POWER TRANSFORMATIONS WHEN THEORETICAL MODELS TO. Raymond J. Carroll and. David Ruppert

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1 POWER TRANSFORMATIONS WHEN FITTING THEORETICAL MODELS TO DATA Raymnd J. Carrll and 2 David Ruppert 1 Running Title: Fitting Theretical Mdels 1 Natinal Heart, Lung and Bld Institute and the University f Nrth Carlina. Grant AFOSR Supprted by the Air Frce Office f Scientific Research 2 University f Nrth Carlina. Supprted by Natinal Science Fundatin Grant MCS Sme key wrds: Transfrmatins, Bx-Cx mdels, theretical mdels, rbustness. ~ AMS 1970 Subject Classificatins: Primary 62F20; Secndary 62G35.

2 -2- A B S T R ACT We investigate pwer transfrmatins in nn-linear regressin prblems when there is a physical mdel fr the respnse but little understanding f the underlying errr structure. In such circumstances and unlike the rdinary pwer transfrmatin mdel, bth the respnse and the mdel must be transfrmed simultaneusly and in the same way. We shw by an asympttic thery and a small Mnte-Carl study that fr estimating the mdel parameters there is little cst fr nt knwing the crrect transfrm a priri; this is in dramatic cntrast t the results fr the usual case that nly the ~ respnse is transfrmed.

3 -3-1: INTRODUCTION t Often in scientific wrk, ne bserves data y and x = (xl.. x p ) and pstulates that these data fllw a mdel (1.1) y. = f(x.~ 8 0 ), i = l~..., N~ t. t. where 8 0 is a k-parameter vectr. The functin f may be derived, fr example, frm differential equatins believed t gvern the physical system which gave rise t the data. The deterministic mdel (1.1) is ften inadequate since the data exhibit randm variatin, but whereas f was derived frm theretical cnsideratins, there is really n firm understanding f the mechanism prducing the randmness. In this case, ne typically assumes that (1. 2) where the {Ei} are i.i.d. N{O,a~). In thse cases in which the data suggest that mdel (1.2) is als unsatisfactry, ne might then assume that the errrs are multiplicative and lg-nrmal, s that (1. 3) The pint here is that mdel (l.l) is equivalent t the mdel h{y.) = h{f{x., 8 0 )) t. t. whenever h{ ) is a mntnic transfrmatin. Therefre (1.2) and (1.3) are based n the same theretical mdel, but they allw variability int the mdel in different fashins.

4 -4- A mre flexible apprach is t take a sufficiently rich family f strictly mntnic transfrmatins h(y~a), indexed by the m-vectr parameter A, and t assume that fr sme value A (1.4) The mdel (1.4) is in the spirit f Bx and Cx (1964), wh suggested the family f pwer transfrmatins with m = 1 and (1.4b) = lg (y) if A= O. Hwever, as we will make clear, ur prpsed mdel (1.4) has greatly different ramificatins than usually assciated with the pwer family. Bx and Cx (1964) used their family in a study f the transfrmatin mdel (1.5) h(y~a ) = x~ e + E. 0 Ntice here that, unlike (1.4), the regressin functin in (1.5) is nt transfrmed. Bx and Cx sught a transfrmatin which achieves 1) a simple, additive r linear mdel, 2) hmscedastic errrs and 3) nrmally distributed errrs. Our mdel is different. Theretical cnsideratins already prvide a regressin functin. We hpe t transfrm the respnse and the regressin functin simultaneusly t btain hmscedasticity and nrmality. There are tw reasns fr using mdel (1.4) instead f simply fitting (1.1) by least squares r sme ther methd. First, estimatin f e based n mdel (1.4) shuld be mre efficient than ther methds. Secnd, it may

5 -5- be necessary t estimate the entire cnditinal distributin f y given x; if the data clearly suggest that the distributins f {y.-f(x.,e )} are nt 1" 1" 0 cnstant, ne must g beynd standard regressin methdlgy. An example, which partly mtivated the research f this paper, cncerns the relatinship between egg prductin in a fish stck and subsequent recruitment int the stck. At least fr sme species, as egg prductin increases, the change in the skewness and variance f recruitment is as large as the change in the median recruitment, and this change in distributinal shape may have imprtant implicatins fr management f the fishery. The utline f the paper is as fllws. Sectin 2 discusses a current cntrversy cncerning the mdel f Bx and Cx. Bickel and Dksum (1981) have shwn that, in mdel (1.5), the ML estimate f e can be much mre variable when A is estimated cmpared t when A is knwn. In Sectin 3, 0 4It we demnstrate fr ur mdel (1.5) an entirely different result: the ML estimate f e in mdel when A is unknwn cmpared t when A is knwn. 0 cnsiderably strnger result. (1.4) turns ut t be nly slightly mre variable In Sectin 4 we prve a By examining a weighted least abslute deviatins estimatr, we prvide a lwer bund f 2/n n the asympttic relative efficiency f the ML estimatr f e in mdel (1.4) when A is unknwn cmpared 0 t the MLE when A is knwn. 2: RECENT STUDIES OF THE BOX AND COX MODEL In Sectin 7 f Bx and Cx's riginal paper they discuss the analysis e f effects after transfrmatin. A They state that, after finding A, ne shuld A -estimate effects (regressin parameters) n the scale A which has been chsen fr analysis and nt n the true but unknwn A scale. 0 Hwever, in discuss- ing interactins, they g n t state that lithe general cnclusin will be A that t allw fr the effect f analysing in terms f A rather than A, the 0

6 -6- residual degrees f freedm need nly be reduced by... the number f cmpnent parameters in A". Bx and Tia (1968) agree, stating that the nly practical effect between using A in the psterir distributin f 8, , is an adjustment in the degrees f freedm. Bickel and Dksum (1981) disagree with this cnclusin. rather than the true Fllwing calculatins fr the lcatin prblem dne by Hinkley (1975) and suggestive Mnte-Carl results f Spitzer (1978) and Carrll (1980), they calculated 2 fr general regressin the large sample infrmatin matrix f A, a and They fund that the large sample variance f 8 is larger, ften much larger, when A is estimated cmpared t when is knwn. They als state that the cnclusin f Bx and Tia is nt crrect. On a technical level, part f the (A) (1..)/ (0)1..-1 z = y y, where y is the gemetric mean f the {Yi}' Hwever, Hinkley and Runger (1982) fund z(a) unsatisfactry in several respects. The differences may als be cntextual; at the null hypthesis f n interactin effects, ne can act as if were knwn, with an apprpriate change in the degrees f freedm. See Carrll (1982) and Dksum and Wng (1981). Since pwer transfrmatins have been used ften and with real satisfactin by applied statisticians, the findings f Bickel and Dksum were surprising and led t further research. Hinkley and Runger argue that the parameter 8 0 in (1.5) is nt physically meaningful; it is defined in an unknwn scale s that a unit change in x is nt easily interpreted by 8 0 alne. Instead, ~ they argue that in practice, the relevant distributin is the cnditinal A distributin f e given A. As N ~ 00, the cnditinal variance f e given A

7 -7- and the variance f ewhen A is knwn cnverge t the same matrix. They then argue that, when analyzing 6 0 ' n adjustment need be made fr the fact that A was estimated. This appealing behavir is smewhat cunter-balanced by difficulties with the cnditinal mean in hypthesis testing in unbalanced designs, as pinted ut by Carrll (1982). Carrll and Ruppert (1981) als nticed the difficulty with interpreting 6 0 and studied predicting the median f Y n the riginaz data scale by t'" backtransfrming x 6. the prblems f definitin inherent with 6 0 data dependent scale. effect f nt knwing A This idea f lking at the respnse surface avids being defined in an unknwn r They fund that when predicting the median f Y, the can be large but is in general similar t the effect f adding ne mre regressin parameter, and it is certainly much less severe than the effect when estimating 6. 0 The abve discussin establishes the extent t the cntrversy surrunding the Bx and Cx mdel applied (1.5). We believe (1.4) entirely avids this cntrversy. First, the parameter 6 0 has physical meaning even if A is unknwn, since f{x ij 6 ) is the median f Yi n matter what the 0 true scale. Secndly, the large sample analysis t fllw indicates that '" 6 is nly slightly mre variable when A is estimated than when A is knwn. 3: LIKELIHOOD ANALYSIS The likelihd analysis prceeds as fllws: define z. = dh{f.{6 ),A )ld6 '/- '/-000 f (6) = f{x. J6), f = f (8 0 )' '/- '/- '/- '/- h (y) = h (YJ A) = dh{yj A)ldYJ and h{y) = h{yj A). Y Y Let h A (y) and h AA (y) be the gradi ent vectr and Hessi an f h{yj A) with respect t A. By simple algebra we find the jint infrmatin matrix f (6 0,O,A )

8 -8- as (all summatins are frm 1 t N) (3.1) s/ L 0 C 1 / N- 1 I = C /a 2 lf 1/(20 1f 0 ) 0 C /0 2 J 0 where (3.2) C 2 = -N-1ELEi[hA(Yi) - ha(fi)]t C J = N-1EL{IhA(Yi) - ha(f i )] [ha(yi) - ha(fi)t t + Ei[hAA(Yi) - haa(f i )] + (a/aa}(a/aa) lg[hy(yi)]}. Using the wrk f Hadley (1971), it is straightfrward, thugh perhaps smewhat tedius, t establish cnditins sufficient that (8, 0 2, ~) is cnsistent and asympttically nrmal. We will nt pursue this matter further, but rather we will assume that ( 8, 0, A ;s apprximately N 8 0 A), I -1) and we will study I -1. A "2 ") t ( ( 2 t 000 In general, C 1 and C 2 are nt zer and the asympttic distributin f (~, 0 2 ) when A ;s estimated differs frm when A is knwn. At least t this 0 pint then, the analysis ;s similar t thse dne in the usual Bx-Cx mdel (1.5). The key questin, f curse, is whether r nt C 1 and C 2 are " sufficiently different frm zer t seriusly affect the distributin f A. The expressins C 1, C 2 and C J are cmplex even when f i (8 0 ) has a nice frm such as simple linear regressin. T simplify matters sufficiently that we can gain sme insight abut the difference between knwing and estima-

9 -9- ting A ' we fllw Bickel and Dksum and thers and let 0 0 ~. While Bickel and Dksum let N ~ 00 and simultaneusly, we let N + 00 and then 0 +. There is n essential difference between the tw appraches. Our is very suitable fr heuristic arguments. It shuld be emphasized that we are nt cncerned nly, r even primarily, with small 0 0 In fact, the need fr transfrmatin is greater when 0 0 is large. The small 0 asympttics d, hwever, lead t majr simplificatins, and the Mnte-Carl results presented later agree with them. (3.3) Taylr expansins shw that under mild regularity cnditins Standard calculatins shw that when A is knwn, (3.4) N~ Cvariance [(8 - e )/0, ( )/0 2 1A knwn] A -1 = [-1 S 0] 02. Let D = Diag(, 0 2, 1). Then, t find this limiting cvariance matrix when 0 A is unknwn, we must find the upper left (k + 1) x (k + 1) crner f DID = s C 1 /O C / C /0 2 ;) 0 which by standard results n inverting partitined matrices is A- 1 + FE- 1 F t where A- 1 is given in (3.4), t E = C3/0~ - B A B, -1 F = A B,

10 -10- and B = Clearly, F= and In rder t btain simple asympttics, we will assume that fr 00 fixed, c1/a~, C2/0~, and CJ/~ cnverge as N + 00, and that these, in turn, have limits D1~ D2~ and D J respectively as We als assume that t S + S (psitive definite) as N If D J - 2D 2 D 2 is nnsingular, then Urn Urn N + 00 Urn + 0 Urn N+ S-l 0 ] [ = W THEOREM 1. Assume that the limits D 1 ~ D 2 ~ D J ~ S mentined abve exist and that D J _ 2D 2 D 2 is nnsingular. As N + 00 and then + O~ the limit A distribu~in f 6 is the same whether A is knwn r unknwn. The limit distributin f 0 depends n whether A is knwn r unknwn. As an example cnsider multiple linear regressin and the pwer transfnnatin family, i.e., h(y~a) is given by (l.4b) and h(y.~a) t =x. e + E '/; '/; 0

11 -11- where xl'.., x n are knwn k x 1 vectrs. Als, suppse that A = 0, A -1. i.e., the lg transfrmatin is needed. Then hy{y) = Y -, ha{y) = (lg y)2/2, and haa(y) = (lg y)3/3. We find that and -1 t t 2 A = N \ x.x./{x.e ) L 1,1, 1,0 C 1 = -(2N)-lE\[x./{x~e )J{[lg(x~e ) + E.J2_[lg{x~e )]2} L 1, 1, 0 1, 0 1, 1, 0 = -002{2N)-1\ x./{x~e ), L 1, 1, 0 1 t 2 t 2 C 2 = -(2N)- ELEi{[Zg{xie) + Ei] - [Zg{x i 80)] } = _N- 1 \ lg{x~e ) 0 2 L 1, 0 0' ė -1 t 3 t 3 + (3N) E\ E.{[lg(x.e ) + E.] - [lg(x.e )] } L 1, 1,0 1, 1,0 t 2 =?/ /N \[lg{x.e)] 0 L 1, 0 Therefre, D 2 = lim N+ N- 1 I Zg(x~e)' and prvided the abve limits exists. Thus, the 1 x 1 matrix D 3-2D;V 2 is twice the 1imi t f the vari ance f lg(x;e),..., lg (x;e ), and wi 11 be nnsi ngu 1ar except in degenerate situatins. There is thus a fundamental difference between the mdels (1.4) and (1.5). A small simulatin study is utlined in Sectin 6 and helps back up Therem 1. This result can be extended t nn-nrmal errr distributins as

12 -12- well as the rbust methds f Carrll (1980) and Bickel and Dksum (1981). The details are nt instructive. ė 4: A LOWER BOUND ON THE EFFICIENCY OF THE MLE. Let e(~) and 8(~) dente the ML estimatr with A estimated and knwn respectively. Let ARE(81~82) be the asympttic relative efficiency f 8 1 t 8 2. Fr fixed 00' it is difficult t find ARE(8(A),e(~)) and, in fact, this may depend n e, A ' the {x.} and the crdinate f e being estimated. 1-- that can be said fr certain is that this ARE All is at least ne and cnverges t ne as In this sectin we will define a weighted L 1 r least abslute deviatin estimatr 8(w)and shw that ARE(8(A )' e(w)) ~ rr/2. Under reasnable regularity cnditins, thi's means that ARE(8(A ), 8(~)) is bunded between ne and rr/2~ in vivid cntrast t the Bx and Cx mdel (1.5) in which this last ARE can apprach infinity. We first lk at general weighted L 1 estimatrs. wn right. The results stated here seem t be new and are f interest in their Let w1~... ~ w N be psitive numbers and let e(l) be any pint which minimizes the expressin I w. IY -!.(8(L))\ Under (1.4),!.(e ) is the unique median f Y.~ s we can expect 8(L) t be cnsistent. The unweighted L 1 estimate fr linear mdels was studied by Ruppert and Carrll (1980). Thse results suggest that

13 -13- (4.1) 0;' LW. sign (y. - f.(8(l))s... ~ ~ ~ ~ s. = df.(e )/de. ~ ~ 0 Define r. = y. - f.(e ) and let m. be the density f r.. ~ ~ ~ 0 ~ ~ By a generalizatin f the strng law, fr example Therem 7.1 f Carrll and Ruppert (1982) which itself generalizes Lemma 4.2 f Bickel (1975), (4.2) ~ lw. {sign(y. - f.(8(l))) -sign(r.)}s. ~ ~ ~ ~ ~ -(ELw.{sign(y. - f.(e))-sign(r.)}s.)! e=8(l)). ~ ~ ~ ~ ~ Nw, as E ~ 0.. we btain that ė (4.3) E(sign(r. + E) - sign(r.)) -2Em.(O) ~. ~ ~ ~ Cmbining (4.1)-(4.3) we get t rder (N- Yz ), (4.4) Nw, since fr mdel (1.4) E. = h(f.(e )+ r... A ) - h(f.(e ), A ), ~ ~ ~ 0 ~ 0 we then have (4.5) m. (0) ~

14 -14- Thus, if we chse (4.6) w. = h (f. (e ), A ), ~ y ~ 0 0 we have by (4.4)-(4.6) and the Central Limit Therem that Nw 8(L) is nt a bna fide estimatr ~ALl N 2 (e(l) - e )/ 0 -> N(O,(n/2) S- ). 0 since w. in (4.6) requires A, e t be knwn. Hwever, if in (4.6) ne plugs in any N Yz cnsistent estimatrs A f e and A and calls the L 1 estimate based n these new weights e(w), then 0 using Therem 7.1 f Carrll and Ruppert (1982), ne can als shw that. ~ Nw, because it then fllws that ~ A L -1) N 2 (e(a )- e )/0 -> N(O, S (4.8) A A ARE(e(A ), e(w)) = n/2, ARE(8(A ), e(~)) ~ n/2. Therem 1 and the Mnte-Carl results t fllw indicate that the upper bund in (4.8) is quite cnservative. it is a bund that des nt depend n 0 0, estimatr f e prvided that Ei is nt needed. The beauty f (4.8) is that The weighted L 1 estimatr may well be useful fr example if in (1.4) ne suspected that the errrs {E.} are nt nrmal. 1- is the unique median f E It is a cnsistent 1- Symmetry f

15 -15-5: THE K-SAMPLE PROBLEM Our mdel (1.4) and Therem 1 prvide sme useful insight int the k-samp1e prblem under the frmulatin (1.5) f Bx and Cx. In their mdel, fr each f k ppulatins we have (5.l) h{y. " A ) = fl. + E.. j = 1, 1-J 0 J 1-J... ~ k; i = 1,...,N. 1- The equivalent frmulatin frm ur viewpint is (5.2) e Here ~. is the median f y. n the riginal scale and fl. is the expected J 1-J J value f y in the A scale. The results f Carrll and Ruppert imply 1-J 0 that f~ estimating the ~IS, there is little cst in nt knwing A, while fr estimating the fl's, Bickel and Dksum shw that the cst f nt knwing A can be enrmus. Since there shuld be little cst in testing fr equality f means when 1. 0 is unknwn. These heuristics are frmally prven by Carrll (1982) and Dksum and Wng (1981). 6: MONTE-CARLO. " T study 8 when N is finite and a tk a small simulatin f the mdel is nt necessarily small, we under (6.1 ) h{y.,a ) = h{ x.,a ) + a E.,

16 -16- where h( ) is the Bx and Cx pwer family (l.4b). In ur simulatins, were nrmally distributed with mean zer and variance ne and 8 1 = 7, 8 2 = 2. We cnsidered three estimatrs: 1) 2) 3) ML estimatr, A knwn (KNOWN) ML estimatr, A unknwn (MLE) The rdinary least squares estimatr (LSE) withut any transfrmatin. The median f y is x, s that LSE frms an especially plausible estimatr f the slpe 8 2 (fr which it is cnsistent). We chse three values f 0 0 : = 0.05, 0.10, and We present results in Tables 1 and 2 fr A = 0 (lg-nrmal data) and A = There were 600 replicatins f the experiment fr each (A,E ) 0 0 and each estimatr, all generated frm a cmmn set f randm numbers. The nrmal randm deviates were generated frm the IMSL ruti:ne GGNPM. N = 50, the design pints {x.} were equally spaced n [-1, 1], the errrs 'Z.- Estimatin f (8 1, 8 2 ) fr each A was dne by the IMSL rutine ZXSSQ while ZXGSN was used t estimate A. The results fr the ML estimatr with A unknwn (dented MLE) are very encuraging. The mean square errrs fr MLE are quite clse t thse fr KNOWN, the ML estimatr with A knwn, especially fr the slpe 8 2. These results agree with ur small 0 thery and indicate the minimal cst fr nt knwing A. The relative efficiencies f MLE t KNOWN are always well abve ~ the lwer bund f 2/n. T appreciate hw well MLE des relative t KNOWN

17 -17~ (line 2 f Tables 1 and 2), it is enlightening t study Table 5 f Bickel and Dksum (1981); in their mdel which we call (1.5), they have ratis MLE(A estimated)/known(a knwn) always at least 1.5 and as large as 211, 0 while urs never exceed 1.2. The ther valuable pint learned frm Table 2 is that when estimating the slpe 8 2, the ML estimatr MLE with A unknwn tends t dminate the LSE, especially fr larger values f a. In ther wrds, fr ur mdel (1.4), there is real value t transfrmatin when it is apprpriate. ė

18 ..18- REF ERE N C E S BICKEL, PETER J. (1975). One step Huber estimates in the linear mdel. J. Amer. Statist. Assc.?O~ BICKEL, P.J., and DOKSUM, K.A. (1981). An analysis f transfrmatins revisited. J. Am. Statist. Assc.?6~ BOX, GEORGE E.P. and COX, DAVID R. (1964). An analysis f transfrmatins. J. Ry. Statist. Sc. Sere B 26~ BOX, G.E.P., and TIAO, G.C. (1973). Bayesian Inference in Statistical Analysis. Reading, Mass. Addisn-Wesley. CARROLL, R.J. (1980). A rbust methd fr testing transfrmatins t achieve apprximate nrmality. J. Ry. Statist. Sc. Series B 42~ CARROLL, R.J. (1982). Tests fr regressin parameters in pwer transfrmatin mdels. Tentatively accepted by the Scand. J. Statist. CARROLL, R.J., and RUPPERT, D. (1981). Predictin and the pwer transfrmatin family. Bi,metrika 68~ CARROLL, R.J. and RUPPERT, DAVID (1982). Rbust estimatin in heterscedastic linear mdels. T appear in Ann. Statist. DOKSUM, K.A., and WONG, C.W. (1981). Statistical tests after transf~atins. Manuscript. HINKLEY, D.V., and RUNGER, G. (1981). Analysis f transfrmed data. T appear in J. Am. Statist. Assc. HOADLEY, B.A. (1971). Asympttic prperties f maximum likelihd estimatrs fr the independent nt identically distributed case. Ann. Math. Statist. 42~ RUPPERT, D. and CARROLL, R.J. (1980). Trimmed least squares estimatin in the linear mdel. J. Am. Statist. Assc.?5~ SPITZER, J.J. (1978). A Mnte-Carl investigatin f the Bx-Cx transfrmatin in small samples. J. Am. statist. Assc.?3~

19 TABLE #1,. Results f the Mnte-Carl study described in the text. These results are fr the INTERCEPT. The median respnse is linear with intercept = 7 and slpe = 2. KNOWN ML estimate with A knwn. MLE = ML estimate with A unknwn. LSE = rdinary least squares estimate cr BIAS OF KNOWN MSE OF KNmJN BIAS OF MLE MSE MSE OF MLE OF KNOWN MSE OF MLE MSE OF KNOWN S.E. OF ABOVE DIFF BIAS OF LSE MSE MSE OF MLE OF LSE t1se OF MLE MSE OF LSE S.E. OF ABOVE DIFF In these calculatins, the mean square errr (MSE) and S.E. f difference terms are multiplied by T**2. Here T = 10 if cr ~ 0.10, T = 1 if cr = 0.50.

20 TABLE #2 Results f the Mnte-Carl study described in the text. These results are fr the SLOPE. The median respnse is linear with intercept = 7 and slpe = 2. KNOWN = ML estimate with A knwn. MLE ML estimate with A unknwn. LSE = rdinary least squares estimate. A a BIAS OF KNOWN MSE OF KNOWN BIAS OF MLE MSE OF MLE MSE OF KNOWN e.. MSE OF MLE - MSE OF KNOWN S.E. OF DIFF , BIAS OF LSE MSE OF MLE MSE OF LSE MSE OF MLE - MSE OF LSE S.E. OF DIFF ~ In these calculatins, the mean square errrs (MSE) and S.E. f di fference terms are multiplied by T**2. Here, T = 10 if a s 0.10, T = 1 if e a = 0.50.

Bootstrap Method > # Purpose: understand how bootstrap method works > obs=c(11.96, 5.03, 67.40, 16.07, 31.50, 7.73, 11.10, 22.38) > n=length(obs) >

Bootstrap Method > # Purpose: understand how bootstrap method works > obs=c(11.96, 5.03, 67.40, 16.07, 31.50, 7.73, 11.10, 22.38) > n=length(obs) > Btstrap Methd > # Purpse: understand hw btstrap methd wrks > bs=c(11.96, 5.03, 67.40, 16.07, 31.50, 7.73, 11.10, 22.38) > n=length(bs) > mean(bs) [1] 21.64625 > # estimate f lambda > lambda = 1/mean(bs);

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