This paper discusses the nonparametric bootstrap method for evaluating the

standard errors of the parameter estimates of orthogonal regression. The

percentile, bias-corrected, the bias-corrected and accelerated (BCA) , and the calibrated or

iterated BCA method were considered for confidence intervals for the parameters

of the model. Based on simulation studies, it was found that the iterated

BCA method produced a more reliable confidence interval than the other

methods

Abdullah, Mokhtar

Universiti Putra Malaysia Institutional Repository

Pertanika J. Sci. & Techno!. 3(2): 349-359 (1995)ISSN:0128-7680© Penerbit Universiti Pertanian MalaysiaOn Bootstrap Methods in OrthogonalRegression ModelMokhtar bin AbdullahDepartment of StatisticsFaculty of Mathematical SciencesUniversiti Kebangsaan Malaysia43600 Bangi, Selangor, MalaysiaReceived 30 June 1994ABSTRAKKertas ini membincangkan kaedah butstrap tak berparameter bagi menentukanralat piawai anggaran parameter model regresi ortogon. Kaedah butstrappersentil, persentil pincang dibetulkan, persentil pincang dibetulkan secara pantas(BCA) dan BCA terlelar digunakan bagi membina selang keyakinan bagiparameter model tersebut. Daripada kajian simulasi yang dijalankan didapatiselang keyakinan berdasarkan kaedah BCA terlelar mememuhi ciri-eiri selangkeayakinan yang dikehendaki.ABSTRACTThis paper discusses the nonparametric bootstrap method for evaluating thestandard errors of the parameter estimates of orthogonal regression. Thepercentile, bias-corrected, the bias-corrected and accelerated (BCA) , and the calibrated oriterated BCA method were considered for confidence intervals for the param-eters of the model. Based on simulation studies, it was found that the iteratedBCA method produced a more reliable confidence interval than the othermethods.Keywords: bootstrap method, orthogonal regression, bootstrap confidence intervalsINTRODUCTIONConsider a pair of variables (~, 7J) satisfying a linear relationship 7J = a + f3~ ,with a and {3 to be estimated; (~, 7J) cannot be observed directly. Instead,(~,7J) are both observed with errors, i.e., we observe the pair (xi' y) where(1)Yi = 7J; + c;with errors 0i and c;, respectively. The g;} are (fixed) mathematicalvariables. The model (1) is known as a linear functional relationship (LFR).We shall consider the special case in which the variance ratio A= V(c; ) / V(0;) = l.With this specification, the LFR in (1) is better known as the orthogonalregression model.(2) (i)Mokhtar bin AbdullahTwo approaches to the estimation of the parameters have been dis-cussed in the literature. In the maximum likelihood estimation (MLE) (Kendalland Stuart 1973: Chapter 29; Fuller 1987) it is assumed that the observa-tions (Xi' y) are independent N(gj,o-2) and N( a+ f3~i,(J2) variates, respec-tively, for i=l, ... ,n. A more general formulation for estimating the param-eters in (1) is the generalized least squares estimation (GLSE) (Sprent 1966).In the GLSE approach no assumptions are made about the distributionsof the observations. The GLSE approach chooses the estimators of a and{3 which minimizeBoth the MLE and GLSE estimations yield identical estimators of a and {3, i.e.,h (Syy - Sxx)+ ~( Syy - Sxx)2 + 4S;yf3 = -------'------2SXYwhere- -1...,x= n ~Xi - -I...,,y= n ~Yi(2) (ii)Patefield (1977) derives the asymptotic variance-covariance matrix ofthe maximum likelihood estimators of a and {3. When cr is unknown, aconsistent estimator of the variance-eovariance matrix is( A)-2Aj S [X(l+i)+SxY 1~-X(l+i)]1+ p (j p n xy -x(l+i) (I+i)where(3)and - 2 2na-2(J =---(n- 2)is the consistent estimator of (J2 and a-2 is the maximum likelihood estima-tor of (J2.Based on the normality assumptions Kendall and Stuart (1973) con-structed a 100(1- y) % confidence interval for {3. The confidence limits for{3 are given by350 Pertanika J. Sci. & Techno!. Vo!. 3 No.2, 1995On Bootstrap Methods in Orthogonal Regression Model(4)where tn- 2,r/2 is the (1--y / 2) percentile point of the t distribution with n-2degrees of freedom,Our interest is to examine an alternative approach to the parametricconfidence interval in (4) which does not rely on the normality assump-tion, This paper discusses, via a simulation study, the use of thenonparametric bootstrap method to assess the standard error and confi-dence intervals for the parameters of the model. The use of thenonparametric bootstrap method is justified since the estimators in (2) (i)-(ii) can be considered as being derived from a general formulation whichmakes no normality assumptions about the observations.BOOTSTRAPPING THE ORTHOGONAL REGRESSIONLet the model (1) be written in the formwhereLet F denote the common distribution function of the Zi and the param-eter vector e = (a, (3)T. As shown in the previous section, existing methodsfor estimating the statistical accuracy of the estimators are largely asymp-totic, and may not apply in finite samples. The bootstrap method, how-ever, may overcome this difficulty as it automatically produces accuracy ofthe estimates and it can be applied in a wide range of situations.BOOTSTRAP STANDARD ERRORSThe bootstrap method advocated by Efron (1979) works by sampling fromthe empirical distribution function of F, denoted by Fn, and then estimatingthe parameter e(F) by e(F). The sampling distribution of e(F) is esti-mated by simulating that of e(Fn)' This is done by repeatedly drawing'resamples' from the original sample 'with replacement' and for eachresample calculating a value of e(Fn) .Pertanika J. Sci. & Technol. Vol. 3 No.2, 1995 351Mokhtar bin AbdullahSuppose we are interested in obtaining a bootstrap distribution ofA ( ")T "() = a,f3 where a and 13 are given by (2)(i)-(ii), respectively. We mayproceed by calculating Monte Carlo approximations baseq on tqeTcom-plete observation vector z;, The bootstrap distribu tion of () = ( a,f3) maybe approximated by drawing B samples of size n fromFn: mass lin at Zi = C:) i= 1,... , n, each time creating pseudo-data setz; =(~D from which i/ =(aO,~ot is calculated from(S Ob SOb) (S*b _S*b)2 + 4S*b213"°b = yy - xx + yy xx xy2S*b ,b = I,K ,Bxy"*b -*b f30b-*ba = y - x(2) (i)(2) (ii)where_Ob -1" *bX = n L.J X j -*b -1" *b,y = n L.J YjandS;; = I (Xj*b - XOb )2 ,S;~ = I (y;b - yObr ,S;; = I (X;b _X*b)( y;b _yOb).After drawing B bootstrap samples, we use the resulting bootstrap estimatesi/ = (a 0, ~*t to calculate the standard errors of the estimates e= (a.pr. i.e.r- 211/2I -Ob -s.e ( e) = (() - e) where~ (B-1)NONPARAMETRlC BOOTSTRAP CONFIDENCE INTERVALSIn a series of papers, Efron (1979, 1982, 1987) and Efron and Tibshirani(1993) have developed procedures for constructing approximate confi-dence intervals for a statistic of interest. The procedures rely on estimatingthe sampling distribu tion of a statistic or an approximate pivot. We shallconsider four popular methods, namely the percentile, the bias-corrected(BC) percentile, the bias-corrected and accelerated (RCA), and the iter-ated BCA methods.352 Pertanika J. Sci. & Technol. Vol. 3 No.2, 1995On Bootstrap Methods in Orthogonal Regression ModelThe percentile method takes the interval 100')' and 100(1-')') percen-.* (. * .*)T .tiles of the bootstrap distribu tion of e = ex , f3 , G< s). Ifwhere Pr* indicates probabili ty compu ted according to the bootstrapdistribution of f/ =(cX',P'r, then 100(1-2)')% approximate interval for6=(a,I3)T is[8-1(y), 8-1(1- y)]The bias-corrected (BC) method is given by(5 )(6)where Zo is the bias correction factor and both z and Zo are standardnormal distribution functions. If zo=O, then the BC method reduces to thepercentile method. The disadvantage of both the percentile and the BCmethods is that they have less satisfactory coverage properties.An improved version of the percentile and the Be methods is the bias-corrected and accelerated (BCA) method. The BCA has better coverageproperties because it is second-order accurate. This means that for a central(1-2)') confidence interval (BL,BU) its errors in matching the probability aof not coverinpth~ true value of efrom above (i.e., pr{e> au} =a) or frombelow (i.e., Prt e> eL } = 0:1 go to zero at rate 1In, for a sample of size n.The percentile and the {)C methods are only first-order accurate becausetheir errors in matching a go to zero at a slower rate, i.e., 1I yn (Efronand Tibshirani 1993: 187). The BCA intervals are transformation respecting,meaning that the BCA endpoints transform correctly if a parameter ofinterest e is changed to some function of e.In the BCA method the percentiles of the bootstrap distribution arealso used to form the endpoints of the intervals. However, the percentilesused are now determined by the bias-correction and acceleration factors.Let a denote the acceleration factor, then the BCA interval with (1-2)')coverage is given byPertanika J. Sci. & Techno!. Vo!. 3 No.2, 1995(7)353Mokhlar bin AbdullahwhereYt = ¢-l {zo + (zo + z(r)) /(1- a(Zo + z (r))))Y21 =¢-l{zo+(zo+z(l-r ));(I-a(zo =z(l-r )))}zo= ¢-l{(no. of (}*s < e)/ B}11 ~ ~ 3 {II ~ ~ 2}3/2a = i~(8(.) - 8_i) / 6 i~(80 - B_i ) ,eO = n-1 IB_iwhere <t> (.) is the standard normal cumulative distribuLion function and (}-i isthe estimate with the i-th observaLion deleted.As pointed out by Efron and Tibshirani (1993), the actual coverage ofa bootstrap confidence procedure is rarely equal to the desired (nominal)coverage and is often substantially different. One way to achieve thecoverage is by use of calibration. The idea of calibration of the bootstrapwas first discussed by I-Iall (1986, 1987) and Loh (1987,1991). Booth andHall (1993) discussed the calibraled confidence interval which is alsoknown as the iteraled confidence interval in the context of function errors-in-variables model.The iterated or calibrated confidenCe interval can be constructed asfollows; Compute A.-level confidence points(8)for a grid of values of A.. For example, these might be the normalconfidence points·(I_.l.)s.e(e*)] ,b=I,K ,8For each A. computeP(A) ={no. of e::; 9~(b)11 Band354 Penanika J. Sci. & Techno!. Vo!. 3 No.2, 1995On Bootstrap Methods in Orthogonal Regression ModelFind the value of -y that satisfiesp{A) =P{1- A) =a / 2The calibration process can be applied to any bootstrap method. Inthis paper we consider the calibration on the RCA method. With thecalibrated BCA method the resulting confidence interval has the desiredproperties, i.e., it is second-order accurate, transformation-respecting andalso has the correct nominal coverage.EXAMPLEThe data are from Miller (1980) and have been analysed by Kelly (1984)using errors-in-variables model. They consist of simultaneous pairs ofmeasurements of serum kanamycin levels in blood samples drawn from 20babies. A heelstick method on umbilical catherer was used to measure thelevels. It was reasoned that the assumption -Y= 1 was correct, A scatterplotof these twenty pairs of observations is illustrated in Fig. 1.2o..,'"NoN15 20heelstick25 30Fig. 1. Serum kanamycin levels (calherer us heelslick)Penanika J. Sci. & Techno!. Vo!. 3 No.2. 1995 355Mokhtar bin AbdullahThe estimates based on (2) (i)-(ii) werea== -1.16, f3 == 1.07The consistent estimate of cr is 0- 2 == 1.53. The standard error of theestimated slope and the corresponding 90% confidence interval based onthe exact normal (asymptotic) theory and the bootstrap methods arepresented in Table 1. In bootstrap methods we used B == 1000.Table 1 displays summary statistics pertaining to the bootstrap analy-ses. The bootstrap standard error is relatively close to that of the normaltheory. Two properties associated with aU the confidence intervals aretheir lengths and shapes, respectively (Efron 1987). "Shape" measures theasymmetry of the interval about the point estimate. An interval is said tobe symmetrical if shape == 1. Shape> 1 indicates asymmetry with greaterdistance from the upper limit to the point estimate than from the pointestimate to the lower limit. The bias-corrected interval is shorter than theother intervals and the iterated nCA produces the longest confidenceinterval among those considered. AU the confidence intervals, except theiterated BCA, indicate sOll}e degree of ~symmetry with greater distancesfrom the upper limits to f3 than from f3 to the lower limits.TABLE IStandard errors of f3 and 90% confidence interval for 13Standard error of f3Exact (normal-theory) 0.160Bootstrap (B==1000) 0.18290% Confidence Interval for 13Lower Upper Length ShapeExact (NormaHheory) 0.811 1.418 0.607 1.347Percentile 0.797 1.420 0.623 1.281BC 0.855 1.438 0.583 1.713BCA 0.871 1.493 0.622 2.130Iterated BCA 0.710 1.493 0.783 1.178Tables 2-4 iUustrate the exact (normal-theory), the percentile, the BC,the ECA, and the iterated BCA confidence intervals for the slope parameter.The exact confidence interval is given by (4) and the booL~trap intervals aregiven by (5)-(8), respectively. Tabulated characteristics for confidence inter-vals are average values of lower and upper endpoints, lengths, shape, andalso estimated coverages of the in tervals (with nominal coverage 90%).356 Pcrtanika J. Sci. & Techno!. Vol. 3 No.2, 1995On Bootstrap Methods in Orthogonal Regression ModelIt is clear from Tables 2-4 that in most cases the percentile, the BC,and the BCA intervals suffer from moderate undercoverage when theunderlying population is non-normal. The exact confidence intervals alsosuffer from moderate undercoverage even in the case of normal popUla-tion. The strength of the iterated BCA method is that it yields confidenceintervals that have coverage equal to the desired nominal 90% coverage.However, the iterated BCA intervals tend to show some degree of asymme-try and are slightly longer than the other intervals.TABLE 2Exact and bootstrap confidence intervals, n=20Error-distr. MethodNormalD-exp.t(3)ExactPercentileBCBCAIter. BCAExactPercentileBCBCAIter. BCAExactPercentileBCBCAIter. BCALower Upper Length Shape Coverage0.914 1.108 0.194 1.102 0.850.907 1.097 0.189 1.167 0.900.903 1.090 0.187 1.034 0.900.888 1.094 0.207 J.ll6 0.910.891 1.130 0.239 1.214 0.900.918 1.112 0.194 1.102 0.920.918 1.101 0.127 1.127 0.870.916 1.098 0.182 1.109 0.840.919 1.100 0.181 1.169 0.840.902 1.136 0.234 1.197 0.900.862 1.227 0.364 1.197 0.870.853 1.213 0.360 1.202 0.870.859 1.210 0.355 1.163 0.850.853 1.210 0.357 1.216 0.880.817 1.320 0.502 1.486 0.90CONCLUSIONThe existing methods for evaluating the statistical accuracy of estimates ofthe parameters of orthogonal regression model are largely asymptotic andmay not apply in finite samples. The non parametric bootstrap method hasfacilitated the evaluations of standard errors and confidence intervals forthe parameters of the model. A limited simulation study presented in thispaper shows that the iterated RCA method, in particular, provides areliable method for constructing a non parametric confidence interval.The method produces a confidence interval that has the most desirableproperties, i.e., it is second-order accurate, transformation-respecting, andhas a correct nominal coverage.Pertanika J. Sci. & Techno!. Vo!. 3 No.2, 1995 357358Mokhtar bin AbdullahTABLE 3Exact and bootstrap confidence intervals, n=30Error-distr. Method Lower Upper Length Shape CoverageNormal Exact 0.942 1.043 0.101 1.052 0.87Percentile 0.952 1.048 0.096 0.988 0.89BC 0.951 1.047 0.096 0.973 0.91BCA 0.944 1.049 0.104 1.051 0.91Iter. BCA 0.935 1.067 0.132 1.123 0.90D-exp. Exact 0.951 1.052 0.102 1.052 0.90Percentile 0.952 1.051 0.099 1.028 0.89BC 0.948 1.048 0.100 0.951 0.85BCA 0.941 1.051 0.110 1.025 0.85Iter. BCA 0.937 1.065 0.128 1.068 0.90t(3) Exact 0.951 1.051 0.186 1.097 0.89Percentile 0.911 1.095 0.183 1.031 0.84BC 0.912 1.096 0.183 1.061 0.84BCA 0.900 1.099 0.200 1.138 0.86Iter. RCA 0.889 1.130 0.241 1.118 0.90TARLE 4Exact and bootstrap confidence intervals, n=50Error-distr. Method Lower Upper Length Shape CoverageNormal Exact 0.978 1.025 0.047 1.024 0.84Percentile 0.979 1.023 0.044 0.960 0.86BC 0.979 1.023 0.044 0.010 0.83BCA 0.968 1.024 0.063 1.050 0.83Iter. RCA 0.970 1.028 0.058 1.061 0.90D-exp. Exact. 0.978 1.024 0.046 1.023 0.94Percentile 0.979 1.022 0.043 1.142 0.88BC 0.976 1.019 0.044 0.877 0.86BCA 0.928 1.020 0.092 1.886 0.85Iter. RCA 0.974 1.032 0.057 1.186 0.90t(3) Exact 0.958 1.040 0.082 1.042 0.88Percentile 0.959 1.035 0.076 1.000 0.87BC 0.959 1.035 0.076 1.022 0.87BCA 0.951 1.037 0.085 1.114 0.90Iter. BCA 0.949 1.050 0.101 1.085 0.09Pcrtanika J. Sci. & Technol. Vol. 3 No.2, 1995On Bootstrap Methods in Orthogonal Regression ModelACKNOWLEDGEMENTThe author wishes to acknowledge the helpful and constructive commentsby the referees.REFERENCESBOOTH,j.G. and P. HALL. 1993. Bootstrap confidence regions for functional relationshipsin errors-in-variables models. Annals ojStatistics 21: 1780-1791.EFRON, B. 1979. Bootstrap methods: Another look at the jackknife. Annals oJStatistics 7: 1-26.EFRON, B. 1982. TheJackknife, the Bootstrap and Other Resamp/ing Plans. Phil: SIAM.EFRON, B. 1987. Better bootstrap confidence intervals. Journal oj the American StatisticalAssociation 82: 171-200.EFRON, B. and RJ. TlBSHlRANI. An Introduction to the Bootstrap. New York: Chapman and Hall.FULLER, WA. 1987. Measurement Error ModeL5. New York: Wiley.HALL, P. 1986. On the bootstrap and confidence in lcrvals. Annals oJStatisties 14: 1431-1452.HALL, P. 1987. On the bootstrap and likelihood-based confidence intervals. Biometrika 74:481-493.KELLY, G.E. 1984. The inOuence function in the errors in variables problems. Annals ojStatistics 12: 87-100.KENDALL, M.G. and A. STUART. 1973. The Advanced Theory oj Statistics. London: Griffin.LOH, W.-Y. 1987. Calibrating confidence coefficien L~. Journal ojAmerican Statistical Associa-tion 82: 15~162.LoH, W.-Y. 1994. Bootstrap calibration for confidence construction and selection. StatisticaSinica 82: 15~162.MILLER, R.G.Jr. 1980. Kanamycin levels in premature babies. Biostatistics Casebook, III: 127-142. Technical Report No. 57, Division of Biostatistics, Stanford University.PATEFlELD, W.M. 1977. On the information matrix in the linear functional relationshipproblem. Applied Statistics 26: 69-70.SPRENT, P. 1966. A generalised least squares approach to linear functional relations. Journaloj the Royal Statistical Society B 28: 278-297.Penanika J. Sci. & Technol. Vol. 3 No.2, 1995 359

On Bootstrap Methods in Orthogonal Regression Model

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