On The Estimation of Two Missing Values in Randomized Complete Block Designs
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1 Mahemaical Theoy and Modeling ISSN (Pape ISSN 505 (Online Vol.6, No.7, 06 On The Esimaion of Two Missing Values in Randomized Complee Bloc Designs EFFANGA, EFFANGA OKON AND BASSE, E. E. DEPARTMENT OF STATISTICS, UNIVERSIT OF CALABAR, CALABAR. NIGERIA. DEPARTMENT OF STATISTICS, UNIVERSIT OF CALABAR, CALABAR. NIGERIA. ABSTRACT: This pape eviews he wo of Bhad and Ahmed (0, idenify flaws in hei model fomulaion and mae coecions accodingly. Fuhemoe, he fomula fo he esimaion of wo missing values in andomized complee loc designs is deived and is applied o he example given in Bhad and Ahmed (0. The esuls oained fom ou fomula poduces a ee esimae of missing values han ha of Bhad and Ahmed (0. Key wods: Randomized Complee Bloc Design, Sum of Squaes of Eos, Missing Values, Esimaion, and Souces of Vaiailiy.. Inoducion Occasionally one o moe values ae missing in a andomized complee loc design (RCBD due o some easons. Such missing values would have o e esimaed efoe pefoming ANOVA ecause of loss of ohogonaliy. Reseaches on he mehodology fo he esimaion of missing values aound in lieaue. Fomula fo he esimaion of a single missing value in RCBD oiginaed fom he wo of ae (933. Mongomey (976 used he fomula fo single missing value ieaively o esimae wo o moe missing values. Ohe mehods fo esimaing missing values in RCBD can e found in Dempse & e al(977, Jae (978 and Muay (986. Bhad and Ahmed (0 developed a mahemaical pogamming model o esimae missing values in he design wih seveal souces of vaiaion and illusaed hei model wih RCBD wih wo missing values. Thei pape conains some flaws in he funcional consains and hus need o e coeced. Fuhemoe, he ieaive pocedue fo esimaing wo o moe missing values may ae seveal ieaions o convege and hus equie anohe mehod of achieving he same o appoximae esul in a vey sho ime and wih less compuaional effo. In his pape, we coec flaws in Bhad and Ahmed (0 and hen deived a fomula fo he esimaion of wo missing values in RCBD.. Review of Bhad and Ahmed (0 The geneal mahemaical pogamming model inoduced in Bhad and Ahmed (0 is as follows: M: Minimize {f(xi SSE} ( Sujec o: Vaiance(S i i ( x i 0 (3 47
2 Mahemaical Theoy and Modeling ISSN (Pape ISSN 505 (Online Vol.6, No.7, 06 Whee S i is he i h souce of vaiaion and he missing values. i is he vaiance of he i h souce of vaiaion wihou consideing The model M was applied o andomized complee loc designs (i.e. design having wo souces of vaiaions, loc and eamen. Thei model which consideed missing values is given as follows: M: Minimize {SSE x. j x i... i i Sujec o: (4 ( i. xi [ ( i. xi ] (5 ( j xi [ ( j xi ].. (6 x i 0 ( i j xi [ ( ij xi ] T (7 Whee, and T ae he fis souce vaiailiy, second souce vaiailiy and he oal vaiailiy.. Flaws in Bhad and Ahmed (0 and hei coecions In model M hee is no disincion eween he suscip used fo he missing values and souces of vaiaions. Also, he i h souce of vaiaion is no esiced o whee he missing value is siuaed as illusaed in he example. If an osevaion is missing in he l h level of vaiaion souce m, hen consains ( could have een expessed as follows: Va(S, m =,,..., n (8 Whee S is l h level of souce m whee a missing value occus, and souce m excluding he missing value. is he vaiance of he l h level of Wih he aove definiions, he model M should have een wien as M3: Minimize {f(xi SSE} Sujec o: Vaiance(S, m =,,..., n; fo each l. x i 0, i,,..., The esiced model M should have een fomulaed as follows: Le ij e he value in he i h level of souce and jh level of souce, whee i =,,..., ; j =,,...,. Suppose he value is missing, whee p is he level of souce and q is he level of souce, hen he vaiance of he p h level of souce, qh level of souce and he oal ae given, especively, as 48
3 Mahemaical Theoy and Modeling ISSN (Pape ISSN 505 (Online Vol.6, No.7, 06 Va(S Va(S p q Va(Toal pj pj (9 j q jq iq iq (0 i p ip ij ij ( i p j q ip jq The sum of squaes of eos (SSE should een expessed as follows: Whee SSE. q p... C ( p. = Level p oal of souce vaiaion excluding he missing value.q = Level q oal of souce vaiaion excluding he missing value.. = Gand oal excluding he missing value C = Tems independen of he missing value The coeced vesion of mahemaical pogamming model (M is heefoe:. p. Min{SSE.. q M4: } (3 Sujec o: jq pj jq pj p (4 iq iq q (5 i p ip ij ij T (6 i p j q ip jq 49
4 Mahemaical Theoy and Modeling ISSN (Pape ISSN 505 (Online Vol.6, No.7, (7 3. Deiving Compuaional fomula fo esimaing wo missing values Suppose wo values s and ae missing in a andomized complee loc design. We esimae he missing values y solving he unconsained opimizaion polem Mininmize {SSE = SST 0 SSB SST = f( s, } (8 whee, SSE, SST 0, SSB and SST ae he sum of squaes of eos, sum of squaes of oal, sum of squaes of locs, and sum of squaes of eamens, especively. SSE i j ij.. i i... j. j.. i, p j s,q ij s i, p. p. q. s. i.. j js,q.. i, p j s,q ij s i. i, p (. s ( p. js,q. (. (. ( s s q.. j s R s (. s ( p. ( ( q. s s. (.. s (9 Whee., p.,.s,.q and.. ae he especive oals excluding he missing values; R epesen ems no involving missing values. Diffeeniaing SSE in equaion (9 wih espec o e and especively and seing each equal o zeo we oain ( s..s.. (0 s ( p..q.. ( Now solving equaions (0 and ( simulaneously we oain s ( ( ( [(( ] (..s.. p..q.. ( 50
5 Mahemaical Theoy and Modeling ISSN (Pape ISSN 505 (Online Vol.6, No.7, 06 ( ( ( [(( ] ( p..q....s.. (3 Example We apply ou fomula o he example in Bha and Ahmed (0. The ale elow shows a andomized complee loc design wih wo missing values and 35. Bloc Teamen i j In ou fomulas (equaions 0 and, = 4, = 6, =, s =, p = 3, q = 5,. = 68.4, 3. = 89.3,. = 50.9,.5 = 46.4,.. = So, 35 ( ( (.... ( [(( ] (4 x x (4 x x ( ( ( ( [(( ].... 5(4 x x (4 x x Compaison of wo mehods The esuls oained y he wo mehods ae summaized in he ale elow: Missing Value Bhad and Ahmed (0 Fomula SSE
6 Mahemaical Theoy and Modeling ISSN (Pape ISSN 505 (Online Vol.6, No.7, Conclusions As can e seen in he aove ale he esimae of missing values oained y ou fomula poduces a lowe sum of squaes of eos and hence a ee esimae han ha of Bhad and Ahmed (0. 5
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