DIAGNOSTIC IN POISSON REGRESSION MODELS. Zakariya Y. Algamal *

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1 Electroc Joural of Appled Statstcal Aalyss EJASA (0), Electro. J. App. Stat. Aal., Vol. 5, Issue, e-issn , DOI 0.85/ v5p76 0 Uverstà del Saleto DIAGNOSTIC IN POISSON REGRESSION MODELS Zakarya Y. Algamal * () Departmet of Statstcs ad Iformatcs, Mosul uversty, Iraq Receved 08 March 0; Accepted 0 Aprl 0 Avalable ole 4 October 0 Abstract: Posso regresso model s oe of the most frequetly used statstcal methods as a stadard method of data aalyss may felds. Our focus ths paper s o the detfcato of outlers, we maly dscuss the devace ad Pearso χ as dagostc statstcs detfcato. Smulato ad real data are preseted to assess the performace of the dagostc statstcs. Keywords: Posso regresso, devace, Pearso χ, outlers.. Itroducto Posso regresso models have receved much atteto ecoometrcs ad medce lterature as model for descrbg cout data that assume teger values correspodg to the umber of evets occurrg a gve terval. The Posso regresso model s the most basc model, where the mea of the dstrbuto s a fucto of the explaatory varables. Ths model has the defg characterstc that the codtoal mea of the outcome s equal to the codtoal varace [5]. Outlers are observatos that do ot follow the statstcal dstrbuto of the majorty of the data. Outler detecto s a prmary step regresso aalyss ad has attracted eormous atteto the lterature over may years cludg Cook ad Wesberg (98), ad Rousseeuw ad Leroy (987). There are a umber of dfferet statstcs used by statstca to ordary least squares regresso. Leverage Posso regresso s assessed by the hat values h. DFBETA ad DFFIT are helpful for detectg fluece Posso regresso. DFBETA s calculated by fdg the dfferece a estmate before ad after a partcular observato s removed. The same DFFIT except the calculatg dfferece wll be predcted values. The devace plays a mportat role assessg the ft of the model ad statstcal tests for parameters the model, ad also provdes oe method for calculatg resduals that ca be used * E-mal: zak.sm_stat@yahoo.com 78

2 Algamal, Z.Y. (0). Electro. J. App. Stat. Aal., Vol. 5, Issue, for detectg outlers [4]. Gura ad Roy (008) used the devace for detectg outlers logstc regresso. Our focus ths paper s o the detfcato of outlers, we maly dscuss the devace ad Pearso χ as dagostc statstcs detfcato. The structure of the paper s the followg. We brefly preset secto the estmato of the Posso regresso parameters for both deleted ad udeleted observato. I secto 3, we troduced the devace ad Pearso χ crtera to detect the outlers. Smulato ad real data examples are covered secto 4 ad 5 respectvely. Secto 6 shows the cocluso.. Backgroud ad Notato for the Posso Regresso Models I Posso regresso model, hereafter PR, the umber of evets y has a Posso dstrbuto wth a codtoal mea that depeds o dvdual characterstcs accordg to the structural model: θ = E(y x ) = Exp(xʹ β) () x β forces the expected cout µ to be postve, whch s requred for Takg the expoetal of the Posso dstrbuto [5]. If a dscrete radom varable y follows the Posso dstrbuto, the: e p(y = y) = θ θ y! y, y = 0,,,... () I order to estmate the PR estmator, we use the maxmum lkelhood estmato. By takg the log-lkelhood wth θ = Exp(xʹ β), we get:, y,..., y β,x,x,...,x ) = logp(y = y β,x ) = log L(y (3) { Exp(xʹ β ) + y(xʹ β ) log(y! )} log L( β) = (4) = The maxmum lkelhood estmator s the defed as: ˆ ML β = argmaxβ logl( β) (5) So, the maxmzg value for β s foud by computg the frst dervatve of the (4) ad settg t equal to zero: 79

3 Dagostc Posso Regresso Models logl( β) = β [ y Exp(x ʹ β ) ] x = 0 = (6) The secod dervatves, Hessa matrx, s gve below: logl( β) = β βʹ = Exp(xʹ β)x xʹ (7) Sce the equato (6) s olear β, oe must use a teratve algorthm. A commo choce that work well s the Newto-Raphso method as: (m+) (m) (m) β = β ( ML ML H ) (m) S (8) Where S s the frst dervatve of the log-lkelhood [0],[]. To see the fluece of the deleto of the k th observato o the PR, we cosder the log-lkelhood fucto as followg: k = ) =, k { Exp(xʹ β ) + y (xʹ β ) log(y! } log L( β) (9) the = [ Exp(xʹ β ) ] x (0) =, k S y H = Exp(xʹ β ) xxʹ =, k Startg wth a tal soluto the the Newto-Raphso become: () =β (m+ ) (m) β ( ML ML H ) S () 3. Sgle Case Deleto Dagostc To show the amout of chage PR estmates that would occurred f the k th observato s deleted. Two dagostc statstcs are proposed, chage devace ad chage Pearso χ to detect the outlers. Such dagostc statstcs are oe that exame the effected of deletg sgle case o the overall summary measures of ft. Let χ p deotes the Pearso χ statstcs ad χp deotes the statstc after the case k s deleted. Usg oe-step lear approxmatos gve by 80

4 Algamal, Z.Y. (0). Electro. J. App. Stat. Aal., Vol. 5, Issue, Pregbo (98) [7], t ca be show that the decrease the value of the χ p deleto of the k th case s: Δ χ ( k) p = χp χp,k =,,3,..., statstc due to (3) The χ p s defed as [6]: χ p = ( y = Exp(xʹ β)) / var(ˆ) µ (4) Ad the χ p for the th k deleted case s: χ ( k) p = ( y =, k Exp( xʹ β)) / var( µ ˆ ) (5) The oe-step lear approxmato for chage devace whe the k th case s deleted s: ( k) Δ D = D D (6) Because the devace s used to measure the goodess of ft of a model, a substatal decrease the devace after the deleto of the k th observato s dcate that s observato s a msft. The devace of the PR model wth ad wthout the k th observato s respectvely [4]: y D = { y log( ) (y µ ˆ )} µ ˆ = (7) where µ ˆ Exp (x ˆ = ʹ β) : D ( k) y ) (y ˆ ( k) = { y log( µ )} (8) ˆ ( k) µ =, k A large value of Δ D dcates that the k th observato s a outler. 4. Smulato Study Results 8

5 Dagostc Posso Regresso Models A smulato study was coducted to vestgate the behavor of the devace ad ch-square Pearso dagostc statstcs uder varous modelg scearos. We cosdered data smulated from a PR wth sample sze ad p explaatory varables for the cases (,p)=(0,),(5,) ad (50,). The frst case represets a smple PR wth X followg uform [0,] dstrbuto ad β=(0,). The secod ad thrd case represet a multple PR wth x ad x have uform [0,] dstrbuto ad β=(0,). The percetage of cotamato was set to be 0%, 4% ad 4% respectvely order to make oe or two observatos from the respose varable sever from shft-mea outler (the 0 th observato from the frst case, the 5 th observato from the secod case, ad observatos 9 ad 48 from the thrd case). For brevty, the Δ D ad Δ χp are preseted oly summary table for the frst case, whle the rest case results are show fgure ad fgure 3. Δ D ad Δ χp statstcs for case oe. Observato Δ D Δ χ Table. The p Fgure. Δ D ad Δ χp for the frst case. 8

6 Algamal, Z.Y. (0). Electro. J. App. Stat. Aal., Vol. 5, Issue, The devace for the full model was (5.37). The 0 th observato wll be outler sce t has D ( k) Δ = 8.64 ad Δ χ p = Fgure shows the Δ D ad Δ χp for ths case. Fgure. Δ D ad Δ χp for the secod case. From fgure () we ca coclude that the observato 5 wll be outler sce ts deleto wll decrease the devace ad Pearso χ by (0.3) ad (3.4). Aga from fgure (3), we ca cosdered observatos 9 ad 48 are outlers sce they have large Δ D ad Δ χ p amog the rest of the observatos. 83

7 Dagostc Posso Regresso Models Fgure 3. Δ D ad Δ χp for the thrd case. 5. Numercal Results The performace of the delta devace ad delta Pearso χ dagostc statstcs was studed a real data example. Aderse (008) [] descrbed data for Caada Equalty, Securty, ad Commuty Survey of 000. He used oly Quebec respodets the aalyss where (=949). The respose varable s the umber of volutary assocatos to whch respodets beloged. The explaatory varables are geder (wth wome as the referece category), Caada bor (the referece category s "ot bor Caada"), ad laguage spoke the home (dvded to Eglsh, Frech, ad other, wth Frech coded as the referece category). He used Cook's dstace whch dcates that there are two observatos (3 ad 786) may be partcularly problematc. Also, he poted that the aalyss of the DFBETA dcates that the fluece of these two observatos s largely wth respect to the effect of Caada bor varable. To assess our dagostc statstcs performace, we used ths example. Table () shows the Δ D ad Δ χp oly for the two observatos (3 ad 949), where the full model devace s Fgure 4 shows the overall look about our aalyss. Table. The Δ D ad Δ χp for the real data. Observato Δ D Δ χ p

8 Algamal, Z.Y. (0). Electro. J. App. Stat. Aal., Vol. 5, Issue, Fgure 4. Δ D ad Δ χp for the real data. 6. Cocluso All of the dagostc statstcs descrbed ths paper use oe-step approxmatos to measure the effect of sgle case deleto o the Posso regresso model parameters. As we ca see from fgure that the observato 0 cosdered outler by our dagostcs statstcs ad ths correspods the fact that case 0 has already sever from shftg mea comparg wth the rest observatos. The same wth observato 5 from fgure ad observatos 9 ad 48 from fgure 3. As metoed secto 5 ad from fgure 4, the observatos 3 ad 786 cosdered outlers usg delta devace ad delta Pearso χ dagostc statstcs, ths s the same decso that made by Aderse (008) usg DFBETA. It should be oted here that the DFFIT poted out that the observato 3 has the value 0.04 whch s less tha the tradtoal cut-off value 0.9 ad that the observato 786 has the value 0.3 whch s greater tha the tradtoal cut-off value. Here we could coclude that our dagostc statstcs well doe detfyg outlers. Refereces []. Aderse, R. (008). Moder methods for robust regresso. New York: SAGE Publcato. []. Cook, R. D. ad Wesberg, S. (98). Resduals ad fluece regresso. New York: Chapma & Hall. [3]. Gura, S. ad Roy, S. S. (008). Dagostcs logstc regresso models. Joural of the Korea Statstcal Socety, 37,

9 Dagostc Posso Regresso Models [4]. Jog, P. ad Heller, G. Z. (009). Geeralzed lear models for surace data. Lodo: Cambrdge Uversty Press. [5]. Log, J. S. (997). Regresso models for categorcal ad lmted depedet varables. New York: SAGE Publcato. [6]. McCullagh, P. ad Nelder, J.A. (989). Geeralzed lear models. d ed. Lodo: Chapma & Hall. [7]. Pregbo, D. (98). Logstc regresso dagostcs. The Aals of Statstcs, 9, 4, [8]. Rousseeuw, P. J. ad Leroy, A. M. (987). Robust regresso ad outlers detecto. New York: Joh Wley. [9]. Seber, G. A. F. ad Lee, A. J. (003). Lear regresso aalyss. d ed. New Jersey: Joh Wley. [0]. Wkelma, R. (008). Ecoometrc aalyss of cout data. 5th ed. Lepzg: Sprger. []. Ya, X. ad Su, X. G. (009). Lear regresso aalyss, theory ad computg. Sgapore: World Scetfc Publshg. Ths paper s a ope access artcle dstrbuted uder the terms ad codtos of the Creatve Commos Attrbuzoe - No commercale - No opere dervate 3.0 Itala Lcese. 86