on the reeated meaure deign. Education and ychological Meaurement 3, Maden, K. S A growth curve model for tudie in morohomatric. Biometric 33, Rahman, M. M. and Miah, A. B. M. 99. Combined analyi of a grou of reeated meaurement exeriment with heterocedutic model. Journal of Statitical Studie, 9-6. Rahman, M. M Comarion of method for the analyi of reeated meaurement exeriment. Unublihed [hd Diertation], Aberdeen Univerity, UK. 59