Research Article On the Performance of the Measure for Diagnosing Multiple High Leverage Collinearity-Reducing Observations

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1 Hndaw Publshng Corporaton Mathematcal Problems n Engneerng Volume 212, Artcle ID 53167, 16 pages do:1.1155/212/53167 Research Artcle On the Performance of the Measure for Dagnosng Multple Hgh Leverage Collnearty-Reducng Observatons Arezoo Bagher 1 and Habshah Md 1, 2 1 Laboratory of Computatonal Statstcs and Operatons Research, Insttute for Mathematcal Research, Unverst Putra Malaysa, 434 Serdang, Selangor, Malaysa 2 Department of Mathematcs, Faculty of Scence, Unverst Putra Malaysa, 434 Serdang, Selangor, Malaysa Correspondence should be addressed to Habshah Md, habshahmd@gmal.com Receved 2 August 212; Revsed 9 December 212; Accepted 9 December 212 Academc Edtor: Stefano Lenc Copyrght q 212 A. Bagher and H. Md. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. There s strong evdence ndcatng that the exstng measures whch are desgned to detect a sngle hgh leverage collnearty-reducng observaton are not effectve n the presence of multple hgh leverage collnearty-reducng observatons. In ths paper, we propose a cutoff pont for a newly developed hgh leverage collnearty-nfluental measure δ D and two exstng measures δ and l to dentfy hgh leverage collnearty-reducng observatons, the hgh leverage ponts whch hde multcollnearty n a data set. It s mportant to detect these observatons as they are responsble for the msleadng nferences about the fttng of the regresson model. The mert of our proposed measure and cutoff pont n detectng hgh leverage collnearty-reducng observatons s nvestgated by usng engneerng data and Monte Carlo smulatons. 1. Introducton Hgh leverage ponts are the observatons that fall far from the majorty of explanatory varables n the data set see 1 4. It s now evdent that hgh leverage pont s another prme source of multcollnearty; a near-lnear dependency of two or more explanatory varables 2. Had 5 ponted out that ths source of multcollnearty s a specal case of collnearty-nfluental observatons; the observatons whch mght nduce or dsrupt the multcollnearty pattern of a data. Hgh leverage ponts that nduce multcollnearty are referred as hgh leverage collnearty-enhancng observatons whle those that reduce multcollnearty n ther presence are called hgh leverage collnearty-reducng observatons

2 2 Mathematcal Problems n Engneerng 6 1. Collnearty-nfluental observatons are usually ponts wth hgh leverages, though all hgh leverage ponts are not necessarly collnearty-nfluental observatons 5. It s very mportant to detect collnearty-nfluental observatons because they are responsble for msleadng concluson about the fttng of a regresson model, whch gves wrong sgn problem of regresson coeffcents and produces large varances to the regresson estmates. Not many studes have been conducted n the lterature on collnearty-nfluental measures and we wll dscuss these methods n Secton 2. Nonetheless most of the exstng methods are not successful n the detecton of multple hgh leverage collnearty-nfluental observatons although ther performances are consdered good for the detecton of a sngle observaton. Moreover these measures do not have specfc cutoff ponts to ndcate the exstence of collnearty-nfluental observatons 1. These shortcomngs motvated us to propose a new detecton measure n such stuaton. Notably, the proposed measure s based on the Dagnostc Robust Generalzed Potental DRGP method developed by Habshah et al. 11 and wll be presented n Secton 3. Secton 4 exhbts the development of the collneartynfluental observatons that can be classfed as hgh leverage collnearty-enhancng or collnearty-reducng observatons. Bagher et al. 1 presented numercal examples and a smulaton study to propose a novel hgh leverage collnearty-nfluental measure and a cutoff pont for the detecton of hgh leverage collnearty-enhancng observatons. The authors also recommended cutoff ponts for collnearty-nfluental measures ntroduced by Had 5 and Sengupta and Bhmasankaram 12. It s also mportant to dentfy hgh leverage collnearty-reducng observatons. However, these observatons are more dffcult to dagnose because they hde the effect of multcollnearty n the classcal analyss. Followng Had 13, Imon 14, and Habshah et al. 11, nsecton 5, we propose a cutoff pont for Bagher s et al. 1, Had 5, and Sengupta and Bhmasankaram 12 s measures to dentfy hgh leverage collnearty-reducng observatons. A numercal example and smulaton study are performed n Sectons 6 and 7, respectvely, to evaluate the performance of our proposed measure δ D and compare ts performance wth Had 5 and Sengupta and Bhmasankaram 12 s measures δ and l. Concluson of the study wll be presented n Secton Collnearty-Influental Measures Let consder a multple lnear regresson model as follows: Y Xβ ε, 2.1 where Y s an n 1 vector of response or dependent varable, X s an n p matrx of predctors n >p, β s a p 1 vector of unknown fnte parameters to be estmated and ε s an n 1 vector of random errors. We let X j denote the jth column of the X matrx; therefore, X X 1,X 2,...,X p. Furthermore, multcollnearty s defned n terms of the lnear dependence of the columns of X. Belsley et al. 15 proposed the sngular-value decomposton of n p X matrx for dagnosng multcollnearty as follows: X UDV T, 2.2

3 Mathematcal Problems n Engneerng 3 where U s the n p matrx n whch the columns that are assocated wth the p nonzero egenvalue of X T X s n p, V the matrx of egenvectors of X T X s p p, U T U I, V T V I, andd s a p p dagonal matrx wth nonnegatve dagonal elements, k j, j 1, 2,...,p, whch s called sngular-values of X. Condton number of X matrx denoted as CN s another multcollnearty dagnostc measures whch s obtaned by frst computng the Condton CI of the X matrx and s defned as k j λ max λ j, j 1, 2,...,p, 2.3 where λ 1,λ 2,...,λ p are the sngular values of the X matrx. The CN corresponds to the largest values of k j. To make the condton ndces comparable from one data set to another, the ndependent varables should frst be scaled to have the same length. Scalng the ndependent varables prevents the egen analyss to be dependent on the varables unts of measurements. Belsley 16 stated that CN of X matrx between 1 to 3 ndcates moderate to strong multcollnearty, whle a value of more than 3 reflects severe multcollnearty. Had 5 noted that most collnearty-nfluental observatons are ponts wth hgh leverages, but not all hgh leverage ponts are collnearty-nfluental observatons. He defned a measure for the nfluence of the th row of X matrx on the condton ndex denoted as δ, δ k k, 1, 2,...,n, 2.4 k where k s computed by the egenvalue of X and when the th row of X matrx has been deleted. Due to the lack of symmetry of Had s measure, Sengupta and Bhmasankaram 12 proposed a collnearty-nfluental measure for each row of observatons, defned as l log ( k k ), 1, 2,...,n. 2.5 Unfortunately, they dd not propose practcal cutoff ponts for δ and l and only mentoned the condtons for collnearty-enhancng and collnearty-reducng observatons. To fll the gap, Bagher et al. 1 suggested a cutoff pont for δ and l for detectng collneartyenhancng observatons as cut CEO Medan θ 3MAD θ, 1, 2,...,n, 2.6 where cut CEO s the Collnearty-Influental Measure cutoff pont for the dentfcaton of collnearty-enhancng observatons whereby θ can be δ or l. θ cut CEO for θ < s an ndcator that the th observaton s a collnearty-enhancng observaton.

4 4 Mathematcal Problems n Engneerng 3. Dagnostcs Robust Generalsed Potental for Identfcaton of Hgh Leverage Ponts The th dagonal elements of the hat matrx, W X X T X 1 X T, s a tradtonally used measure for detectng hgh leverage ponts and s defned as w x T ( X T X) 1x, 1, 2,...,n. 3.1 Hoagln and Welsch 17 suggested twce-the-mean-rule 2 p 1 /n cutoff ponts for the hat matrx. Had 13 ponted out that the leverage dagnostcs may not be successful to dentfy hgh leverage ponts and ntroduced a sngle-case-deleted measure, known as potental, and s defned as p x T ( X T X ) 1x, 1, 2,...,n 3.2 or p w 1 w, 1, 2,...,n, 3.3 where X s the data matrx X wth the th row deleted. Imon 14 ponted that potentals may be very successful n the dentfcaton of a sngle hgh leverage pont, but they fal to dentfy multple hgh leverage ponts. To rectfy ths problem, Imon 14 proposed a group deleton verson of potentals GP, known as generalzed potentals. Pror to defnng the GP, Imon 14 parttoned the data nto a set of good cases remanng n the analyss and a set of bad cases deleted from the analyss whch were denoted as R and D. Nonetheless, Imon s measure has drawbacks whch are due to the neffcent procedure that he used for the determnaton of the ntal deleton set D. To overcome ths shortcomng, Habshah et al. 11 proposed the dagnostc robust generalzed potental DRGP where the suspected cases bad cases were dentfed by Robust Mahalanobs Dstance RMD, based on the Mnmum Volume Ellpsod MVE. Rousseeuw 18 defned RMD based on MVE as follows: RMD X T R X T C R X 1 X T R X for 1, 2,...,n, 3.4 where T R X and C R X are robust locatons and shape estmates of the MVE, respectvely. In the second step of DRGP MVE, the GPs are computed based on the set of D and R obtaned from RMD MVE. The low leverage ponts f any are put back nto the estmaton data set after nspectng the GP proposed by Imon 14 whch are defned as follows: p w D for D, w D 1 w D for R, 3.5

5 Mathematcal Problems n Engneerng 5 where w D X T XT R X R 1 X. He suggested the cutoff pont of p as p > Medan( p ) ( cmad p ), 3.6 where c can be taken as a constant value of 2 or 3. The DRGP MVE have been proven to be very effectve n the dentfcaton of multple hgh leverage ponts. 4. The New Proposed Hgh Leverage Collnearty-Influental Observatons Measures As already mentoned n the precedng secton, the man reason of developng a new measure of hgh leverage collnearty-nfluental measure s due to the fact that the commonly used measures faled to detect multple hgh leverage collnearty-nfluental observatons. In addton, not many papers related to ths measure have been publshed n the lteratures. It s mportant mentonng that the collnearty-nfluental measure whch were proposed by Had 5 and Sengupta and Bhmasankaram 12 are related to the Had s sngle-casedeleted leverage measure 13. Snce the robust generalzed potentals that was developed by Habshah et al. 11 was very successful n the dentfcaton of multple hgh leverage ponts compared to other wdely used methods, Bagher et al. 1 utlzed a smlar approach n developng multple Hgh Leverage Collnearty-Influental Measure HLCIM. The proposed measure s formulated based on Sengupta and Bhmasankaram 12 s measure wth slght modfcaton whereby almost smlar approach of DRGP MVE 11 was adapted. Hence t s referred as HLCIM DRGP and denoted as δ D. Ths new measure s defned as follows: δ D log log log ( k D k D ( k k ) ( k D k D ) ) f D, #{D} / 1, f #{D} 1, D, 1, 2,...,n, f R, 4.1 where D s the suspected group of multple hgh leverage collnearty-nfluental observatons dagnosed by DRGP MVE, p,#{d} s the number of elements n D group, and R s the remanng good observatons. As such, followng Habshah et al. 11 approach, three condtons should be consdered n defnng δ D. Bagher et al. 1 summarzed the algorthm of HLCIM DRGP n three steps as follows. Step 1. Calculate DRGP MVE, p,for 1, 2,...,n. Form D as a hgh leverage collneartynfluental suspected group whereby ts members consst of observatons whch correspond to p that exceed the medan p 3MAD p. Obvously the rest of the observatons belong to R, the remanng group.

6 6 Mathematcal Problems n Engneerng Step 2. Compute hgh leverage collnearty-nfluental values, δ D, as follows. If only a sngle member n the D group, the sze of R s n 1, andd, calculate log k /k where k ndcates the condton number of the X matrx wthout the th hgh leverage ponts. In ths way, δ D l. If more than one member n the D group, calculate log k D /k D where k D ndcates the condton number of the X matrx wthout the entre D group mnus the th hgh leverage ponts, where belongs to the suspected D group. For any observaton n the R group, compute log k D /k D where k D refers to the condton number of the X matrx wthout the entre group of D hgh leverage ponts plus the th addtonal observaton of the remanng group. Step 3. If any δ D values for 1, 2,...,n does not exceed the cutoff ponts n 2.6, put back the th observaton to the R group. Otherwse, D group s the hgh leverage collneartyenhancng observatons. Bagher et al. 1 only defned the cutoff pont for θ to ndcate hgh leverage collnearty-enhancng observatons and they dd not suggest cutoff pont for collneartyreducng observatons. The authors consdered θ to be hgh leverage collnearty-enhancng observatons f θ s less than the cutoff ponts; that s medan θ 3mad θ for θ <, where c s a chosen value 3 and θ may be δ D, δ or l. Snce hgh leverage collnearty-reducng observatons are also responsble for the msleadng nferental statements, t s very crucal to detect ther presence. In the followng secton, we propose a cutoff pont for dentfyng hgh leverage collnearty-reducng observatons. It s mportant mentonng that not all δ D whch exceed the cutoff pont are hgh leverage ponts. Ths s true for the stuaton when δ D exceeds the cutoff pont but belongs to the remanng group, R. In ths stuaton, the observaton s consdered as collneartynfluental observatons snce they are not hgh leverage ponts. 5. The New Proposed Cutoff Pont for HLCIM (DRGP) Had 5 and Sengupta and Bhmasankaram 12 mentoned that a large postve value of ther collnearty-nfluental measures, δ and l, respectvely, ndcates that the th observaton s a collnearty-reducng observaton. However, they dd not suggest any cutoff ponts to ndcate whch observatons are collnearty-enhancng and whch are collnearty-reducng. Bagher et al. 1 proposed a nonparametrc cutoff pont for hgh leverage collneartyenhancng observatons. Ther work has nspred us to nvestgate hgh leverage collneartyreducng observatons among the observatons that correspond to postve values of hgh leverage collnearty-nfluental measures. Fgure 1 presents the normal dstrbuton plot of θ. Based on ths fgure, any value that exceeds medan θ 3MAD θ can be utlzed as a cutoff pont for θ. Hence, we propose the followng cutoff pont: cut CRO Medan θ 3MAD θ, 5.1

7 Mathematcal Problems n Engneerng 7 Collnearty-enhancng observatons Collnearty-reducng observatons 3Mad (θ ) Medan (θ ) +3Mad (θ ) Fgure 1: Normal dstrbuton plot of hgh leverage collnearty-nfluental measure. where cut CRO s the Collnearty-Influental Measure cutoff pont for Collnearty- Reducng Observatons. θ can be δ D, δ or l. θ cut CRO for θ > s an ndcator that the th observaton s a collnearty-reducng observaton. 6. A Numercal Example A numercal example s presented to compare the performance of the newly proposed measure δ D wth the exstng measures δ and l. An engneerng data taken from Montgomery et al. 19 s used n ths study. It represents the relatonshp between thrust of a jet-turbne engne y and sx ndependent varables. The ndependent varables are prmary speed of rotaton X 1, secondary speed of rotaton X 2, fuel flow rate X 3, pressure X 4, exhaust temperature X 5, and ambent temperature at tme of test X 6. It s mportant mentonng that, the explanatory varables of ths data are scaled before analyss n order to prevent the condton number to be domnated by large measurement unts of some explanatory varables. Pror to analyss of ths data, the explanatory varables have been scaled followng Stewart s 2 scalng method as x j x j Xj, 1,...,p, j 1,...,n. 6.1 There are other alternatve scalng methods whch can be found n Montgomery et al. 1, Stewart 2, andhad 5. The matrx plot n Fgure 2 and the collnearty dagnostcs presented n Table 1 suggest that ths data set has severe multcollnearty problem CN Wewouldlketo dagnose whether hgh leverage ponts are the cause of ths problem. As such, t s necessary to detect the presence of hgh leverage ponts n ths data set. The ndex plot of DRGP MVE presented n Fgure 3 suggests that observatons 6 and 2 are hgh leverage ponts. By deletng these two observatons from the data set, CN ncreases to It seems that these two hgh leverages are collnearty-reducng observatons. Theeffect of these two hgh leverage ponts on collnearty pattern of the data s further nvestgated by applyng δ D, δ and l wth ther respectve new cutoff pont ntroduced n 5.1 for detectng hgh leverage collnearty-reducng observatons. Fgure 4 llustrates

8 8 Mathematcal Problems n Engneerng X 1 X 2 X 3 X 4 X 5 X 6 Fgure 2: Matrx plot of jet turbne engne data set. Table 1: Collnearty dagnostcs of jet turbne engne data set. Dagnostcs r r r r r 16.7 r Pearson correlaton coeffcent r r r 26.2 r r r r r r 56.3 VIF > Condton ndex of X matrx > the ndex plot of these measures. Accordng to ths plot, all these three measures have ndcated that observatons 6 and 2 as hgh leverage collnearty-reducng observatons. Nevertheless, besdes observatons 6 and 2, they detect a few more observatons as collnearty-reducng observatons. It s nterestng to note that none of the observatons are detected as hgh leverage collnearty-enhancng observatons or collnearty-enhancng observatons. It s worth mentonng that we do not have any nformaton about the source of the two exstng hgh leverage collnearty-reducng observatons cases 6 and 2. Therefore, we cannot control the magntude and the number of added hgh leverages ponts to the data n order to study the effectveness of our proposed measures. In ths respect, we have modfed ths data set n two dfferent patterns followng 7. Habshah et al. 7 ndcated that n the collnear data set, when hgh leverages exst n just one explanatory varable or n dfferent postons of two explanatory varables; these leverages wll be collnearty-reducng observatons. Thus, the frst pattern s when we replaced observatons 5, 6, 19, and 2 of X 2 wth a fxed large value of 5. The second pattern s created by replacng the large value of 5 to X 2 for observatons 5, 6 and observatons 19, 2 of X 3. The DRGP MVE ndex plot for Fgure 5 reveals that observatons 5, 6, 19, 2 are detected as hgh leverage ponts for modfed jet turbne engne data set.

9 Mathematcal Problems n Engneerng 9 DRGP (MVE) Fgure 3: DRGP MVE ndex plot of jet turbne engne data set. Fgures 6 and 7 present the ndex plot of δ D, δ and l for the frst and the second pattern of the modfed jet turbne engne data set. The results of δ D n these fgures agree reasonably well wth Bagher s et al. 1 fndngs that when hgh leverage ponts exst n just one explanatory varable frst pattern or n dfferent postons of two explanatory varables second pattern n collnear data sets, these observatons are referred as collnearty-reducng observatons. For both patterns, δ D correctly dentfed that observatons 5, 6, 19, and 2 are hgh leverage collnearty-reducng observatons. However, for the frst pattern, both δ and l are not successful n detectng all of observatons; 5, 6, 19, and 2 as hgh leverage collneartyreducng observatons. In the frst pattern, they only correctly detected observatons 19 and 2 as hgh leverage collnearty-reducng observatons. However, none of the added hgh leverage collnearty-reducng observatons can be detected by these two measures n the second pattern. It s mportant to note that for the frst and the second patterns, the values of δ and l for the observatons 5 and 6, and observaton 19, respectvely are becomng negatve. Ths ndcates that for both patterns, δ and l have wrongly ndcated these observatons as suspected hgh leverage collnearty-enhancng observatons. 7. Monte Carlo Smulaton Study In ths secton, we report a Monte Carlo smulaton study that s desgned to assess the performance of our new proposed measure δ D n detectng multple hgh leverage collnearty-reducng observatons and to compare ts performance wth two commonly used measures δ and l. Followng Lawrence and Arthur 21, smulated data sets wth three ndependent regressors were generated as follows: x j ( 1 ρ 2) z j ρz 4, 1,...,n; j 1,...,3, 7.1 where the z j, 1,...,n; j 1,...,3 are Unform, 1. The value of ρ 2 whch represents the correlaton between the two explanatory varables are chosen to be equal to.95. Ths amount of correlaton causes hgh multcollnearty between explanatory varables. Dfferent percentage of hgh leverage ponts are consdered n ths study. The level of hgh leverage ponts vared from α.1,.2,.3. Dfferent sample szes from n 2, 4, 6, 1, and

10 1 Mathematcal Problems n Engneerng δ (D) a δ l b 2 c Fgure 4: plot of collnearty-nfluental measures for orgnal jet turbne engne data set. 3 wth replcaton of 1, tmes were consdered. Followng the dea of Habshah et al. 7, twodfferent contamnaton patterns were created. In the frst pattern, 1 α percent observatons of one of the generated collnear explanatory varables were replaced by hgh leverages wth unequal weghts. In ths pattern the explanatory varable and the observaton whch needed to be replaced by hgh leverage pont were chosen randomly. The second pattern s created by replacng the frst 1 α/2 percent of one of the collnear explanatory varable and the last 1 α/2 percent of another collnear explanatory varable wth hgh

11 Mathematcal Problems n Engneerng 11 DRGP (MVE) a DRGP (MVE) b Fgure 5: plot of DRGP MVE for modfed jet turbne engne data set, a pattern1, b pattern2. leverages wth unequal weghts. The two ndependent varables are also randomly selected and the replacement of the hgh leverage pont to the observatons n dfferent postons of explanatory varables was also performed randomly. Followng Habshah et al. 11 and Bagher et al. 1, the hgh leverage values wth unequal weghts n these two patterns were generated such that the values correspondng to the frst hgh leverage pont are kept fxed at 1 and those of the successve values are created by multplyng the observatons ndex, by 1. The three dagnostc measures δ D, δ and l wth the proposed cutoff pont were ntroduced to 5.1 and were appled to each smulated data. The results based on the average values are presented n Table 2. The α and HLCIO n Table 2 ndcate, respectvely, the percentage and the number of added hgh leverage collnearty-reducng observatons to the smulated data sets. Furthermore, the number of hgh leverage ponts whch s detected by DRGP MVE s denoted as HL. It s nterestng to pont out that the percentage of the hgh leverage pont, p detected by DRGP MVE denotedashlntable 2 s more than the percentage of the added hgh leverage collnearty-reducng observatons to the smulated data sets, α. However, by ncreasng the sample sze and the percentage of added hgh leverage ponts to the smulated data, both percentages became exactly the same. The CN1 and the CN2 ndcate the condton number of X matrx wthout and wth hgh leverage collnearty-reducng observatons, respectvely. Moreover, Cut θ 1 and Cut θ 2 represent the number of hgh leverage collnearty-reducng observatons and the number of collnearty-reducng observatons whch have been detected by cutoff θ. Table 2 clearly shows the mert of our new proposed measure for hgh leverage collnearty-nfluental measure exhbted n 4.1. It can be observed that no other measures that were consdered n ths experment performed satsfactorly except for our proposed measure. The smulated data sets have been created collnearly whch produced large values of CN 1, condton number of smulated data sets wthout hgh leverage ponts CN 1 > 3. The added multple hgh leverage collnearty-nfluental observatons reduces multcollnearty among the smulated explanatory varables; ths reducton may result from the smaller values of CN 2 compared to CN 1. It s mportant mentonng that the reducton of the CN 2 values for the second pattern was much more sgnfcant compared to CN 2 for the frst pattern. We can conclude that the nfluence of the added hgh leverage ponts to dfferent

12 12 Mathematcal Problems n Engneerng δ (D) a.15.1 δ l b 19 2 c Fgure 6: plot of collnearty-nfluental measures for the frst modfed pattern of jet turbne engne data set. postons of two explanatory varables for changng the multcollnearty pattern of smulated data, s more sgnfcant compared to the added hgh leverage ponts to only one explanatory varable. The results of Table 2 for the frst pattern of smulated data sets ndcate that for small sample szes n 2 our proposed measure could not ndcate the exact amount of hgh leverage collnearty-reducng observatons. However, by ncreasng the sample sze and the percentage of added hgh leverage ponts to the smulated data sets, the measure s capable

13 Mathematcal Problems n Engneerng 13 Table 2: Collnearty-nfluental measures for smulated data sets. a Measures n 2 n 4 Pattern1 Pattern2 Pattern1 Pattern2 α HLCIO HL CN CN Cut δ D Cut δ D Cut δ Cut δ Cut l Cut l b Measures n 6 n 1 Pattern1 Pattern2 Pattern1 Pattern2 α HLCIO HL CN CN Cut δ D Cut δ D Cut δ Cut δ Cut l Cut l c Measures n 3 Pattern1 Pattern2 α HLCIO HL CN CN Cut δ D Cut δ D Cut δ Cut δ Cut l Cut l

14 14 Mathematcal Problems n Engneerng δ (D) a δ b l c Fgure 7: plot of collnearty-nfluental measures for the second modfed pattern of jet turbne engne data set. of detectng the exact amount of added hgh leverage collnearty-reducng observatons. It s evdent by lookng at the value of Cut δ D 1 s exactly the same as HLCIO. On the other hand, the other two collnearty-nfluental measures, δ and l, faled to ndcate the exact amount of hgh leverage collnearty-reducng observatons. It s worth notng that all of these three measures also detect some ponts as collnearty-reducng observatons see the Cut θ 2 n Table 2, where θ s δ D, δ or l. Smlar results wll be obtaned f pattern 1 can be drawn for the second pattern of the smulated data sets. Compared to the frst contamnaton pattern, t s clearly seen that δ and l almost completely faled to detect ether

15 Mathematcal Problems n Engneerng 15 hgh leverage collnearty-reducng observatons or collnearty-reducng observatons. Our proposed measure dd a credble job where t s successfully detect hgh leverage collneartyreducng observatons for both contamnated patterns. 8. Concluson The presence of hgh leverage ponts and multcollnearty are nevtable n real data sets and they have an unduly effects on the parameter estmaton of multple lnear regresson models. These leverage ponts may be hgh leverage collnearty-enhancng or hgh leverage collnearty-reducng observatons. It s crucal to detect these observatons n order to reduce the destructve effects of multcollnearty on regresson estmates whch lead to msleadng concluson. It s easer to dagnose the presence of hgh leverage ponts whch ncrease the collnearty among the explanatory varables compared to those whch reduce collnearty. In ths respect, t s very mportant to explore a suffcent measure wth an accurate cutoff pont for detectng hgh leverage collnearty-reducng observatons. In ths paper, we proposed a precse cutoff pont for a novel exstng measure to detect hgh leverage collneartyreducng observatons. By usng an engneerng data and a smulaton study, we confrmed that the wdely used measures faled to detect multple hgh leverage collnearty-reducng observatons. Furthermore, our proposed cutoff pont successfully detects multple hgh leverage collnearty-reducng observatons. References 1 D. C. Montgomery, E. A. Peck, and G. G. Vvng, Introducton to Lnear Regresson Analyss, John Wley & Sons, New York, NY, USA, 3rd edton, Md. Kamruzzaman and A. H. M. R. Imon, Hgh leverage pont: another source of multcollnearty, Pakstan Journal of Statstcs, vol. 18, no. 3, pp , M. H. Kutner, C. J. Nachtshem, and J. Neter, Appled Lnear Regresson Models, McGraw-Hll, New York, NY, USA, S. Chatterjee and A. S. Had, Regresson Analyss by Examples, John Wley & Sons, New York, NY, USA, 4th edton, A. S. Had, Dagnosng collnearty-nfluental observatons, Computatonal Statstcs & Data Analyss, vol. 7, no. 2, pp , M, A. Habshah, Bagher, A. H. M. R, and Imon, The applcaton of robust multcollnearty dagnostc method based on robust coeffcent determnaton to a non-collnear data, Journal of Appled Scences, vol. 1, no. 8, pp , M. Habshah, A. Bagher, and A. H. M. R. Imon, Hgh leverage collnearty-enhancng observatons and ts effect on multcollnearty pattern; Monte Carlo smulaton study, Sans Malaysana, vol. 4, no. 12, pp , A. Bagher, H. Md, and A. H. M. R. Imon, The effect of collnearty-nfluental observatons on collnear data set: A monte carlo smulaton study, Journal of Appled Scences, vol. 1, no. 18, pp , A. Bagher and H. Md, On the performance of robust varance nflaton factors, Internatonal Journal of Agrcultural and Statstcal Scences, vol. 7, no. 1, pp , A. Bagher, M. Habshah, and R. H. M. R. Imon, A novel collnearty-nfluental observaton dagnostc measure based on a group deleton approach, Communcatons n Statstcs, vol. 41, no. 8, pp , M. Habshah, M. R. Norazan, and A. H. M. R. Imon, The performance of dagnostc-robust generalzed potentals for the dentfcaton of multple hgh leverage ponts n lnear regresson, Journal of Appled Statstcs, vol. 36, no. 5-6, pp , D. Sengupta and P. Bhmasankaram, On the roles of observatons n collnearty n the lnear model, Journal of the Amercan Statstcal Assocaton, vol. 92, no. 439, pp , 1997.

16 16 Mathematcal Problems n Engneerng 13 A. S. Had, A new measure of overall potental nfluence n lnear regresson, Computatonal Statstcs and Data Analyss, vol. 14, no. 1, pp. 1 27, A. H. M. R. Imon, Identfyng multple hgh leverage ponts n lnear regresson, Journal of Statstcal Studes, vol. 3, pp , 22, Specal Volume n Honour of Professor Mr Masoom Al. 15 D. A. Belsley, E. Kuh, and R. E. Welsch, Regresson Dagnostcs: Identfyng Influental Data and Sources of Collnearty, John Wley & Sons, New York, NY, USA, D. A. Belsley, Condtonng Dagnostcs-Collnearty and Weak Data n Regresson, John Wley & Sons, New York, NY, USA, D. C. Hoagln and R. E. Welsch, The Hat Matrx n regresson and ANOVA, Journal of the Amercan Statstcal Assocaton, vol. 32, no. 1, pp , P. Rousseeuw, Multvarate estmaton wth hgh breakdown pont, n Mathematcal Statstcs and Applcatons, pp , Redel, Dordrecht, The Netherlands, D. C. Montgomery, G. C. Runger, and N. F. Hubele, Engneerng Statstcs, John Wley & Sons, New York, NY, USA, 5nd edton, G. W. Stewart, Collnearty and least squares regresson, Statstcal Scence, vol. 2, no. 1, pp. 68 1, K. D. Lawrence and J. L. Arthur, Robust Regresson; Analyss and Applcatons, Marcel Dekker, New York, NY, USA, 199.

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