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An Exact Algorithm for Estimating Breakpoints in Segmented Generalized Linear Models

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An Exact Algorithm for Estimating Breakpoints in Segmented Generalized Linear Models Küchenhoff: An exact algorithm for estimating breakpoints in segmented generalized linear models Sonderforschungsbereich 386, Paper 27 (1996) Online unter: http://epub.ub.uni-muenchen.de/ Projektpartner An exact algorithm for estimating breakp oints in segmented generalized linear mo dels Helmut Kuc henho University of Munich Institute of Statistics Akademiestrasse D Munc hen Summary We consider the problem of estimating the unknown breakp oints in seg mented generalized linear mo dels Exact algorithms for calculating maxi mum likeliho o d estimators are derived for dierenttyp es of mo dels After discussing the case of a GLM with a single covariate having one breakp oint a new algorithm is presented when further covariates are included in the mo del The essential idea of this approach is then used for the case of more than one breakp oint As further extension an algorithm for the situation of two regressors eachhaving a breakp oint is prop osed These techniques are applied for analysing the data of the Munichrental table It can b e seen that these algorithms are easy to handle without to o much computational eort The algorithms are available as GAUSSprograms Keywords Breakp oint generalized linear mo del segmented regression Intro duction In many practical regressiontyp e problems we cannot t one uniform re gression function to the data since the functional relationship b etween the resp onse Y and the regressor X changes at certain p oints of the domain of X These p oints are usually called breakp oints or changep oints One imp or tant example is the threshold mo del used in epidemiology see Ulm Kuchenho and Carroll where the covariate X typically an exp osure has no inuence on Y eg the o ccurrence of a certain disease up to a cer tain level Thus the relationship b etween X and Y is describ ed by a constant up to this level and for values of X greater than this level it is given byan increasing function In such situations we apply segmented or multiphase regression mo d els which are obtained by a piecewise denition of the regression function E Y jX xonintervals of the domain of X Anoverview concerning this topic can b e found in Chapter of Seb er and Wild Assuming a gen eralized linear mo del and a known numb er of segments wehave G x if x G x if x E Y jX x G x if x K K K and y b cy f y j exp Here is the nuisanceparameter and b E Y jX x see Fahrmeir and Tutz Seb er and Wild G is the linkfunction eg logistic identityetcand f denotes the density function of Y given X xFor the threshold mo del mentioned ab ove for instance there are two segments where G is the logistic link and The endp oints of the intervals denote the i breakp oints Since they are typically unknown they have to b e estimated For theoretical but also practical reasons the breakp oints are assumed to ly between the smallest and the largest sample value x i n i We further assume that the regression function is continuous ie i K i i i i i i Thus the mo del can b e stated in another parameterisation K X E Y jX x G x x i i i t if t where t if t From this representation it can b e seen that is a usual generalized linear mo del if the breakp oints are known Therefore the MLestimation can i be performed by a gridsearchtyp e algorithm in case of two segments see Stasinop oulos and Rigby For the linear mo del an exact algorithm for the least squares estimator was given by Hudson see also Schulze or Hawkins In Section it is shown that this algorithm also works for the GLM with one breakp oint In Section the algorithm is extended to mo dels with further covariates In Section the ideas of Section are used to derive the algorithm for fairly general mo dels with more than one breakp oint and more than one co variate with breakp oints Giving an algorithm for such general mo dels we ll a gap existing so far in the literature In Section an example is considered Weinvestigate the relationship b etween the net rent of ats in Munich and the at size as well as the age of the ats based on data of the Munichrental table Finally problems concerning the computing time are discussed and some interesting additional asp ects are p ointed out Exact MLestimation for mo dels with one breakp oint We consider a GLM with one breakp oint and density The regression function can then b e written as E Y jX xG x x witht t Here is the slop e parameter of the rst segment and is the slop e in segment The loglikeliho o d function of one observation conditioned on X isgiven by y b G y x x cy where the nuisance parameter is assumed to b e constantover the segments If G is the natural link function then x x Having iid observations x y the loglikeliho od function to i i in b e maximized in is n X G y x x i i i i Since this function is not dierentiable in at x we rst calculate the prole i likeliho o d That is we maximize with resp ect to all other parameters and get n X P max G y x x i i i i Obviously mo del is a GLM for xed Thus the calculation of the prole likeliho o d corresp onds to the MLestimation of a GLM Since is continuous in it can b e maximized by a grid search see Ulm For a GLIMmacro see Stasinop oulos and Rigby Though these algorithms give reliable results if the grid is appropriately chosen it would b e desirable to have an exact algorithm at ones disp osal We derivesuch an exact algorithm for maximizing following the ideas of Hudson Since wehave assumed that the nuisance parameter is constant for the two segments it can b e neglected in maximizing Let the observations x y b e ordered with resp ect to x suchthatx x for i i in i i j i j The loglikeliho o d is dierentiable with resp ect to everywhere except for those values of with x for one ifng i Therefore the algorithm has to b e divided into roughly two steps according to the dierentiability of the loglikeliho o d In the rst step the p oints of dierentiability ie the case x x k k for some k are considered Denoting the partial derivativeofG with resp ect to the second argumentby G we therefore get n n X X C B x i C B G y glp G y glp i i i i A x i i i I I fx g fx g i i where glp is the broken linear predictor glp x x i i i and I denotes the indicator function Let b e a zero of with x x and The k k system of equations obtained by equating to results after some algebra in k X G y x i i i k X G y x x i i i i n X G y x i i ik n X G y x x i i i ik with j j j From equations and and we conclude that and resp ectively are MLsolutions of the regressions in the two seg ments Since is uniquely determined by the continuity condition p ossible zeros with x x can b e determined by estimating the parameters sep k k and x y arately in the two segments based on x y k i i ik n i i i which yields and resp ectively The estimator for is then obtained from If x x it is a zero of In this case the estimator is given by k k If x x we deduce from that there is no lo cal maximum k k with x x k k The ab ovementioned pro cedure is p erformed for the nite number of intervals x x where these intervals havetobechosen such that the k k MLestimators exist in the corresp onding segments In the second step we calculate the prole likeliho o d P x i n i obtaining the maximum of the loglikeliho o d at all p oints of nondieren tiability Finally the global maximum of the loglikeliho o d is given bythe maximum of this nite n umb er of lo cal maxima Conducting this algorithm the estimation of at most m m GLMs is needed if there are m observations with dierentvalues of x Mo dels with covariates In many practical situations there will b e further covariates in the regression mo del which leads to the following extension of mo del E Y jX x Z z G x x z where Z is the vector of covariates with vector of parameters As in Sec tion x y z denote the corresp onding observations with x x i i i in i j for ij Toderive the MLestimator we rst consider again the case x x k k Then the derivative of the loglikeliho od with resp ect to is C B n n x i X X C B C B G y glp G y glp x i i i i i C B A i i I I fx g fx g i i z i with glp x x z i i i i Analogously to we get the following system of equations with denoting a zero of k X G y x z i i i i k X G y x z x i i i i i n X G y x z i i i ik n X G y x z x i i i i ik k n X X G y x z z G y x z z i i i i i i i i i ik with j j j Equations corresp ond to a generalized linear mo del with an analysis of covariancetyp e design matrix x z B C B C B C B C x z k k B C D k B C x z k k B C B C A x z n n Therefore we obtain zeros of by tting a generalized linear mo del with design matrix D which again yields b ecause of the continuity condition k If x x wehave found a lo cal maximum otherwise there is no max k k imum with x x k k The remaining part of the algorithm is now completely analogous to that presented in Section Further extensions Mo dels with more than twosegments Let us now consider the case of K segments ie Mo del In practical
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