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OPTIMAL STRATEGIES FOR REGIONAL YIELD TESTS OFCULTIVARS Ross S. Kingwell Adviser, Division of Agricultural Economics and Marketing, Department of Agriculture t western AUstralia. 2 In the plant breeding programmes of many countries field assessment of potential new cult1vars involves comparing their characteristics (for example~ yield and disease resistance) to those of one or more standard cult1vars. Cultivar comparison usually involves a series of field trials replicated over sites and years. The usual aim of these trials is to identify which potential new cultivars are genuinely superior to the standard or convnonly grown cultivars. However. the very nature of both the cultivars and the environments in which field testing ocCUrs commonly introduces risks in selecting or identifying "trulY" superior cultiva.rs~ Cultivarsrarely perform uniformly across locations and years. Consequently, the ranking ofcultivars 1s seldom the same at each location or in each year. The interaction of cultivars with their test environment (genotype x environment interaction) means that it is unlikely for superior cultiVars to be identified on the bas.is of a single year's testing at a sin91esite. Hence the question often arises; Over hoW many years, locations and plot replications should the field testing of cultivars take place before one lsreasonably sure that a ··truly" super10rcultivar has been identified. This question has almost solely concerned biologists and biometricians (Sprague and Federer; Finney; Patterson ,!:It ali Talbot). rather than agri.cultural economists. The work of the f\lrmer group, perhaps to be expected, has involved analyses that ignore financial costs (Finney; Patterson ~t ali Talbot), capture only some of the costs of field testing (Spra9ueand Federer) or that consider mainly technical restrictions on the cQmponentsof a field testing progranvne. None of the studies explicitly accounts for the opportunity costs and actual costs of foresone cultivar superiority. As examples of suoh costs: (a) field test results may falsely indicate anewcultivar is higher yielding than standard or conwnonlygrown cultivars. Hence, when the new cult1var is recommended and .adopted, farmers will likely forego some yield~ (b) field test results maY falsely indicate a neW cultivar yields no more than the standard or commonly grown cultivat"~h Fai lure to recommend and adopt this superior cultivar involves an oppOrtunity cost of foregone yield. In this article field testing for the purpose of identifying high yielding cultivars 1.s examined. Adecis10n model that firstly accounts for costs (including foregone yield) associetedwithfield testing and which secondly identifies optimal field test strategies is described and applied. In the f1rstsect1on the decision model is described. Then it is applied to a case study of' wheat breeding in the south-east ·of western Australia. The final section presents some caveats of the analysis and conSiders further research issues. 3 Field Testing Model The model described here focuses solely on field testing for yield and 1s couched in terms of a non-linear programning problem . The model assumes the sole purpose of field testing is to identify if a potential new cultivar, when grown at sites representative of some region, does in fact on average out yield some standard cultivar(s).In the model other cultivar characterist i cs (for example, baki ng qual i ty) are ignored. The purpose of the model is to identify which testing st."atagy is optimal, given that test costs and costs of incorrect decisions basad on test results should be minimized, and that certain technical restrictions .may apply to field testing. The objective fUnction of the model is to minimize the combined costs of yield testing and wrong inferences about cultivar Yields. Algebraically, the objective function is to 2 ('1) Minimize .£ = ~ +. I (E[~~ 1=1 IQJ P[Tj. IDl) , where Q is the total cost of field testing, A is the direct cost of field testing ·s given in equation (2). E[1; IQ.] is the expected revenue loss associ'it r ,! with error type i given the actual yield difference between the new and .. t ndard cult;vars is Rand P[li IQ] is the probabi lity of error type igiven the yield difference Q. The direct cost of field testing, A, comprises (2) A = pyr(n + s)fi. + PY!i + t.z where p, y and r are the number of places. years and plot replicates respectively in the testing programme., nand s are the numbers of new and standard cultivars in the testing prograJmle, !! and ~are the average variable plot and site costs and finally, f is the fixed cost of the progranroe. Equation (1) may also be subject to several technical or logistic restrict'tons such as (3) pr(n + s) <= ~1 (4) (5) r(n t + .s) <= !2 , n<= ~3 t where!1 1s the total number of plots permissable 1n the program."I1e due to staff and eqUipment limitations, !.2 is the number of plots permissable at anyone s1teand ~3 is the maximum number of new cultivars that the testing progranvne can accommodate. Solving equation (1), subject to constraints (3) to (5), reveals .optimal field test. strategies, in terms of the necessary numbers of years, sites and plot replications, for various preconditions such assizes of Q, 4 tstlmating Probabilities of Decision Errors Before applying the model, the probabilities of decision errors of types I and II need to be estimated. The nature of type I and II errors and their estimation can be illustrated by a Simple comparison of two cultivars using independent, d1.fferently sized .samples and assuming unequal population variances. The null and alternate hypotheses are H1 Xi ) Xs where Xi and Xs are the "true'l or population yields of the new cultivar i and the standard cultivar s. The test of the null hypothesis is based on the modified! statistic (~) which is an approx.imat10n to Student's ~ (Steel and Torrie, p.l06). The test of the alternate hypothesis, however, is based on the non-central In statistic (Hays and Winkler). The test statistics, !at and !:n ' are random variables whose values are sample dependent. Associated with the distributions of ~ and tn are regions of rejection or critical regions which define the ranges of !m and In that lead to rejection of the null and alternate hypotheses respectively .. Because the test statistics are sample dependent their decision rules are not 8 guarantee of arriving at a correct decision. Two types of error arise. A type I error occurs when the null hypothesis is true yet it is rejected (cultivar it although the same or lower yielding than the standard cultivar s,is accepted .as being higher yielding). A type II error occurs when the alternate hypothesis is true yet it is rejected (cultivar i, although higher yielding than the standard, is accepted as yielding the same or less than the standard). The probability of a type I error is the probability of rejecting hypothesis when it is true. Commonly in agricultural experiments probability is set in advanc~ at 0.05. Following Cochran, in the cultivar case already outlined, the probability of a type I error stated as alpha = 0.05 = Px =x i s [!m> QJ the null this two can be where and where !i and ~$ are the Student 1:.. values for pyr1 and pyrs degrees of freedom and prObability of 0.05; p,y,ri and rs refer the numbers of places, years and plot replicates for cultivars i and s respectively and !;2and ~2 are sample variances for cultivars ; and s. Calculation of the probability ofa type II error involves recourse to the non-central t...stat1stic. The probabilities of t.n values are usually 5 tabled as functions of delta (the non-centrality parameter, see Johnson and Katz. chp. 31) and the degrees of freedom (Neyman et all or as functions of other factors that produce very close approximations to !n values (Resnikoff and Lieberman, Locks et al, OWen). Recourse to the non-central In. statistic causes estimation of type II error to depend on several factors: (a) tha discrepancy between the null hypothesis and the true relationship. The larger the discrepancy the smaller is the type II error. (b) sample size. The larger the sample size (more sites, years and plot replicates) the smaller is th~ error. (c) the standard deviation of the difference in mean yields. The reliability of the estimate may be improved by altering the mix of sites, years and replicates. (d) the level of significance of the test of the null hypothesis. The larger alpha is the smaller the type II error. Unlike alpha (the probability of type I error) which tends to be set in advance, the probab1"ity of a type II error is not pre-set. Rather it is conditional on seve'al factors some of Which are interrelated. Hence, calculating type] . error using known formulae (for example, Johnson and Kotz) , gener.ates ,value uniquely associated with a particular set of precondi.tions ($ .lch as the .size of alpha). In applying the model described in equations (1) to (6) various sets of preconditions were examined and the associated probabilities of type II error were estimated using an accurate approximation method developed by Hodges and Lehmann. The sets of preconditions were t 1n effect, different field test strategies. For example, sample size was determined by the number of cultivars. places, years and replicates to be considered and these in turn affected the estimate of the standard deviation of the difference inmaan yields of .a new and standard cultivars which in turn affected the probability of type II error. These effects are described in more detail below. In practice, the field testing of cultivars rarely involves the comparison of a single new cultivar with a standard. Usually several new cultivars are compared to a standard cultivar. For any new cultivar nominated in advance of any statistical examination of yields, Yi is an unbiased estimator of xi. Following Finney, the distribution of Y1 ;s and (6) where Yi is the mean of pyr plots of cultivar 1, Xi and Xs are the true yielding capacities of cultivar i and the standard respectiVely, Ys is the mean of pyrk plots of the standard and ve t vee 1l.ld vf are variance components relating to environmental effects, cult1var x environment interaction and residual error effects respectively. The Variance components are generally defined by Vp (8) ve == - .... Vy Vpy +-- + y py , 6 (9) Ycp "c.y VCpy vce = _. + - . + P Y py (10) Vf J 'Ie =--rpy where variance components are further disaggregsted into the components 'lip 'IVy ,Vpy ,v ep ,'Icy ,V epy and Yf whieh relate to variance associated with places, years, place x Yearintersction, cultivar x place interaction, cultivar x year interaction, cultivar x place x year interaction clOd residual error respectively. Given that field testing of cultivars usually fol' '\>/s some experimental design (for example, randomized blocks or Latin squares) analys1sof variance can usually be applied to the field data to generate estimates of these variance components. These variance components di rectly af~'ect the variance of yield difference in equation (7). By substitutinJl for various values of p,y,rand k in equations (1) to (10), the impact (If various field test strategies on the estimate of the variance of the yielc difference in equation (7) can be assessed and this in turn affects theorobab11 ity of a type II error. Estimating Costs of Decision Errors The costs of type I and II decision errors crucially depend on the degree to which the adoption response of farmers depends on results from cultivar field trials.. Brennan and Cullis have examined regional adoption and disadopti.on of wheat cultivars in Australia and found that relative yield advantage. as recorded in field trials, was a significant explanator of adOPtion response. Their approach was to fit inverse polynomia1s (t~elder) to adoption and disadoptlon responses for many cultivars. The fitted inverse polynomials were of form (11) P;y = yliai + bi Y + el y2 ), where PlY is the proportion of the total wheat area sown to cultivar i 1n year y. .From equation (11), the year (Yim) in which Pim is maximized is (12) Vim = (81/c1o.5), and the maximum proportion of total wheat area in year Yim is ValuesofPim and Vim were treated as observations of the dependent variables Pm and Yro in regression analyses of adoption response. From estimates of the expected values of Pm and Ym values of a, band c were calcu'lated. Following the predictton of adoption and disadoption response based on fie1d tria.1 results, itisposs1ble to estimate the costE> of type I and errors.. The cost pf type. II errors can be estimated by contrasting the II adopti.on response which would have occurred if field trial results truly indicated yield relativities versus current adoption response (the decision rule accompanying a type II error would be to not release the naw cultivar)" The cost of type I errors can be estimated by contrast·ing current adoption response to that associated with the wrong release of an inferior cultivar . lnthis paper, .although regression analysis was used in predicting adoption and disadopt1on response, the form of the ana·lysis differed from that of Brennan and Cullis.The approach Was firstly. to compare the yield relatiVities of historical and current wheat cultivars using special field trial data. Secondly, truncated binomial distributions were fitted to adoption patterns of these cult1vars. Ttli rdly, feature.s of these distributions were regressed 8gainstappropriate yield relativities of the cultivars. Using the regression results, adoption patterns for cultivars of known or hypothsticaly1eld superi.ority were predict$d~ Fourthly, the adoption patterns were converted into wheat product.ion and revenue estimates. The production and revenue estimates became the basis for estimates of foregone yield and revenue respectively. These latter estima.tes wefe the costs of type land II errors. In the cassof a type I errQr it is probably sensible to not rely on the regressionrelat1onships as accurate predictors of the adoption response of the wrongly recommended i.nferiorcult1var t if only because it means extrapolating from data on which the regression or curve,...f1tting is based. Also, in practice farmers through their own trial sowings and from those on neighbouring farms.• would fairly quickly discern acultivar t s yield inferiority. Hence, usually one would expect type I .costs to be relatively small. For a type II error however, fanners have no access to the rejected cultivar and sornayunknowingly forego a yield advantage over .many years. Hence, type II errors are potentially more costly. Application of the Hodel The modal was applied toy; e 1d testing of wheat cult ivars in the south-east of Western Australia. To Simplify theapp11cation, the technicalaod logistlc restrictions of equations (3) to (5) were assumed to not apply. Hence, the application involved estimating the var10usparameters in equations (1) and (2), setting preconditions and then, by calculation, finding Wllich values of P,Y and renabled £to be minimized in equation (1) .. Firstly, values for !!, g. and t in equation (2) were obtained from staff administering the cultivar field tria1s. These values are given in Table 1 and. together with various values of p, y and r, provide values forA in equation (1). Crable 1 about her'e) .secondly, data wereco'llected on the yield performance of a set of wheat cult'ivarsgrown in the same cultivar trials at four sites over four years in, the south-east of Western Austral ia. An ana lysis of var'iance of these data provided estimates of yield variance components given in Table 2. 8 (Table 2 about here) n (7). Also The variance components in Table 1 sUbstitute 1ntoe quatio (1) to (10) to ns uat1o intoeq tuted substi are various values ,ofp,Y ,r and k te of the estima the on gies strate test fie.ld s variou of . show the impact of the tes estima The (7). variance of the yield difference in equation type II of bility proba the affect turn in ence variancaof the yield differ in error 1.I error. A150 affecting the probab'11 ity and cost of 3)type such .as the level of equation (1) are various preconditions (see Table ence (Q) and the differ yield actual likely the alpha, thewheatpr1ce, tivars a discount rate.. To calcul ate the effect of Ron the adoption ofcul regression approach previously described was applied. equation (1) reveals Given apart icu1a rset of preconditions, minimizingprogram me associated with trial eld costf1 the .chara cteris tics of the leastpreconditions are and tics cteris chara vne progran Suoh s. those precondition given in Table 3. (Table 3 about here) sets of preconditions, A consis tent result in Table 3 is that across all ates. Given the size of least""cost field trial programmes include 3 replic effect s) in Table 2, plot es includ va (the estimate of p-Qolederror which finding in usual a is nd such a ne1)d for repl1c at1oh is not surpri.tJins;a .. kes) Kalts1 ~, (Finne~ s system studie s of agricu ltural trial s ,Whenever yield Another result is that, in the 9~ 'fen sets of precon dition a then a 15ks/h to equal or tnan less bb to ed differences (Q)ar e expect 3 replic ates is testin g programme of at least 4 sites, 4 years and edly high (~ 175 expect are nces differe yield when er, Howev necessary.. although a few mte, prograr g kg/ha) there is no uniform 1east cost testin trends emerge. ranges Firstl y, when the real discount rate is 10 percent then in the given t withou almost mt3 program g 3 testin of wheat price and alpha, the least- cost t highas the at ly, Second .. est1ng yieldt of years 3 exception only includes lly genera is sites test level of yield difference (200 kg/ha) the number of reduced to 3 sites . tics of leastAt the higher discount rate (20%) trends Hl the chara cteris values of higher the At clear. less cost yield testin g progra nJnesa re of years of yield alpha., wheat pr1ce and Yield difference, thanumber 3 years. In casas testin g falls to .2 years and 1n almost all other mes ~'isross various program cost least~ summarY, th$ chara cteris tics of (b) never ates. replic 3 least eat inolud always (a) to: are preconditions g testin e involv al1y (c)usu rely on single year or s1ngl esita testin g . under gh althou sites, 4 over g testin e requir y over 3 years and (d) usuall some oircumstances testin g over 30r 2 sites is possible. Table Sare presented The costs of the least- cost field trial programmes asin alpha decreases so the that in Table 4.. The result s in Table 4 indicate is due to This ses. increa mes program g present value cost .of the testin se in the increa an with ated associ being alpha in ion reduct firstl Y,the 9 probabi11tyof type II errors and secondly. due, to the relatively greater expense of typelI errors relative to type I errors (see the footnote to Table 4) .. (Tabla 4 about hare) Although results in Table 4 suggGst that relaXing of the significance level (that is., la.rgeralphas) in comparisons of cultivar yie.lds reduces test ~O$.ts. in practice such actions will incur additional costs not included here.. For ex.wnple, a m.arked relaxation of .alphaw111 increase the probability or lower yield1ng cultivars being released to farmers. Even if the on...farm costs of fanners initially trying and subsequent.ly rejecting such cul,tivarsis relatively small , the organisation recomnendingthe cultivars w111 likely lose soma of their credibility causing farmers· to be mere cautious 1n their adoption of even truly higher yie1ding cultivars recOll1mended by the.organisation. A lesser rats of adoption of such higher yielding cultlvars w111obviously involve .significant costs of foregone yield. Further., the organisation's .cereal bree.ding activity may suffer if 'its funding source. is by plant. variety rights or industry funds; in Australia the latter'sallQcat10nis influenced by the credibility growers attach to the Qrganisation.. Such a reduction in breed1ngeffort may cause the rataofinctease in wheat yield on farms to be less than m19htoccur if breed1ogafforts were not curtailed~ However, nonethe less the results do suggest. thAt strict adherence to low alpha levels, say 0.05 or less, is not warranted in the fieldt.rial progratl1nes considered here because of the costs incurred by such strategies. Resu1t,s in Table 4 also $howthat as. the l1kelyyield difference (Q) increases so the costs aftesting deer(!8se. The reason for this inverse relationship 1s due to the impact ofy1eld differences on beta values. Ths larger the yield difference the smaller is the,probabil ityof a type II error and 'in general the reducti.on in this probabili.ty more than offsets the cost increasing effect of the larger yield difference. Other results in Table 4 are that as the price of wheat decreases or as the discot,mtrate 1ncrel'ses, then the cost of least cost testing decreases. ln~ r;ffeet oftt.efal1 in the wheat price is s1m~ly to make the cost of decision error, less and thereby reduce the combined cost Qf decision errorssnd fie·ld t~ial costs. Conclusions This paper describes a model that identifies the cost and characteristics of least-cost field trial progranvnes for yield tests of wheat cult1vars 1n a region of Western Australia. The least-cost progrSfM\SS developedalway.s involve at least 2 years of testing (conrnonly 3 year test1ngis selected), usually over 3 or 4 sites and always with at least 3 rep.licates. The cost of anyprogranrne depends mainly on the pteconditions set for the model. Fc~ .Bxample, if thepr1ce of wheat decrea.sescr real discount rates increaSe then the cost of least-cost testing t4-aoreases. Further, if cultiVars within the progranvne are likely to be .at least moderately higher yie.1dingthan the standard cultivar (for example. higher yielding by at least 150 ks/ha) then the costs of testing decrease relative to the case where only small yield differences are expected.. Also if theground$ for 10 accepting&.. cultivaras higher yielding than the standard are relaxed (for Qxarnpliil, alpha is set at 0.1 rather than 0.05) then the cost of 1east-cost programmes falls. References Brennan, J.P. and B.. R. Cullis. "Estimating the Adoption and 01sadopt1onof Wheat Cultivars." A Paper presented to the 31st Annual Conference of the Australian Agricultural Economics .society, Univ. of Adelai.de, South Australis, February, 1981 . F1nneY)O.J •. °The Replicat'ion of Variety Trials". Siom. 20(1964):1-15. Hays, W.L. and R.. L. Winkler .. StatisticS Vol 1: Probability Inference and DeciSion. Holt, Rinehart and W1nstor} Incorp., New York, USA. Hodges, J .. L., and E.L .. Lehmann .. "A Compact Table for Power of the 1-1est/' Annal. Math. stat •.39(1968): 1629-1631 .• Johnson, N.L. and $. Kott. continuous UnivariatE:) Distributions-2. Houghton Mifflin Company," Boston, USA. Kaltsikes,P.J~ uGenotype-Environment Interaction Variances in Yield Tria.l$ of Fall Rye. to Can. J. Plant Sci •.50(1970)t71.... S0. Kingwe.ll. R.S. Economically OPtimal strate.9ies for Regional Yield Tests of Cultivars. Unpublished H.Sc. thesis, Univ. Qf Western Australia t Nedlands, WA, 1987 . Locks, M.O. t AleXander, N.J. and B.. J. Byar!j. "New Tables of the Non-Central S-Distribution.·\ Report ARL6j-19(1963) t WrightPatterson A1r Force Base . Nelder.J",A. uInverse Polynom1als,8 Useful Group. of Multi-Factor RE)spond Functions. "Siom. 22(1966): 128-141 .. Neyman, \ t Iwaszkiewicz, K. and S,. Kolodziejczyk. "Statistical problems in Agricultural Experimentation." J. Royal stat. Soc., Series B. 2(1935):107-180. Owen, 0 .. ,8. itA Survey of Properties and Appl ications of the NonCentral t-Distrtbution. u Technomet. 10(1968):445-478. PatiJ.f.son, H.O., Silv.ey, Vq Talbot.M~ and S. T.C. Weatherup • .tV.ariab111ty of Yields of Cereal Varieties in UK Tria1s. J. Agricl! Sci. Carob. 89(1977):239-245. to Resn1koff,G.J .. and G.J. Lieperlltan. Tables of the. Non-Central Distribution ...stanford Univ.ersity Press, Stanford, USA. ~ 11 ..iteel,R.G.O. and J .. H. Torria .. Principles and Procedures of Statistics: a B10metrical ApprQach 2nd. Ed. McGraw-Hill Kogskusha Incorp.) Tokyo, Japan. Sprasue,G.F. and W.T .. Federer. uA Cotnparison of Variance Components in Corn Yield Trials:!I Error. Year x Variety, Location x Variet.y and Variety Components. Agron. J .• It 43(1951):535-541. Talbot, M. ··Yield Variability of Crop Varieties in the U.. K. Agrie. Sci~ 102(1984):31S-321~ il J. 12 1 Patterson ~t a1 and Talbotimpl1c1tly recognize the importance of such costs in their concepts of critical difference and acceptance regions. 2 .Ingeneral the calculation of Sy - Y depends on whether (a) the i s two populations have a common variance, (b) the COfMlOn or individual variances are known or estimated, (e) the two samples are of the same size and (d) yie1d observations are paired. S Kingwel1 (1981) has also considered the impact of downward or upw.ard trends in the wheat price on the characteristics of leastr ' c o s t field trial programmes. ~. [ r ,~ K~ [ [, ~. 13 Table 1: Field Trial Operating Costs Cost Item Unit Average plot c~st (8) Average site cost (G) Fixed cost (F) $/plot 8.98 $/site 782 $/year 2843 a Cost a All costs are in constant 1986 dollar terms. Table 2: Estimates of Vield Variance Components in the Wheatbelt of Western Australia Sl~~th-east ---~-~-------~-------~-----~~---~-~--------~~~--~-,~----~-~----~-~--~-- Source of variation df SS MS MS a VR symbol F ratio probab. ~------~---~---~~--~~------------------~------------~------~-----~-- Year(y) Place(p) YXP Reps in y & P CUl t.1var(c) YX c P Xc PX YXc residual Grand Total 47854552 33648984 40698352 3420444 3 2778109 9 2194130 9 605758 3002208 27 96 1535528 191 135738080 3 3 9 32 15951518 11216328 4522.039 106889 926036 243792 6730t, 111193 15995 Vy vp Vpy Vc Vcy vcp Vcpy ve 149.2 104.9 42.3 <0.001 <0 .. 001 <0.001 57.9 15.2 4.2 7.0 <0.001 <0.001 <0.001 <0.001 ~~~~~~--~~-~-~--~--~~-------------~---~---~~~--~~-------,--------------~ a These are the same variance components and symbols as given in equations (8) to (10)!! 14 Table .3: Characterist,ics of Least-Cost Field Trial Programmes a b for Various Preconditions. _ _ ... _ ...... _ ...._ _ _... _ ... _ ..... _ _ ,..., _ _ _ .... _ _ _ _ ~ _ _ _ _ . _ _. __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ .... _ _ ...-r OR Wheat Alpha price (%) ($/t) ----------Vield difference(kg/ha)-------25 50 15 100 125 150 175 200 ---~~~~-~--~---~------~~----------~-~~---~~--------~~-------- 10 130 0.1 0.05 0.025 0 .. 01 115 20 130 115 0.1 0.05 0.025 0.01 0.1 0.05 0.025 0.01 0.1 0.05 0 •. 025 0.01 443 443 443 443 443 41 443 443 443 443 443 443 443 443 443 443 443 443 443 443 443 443 443 443 443 443 443 443 443 443 443 443 443 443 443 443 443 443 443 443 443 443 443 443 443 443 443 443 443 443 443 443 443 443 443 443 343 443 443 443 343 443 443 443 433 443 443 443 433 343 443 443 433 343 343 443 .;:3.3 433 343 443 433 343 433 433 433 343 433 433 333 433 433 433 333 333 33.3 433 433 333 333 433 3~3 433 423 42.3 333 423 333 333 433 243 333 423 433 333 423 433 333 333 4.33 433 243 433 433 433 343 433 433 433 ~-~---~-~---~----~-~~-~--~~-------~-~-----~---~----~---~~- a Theprogramrnes are denoted by figures such as 443 where the first digit (4) refers to the number of trial sites, the second digit (4) refers to the number of years .and the last digit (3) refers to the number of plot, replicates" b The precondit.ions relate to values ·of alpha, the discount rate, wheat price and likely yield difference.., 15 Table 4: The Cost of Least-Cost Yield Testing Programmes a ($m) ~~----------------~--~------------------------------------~---~- DR Wheat Alpha price (~) ($/t) ------------Vield d1fference(kg/ha)---------75 100 125 25 150 175 200 50 ~-~---------~~------------~-----~~-----------~--~------------- 10 130 115 20 130 115 0.1 0.05 0.01 0.1 0.05 0.01 0.1 0.05 0.01 0.1 0.05 0 .• 01 1.92 2.38 3.04 1,,72 2.12 2.71 0.93 1.14 1.44 0.84 1.02 1.29 1.21 1.96 3.97 1.09 2.85 4 .. 79 0.60 0.92 1.79 0.54 0.83 1.61 0.32 0.53 1.65 0.31 0.49 1.48 0.21 0.29 0.75 0.20 0.28 0.68 0.21 0.20 0.34 0.21 0.20 0.32 0.15 0.16 0.21 0.15 0.16 0.21 0.13 0.16 0.22 0.13 0.16 0.22 0.10 0.12 0.17 0.10 0.12 0.16 0 .. 11 0.14 0.17 0.11 0.14 0.16 0.09 0.11 0.11 0.08 0.11 0.11 0.11 0.11 0.14 0.11 0.11 0.14 0.08 0.09 0.11 0.08 0.09 0.11 0.08 0.08 0.11 0.08 0.08 0.11 0.06 0.06 0.09 0.06 0.06 0.09 ----------~----------------------~-------~--,----------------~ a These costs are in present value 1986 dollar terms and assume it is reasonable to attribute no cost to type I decision errors because of farmers' likely quick rejection of such lower yielding and wrongly recommended cult1vars.