Estimation of Lag Time Between Onset of and Death from an Occult. Tumor

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1 Esimaion of Lag Time Beween Onse of and Deah fom an Occul Tumo Hongshik Ahn Hoin Moon and Ralph L. Kodell Depamen of Applied Mahemaics and Saisics Sony Book Univesiy Sony Book NY Depamen of Mahemaics and Saisics Califonia Sae Univesiy Long Beach CA Depamen of Biosaisics Univesiy of Akansas fo Medical Sciences Lile Rock AR 705 Running ile: Esimaion of Lag Time fom an Occul Tumo * Coesponding auho: Hongshik Ahn Ph.D. Depamen of Applied Mahemaics and Saisics Sony Book Univesiy Sony Book NY Tel: hahn@ams.sunys.edu

2 Asac A new saisical mehod fo esimaing he lag ime eween onse of and deah fom an occul umo is poposed fo daa wihou cause-of-deah infomaion. In his mehod he suvival funcion fo ime o umo onse umo-specific suvival funcion and compeing isks suvival funcion ae esimaed using he maximum likelihood esimaes of he paamees. The poposed mehod uilizes he esimaed suvival funcions and saisically impued faal umos o esimae he lag ime. This appoach is developed fo oden umoigeniciy assays ha have a leas one ineim sacifice and a eminal sacifice. If he daa conains cause-of-deah infomaion given y pahologiss and i is elieved o e eliale i may e used fo esimaing he lag ime. The poposed mehod is illusaed using a eal daa se. The accuacy of he esimaion of lag ime is evaluaed via a Mone Calo simulaion sudy. Ou sudy shows ha he esimaed lag ime is quie eliale. KEY WORDS: Bioassay Cause of deah Likelihood Sacifice Tumo lehaliy. Inoducion Animal cacinogeniciy expeimens ae employed o es he cacinogenic poenial of dugs and ohe chemical susances used y humans. Many mehods have een poposed fo analyzing umo incidence daa fom animal ioassays. Kodell and Nelson (980) pesened a paameic model o esimae he aveage ime o deah fom a specific umo afe onse fo daa wih cause-of-deah infomaion and ineim sacifices. Howeve he elapsed-ime (lag-ime) esimaion has no een done exensively y ohe

3 eseaches. In his pape a mehod is poposed fo finding a epesenaive esimae of he lag ime in conuncion wih he esimaes of he suvival funcions and impued nume of faal umos poposed y Ahn Kodell and Moon (000) and modified y Moon Ahn and Kodell (00). The poposed mehod fo lag-ime esimaion is ased on a nonpaameic model and can e applied o daa wih eihe ineim sacifices. I is designed fo daa lacking cause-of-deah infomaion alhough i can uilize he cause-ofdeah assigned y pahologiss. A suvival/sacifice expeimen wih enzidine dihydochloide was conduced a he Naional Cene fo Toxicological Reseach o sudy sain and sex diffeences wih egad o chemically-induced live umos in mice. The daa wee peviously inoduced and analyzed y Kodell and Nelson (980). Male and female mice fom wo sains (F and F) wee used. The daa fom dose goups 60 ppm 0 ppm 00 ppm and 400 ppm wee epoed in Kodell and Nelson (980). The couns of sacifices and deahs wih and wihou he umo of inees ae given in Tale. Figue shows he diffeence eween he ime-o-umo-onse suvival funcion S() and he umo-specific suvival funcion P () fo each sain and sex. These suvival funcions will e discussed in Secion. The lag-ime esimaion equies cause-of-deah infomaion. Cuenly cause-ofdeah infomaion is oained fom case-y-case assignmen of conex of osevaion y pahologiss in animal cacinogeniciy expeimens. In pacice howeve accuae deeminaions of he cause of deah ae no easy and classificaion eos can poduce iases (Lagakos 98; Racine-Poon and Hoel 984; Lagakos and Louis 988). In his sudy he impued cause of deah y Moon e al. (00) is used as a eplacemen of he cause-of-deah assignmen y pahologiss.

4 In esimaing lag ime he conol and dose goups ae pooled fo esimaing he paamees since i is desiale o esimae one epesenaive lag ime fo a specific umo. We expec o oain a ee lag-ime esimae y pooling he daa y using moe infomaion. The analysis y Kodell and Nelson (980) showed ha he aveage umo onse ime vaies fo he diffeen dose goups u he aveage ime o deah fom a specific umo afe onse does no change susanially fo diffeen dose goups. In his pape he maximum likelihood esimaes of he paamees oained y Kim e al. (007) and he impued nume of faal and incidenal umos oained y Ahn e al. (000) ae used in he poposed mehod fo esimaing he lag ime. Alenaively he nume of faal and incidenal umos assigned y pahologiss can e used fo he lag-ime esimaion alhough we ecommend using he saisically impued numes.. Esimaion of Time o Onse Fo he pupose of esimaing he lag ime comine he daa in conol and all he dose goups. Divide he ime scale ino discee inevals wih he h ineval given y ( ] = L J whee = 0 0 and J denoe he eminal sacifice ime. In he case of an expeimen wih muliple sacifices sequenial sacifice is pefomed a he end of each ineval. Fo daa wih a single eminal sacifice pe-deemined inevals such as he NTP (Naional Toxicology Pogam) inevals (Baile and Poie 988) can e imposed. Using he noaion of Moon e al. (00) define S () as he suvival funcion fo he disiuion of ime o umo onse i.e. S ( ) = P( T > ) whee T is he andom vaiale epesening ime o onse of he umo of inees. Time o onse is defined as 4

5 he ime when an occul umo fis ecomes lage enough o e deeced hisologically. Define P () as he suvival funcion wih espec o deah caused y he umo of inees i.e. P( ) = P( T > D ) whee T D epesens he oveall ime o deah fom he umo of inees. Define Q () as he suvival funcion wih espec o deah fom compeing isks i.e. Q( ) = P( X > C ) whee X C is he andom vaiale epesening ime o deah fom a cause ohe han he umo of inees. Moon e al. (00) fomulaed he coniuions o he likelihood funcion in ems of S () P () and Q () fo muliplesacifice daa as shown in Tale as if complee infomaion on cause of deah is availale fo he expeimenal animals. Following Malani and Van Ryzin (988) (see also Kodell and Ahn 997) hey assumed ha T and T D occu efoe ha poailiies involving T T D and Fo expeimens wih a single eminal sacifice X C in an ineval and X C can only e evaluaed a he end of inevals. a and ae zeo excep fo he las ineval. Fuhe deails on hese likelihood coniuions ae given in Moon e al (00). Fo all le π = S ) / P( ) p P ) / P( ) and q Q ) / Q( ). Then ( he suvival funcions can e expessed as k= k= = ( = ( P( ) = p Q( ) = q and S( ) = π P( ) = L J. () k k Afe he epaameeizaion Kim e al. (007) oained he log-likelihood funcion as J l = [{ N i= (log p {log p {log p log q log( q log q ) ( d ) logπ } a logπ }] consan a )log{ p {log p ( q log q ) p ( π )} log( π )} 5

6 6 whee N is he nume of live animals a. The monooniciy of ) ( ) ( P S and ) ( Q povides he following inequaliy condiions: and p q p π π π fo. J L = () Kim e al. (007) analyically deived he consained MLEs of π p and q as follows: Case If a d a hen he MLEs ae = = = * max N N q p p N N π π Case If a a d hen he MLEs ae = = = } ) ( { max * * * q q N N p N N q p a π π π π π Hee we define * N as he nume of live animals igh efoe. Thus a N N * = and. * N a d N = Fuhe deails ae given in Kim e al. (007).. Esimaion of Lag Time

7 In his secion a new mehod is developed fo calculaing a epesenaive esimae of he elapsed ime (lag ime) eween he ime o onse of an occul umo and he ime ha i causes he deah of an animal. To esimae he lag ime seveal noaions will e inoduced. Le nume of umo eaing animals a he h ineval; hen a e he can e expessed as d a N Q( ){ P( ) S( )}. Le To e he nume of umos ha occued a he h ineval and le Te e he nume of deahs o sacifices wih umo a he h ineval. Define lag ( k) as he nume of deahs a he h ineval fom he umo of inees having ineval lag k (umo occued a he ( k) h ineval) k = 0 L. Fo he fis ineval i is clea ha To is he same as and fo = To Te = To N Q( ){ P( ) S( )}..() Fom Equaion () To can e oained as N Q ){ P( ) S( )}. To ( avoid having a negaive esimae of To should e lage han N Q( ){ P( ) S( )}. In ode o make an adusmen of and To fis define A = min{ N Q( )[ P( ) S( )] } = m. Fo he las ineval find = a d a N Q( ){ P( ) S( )} and calculae m m m m m m m m To m = m Am. Saing fom ineval m oain and hen A and To A. = a d a = To as Fo he fis ineval oain = a d a A and To =. This appoach guaanees nonnegaive esimae of To = L m. 7

8 The esimao of lag ( k) is given as lag ( k) = d To To k = k k fo k = 0 L. The esimaed nume of deah wih faal umos fom Secion can susiue fo esimaion of lag. d in he aove equaion. Appendix A. povides fuhe deails of he Le u e he lengh of he h ineval = L m. If an animal died a he h ineval fom he umo of inees which occued a he same ineval hen he expeced lag ime of he umo fo ha animal is u / 4..(4) if we assume ha oh he ime of umo onse and he ime of deah fom he umo ae disiued as a unifom in he ineval. If u = L = um = hen he expeced lag ime is u u / 4. If an animal oained he umo a he h ineval and died fom he umo a he h ineval such ha > hen he expeced lag ime of he umo fo ha animal is u u us s= = L m...(5) If u = L = um = hen he expeced lag ime is ( ) u. Fuhe deail of he u individual lag ime is given in Appendix A.. Now we develop an esimaion of he lag ime eween he ime o onse of an occul umo and he ime ha i causes he deah of an animal. Fo he faal umos whose onse occued a each ineval a mehod of esimaing he aveage suvival ime 8

9 afe onse is inoduced. Using hese esimaes he aveage lag ime fo he umo i.e. he aveage of T D T is oained. Define lag as he aveage suvival ime fo he animals wih faal umos whose onse occued a he h ineval. If = m fom (4) lag = u / 4. Fo he faal umos whose onse occued a he h ineval he nume of deahs is lag ( ) if he umo caused deah a he h ineval. If < m using (4) and (5) we oain m m lag u lag (0) / 4 [{( u = = m m = u ) / } lag ( ) s= us ] lag ( ). (6) If u = L = um = hen he aveage suvival ime is u lag u m { lag (0) / 4 ( ) lag ( ) } = = m = lag ( ) A epesenaive esimae of he lag ime eween onse of and deah fom an occul umo can e esimaed y aveaging he aveage suvival imes oained aove. Le L e he oveall aveage lag ime hen i can e esimaed as whee w = u / u. m = L = m = w lag (7). 4. Example: Benzidine Daa The poposed mehods ae applied o analyze he Benzidine dihydochloide daa discussed in Secion. The fou dose goups ae comined in ode o esimae he lag ime. Howeve he F sain Male 60 ppm dose goup is no included due o insufficien 9

10 osevaions fo oaining esimaes. The ohe hee dose goups ae comined fo his one. The nume of deahs fom faal umo was esimaed using he poposed mehod and compaed wih he numes assigned y he pahologiss. Fequency daa and esimaed quaniies of inees along wih he esimaes of and To ae given in Tale. The cause of deah (COD) assigned y pahologiss and he esimaed COD ae compaed in his ale. Tale 4 pesens he esimaes of he ime-o-onse suvival funcion S () fom Tale along wih esimaes of a simila umo onse suvival funcion oained fom a discee cumulaive umo incidence funcion poduced y he mehod of Kodell and Ahn (997). Noe ha he mehod of Kodell and Ahn does no involve cause of deah. This ale shows ha ou esimaes of S () ae vey close o ha of Kodell and Ahn. A compaison of hese esimaes seves as a validiy check on he mehod developed in his pape ecause he mehod of Kodell and Ahn is compleely nonpaameic and makes fewe assumpions. The good coespondence eween he wo ses of esimaes lends cediiliy o he esimaes of he lehaliy and he umo onse suvival funcion S (). Tale 5 compaes he esimaed lag ime oained using he poposed mehod and he mehod y Kodell and Nelson (980). The esimaed lag ime y ou mehod is appoximaely 4 o 6 monhs afe onse. The esimaed lag imes fom ou mehod ae quie close o hose y Kodell and Nelson excep fo he F male goup. Noe ha d = in he F male goup while d 0 in he ohe hee goups accoding o oh ou = mehod and he assignmen y he pahologiss. When we compae F and F male goups hee is a susanial diffeence in lag while lag and lag ae close eween 0

11 he wo goups. This diffeence is caused y he deah fom he umo occued in he fis ineval of he F male goup. In he F male goup lag is calculaed as 0(.8966) 0(.8) 50(.46) lag = = y susiuing lag ( ) fom Tale 6 ino Equaion (6). In a hypoheical case ha d = 0 insead of d = in his goup he value of lag ecomes 0(.8) 50(.46) lag = = which is vey close o he value of lag in he F male goup. Fuhe L = (4.480 / ) ( ) / 4 = 7.0 ecomes close o he esimae of he oveall aveage lag ime 6. in he F male goup. Ou esimaes appea o e eliale ased on he esuls in Tale 4 and he simulaion esuls given in Secion 5. The umo oseved in his example can e consideed highly lehal accoding o oh he assignmen y pahologiss and ou esimaion mehod. Ou esimaed lag ime is close o he acual lag ime accoding o ou simulaion sudy especially fo inemediae and highly lehal umos. 5. Simulaion A Mone Calo simulaion sudy was conduced o evaluae he accuacy of he esimaion of he lag ime. A ioassay design wih fou dose goups (0 4) of 50 animals each was consideed. The poposed pocedue was simulaed o have he NTP inevals ( and 9-04 weeks) wih ineim sacifices a 5 78 and 9 weeks and a eminal sacifice a 04 weeks which is he nomal em of a chonic woyea sudy in odens. The ineim sacifices wee implemened y escheduling a poion

12 of he 50 animals pe goup fo ealy sacifice which ae nomally scheduled fo a single eminal sacifice in he cusomay lifeime oden ioassay. Six animals wee andomly pe-seleced o e sacificed a he end of each ineval. All he emaining live animals wee sacificed a he end of he expeimen. I was assumed ha hee independen andom vaiales compleely deemined he oseved oucome fo each animal. The andom vaiales wee he ime o onse of umo T he ime afe onse unil deah fom he umo T and he ime o deah fom a compeing isk X C. Noe ha T T = T whee T D epesens he oveall ime o deah D fom he umo of inees. Thus he umo of inees was pesen in an animal a he ime of deah if T min{ X C X } whee X S denoes an animal's scheduled sacifice ime. An S animal died fom he umo of inees if T min{ X X }. Ohewise i died fom a D compeing isk including sacifice. Fo he animals die fom he umo of inees he lag ime is T = T. Theefoe in his simulaion he esimaed lag ime of he umo D T was compaed wih he aveage of he simulaed values of T fo each simulaion daa se. Disiuions of ime o onse ime o fom he umo of inees and ime o deah fom compeing isks wee of he fom used y Moon e al. (00). The poailiy of he ackgound umo onse poailiy (fo he conol goup) y 04 weeks was chosen o e 0.05 (ae umo) 0.5 o 0. (common umo). The poailiy of umo poailiy of umo onse was chosen such ha umo onse poailiy in he highes dose goup y 04 weeks was 5 and imes he ack gound umo aes and 0. especively. The umo lehaliy aes wee chosen o e appoximaely 5% 5% and 60%. C S

13 Lag ime was calculaed afe pooling he fou dose goups. Five housand simulaed daa ses wih Weiull umo onse disiuions (wih shape paamee ) wee geneaed wih he compeing isks suvival ae 0.5 and wo umo lehaliy aes. Tale 7 shows he simulaion esuls. The aveage of he ue lag imes and he aveage of he esimaed lag imes ove he 5000 simulaion daa ses ae calculaed fo each configuaion. I is difficul o oain an accuae esimaion of lag ime if he umo is mosly incidenal ecause of a lack of infomaion on deah caused fom he umo. Howeve ou esimaed lag ime appeas o have an ageemen wih he acual lag ime in his ale as long as he umo is modeaely o highly lehal. 6. Concluding Remaks I is no an easy ask o esimae he umo onse ime fo an occul umo. Kodell and Nelson (980) developed a semi-makov model fo desciing he developmen of a umo which caused a deah. Using his model hey povided he aveage ime o umo and ime o deah fom umo afe onse. This mehod equies cause-of-deah infomaion. In many animal ioassays fo cacinogeniciy howeve cause-of-deah infomaion is no availale. Even when cause of deah is availale i is suec o eo poenially leading o misepesenaive esimaes and ess of umo aes. The pesen pape develops a mehod of esimaing he lag ime eween he umo onse and he deah fom ha umo wihou he need fo cause-of-deah daa. The poposed esimaing mehod uilizes he disiuions of ime o umo and ime o deah fom he umo in

14 conuncion wih he disiuion of compeing isks given y Moon e al. (00). The cause of deah is saisically aiued y he mehod of Ahn e al. (000). Mos of he exising esing pocedues do no povide accompanying esimaes of he umo onse disiuion funcion while such an esimae is paiculaly useful fo gaphical displays o illusae he esuls of he saisical ess. The poposed mehod povides he umo onse disiuion as well as he suvival funcion fo he umo. The poposed lag ime is esimaed ased on hese suvival funcions and i is easonaly accuae especially when he umo is modeaely o highly lehal since he daa conain enough infomaion on animals died fom he umo. Acknowledgmens Hongshik Ahn s eseach was paially suppoed y he Faculy Reseach Paicipaion Pogam a he NCTR adminiseed y he Oak Ridge Insiue fo Science and Educaion hough an ineagency ageemen eween USDOE and USFDA. Hoin Moon s eseach was paially suppoed y he Scholaly and Ceaive Aciviies Commiee (SCAC) Awad fom CSULB. 4

15 Refeences Ahn H. Kodell R. L. and Moon H. (000). Aiuion of umo lehaliy and esimaion of ime o onse of occul umos in he asence of cause-of-deah infomaion. Applied Saisics 49: Baile A. J. and Poie C. J. (988). Effecs of eamen-induced moaliy and umoinduced moaliy on ess fo cacinogeniciy in small samples. Biomeics 44:47-4. Kim W. Ahn H. and Moon H. (007). A dose-esponse es via closed-fom soluions fo consained MLEs in suvival/sacifice expeimens. Saisics in Medicine 6(): Kodell R. L. and Ahn H. (997). Age-adused end es fo he umo incidence ae. Biomeics 5: Kodell R. L. and Nelson C. J. (980). An illness-deah model fo he sudy of he cacinogenic pocess using suvival/sacifice daa. Biomeics 6: Lagakos S. W. (98). An evaluaion of some wo-sample ess used o analyze animal cacinogeniciy expeimens. Uilias Mahemaica B:9-60. Lagakos S. W. and Louis T. A. (988). Use of umou lehaliy o inepe umoigeniciy expeimens lacking cause-of-deah daa. Applied Saisics 7(): Malani H. M. and Van Ryzin J. (988). Compaison of wo eamens in animal cacinogeniciy expeimens. Jounal of he Ameican Saisical Associaion 8:

16 Moon H. Ahn H. and Kodell R. L. (00). Exension of Peo's es y aiuion of umo lehaliy in he asence of cause-of-deah infomaion. Biomeical Jounal 44: Racine-Poon A. and Hoel D. G. (984). Nonpaameic esimaion of he suvival funcion when cause of deah is unceain. Biomeics 40:5-58. Appendix A.. Esimaion of lag ( k). Fo = lag(0) = d ecause = To.. Fo = a. If = To (i.e. = Te ) hen lag (0) = d and lag () = 0.. If > To (i.e. > Te ) hen among he umo eaing animals in his ineval To / of hem oained umo in he second ineval and To ) / ( of hem oained umo in he fis ineval. Theefoe To lag (0) = d lag To () = d = d NQ( ){ P( ) S( )} / fom (). Noe ha To = P(Tumo occuing a Tumo eaing a ) = P( < T T TD > X c > ) To = P(Tumo occuing efoe Tumo eaing a = P( T T TD > X c > ). ) 6

17 . Fo = a. If = To hen lag (0) = d lag() = 0 and lag () = 0.. If > To i. If = To ii. If > To To lag(0) = d To lag () = d lag () = 0. To lag(0) = d To lag () = d To To lag () = d To. c. In geneal fo he h ineval To lag (0) = d To lag () = d To To lag () = d To To. 7

18 To To k lag ( k ) = d k = 0 L. = k k A.. Esimaion of individual lag ime. Suppose a umo and deah (fom he umo) occued a he h ineval fo an s s= animal. Define V = T u = s and W = T D u s hen W V = T D T is he lag ime of he umo fo ha animal. We assume ha umos pecede deahs u he exac ime of he umo onse and deah ae no known. Thus we can assume V is disiued as U 0 u ) and given V W is disiued as U V u ) ( ( whee U ( a ) denoes he unifom disiuion on he ineval ( a ). The oin pdf of W and V is given as f w v) = f ( w v) f ( v) = /[ u ( u v)] 0 < v < w < u W V ( W V V Le X = W V hen y an appopiae ansfomaion of he vaiales we oain he pdf of x given y and he expecaion E X ) = u / 4. x f ( x) = log 0 < x < u u u (. Suppose a umo occued in an animal a he h ineval and he animal died fom he umo a he h ineval such ha >. Define V = T u s= s and s= W = T D u s hen W V = T T is he lag ime of he umo fo ha D. animal. Assume ha V is disiued as U 0 u ) and W is disiued as U ( u s us ). The esimaed lag ime is s= s= ( 8

19 E( W u u V ) = E( W ) E( V ) = u s= s = L m. 9

20 Tale : Fequency daa fom Benzidine daa Female Male Naual Sacifice Naual Sacifice Sain Goup wih w/o wih w/o wih w/o wih w/o F 60ppm * ppm ppm ppm F 60ppm ppm ppm ppm Time inevals - epesen especively weeks. Wih umo; Wihou umo; 4 Insufficien daa 0

21 Tale : Likelihood coniuion of each even Even Likelihood coniuion Coun fo ineval Deah fom faal umo Q( ){ P( ) P( )} d Deah wih incidenal umo { Q( ) Q( )}{ P( ) S( )} a Deah wihou umo S( ){ Q( ) Q( )} Sacifice wih umo Q( ){ P( ) S( )} a Sacifice wihou umo S ( ) Q( )

22 Tale : Couns fo fiing he discee model o daa and esimaed quaniies fom he Benzidine Daa Goup ad a π p q S ( ) P ( ) Q ( ) To FF FM FF FM Time inevals - epesen especively weeks. a d Esimaed nume of faal umos 4 Nume of faal umos assigned y pahologiss d ~ 4 d

23 Tale 4: Esimaed ime-o-umo-onse suvival funcions ~ S ( ) fom () and S ( ) y he mehod of Kodell and Ahn (997) fo he enzidine daa. Ineval F Female F Male F Female F Male a ~ S ) S ( ) ~ S ) S ( ) ~ S ) S ( ) ~ S ) S ( ) a Time inevals - epesen especively weeks

24 Tale 5: Esimaed lag ime (in weeks) in (7) fo he Benzidine daa Sain Sex Concenaion RL Poposed (ppm) Female F * Male Female F Male Esimaed lag ime using he mehod y Kodell and Nelson (980) Insufficien daa o oain esimaes 4

25 Tale 6: The values of lag in (6) and lag ( ) used fo he calculaion of lag in each goup fo he enzidine daa Goup F female F male F female F male lag lag lag lag () lag () lag ()

26 Tale 7: Aveage of he esimaed and acual lag imes fo simulaed daa wih sacifices a he end of each NTP inevals. Sandad eo esimaes ae given in paenheses. Fo each configuaion he aveage is aken fom 5000 ials. The compeing isks suvival ae is 0.5. Backgound Tumo ae Tumo lehaliy Aveage lag ime Esimaed Acual (0.) (0.) (0.08) (0.) (0.) (0.09) (0.0) (0.0) (0.08).7 6

27 F Female F Male Suvival ae P S Suvival ae P S Suvival ime (week) Suvival ime (week) F Female F Male Suvival ae P S Suvival ae P S Suvival ime (week) Suvival ime (week) Figue. Time-o-umo-onse suvival funcions (S) and umo-specific suvival funcions (P) using he poposed mehod fo he Benzidine daa. 7

On The Estimation of Two Missing Values in Randomized Complete Block Designs

On The Estimation of Two Missing Values in Randomized Complete Block Designs Mahemaical Theoy and Modeling ISSN 45804 (Pape ISSN 505 (Online Vol.6, No.7, 06 www.iise.og On The Esimaion of Two Missing Values in Randomized Complee Bloc Designs EFFANGA, EFFANGA OKON AND BASSE, E.