THE ROYAL STATISTICAL SOCIETY 2006 EXAMINATIONS SOLUTIONS GRADUATE DIPLOMA APPLIED STATISTICS PAPER I

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1 THE ROYAL STATISTICAL SOCIETY 006 EXAMINATIONS SOLUTIONS GRADUATE DIPLOMA APPLIED STATISTICS PAPER I The Socety provdes these solutons to assst canddates preparng for the examnatons n future years and for the nformaton of any other persons usng the examnatons. The solutons should NOT be seen as "model answers". Rather, they have been wrtten out n consderable detal and are ntended as learnng ads. Users of the solutons should always be aware that n many cases there are vald alternatve methods. Also, n the many cases where dscusson s called for, there may be other vald ponts that could be made. Whle every care has been taken wth the preparaton of these solutons, the Socety wll not be responsble for any errors or omssons. The Socety wll not enter nto any correspondence n respect of these solutons. Note. In accordance wth the conventon used n the Socety's examnaton papers, the notaton log denotes logarthm to base e. Logarthms to any other base are explctly dentfed, e.g. log 10. RSS 006

2 Graduate Dploma, Appled Statstcs, Paper I, 006. Queston 1 () The AR(p) tme seres model s Yt = φ0 + φ1yt 1+ φyt φpyt p + εt where Y t represents the seres at tme t and φ 0, φ 1,, φ p are constants. The {ε t } are "pure error" or "whte nose" random terms, ndependently dentcally dstrbuted N(0, σ ε ), and not correlated wth {Y t }. The MA(q) tme seres model s Y t = ε t + θε 1 t 1+ θε t θqεt q where {Y t }, {ε t } are as above and θ 1, θ,, θ q are constants. ()(a) Yt 5 εt 0.3ε t 1 = + s a statonary process. [ ] 0 [ ] 5 [ ε ] 0.3 [ ] EY = + E E =5. t t εt 1 Also, agan usng the condtons n part (), ( Y ) ( ) ( ) ( ) E ε = (as n part ()), so t εt εt 1 σε Var = 0 + Var Var = The autocovarance s ( Y Y ) ( ) γ = Cov, = Cov 5 + ε 0.3 ε, 5 + ε 0.3ε k t t k t t 1 t k t k 1 ( ) 0.3Var εt 1 = 0.3σε f k = 1 = 0.3Var ( εt ) = 0.3σε f k = 1 0 otherwse t Thus the autocorrelaton ρ k s 0.3 = 0.75 for k = ± 1, and 0 otherwse So the partal autocorrelaton functon decays to 0 and s negatve. Soluton contnued on next page

3 ()(b) Y t = Y t + ε t s statonary (ths s gven n the queston). Also, E[Y t ] = E[Y t ], so [ ] 34 EY t = = ( ) = ( ) ( ) + σ, so ( Y ) Var Yt 0.1 Var Yt ε For the autocovarance, consder σ ε Var t = = 1.01σ ε. 1 (0.1) ( Y Y ) = ( Y Y + ε ) 0.1 cov (, ) Cov, Cov, t t k t t k t k =. Yt Yt k γ k 0.1γ k = and ρk = 0.1ρ k. However, ρ 1 = 0 and ρ = 0.1. Hence we have ρ 3 = ρ 5 = ρ 7 = = 0, and also ρ = 0.1, ρ 4 = (0.1), ρ 6 = (0.1) 3, ρ 8 = (0.1) 4,. The partal autocorrelaton functon cuts off sharply after k = (wth a negatve spke at k = ). () Yt = 54 0.Yt 1 + εt. ( ) 1+ 0.LY= 54+ where L represents the "backward shft" operator. ( 1+ 0.L) t εt ( 1 0. ( 0.) ( 0.3 ) )( 54 + εt ) 54 + εt Y = = L+ L L + t ( ) ( ) 3 t t 1 t t 3. = 54 + ε 0.ε + 0. ε 0. ε +...

4 Graduate Dploma, Appled Statstcs, Paper I, 006. Queston () (a) Varance and covarance depend on the unts of measurement, but correlaton does not. These varables are n completely dfferent unts of measurement, and ths would serously nfluence the results f the covarance matrx were used. There s also the ssue (see ()(a)) of whether, for example, length of road should be measured n mles or klometres. It would not be approprate to use the covarance matrx. (b) When the prncpal components are arranged n order of sze of egenvalues, as here, "cumulatve proportons" are found by addng egenvalues from the left, endng wth a total (n ths example) of 4, the number of varables. The "proportons" for each egenvalue here are = 55.75%, 100 = 33.5, 100 = 6.5%, so the cumulatve proportons are 55.75, 89.00, 95.5 and 100%. (c) Most of the varaton s explaned by PC1 and PC, leavng only 11% for PC3 and PC4 together. An explanaton of the data mght therefore be gven n terms of these two. The coeffcents n the components are the weghtngs of the four varables n each one. PC1 s a (weghted) average of all four varables, as often happens when usng the correlaton matrx; populaton s of less mportance than the other three varables. PC s manly a contrast between populaton and length of road. Ths mght reflect the effect of length of road outsde man populaton centres. PC3 seems to be a contrast between number of drvers and (populaton densty, length of road) but does not contrbute much to the total. It mght be some form of measure of the number of drvers outsde man populaton centres. PC4 s a contrast between fuel consumpton and the others, but contrbutes lttle. Perhaps t s manly a contrast between number of drvers and fuel consumpton, and thus could gve a measure of the number of drvers wth hgh fuel consumpton. (Note that components that contrbute lttle and mght reasonably be gnored can sometmes gve nformaton about what s not mportant.) Soluton contnued on next page

5 () (a) Snce the correlaton matrx has been used, changng the unts as suggested would make no dfference to the results. Identcal prncpal components and regresson analyses would be obtaned. The whole analyss s ndependent of the orgnal scales of measurement. (b) A smple explanaton, wth farly hgh R, comes from PC1, and shows that deaths are postvely related to number of drvers, length of road and fuel consumpton, but depend lttle on populaton densty. Addng PC3 and PC4 mproves the explanaton and leads to a very hgh R. Note that the nterpretatons depend on the sgns of the coeffcents. () Unless there are correlatons among the predctor varables (whch of course there often are), there s nothng to gan from usng prncpal components. The prncpal components themselves are uncorrelated, so ths makes model selecton easer f a "sutable" subset s known. However, the prncpal components are not always easy to nterpret as they are purely mathematcal constructs, and thus the resultng regresson models may not be easy to nterpret. (Note for example that the negatve coeffcent of PC4 n the regresson analyss n part () suggests that the number of deaths s postvely related to drvers wth low fuel consumpton, whch seems strange.) It can be dffcult to choose a "sutable" subset of prncpal components. In ths example, PC appeared mportant n the explanaton of the data n part () but dd not enter the regresson analyss n part (). On the other hand, t can happen that components wth low egenvalues are mportant n the regresson as appears to have happened wth PC3 and PC4 here. One way to try to deal wth ths s to nclude the response varable n the prncpal components analyss and select PCs wth hgh coeffcents of the response varable and large egenvalues; but there s no guarantee that mportant regresson varables wll be dentfed.

6 Graduate Dploma, Appled Statstcs, Paper I, 006. Queston 3 () Both methods can be carred out on data from groups of tems for whch there are multple measurements. In cluster analyss the groups are not known already, but are constructed from the data; the analyss nvestgates possble groupngs, based on the multple measurements. Dscrmnant analyss s carred out when the (two or more) groups are known already and the am s to fnd the (lnear) combnaton of the varables whch best separates the tems nto ther groups. The method assumes that the data n each group are from multvarate Normal dstrbutons wth smlar varance-covarance structures. As an example, dscrmnant analyss would be approprate for dstngushng between those anmals whch survve n a hard envronment and those whch do not, usng a range of physologcal measurements on bodes. A sutable dscrmnatory combnaton of these measurements would have been found from several speces for whch t was known whether or not they survve. The same combnaton would then be appled to measurements on a dfferent speces to gve a predcton of whether or not t would be lkely to survve. () (a) For d(x, y) to be a proper dstance measure for X and Y, we requre: d(x, x) = 0; d(x, y) = d(y, x); d(x, y) d(x, z) + d(z, y) for all Z. Certanly d(a, A) = d(b, B) = d(c, C) = 0, from the dagonal entres. Also, d(a, B) = 3.9 = d(b, A); and smlarly for d(a, C) and d(b, C). Fnally, d(a, C) = 5.5, wth d(a, B) + d(b, C) = = 6.9 > 5.5; d(b, C) = 3.0, wth d(b, A) + d(a, C) = = 9.4 > 3.0; and d(a, B) = 3.9 wth d(a, C) + d(c, B) = = 8.5 > 3.9. (b) At stage 1, tems A and H have dstance 1.0 and these form a cluster. Cluster s D and E wth dstance 1.7, so stage has two clusters, AH and DE. The next cluster s BC wth dstance 3.0. The next dstance s 3.7, for AF and for BH; as shown n the dendrogram, we therefore combne ABCFH as a cluster at ths dstance, from whch the cluster DE s stll separate. The next dstance s 4.6, for CD and CG; thus at ths dstance we are down to a sngle cluster. [Note. In any practcal case we may specfy the number of clusters requred as a reasonable summary of the data or we may specfy the maxmum dstance for combnng clusters. As an llustraton of the latter, suppose we specfed dstance 3.1; at ths dstance there are 3 clusters, AH, BC and DE, and the other tems stand on ther own.] (c) The dendrogram appears rather "straggly". A and H seem closely smlar, as do D and E, but otherwse there are no very clear and concse clusters. There s not really any strong evdence of "dstnct categores" by the sngle lnkage cluster analyss used here, though dfferent methods of clusterng mght gve dfferent structures. The dendrogram s on the next page

7 Dstance A H F B C D E G Item

8 Graduate Dploma, Appled Statstcs, Paper I, 006. Queston 4 () (a) The probablty functon of the bnomal dstrbuton B(m, π) wrtten n exponental form s m f y y m y y (, π ) = exp log + logπ + log ( 1 π) log ( 1 π) π log, = log + log 1 + constant 1 π so f ( y π ) y m ( π ). As y s on the rght of ths equaton, t s n canoncal form, and the multpler of y s the natural parameter, whch s therefore π log. 1 π [Ths s the log-odds, or logt.] The generalsed lnear model sets ths lnk functon equal to the lnear predctor. (b) The odds s the rato of probabltes of "success" and "falure" for Y,.e. π /(1 π ). The log-odds s smply the logarthm (base e) of ths, as used n the lnk functon. After the generalsed lnear model has been ftted, the (estmated) value of η s obtaned ths s the estmate of the log-odds. We have π π η = log = exp( η) 1 π 1 π so the estmate of the odds s exp(η ). Gven the necessary standard errors (SE), approxmate 95% confdence lmts are "estmate ± 1.96 SE" for each of odds and logodds. The detals are n ()(c) below. () (a) We are not told how the samplng was carred out, so the ndependence of observatons s not guaranteed; nether s the randomness. The analyss would be approprate f a random sample of data from a larger populaton has been selected, omttng multple brths (twns etc) and only usng a mother once f she has had more than one chld at dfferent tmes [to avod probable lack of ndependence]. Many hosptals would need to be represented n the samplng, as well as home brths. It would not be approprate to use ths analyss f the "group of women" mentoned came from a lmted area, for example by studyng all brths from the local hosptal over a few years. Soluton contnued on next page

9 (b) Step 1 chooses the sngle predctor varable whch reduces the scaled devance as much as possble from the "constant only" model. Clearly ths s GEST, the length of gestaton perod, whch reduces the devance by (on 1 df). We next consder addng AGE, and ths step further reduces the devance by 6.566, also on 1 df; ths s sgnfcant as an observaton from χ wth 1 df, so AGE should be ncluded. So we use AGE and GEST n the model. (c) The codng AGE = 0, GEST = 0 gves ( ˆ η) ˆ η = , SE = The estmate of the odds s exp( ) = % confdence lmts for η are ± ,.e. (.00, 1.51), so the correspondng lmts for the odds are (0.137, 0.05) after exponentatng. The estmate of the probablty of mortalty s ˆ η e ˆ π = = = ˆ η 1+ e Smlarly, the upper and lower lmts of the 95% confdence nterval for ths probablty are as follows. Lower lmt: = ; upper lmt: =. (d) GEST s now to be coded as 1, AGE remanng zero. The log-odds rato for ths group compared to the group n (c) s thus smply the value of the GEST parameter,.e So the odds rato s exp( 3.886) = % confdence lmts for ths log-odds rato are ± ,.e. ( 3.650,.97). Thus the lmts for the odds rato are exp( 3.650) = 0.06 and exp(.97) =

10 Graduate Dploma, Appled Statstcs, Paper I, 006. Queston 5 = + +, where the { ε } ()(a) The approprate model for log(y) s log y log a bx ε ' are ndependent dentcally Normally dstrbuted errors, N(0, σ ). ' Thus for the untransformed data we have lognormal. bx y ae = ε where the ε are ()(b) Denote log(y) by Y and log(a) by A so that Y = A + bx + ε '. The normal equatons are obtaned by mnmsng ( ). S = Y A bx Dfferentatng wth respect to A and B leads to the followng (strctly speakng t should be checked that these are mnmsng values). S = ( Y A bx ). A Settng ths equal to 0 gves  = Y bx ˆ. Y = Aˆ + bx ˆ as one normal equaton. So S = ( ). b x Y A bx Settng ths equal to 0 gves ˆ ˆ Thus x Y = ( Y bx ) x + b x x Y Aˆ x bˆ = + x. ΣxΣy ΣxY ˆ n nσxy ( Σx)( ΣY) b = or ( Σx ) nσx ( Σx) Σx n Soluton contnued on next page

11 bx () Here we smlarly mnmse S = ( y ) ae as follows. S bx = bx ( y ae ) e. a, gvng two normal equatons Settng ths equal to 0 gves bx ˆ ˆ bx ye ˆ = a e. s = bx ( ). b bx y ae axe Settng ths equal to 0 gves bx ˆ ˆ bx ˆ = or aˆ x ye a xe bx ˆ ˆ bx x ˆ ye = a xe. From the frst normal equaton we have Insertng ths n the second gves aˆ = bx ˆ ye e bx ˆ. ˆ bx ˆ ˆ ˆ bx bx bx xye e = ye xe (Ths would requre an teratve method of soluton such as Newton- Raphson.) () (a) Frst model: aˆ = e =.504; b = ˆ Second model: aˆ =.45; bˆ = 1.0. The estmates are very smlar. (b) (c) The stuaton s not clear-cut. The standardsed resduals from the frst model are perhaps more scattered than those from the second, and wth an appearance of becomng less varable as x ncreases. For the second model, there s only one outler among the resduals but the pattern may be slghtly skew. The outler n ths model corresponds to the pont where there s a slght jump n the orgnal scatter dagram, and ths also produces a somewhat hgh standardsed resdual from the frst model. Perhaps there s a very slght preference for the second model. Any exstng knowledge about the system wll be helpful, as wll any pror nformaton on the error structure s t Normal or lognormal (or nether)?

12 Graduate Dploma, Appled Statstcs, Paper I, 006. Queston 6 () The sngle varable whch reduces the resdual sum of squares most from the model wth ntercept alone s x 4, so forward selecton frst ntroduces ths varable. It then examnes addng another varable to x 4. The smallest resdual sum of squares for x 4 and one other varable s when x 1 s taken wth x 4. Next to be entered would be x but ths does not seem to make much dfference compared wth (x 1, x 4 ). Addng x 3 as well does not seem to make a worthwhle dfference ether. Formal tests for sgnfcance at each stage are as follows. (1) Addng x 4. The SS for addng x 4 s = , and the remanng resdual then has 11 df. So the test statstc s =.80, 11 whch s very hghly sgnfcant as an observaton from F 1,11 ; there s very strong evdence that x 4 should be ncluded n the model. () Addng x 1. The SS for dong ths s = , and the remanng resdual has 10 df. The test statstc s = 108., whch s very hghly sgnfcant as an observaton from F 1,10 ; there s very strong evdence that x 1 should also be ncluded n the model. (3) Addng x. The SS for dong ths s = 6.789, and the remanng resdual has 9 df. The test statstc s = 5.03, whch s not (qute) sgnfcant at the 5% level as an observaton from F 1,9 (the 5% crtcal pont s 5.1). Judged at the 5% level, there s no evdence that x should also be ncluded n the model. Thus the model reached by forward selecton has x 1 and x 4. Soluton contnued on next page

13 () Not f there are correlatons among the predctor varables. Startng from the full model and omttng varables can gve very dfferent results from forward selecton. () Mallows' C p s a dagnostc statstc desgned to help n dentfyng a "best subset" model. The defnton of C p s where C p RSS p = p s ( n ) n s the number of data values, p s the number of parameters (ncludng the constant) n the model beng nvestgated, RSS p s the resdual sum of squares for the model beng nvestgated, s s the resdual mean square for the full model usng all possble predctors. It can be shown that, for a satsfactory model, E[C p ] = p. Lookng at the values of C p gven n the queston, ths suggests that the (x 1, x 4 ) model may be not qute "good enough" as t has a C p value of 5.50 as opposed to an expectaton of. All of the three-predctor models seem satsfactory apart from (x, x 3, x 4 ). (v) The two-predctor model (x 1, x ) appears good, both from ts resdual sum of squares and from ts C p value. It also has a hgher R value than any of the other models wth fewer than three predctors. Ths model has not appeared n the forward selecton because nether x 1 nor x was the frst varable chosen, but t appears to be the best parsmonous model. (v) It s always mportant to have nformaton about the practcal stuaton, to avod proposng a model that researchers or practtoners n the feld consder unreasonable. They may do so from ther general techncal knowledge of the feld and/or n the lght of earler work whch has shown that some potental varables n fact have lttle relaton to the response y. Correlatons among the x varables may, mathematcally, lead to such "unreasonable" models. If there s a choce, those varables that are easer and cheaper to measure accurately would often be preferred. Why have the varables x 1, x, x 3, x 4 n the example been suggested?

14 Graduate Dploma, Appled Statstcs, Paper I, 006. Queston 7 () y x There s a consderable amount of scatter but some ndcaton of a weak postve assocaton between y and x. The varance of the y varable looks as f t could be assumed constant there s no apparent pattern (such as a dependence on x). () Any assocaton that exsts between y and x s not obvously curved, and does appear to have a lnear component. So a lnear regresson model seems reasonable. Because there are repeat observatons on y at some of the x- values, a "pure error" term can be extracted from the resdual as the sum of squares between these repeats (see below). The remander of the resdual ("lack of ft") then represents departure from lnearty, whch can be tested aganst the "pure error". Ths should gve a better test of the lnear regresson model. The model s Y j = a + bx + ε j = 1,,, 13 j = 1 or or 3, dependng on the value of. where x s a value of x, Y j represent the (sngle or repeat) observatons taken at x = x, {ε j } are ndependent normal N(0, σ ) random varaton terms wth constant varance. Soluton contnued on next page

15 () (a) At x = 1., we have y =. and 1.9, wth total 4.1. So the pure error 4.1 SS here s = Ths has 1 degree of freedom. (b) (c) At x = 4.1, we have y =.8,.8 and.1, wth total 7.7. So the pure error SS here s = Ths has df. 3 At each x value where there are repeats, a smlar calculaton s carred out. The sums of squares are added to obtan the total pure error SS. The numbers of degrees of freedom would also be added to obtan the total df for "pure error", whch here wll be 9 (ths s needed below, n part (v)). (v) If the "pure error" SS s , the "lack of ft" SS must be =.837, and ths wll have 0 9 = 11 df. Hence the analyss of varance s Source of varaton df Sum of Mean F value squares square Regresson / = 5.50 Lack of ft / = 0.43 Pure error = ˆ σ Total The F value for regresson (note that ths s now a comparson wth the pure error term) s referred to the F 1,9 dstrbuton. Ths s sgnfcant at the 5% level (crtcal pont s 5.1), so there s some evdence n favour of the regresson model. There s no evdence of lack of ft. (It could be argued that the lack of ft and pure error SSs should therefore be recombned to gve the resdual as before, wth 0 df.) We note that R =.673/9.377 = 8.6%, whch s low; despte the absence of evdence for lack of ft, only about 9% of the varaton n the data s explaned by the lnear regresson model. Ths s because the underlyng varablty (estmated by the pure error mean square) s hgh. (v) Resduals after fttng the proposed model can be examned, and any patterns n them noted. Departures from the model can be detected n ths way, such as a need for an addtonal term, or systematc non-constant varance, etc.

16 Graduate Dploma, Appled Statstcs, Paper I, 006. Queston 8 Part () Totals are as follows For A A1: 56; A: 78; A3: 83. For B B1: 145; B: 7. Grand total: 17. Sum of squares of observatons: The (corrected) total sum of squares s 1977 = , wth 9 df. 30 The sum of squares for factor A s = 41.67, wth df The sum of squares for factor B s = , wth 1 df The sum of squares for the nteracton AB s SS for A SS for B= 6.067, wth df. The resdual SS s obtaned by subtracton. Ths has 9 1 = 4 df. A s a fxed factor n both (a) and (b) below; ts nterpretaton s the same n the two parts. B s fxed n (a) and random n (b), so ts nterpretaton s dfferent n the two parts, and ths also apples for the nteracton AB. Soluton contnued on next page

17 (a) The model s ( ) y = μ + α + β + αβ + ε jk j j jk = 1,, 3; j = 1, ; k = 1,,, 5 α = 0, β j = 0, ( αβ ) = 0, ( αβ ) = 0 j. j j j Here μ, α, β j, (αβ) j are all unknown constants, representng the grand mean, the effect of level of A, the effect of level j of B, and the nteracton between A at level and B at level j, respectvely The "error" terms {ε jk } are ndependent Normal random varables, N(0, σ ), wth σ constant. The analyss of varance s as follows. A column of expected mean squares (E[MS]) s nserted n the table. Source of varaton df Sum of squares Mean square E[MS] A σ 5 ( ) B ( ) AB σ ( 5 ) 1 F value + Σ α 0.633/5.67 = 3.9 σ + Σ β j /5.67 = ΣΣ( αβ ) j /5.67 = 5.89 Resdual = ˆ σ Total Tests for the null hypotheses "all α = 0", "all β j = 0", "all (αβ) j = 0" use the gven F values. The upper 5% pont of F,4 s 3.40 and the upper 1% pont s 5.61, so the frst of these s sgnfcant at the 5% level and the thrd at the 1% level. For F 1,4, the upper 0.1% pont s 14.03, so the second s very hghly sgnfcant. There s evdence that both man effects and the nteracton are mportant. As the nteracton s sgnfcant, the nterpretaton should be based on a -way table of AB means. The response for B s much the same at each level of A; however, for B1 the response at A1 s well below the other two, wth A3 a lttle larger than A. Soluton contnued on next page

18 (b) The model s ( ) y = μ + α + β + αβ + ε jk j j jk = 1,, 3; j = 1, ; k = 1,,, 5. Here μ and the α are unknown constants, wth Σα = 0, as before, and the "error" terms {ε jk } are ndependent Normal random varables, N(0, σ ), wth σ constant, as before. The β j are uncorrelated random varables, uncorrelated wth the (αβ) j and wth the {ε jk }, wth mean 0 and constant varance σ B. The (αβ) j represent the nteracton. For each level of A (.e. for each ) they are uncorrelated random varables, uncorrelated wth the β j and wth the {ε jk }, wth mean 0 and constant varance σ AB. For each level of B (.e. for each j) they are constants wth Σ ( αβ ) =. j 0 Assumptons of Normalty for the β j and (αβ) j are added for the formal nferences, so that these become ndependent Normal random varables. The analyss of varance s as follows. A column of expected mean squares (E[MS]) s agan nserted n the table. Source of varaton df Sum of squares Mean square E[MS] σ + σ + Σ α A ( 5 ) B AB AB σ + (5 3) σ B σ + 5σ AB Resdual σ F value 0.633/ = /5.67 = /5.67 = 5.89 Total The null hypotheses are "all α = 0", "σ B = 0" and "σ AB = 0". The expected mean squares ndcate how these are tested. Soluton contnued on next page

19 The F value to test the frst null hypothess s 0.665, whch s referred to F,. Ths s not sgnfcant; there s no evdence aganst ths null hypothess. The F value to test the second s 33.73, whch s referred to F 1,4. Ths s very hghly sgnfcant (see part (a) above); there s very strong evdence aganst ths null hypothess. The F value to test the thrd s 5.89, whch s referred to F,4. Ths s sgnfcant at the 5% level (see part (a) above); there s evdence aganst ths null hypothess. Estmates of σ B and σ AB can be obtaned f requred. The explanaton should be n terms of there beng evdence for varaton among the effects of the levels of B n the underlyng populaton of all such levels, and smlarly for the AB nteracton. Part () Suppose that A1, A, A3 are three alternatve cultvaton treatments n an agrcultural tral, whle B1, B are two stes upon whch that experment s carred out. If B1, B are the only two avalable stes about whch nferences are to be made, (a) s approprate. If B1, B are selected (at random) from several avalable stes and nference s to be made for the whole collecton of stes, (b) s approprate.