RESIDUALS AND INFLUENCE IN NONLINEAR REGRESSION FOR REPEATED MEASUREMENT DATA

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1 Operatons Research and Applcatons : An Internatonal Journal (ORAJ), Vol.4, No.3/4, November 17 RESIDUALS AND INFLUENCE IN NONLINEAR REGRESSION FOR REPEAED MEASUREMEN DAA Munsr Al, Yu Feng, Al choo, Zamr Al Nanjng Unversty of Scence and echnology, P.R. Chna ABSRAC All observatons don t have equal sgnfcance n regresson analyss. Dagnostcs of observatons s an mportant aspect of model buldng. In ths paper, we use dagnostcs method to detect resduals and nfluental ponts n nonlnear regresson for repeated measurement data. Cook dstance and Gauss newton method have been proposed to dentfy the outlers n nonlnear regresson analyss and parameter estmaton. Most of these technques based on graphcal representatons of resduals, hat matrx and case deleton measures. he results show us detecton of sngle and multple outlers cases n repeated measurement data. We use these technques to explore performance of resduals and nfluence n nonlnear regresson model. KEY WORDS: Hat matrx, Cook dstance, Resduals, Nonlnear regresson models. Mathematcs Subject Classfcaton: 6J,6J, 6G5,6J5,6J INRODUCION Data contanng of repeated measurements hold on each of number of ndvduals appear frequently n bomedcal and bologcal mplementatons. hs knd of modelng data generally mples characterzaton of the relatonshp among the measured response of y, measurement factor, or covarate x [11]. In many mplementatons, the relatonshp between y and x s nonlnear n unknown parameters of attenton. he expresson of repeated measurement on an ndvdual requres defnte care n markng the random varaton n the data. It s mportant to recognze random varaton among measurements wthn a gven ndvdual and random varaton among the ndvduals. Inferental methods assst these dfferent varance components n the framework of a proper herarchcal statstcal model. When the relatonshp between x and y n the unknown parameters s lnear, the framework s that of the classcal lnear mxed effects model [1]. In ths case, Bayesan nferental method s provded satsfactory herarchcal lnear model [14]. here s a substantal lterature about herarchcal lnear model, McCulloch, Casella, and Searle (199). Lnear modelng methods for repeated measurement data are qute advanced and developed, and well recorded n statstcal lterature, Crowder and Hand (199), Lndsey(1993), and Dggle, Lang and Zenger(1994). In ths partcular work, we am to ndcate resduals data ponts n nonlnear regresson for repeated measurement data and parameter estmaton. We use Cook.dstacne and Gauss newton method, and we also explore some useful examples for parameter estmaton and Outlers detecton. he organzaton of ths paper s gven as; n secton, we gve some models and parameter estmaton; secton 3 deals wth the dagnostcs methods n case of sngle and multple Outlers detecton by DOI: 1.511/oraj

2 Operatons Research and Applcatons : An Internatonal Journal (ORAJ), Vol.4, No.3/4, November 17 scatterplots and parameter estmaton wth some applcable examples whle secton 4 concludes the paper.. HE MODEL AND HE PARAMEER ESIMAION We ntroduce herarchal nonlnear model that forms the fundamental nferental methods and dscuss the avalable technques for the analyss of repeated measurement data. In the lnear case, ntra and nter ndvdual varaton can assst wthn the two stages model. he frst stage characterzes by a nonlnear regresson model wth a model for ndvdual covarance structure, and nters ndvdual varablty represent n the second stage through ndvdual specfc regresson parameters. Let yj denote the jth response, condtons sum up by the vector of covarates x j, so that a sum of been observed. he vector x j ncludes varables. Suppose that, for ndvdual, the jth response obey the model. j = 1,.., n for th ndvdual, = 1,.., m, taken at a set of y = f ( x, ) + e (1) j j j m N Σ = 1 = n response have Where ej s a random errorexpresson consderng unrelablty n the response, gven the th ndvdual, wth E( e ) = Gettng the response and errors for the th ndvdual nto the ( n 1) vectors ( n 1) vector. j y = where E( e ) =. ' [ y1,..., yn ], and e = ' [ e1,..., en ] y = f ( x, ) + e, (), respectvely, and nterpretng the he model gven n (1) and () descrbes the organzng and random varaton assocaton wth measurement on the th ndvdual. If for nonlnear regresson ε ~ N(, ), then y on the parameter of score functon L( ) observaton nformaton matrx L( ) and fsher nformaton matrx I( ) respectvely. Computatonal of nonlnear least square estmates need to use the teratve numercal algorthm. ( θ L ) =, we may use aylor expanson at pont θ L( θ ) = L( θ ) + L( θ )( θ θ ) + o θ θ = θ = θ + [ L( θ )] L( θ ), = 1,,... (3) 1 Untl θ + θ < δ, δ s an advance fxed value. Gauss newton method has some mportant propertes.

3 Operatons Research and Applcatons : An Internatonal Journal (ORAJ), Vol.4, No.3/4, November SAISICAL DIAGNOSICS FOR NONLINEAR MODELS WIH REPEAED MEASUREMEN DAA In statstcs, Cook's dstance s an often used to estmate the nfluental ponts of a data [1].Data ponts wth huge resduals (outlers) and/or hgh leverage may msrepresent the outcome and accuracy of a regresson. f ( x, ) WhereU = ( j) U U ( j ) ( Dj = Dj ( U U, p σ ) = ) ( )( ) p σ (4), Cook dstance gves squared dstance from to relatve to the fxed geometry ofu U. he values of D ( U U, p σ ) can be converted to a famlar probablty scale by comparng calculated values to the F( p, n p ) dstrbuton. ( ) Cook dstance n multple cases: D = D ( U U, p σ ) = ( ) U U ( ) ( )( )( ) (5) D Can be expressed n multdmensonal analogues of the r, and v. he results are obtaned by frst expressng as a functon of : ( ) 1 ( ) = ( U ( ) U ( ) ) U ( ) Y( ) p σ = (6) 1 ( U U U ) ( U X Y X Y ) he nverse of (6) ( ) [( ) ( ) ( ) ( = U U + U U U I V U U U ) ][ U Y X Y ] = ( U U ) U [ ( I V ) X + ( I + ( I V ) V ) Y ] (7) Substtutng nto (6) ahead to the form: 1 1 U U U I V e ( ) = ( ) ( ) (8) Sngle case Cook dstance: 1 1 e ( I V ) V ( I V ) e D = (9) p σ I 1 ( j) I( j) Lj = + [ ( )] ( ) (1) 1 In ths case, I( ) = U Σ U, and 1 L( ) = U Σ Ue 3

4 Operatons Research and Applcatons : An Internatonal Journal (ORAJ), Vol.4, No.3/4, November 17 Replacng nto (4), we get the form Multple cases Cook dstance Substtutng nto (6), ths form gets D = [ U Σ U ] U Σ e (11) j j j j j j j 1 ( ) = + I( ) L( ) [ ( )] ( ) D = [ U Σ U ] U Σ U e (1) ( ) ( ) ( ) ( ) ( ) ( ) Example 1: We observe the data n table I that taken from a study reported by Kwan et al. (1876) of the pharmacoknetcs of ndomethacn followng bolus ntravenous njecton of the same dose n sx human volunteers, for each subject plasma concentratons of ndomethacn were measured at 11 tmes ntervals regardng from 15 to 8 hours post-njecton[11]. able. : plasma concentratons ( µ g / ml) followng ntravenous njecton of ndomethacn for sx human We consder two examples to calculate Gauss newton method We examne Gauss Newton method y x x = 1 exp( ) + 3 exp( 4 ), 1 4,..., >, (13) = + [ U Σ U] U Σ e Usng MALAB s conventon for representng Jacobn matrx U whch s equal to where = In a known case, and e = y f ( ), f ( ) U = We chose ntal values of, = [.7,.6,.54,.5], after 5 teratons we obtaned = [.75,.65,.5,.45].whch s satsfed under condton <

5 Operatons Research and Applcatons : An Internatonal Journal (ORAJ), Vol.4, No.3/4, November 17 Example : We consder another example to compute Gauss newton method. he result of estmaton of the parameters n based on 11 responses for the ffth subjects are gven n able I. Usng Matlab to calculate G-N method and get parameter estmatons. We choose ntal values, = [1.,1.,-1.1,-1.], then use Gauss newton method to estmate the values of. After 5 teratons, we obtaned ˆ = [1.715, 1.48, -1.37, ], and we satsfed under ths condton < Example.3. We consder able I, we focus on ffth subject to detect sngle case outler. Where = In and e s unobservable error y f ( ). U f ( ) =, Fg.1. Scatter plot for the table I (ffth ndvdual) under model (11). In the above scatterplot, we obtaned cook s dstance and found outler n a set of predcted values. Frst observaton of our data set s an outler whch s ndcated n (fgure.1). 5

6 Operatons Research and Applcatons : An Internatonal Journal (ORAJ), Vol.4, No.3/4, November 17 Example.4 We consder another example to detect multple outlers cases. Fgure.. Scatter plot for the table I under model (1). We obtaned cooks. Dstance and found four values that fall far from other data ponts. So we consder these (3, 56, 45, 1) ponts outlers n 66 observatons data. he outlers are desgnated n (fgure.) cook s dstance plot. 4. CONCLUSION: It s well understood that all observatons of a data set don t play the same role n the result of regresson analyss. For example, the character of the regresson lne maybe determne by only a few observatons, whle most of the data s somewhat gnored. Such observatons that hghly nfluence the results of the analyss are called nfluental observatons.it s mportant, for many causes, to be able to detect nfluental observatons. In ths paper, we establshed Gauss newton method for parameter estmaton and as well we extended rebut verson of Cook. Dstance n sngle and multple cases to detect outlers data ponts for repeated measurement data. REFERENCES: [1] Aynde, K., Lukman, A.F. and Arowolo, O. (15) Robust Regresson Dagnostcs of Influental Observatons n Lnear Regresson Model. Open Journal of Statstcs,vol.5, pp [] Altman, N. & Krzywnsk, M.(16) Analyzng outlers nfluental or nusance..nature methods, vol.13, pp81-8. [3] Law, M. & Jackson, D. (17) Resdual plot for lnear models wth censored outcome data: A refned method for vsualzng resdual uncertanty. Communcaton n statstcs smulaton and computaton,vol. 46, pp [4] Cook, R.D and sa, C.L. (1985) Resdual n nonlnear regresson, Bometrka, vol. 7, No.1, pp3-9. 6

7 Operatons Research and Applcatons : An Internatonal Journal (ORAJ), Vol.4, No.3/4, November 17 [5] Cook R.D. (1979) Influence observatons n lnear regresson, J.Amer.statst.Assoc, vol.74, pp [6] Cook R.D, and presscot. (1981) Approxmaton sgnfcance levels for detectng outler n lnear regresson, echnometrcs, vol.3,pp [7] Ellenberg, J.H. (1976) estng of a sngle outler from a general regresson model, Bometrcs, vol. 3, pp [8] Vonesh, E.F. (199) Nonlnear models for the analyss of longtudnal data, Statstcs n medcne, vol. 11, pp [9] Solomon P.J. and cox D.R. (199) Nonlnear components for varance models, Bometrkka,vol. 79, pp1-11. [1] Cook R.D. (1979) Influence observaton n lner regresson, J.Am.statst.assoc,vol. 74, pp [11] Dggle, P. J. (1988) An approach to the analyss of repeated measurements, Bometrcs, vol. 44, pp [1] PREGIBON, D. (1981) Logstc regresson dagnostcs, Annual of statstcs, vol.9, pp [13] Anscombe, F.J. (1961) Examnaton of resduals, Proc.fouth Berkeley symp vol. 1, pp1-36. [14] MARIE DAIDIAN and DAVID M.GILINAN.march. (1995) Nonlnear models for repeated measurement data. AUHOR Munsr Al, school of scence, department of statstcs Nanjng Unversty of scence and technology, P.R chna. 7