MD. LUTFOR RAHMAN 1 AND KALIPADA SEN 2 Abstract

Transcription

1 ISSN Bangladesh J. Agrl. Res. 34(3) : , September 009 PROBLEMS OF USUAL EIGHTED ANALYSIS OF VARIANCE (ANOVA) IN RANDOMIZED BLOCK DESIGN (RBD) ITH MORE THAN ONE OBSERVATIONS PER CELL HEN ERROR VARIANCES VARY FROM CELL TO CELL MD. LUTFOR RAHMAN 1 AND KALIPADA SEN Abstract It s well nown that classcal analss of varance (ANOVA) s not sutable for heteroscedastc laouts. eghted analss of varance (ANOVA) s the onl wa to deal wth such stuatons. Problems of usual ANOVA n Randomzed. Bloc Desgn (RBD) wth more than one observatons per cell wth nteracton when error varances var from cell to cell are dscussed n ths paper. Ke ords : eghted analss of varance, error varances, cell to cell. Introducton Smple heteroscedastct s dscussed n detal b Sen (1984), Sen & Ponnuswam (1991) and Rahaman and Sen (1995). In ths paper, heteroscedastct s consdered n a RBD assumng error varances var from cell to cell. There are more than one observatons per cell n ths laout. The twowa general heteroscedastc model s consdered here wth nteracton. On the assumpton that the unequal error varances are nown, ANOVA are derved usng usual weghted least square methd. Ths method has some arbtrarness and problems such as: soluton of dependent equatons, mposton of arbtrar constrants both on parameters and on estmators of parameters, the problem of non-testabtt, etc. All such problems are analtcall dscussed n ths paper. General heteroscedastc model for RBD Suppose there are p treatments and q blocs n RBD wth multple observatons per cell whch are ncluded n ths experment. The observatons n such an experment ma be arranged n a two-wa table. Let there are r observatons n each of the pq cells of the table. 1 Assocate Professor, Department of Statstcs, Bostatstcs & Informatcs, Unverst of Dhaa, Professor, Department of Statstcs, Bostatstcs & Informatcs, Unverst of Dhaa, Dhaa, Bangladesh.

2 396 RAHMAN et al. The fxed effect addtve model wth nteracton for the above stuaton ma be taen as follows : µ + α + β + ( αβ ) e + for 1,, p, 1,, q and 1,,, r where the -th observaton of the (,) the cell µ the general mean effect α -th treatment effect β -th bloc effect (αβ) the nteracton effect of the smultaneous occurrence of the -th treatment and -th bloc e s are normall and ndependentl dstrbuted wth mean zero and varance whch vares from cell to cell. α Our man obectve of ths analss s to test the followng gpotheses: H AB : (αβ) 11 (αβ) 1 (αβ) pq 0 aganst H: all (αβ) s are not equal to zero () HA: α 1 α α p aganst H: all α s are not equal (3) HB: β 1 β β q aganst H: all β s are not equal (4) Usual ANOVA and analtcal problems Assumng that the error varances ( 1 α ) are nown wth weght, α for 1,,., p; 1,,, q, the eghted Least Square (LS) estmators of the parameters are obtaned b mnmzng L ( µ α ) wth respect to µ, α, β and (αβ), respectvel, as follows; 0 µ ( µ α ) 0

3 PROBLEMS OF USUAL EIGHTED ANALYSIS OF VARIANCE α 0 β ( αβ ) ( µ α ) ( µ α ) 0 ( µ α ) 0 Thus the normal equatons are as follows: µ : r ˆ.. µ + r. ˆ α + r ˆ. β + r α : r ˆ µ + r ˆ. α + r ˆ.β + r β : r ˆ µ. + r ˆ α + r. β + r αβ ) : r ˆ µ + r ˆ α r ˆ + β + r ( wherew..,. and. 0 0 Snce the equatons are dependent there are nfnte se of solutons of the above normal equatons. Under the followng constrants: ˆ. ˆ.β α 0 (5) for all 1,, 3, p and 1,, 3,, q, we have the followng standard ˆ µ r.. L ˆ α ˆ µ.. r.

4 398 RAHMAN et al. ˆ β r. ˆ µ.. ˆ ˆ ˆ µ α r Here the LS estmators of α s and β s are not unbased. In addton to ther lnear constants are not often unbased and hence hpotheses are not alwas testable. Under the lnear model (1), the sample weghted resdual sum of squares (SS) s ( ) SSE() mn µ α ( ) wth pq(r 1)d.f.. (7) The weghted resdual SS under H AB s S1 () mn( µ α ) subect to H AB (8) ( + ) wth (pqr - p - q 1) d.f. Thus SS due to H AB s SS AB () S ()-SSE () r ( ) wth (p -1) (q -1) d.f. The weghted resdual SS under H A s S (9) ( µ α ( ) ) subect to A mn αβ H ( + ) wth (pqr - pq p -1) d.f. (10) Thus SS due to H A s SSA () S -SSE ().. +

5 PROBLEMS OF USUAL EIGHTED ANALYSIS OF VARIANCE 399 ( ) wth (p -1) d.f. r (11)... Smlarl, t can be shown that SS due to H B s SSB () r ( ) wth (q-1) d.f. (1)... It s nterestng to note that desrable expected values of dfferent SS and ther desrable samplng dstrbutons are not feasble untl one mpose further restrctons on the lnear parameters of the model and ths s shown n the next secton. Calculaton of expectaton of Dfferent SS In order to fnd the expected values of dfferent SS, parametrc constrants are mposed smlar to (5) as follows:. α.β ( αβ ) ( αβ ) 0 Under above constrants, r. SSA () ( ) here, r.. ( µ + α + β + ( αβ ) + e ) µ + e r.. Here, e the weghted average of error values e e r r.

6 400 RAHMAN et al. ( µ + α + β + ( αβ ) + e ) µ + α + e.. r so that SSA ()] ( ). r α + e... e. α Thus E[SSA () r + re[ ( e e ) ] r α. + r(p 1) Smlarl t can be shown that, E[SSB ()] r β + r(q 1). E[SSAB ()] r ( αβ ) + (p 1)(q 1) E[SSE ()] pq (r-1) For testng hpothess of the above sectons, t s essental to mpose the above parametrc restrctons. In secton 3, t has been shown that to estmate the parameters t needs to mpose restrcton on the estmators. Table 1. ANAVA for RBD Sources of Varatons d.f. SS E (SS) A (p-1) SSA () ( p 1) + r α B (q-1) SSB () ( q 1) + r β AB (p-1) (q-1) SSAB () ( p 1)(q 1) + r ( αβ Error (pq (r-1) SSE () pq (r-1) Total (pqr-1) SST () A means B means )

7 PROBLEMS OF USUAL EIGHTED ANALYSIS OF VARIANCE 401 A B means The correspondng ANOVA table s shown as Table 1. B the general theorem of Sen (1984). t can be stated that SSE () and SSA () under H A are ndependentl dstrbuted as central χ wth pq (r-1) and (p-1) d.f. respectvel. Also SSE () and SSB () under H B are ndependentl dstrbuted as central χ s wth pq (r-1) and (q-1) d.f. respectvel. Hence, ANOVA χ can be performed for H A and H B, especall when error varatons are nown. Concluson Usual weghted least squares method provdes a number of arbtrarness and analtcal problems n desgnng models. These are dscussed analtcall n ths paper wth general heteroscedastc model n RBD. New method can be developed n order to avod all such analtcal problems. References Rahaman, L. and Sen, K On a new technque of ANAVA wth heteroscedastc 3-wa model havng nteractons. Dhaa Unverst Journal of Scence 43(1): Sen, K Some contrbutons to hetroscedastc analss of varance (HANOVA), PhD Thess. Department of Statstcs, Unverst of Madras, Inda. Sen, K On a general theorem of ANOVA wth the general heteroscedastc model not of full ran. Journal of Bangladesh Academ of Scences 15(): Sen K. and K.N. Ponnuswam Compartve studes on the adequac of some new test procedures for testng equalt of means when populaton varances are unequal and unnown. Journal of Statstcal Research 5(1&):