Estimation Methods for Multicollinearity Proplem Combined with High Leverage Data Points

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1 Journal of Mathematcs and Statstcs 7 (): , 011 ISSN Scence Publcatons Estmaton Methods for Multcollnearty Proplem Combned wth Hgh Leverage Data Ponts Moawad El-Fallah and Abd El-Sallam Department of Statstcs and Mathematcs and Insurance, Faculty of Commerce, Zagazg Unversty, Egypt Abstract: Problem statement: Least Squares (LS) method has been the most popular method for estmatng the parameters of a model due to ts optmal propertes and ease of computaton. LS estmated regresson may be serously affected by multcollnearty whch s a near lnear dependency between two or more explanatory varables n the regresson models. Although LS estmates are unbased n the presence of multcollnearty, they wll be mprecse wth nflated standard errors of the estmated regresson coeffcents. Approach: In ths study, we wll study some alternatve regresson methods for estmatng the regresson parameters n the presence of multple hgh leverage ponts whch cause multcollnearty problem. These methods are manly depend on a one step reweghted least square, where the ntal weght functons were determned by the Dagnostc-Robust Generalzed Potentals (DRGP). The proposed alternatve methods n ths study are called GM-DRGP-L 1, GM- DRGP-LTS, M-DRGP, MM-DRGP and DRGP-MM. Results: The emprcal results of ths study ndcated that, the DRGP-MM and the GM-DRGP-LTS offers a substantal mprovement over other methods for correctng the problems of hgh leverage ponts enhancng multcollnearty. Concluson: The study had establshed that the DRGP-MM and the GM-DRGP-LTS methods were recommended to solve the multcollnearty problem wth hgh leverage data ponts. Key words: Multcollnearty problem, multple hgh, leverage ponts, alternatve robust estmatons robust generalzed, explanatory varables, dagnostc tool, regresson analyss., hgh leverage, substantal mprovement, leverage ponts, bounded nfluence INTRODUCTION Least squares estmaton s one of the most mportant regresson technques used for estmatng the parameters of a model. Two of the assumptons that make least squares so attractve n terms of general model hypothess and parameter sgnfcance testng, are normalty of error dstrbuton and ndependency of explanatory varables. The normalty assumpton can be volated n the presence of one or more suffcently outlyng observatons n the data set resultng n less relable estmates of the model parameters. The second s multcollnearty, whch s a near-lnear dependency among the explanatory varables (X-drecton). Multcollnearty can cause large varablty n the estmaton of parameters. Sometmes t causes the parameters estmaton to be dfferent from the true values by orders of magntude or ncorrect sgn. It may also nflate the varance of the estmatons. Hgh leverage ponts, the ponts far from the rest of the data n the X-drecton, have hgh potental for nfluencng most of the regresson results such as egenstructure and condton ndex of X. Had (199) noted that collnearty-nfluental ponts are usually the ponts wth hgh-leverage whch tends to pull the model ft to ther drecton and ntroduced these ponts as a new source of multcollnearty problems. Thus, dagnosng the multple hgh leverage ponts and recognzng estmatons, methods whch are resstant to these ponts may mprove regresson estmatons. In ths respect, alternatve robust regresson methods are desgned to be less senstve than least squares to outlers mostly n Y- drecton, resultng n mproved fts to the non-outlyng observatons. In order to acheve ths stablty, alternatve robust regresson methods lmt the nfluence of outlers. Three most mportant propertes of any alternatve robust regresson method are effcency, breakdown pont and bounded nfluence (Andersen, 008). The man objectve of ths study s to propose some alternatve estmators that are able to perform well where multple hgh leverage ponts cause multcollnearty problem n regresson analyss. Nonetheless, the development of such estmators has not been publshed extensvely n the lterature. To Correspondng Author: Moawad El-Fallah, Department of Statstcs and Mathematcs and Insurance, Faculty of Commerce, Zagazg Unversty, Egypt 19

2 acheve ths objectve, dfferent types of related alternatve robust methods have been nvestgated and ther propertes are compared. Among the dfferent types of robust technques, we wll consder the bounded nfluence or Generalzed M-estmators (Marronna and Yoha, 000; Ghaz et al., 010; Ramz. and Vvane, 010) whch attempt to assgn less weght to the hgh nfluence observatons and large resdual ponts. To enhance the GM-estmators, these estmators may be defned as mult-stage estmators where n dfferent stages, dfferent alternatve robust propertes of each technque are appled to combne the desrable propertes of each technque (Smpson et al., 199). MATERIALS AND METHODS Robust regresson methods: Let us consder the followng lnear regresson model as Eq. 1: Y = Xβ+ (1) Where: Y = The n 1 vector of response X = The n P (P = k+1) matrx = The n 1 vector whch has standardzed normal dstrbuton When the Least Squares (LS) method s employed, estmaton of the regresson parameters can be obtaned from Eq. : estmators, where the LMS and the LTS has low and medum effcency value, respectvely (Smpson et al. (199). Huber (1973) proposed a robust M-estmators where ˆβ are obtaned by solvng Eq. 3: y x βˆ n ψ = 1 s (3) It s mportant to pont out that, there are two types of ψ-functons, that s the monotonc ψ-functons (e.g. Huber s ψ-functons) and the redescendng ψ - functons (e.g., bweght ψ functon s, Beaton and Tukey (1974). M-estmators are the smplest hgh effcency robust procedures, both computatonally and theoretcally havng desrable asymptotc propertes. However, the M-estmator s not robust n the X- drecton and has a low breakdown pont, that s equal to (1/n) (Smpson et al., 199; Marronna et al., 006) ntroduced a class of methods whch s called the Generalzed M-estmators (GM-estmators) wth a major am of downweghtng those hgh leverage ponts whch have large resduals. Marronna et al. (006) also, reported that these estmators have hgh effcency and bounded nfluence propertes whch acheve a moderate breakdown pont equal to (1/P). The GM-estmator s the soluton of the normal equaton (4): y x βˆ n ϕ ψ x1 = 0 = 1 sϕ= (4) ˆ 1 (x / x) x / Y β= () Robust regresson procedures are manly am to provde resstant (stable) results n the presence of outlers. The Least Absolute Values (LAV) s one of the frst robust methods that was ntroduced by Armstrong and Kung (1987) wth a hgher effcency than LS by mnmzng the sum of the absolute resduals. The use of ths crteron, rather than ordnary least squares, provdes robustness aganst outlers and s partcularly useful when the dsturbances are generated by fattaled dstrbutons. Rousseuw (1984). Rousseuw et al. (003) ntroduced two robust methods namely the Least Medan of Squares (LMS) and the Least Trmmed Squares (LTS). The LMS attempts to mnmze the medan of e whle the LTS mnmze h e() where e (1) = 1 where, φ are defned to down weght hgh leverage ponts, wth hgh resduals and s s a robust scale estmate. Iteratvely Reweghted Least Squares (IRLS) may be used to solve (4). At convergence, the GMestmator may be wrtten as Eq. 5: β ˆ = ( XWX ) 1 XWY (5) e (),, e (n) are the ordered squared resduals, =1,,n and h s the number of resduals ncluded n the calculaton. Both estmators have hgh breakdown, that 50%. However, they are unbounded nfluence been publshed extensvely n the lterature. Snce hgh 130 GM where, n ths case, the dagonal elements of W are the weghts w defned as Eq. 6: y x ˆ β GM ψ ϕs W = (y x ˆ ) ϕ s GM (6) The man objectve of ths study s to study some alternatve estmators that are able to perform well where multple hgh leverage ponts are the cause of the multcollnearty problem n regresson analyss. In partcular, the development of such estmators has not

3 leverage ponts may be collnearty-enhancng observatons, we attempt to reduce ts nfluence by employng robust estmator whch s known to be resstant to hgh leverage ponts. In ths connecton, we wll consder the bounded nfluence or Generalzed M- estmators wth a major am of down weghtng the hgh leverage ponts whch have large resduals. Hence, n ths study, we propose manly alternatve mult-stage GM-estmators and weghted MM-estmators to remedy the problem of collnearty-enhancng observatons on the parameter estmates of the multple lnear regresson model. Unfortunately, the MM-estmators are also senstve to outlers n X-varables. As a soluton to ths drawback of MM-estmators, an alternatve robust method s developed n secton four. To confrm the advantage of our alternatve proposed methods, these methods compared wth reweghted least square based on LMS (RLS-LMS) defned by Rousseuw and Leroy (003). They computed scale estmator as: 0 5 s = * 1 med(r ) n p 1 where, r s the resdual of LS. The followng hard r rejecton functon for standardzed resduals s S utlzed to compute the followng ntal weghts Eq. 7: w r 1 f.5 0 = S 0 otherwse (7) The robust verson of (8) s Eq. 9: x medan(x) T = (9) Max(x) where, Mad (x) s the normalzed medan absolute devaton about the Medan (x) (Mad= 1.486(medan r -medan (r ) ). When the dstrbuton of the data s normal, T and T are approxmately equal. Any observaton whch has absolute value of T or T greater than 3, s consdered as outler (Marronna et al., 006). Ths method can be used n unvarate regresson models as a dagnostcs rule to detect hgh leverage ponts. Snce n most of the regresson analyss, more than one collnear explanatory varable exsts n the model, nvestgatng some useful methods n these cases seems to be necessary. One of the handest methods can be defned as hat matrx. Hat matrx whch s tradtonally used as a measure of leverage ponts n regresson analyss s defned as W=X (X T X) -1 X T. The most wdely used cutoff ponts of hat matrx s twcethe- mean-rule (k/n) by Hoagln and Welsch (1978). Had (1988) ponted out that the hat matrx may fal to dentfy the hgh leverage ponts due to the effect of hgh leverage ponts n leverage structure. He ntroduced another dagnostc tool as follows Eq. 10: p w = (10) 1 w T T where, w = ( ) x x x x s the dagonal element of W The fnal weghts for RLS-LMS are dentfed by the usage of a hard rejecton for standardzng the LMS w r resduals by new scale as scale = the n w 1 p 1 = weakness of ths method s usng LMS whch s a low effcency estmator. Dagnostc robust generalzed potental statstcs: A tradtonal measure of the outlyngness of an observaton x wth respect to the sample s three-sgma edt rule whch s defned as follows Eq. 8: x x T = (8) S Where: x = The mean s = The standard devaton of collnear explanatory varables 131 and the -th, dagonal potental p can be defned as p = T T 1 x ( x () x ()) x, where X () s the data matrx X wthout the -th row. He proposed a cut off pont for potental values p as Medan (p ) + c Mad (p ) (MADcutoff pont) and c can be taken as constant values of or 3. Ths method also s unable to detect all of the hgh leverage ponts. So, Had (1988) ntroduced another dagnostc tool as generalzed potentals for the whole data set whch s defned as Eq. 11: w p = 1 w for R = w for D ( D) ( D) ( ) (11) Where: D = The deleted set whch corresponds to the suspected outlers R = The remanng set from observatons after deletng d < (n-k) and t contans (n-d) cases

4 Snce there sn t any fnte upper bound for p * s and the theoretcal dstrbuton of them are not easy to derved, he ntroduced a MAD-cutoff pont for the generalzed potental as well. Recently, Habshah et al. (009) developed Dagnostc Robust Generalzed Potental (DRGP) to determne outlyng ponts n multvarate data set by utlzng the Robust Mahalanobs Dstance (RMD) based on Mnmum Volume Estmator (MVE) (RMD-MVE) (defned by Rousseuw (1985) as Eq. 1: 1 RMD = (x T R(x)) C R(x) ( T R(x)) for = 1,...,n (1) where, T R ( ) and C R ( ) are robust locaton and shape estmate such as MCD or MVE. The RMD- MVE has been used to detect the suspected group (D group) n generalzed potental method n (11). The mert of ths method s swampng less good leverage as hgh leverage ponts comparng wth the RMD -MVE. In the next secton we propose robust methods based on DRGP. Alternatve proposed robust methods: In ths secton, some proposed form of φ-weghts n (4) are generated and dscussed. It s mportant to pont out that n the proposed methods, P s the DRGP statstcs wth MADcutoff of ths statstc. In these methods, we wll employ the Tukey s bweght redescendng φ-functon whch s defned as Eq. 13: the calculaton of the φ. The frst two proposed estmators are mult-stage GM-estmators, whle the others are defned based on the M-estmator and MMestmators. The proposed methods wll be computed n three steps and summarzed as follow. GM-DRGP-L 1 : Step 1: Employ L 1 as ntal estmate and then obtan the standardzed resduals of L 1 estmator. Compute MAD = (med r -med (r ). accordng to Marronna and Yoha (000). It s mportant to menton that f MAD s computed from all the resduals of L 1 estmators, the scale estmates wll become too small due to defnng some zero resdual. Thus, non-null resduals have been used to compute the scale estmate. Step : Defnng ( ( )) (( )) ϕ = mn 1, MAD cutoff p / p n (4) and usng functon (13) to assgn fnal weghts to the observatons. Step 3: Compute a one step reweghted least squares as a convergence approach. GM-DRGP-LTS: Step 1: Consder the LTS as ntal estmate and compute the standardzed resduals and scale estmate based on LTS. f t t ψ (t) = t 1 c c f t > c (13) Step : Defne ϕ = mn 1, ( MAD cutoff ( p )) / (( p )) n (4) and usng functon (13) to assgn fnal weghts to the observatons. The Tukey s bweght wth the tunng constant c = wll result a 95% effcency under normal error dstrbuton. Assgnng lower weghts (even zero f the Step3: Compute a one step reweghted least squares as resdual s too large) to large outlers, a redescendng φ- convergence approach. functon s better compared to monotonc functons such as Huber s functon. In ths respect, a M-DRGP: redescendng φ-functons lmts the nfluence of outlers Step 1: Compute the resduals of M-estmates of scale more effectvely than a monotone φ-functon. Gven the by assgnng the ntal weght of W, (DRGP (MVE) fact that teratve technques are computatonally expensve and there s no guaranty to result n better mn 1, ( MAD cutoff ( p )) / (( p )) where P s DRGP estmatons (Smpson, 1995) a one step reweghted LS s (MVE) statstcs. used for most of the proposed methods except DRGP- MM estmator. Smpson et al. (199) enumerated that Step : Defne new weghts as w = r (Mestmator)/scale (M-estmator) and usng a Tukey s the GM and MM-estmators surpass other robust method. In ths connecton, most of the alternatve bweght to assgn fnal weght to the observatons. proposed methods are smlar to that of GM and MM estmators wth alght modfcaton n whch the DRGP proposed by Habshah et al. (009) s ncorporated n Step 3: Compute a one step reweghted least squares. 13

5 MM-DRGP: Ths method s smlar to that of M- DRGP, where on the second and thrd steps the M s replaced wth the MM-estmators. DRGP-MM: Step 1: Compute the ntal weght W, (DRGP(MVE)) whch s defned n the frst step of M-DRGP and usng functon (13) to assgn fnal weghts to the observatons. Step : Compute the weghted MM-estmators by these fnal weghts. weghted multcollnearty dagnostcs. These dagnostcs can be defned as a measure to evaluate whch method s more robust aganst the hgh leverage ponts that are responsble for the multcollnearty. It s mportant to pont out that all hgh leverage ponts are not collnearty-nfluental and vce versa (Had, 199) The weghted correlaton matrx can be computed through the correlaton matrx of X w. The weghted VIF s defnd as follows Eq. 15: w = ( ( )) 1 (15) VIF () 1 R w Weghted multcollnearty dagnostcs: Weghted multcollnearty dagnostcs are defned as practcal tools to nvestgate the source of multcollnearty whch may be the hgh leverage ponts n the data set. Indeed, robust estmators to deal wth multcolnearty problems are largely gnored ssues. Walker (1985) noted that sometmes the weghtng process n robust methods can mprove the multcollnearty of X matrx. An effectve measure of robust methods whch reduce multcollnearty problems due to the presence of multple hgh leverage ponts can be defned as weghted multcollnearty dagnostcs. The two most classcal and practcal multcollnearty dagnostcs are Correlaton X matrx and Varance Inflaton Factors (VIF). In bvarate regresson analyss, when correlaton coeffcent exceeds 0.9, multcollnearty can be detected. However, n the case of more than two explanatory varables model, multcollnearty may occur n less than 0.9 correlaton coeffcents (Rosen, 1999). Snce, ths multcollnearty dagnostcs s smple and easy to compute, t s more preferred (Belsley, 1991, Belsley et al., 1990). Another practcal approach to detect multcollnearty s by usng varance nflaton factors (VIF). VIF s defned as V1F () ( 1 R ( )) 1 w where R s the coeffcent determnaton of regressng each x on the other explanatory varables, whch produced a valuable ndces to detect nflated varances of regresson parameter estmatons (Marquardt, 1970). A cutoff pont of (11) s recommended as a rule of thumb for VIF to detect severe multcollnearty. The weghted lnear regresson can be expressed as a transformed model Eq. 14: Y w = X wβ + Є w (14) where, Y w = W 1/ Y, X w = W 1/ X and Є w = W 1/ Є (Neter et al. (004). The fnal weght of the proposed estmator, whch s expected to be robust aganst hgh leverage ponts, can be used n the computaton of 133 where, R (W) s the coeffcent of determnaton of regressng each X w on the other weghted explanatory varables. It s worth to menton that f the hgh leverage ponts are the source of multcollnearty n the data set, the weghted multcollnearty dagnostcs wll not detect multcollnearty due to these ponts otherwse multcollearty wll be detected easly. RESULTS Numercal example: In ths secton we consder a real data set to evaluate the performance of our proposed robust methods. Chld mortalty data set: Gujarat (00) ntroduced ths data set wth 64 observatons whch ncludes chld mortalty as dependent varable and Gross Natonal Producton (GNP) per capta and Female Lteracy Rate (FLR) as ndependent varables. Table 1 presents the classcal multcollnearty dagnostcs methods such as the correlaton matrx and VIF. The classcal dagnostcs measures of the orgnal data clearly ndcates that the data set doesn t have collnear explanatory varables. The T and F-tests confrm that there exsts relatonshp between the explanatory and response varable. Ths data set has two hgh leverage ponts based on the hat matrx by twcemean-rule cutoff pont, whle DRGP (MVE) can detect 11 observatons as hgh leverage ponts. Table 1: Multco llnearty dagnostcs and least square coeffcents of chld mortalty data set. Cor VIF b 1 b F p- S (e) (x 1,x ) (t p- (t p- value value) value) Orgnal data (0.007) (0.000) Modfed data (0.51) (0.938)

6 Table : Hgh leverage dagnostcs for chld mortalty data set Orgnal data Modfed data ndex Hat(x) (0.09) DRGP(x)(0.11) T 1 (3) T (3) ndex modfed X 1 Hat(x) (0.09) DRGP(x)( , Table 3: Multcollnearty dagnostcs and least square coeffcents of dfferent methods for modfed chld mortalty data set Method Cor (x 1,x ) VIF b 1 (t p-value) b (t p-value) F p-value s (e) Ls (0.51) (0.938) GM-DRGP-L (0.0) (0.34) GM-DRGP- LTS (0.00) (0.00) DRGP- MM (0.00) (0.00) - - MM-DRGP (0.0) (0.0) (0.33) M-DRGP (0.0) (0.33) RLS-LMS (0.00) (0.00) The hgh leverage ponts aren t collnearty-enhancng observatons evdent by the small value of correlaton matrx and VIF (Table 1). It s mportant to note here that the hgh leverage ponts wll be the prme source of multcollnearty when they are n the same observatons wth at least two explanatory varables. The robust three-sgma edt rule (Eq. 9) s shown n Table. The results of Table sgnfy that all the T exceeds the cutoff pont of 3 whch can be consdered as hgh leverage ponts, except observatons 1,5,38,and 54. In order to obtan a large magntude of hgh leverage ponts n x 1 as n x, a modfcaton n x n the ponts 4, 7, 30, 33, 53, 58-6 has been consdered based on the followng formula: ( ) ( ) Modfed x = T mad (x + medan (x ) The modfed x 1 s also dsplayed n Table. It s nterestng to pont out that after the modfcaton, the hat matrx can't detect all of these modfed observatons as hgh leverage ponts, whle the DRGP (MVE) statstcs dentfed them as hgh leverage defned estmators. 134 ponts. The results of Table 1 suggests that there s a strong multcollnearty n the modfed data set. Moreover, the non-sgnfcant of the t-statstcs and the sgnfcant of the F-statstcs of two coeffcent estmatons confrmed the presence of multcollnearty n the modfed data. The presence of multcollnearty has produced larger standard devaton of the errors for the modfed data. Table 3 presents the multcollnearty dagnostcs and least squares coeffcents of the modfed chld mortalty data set for proposed robust methods and the exstng robust methods whch were ntroduced prevously. The results of Table 3 pont out that the F-statstcs can t be obtaned for DRGP- MM estmator because t s not a one step reweghted estmator. It can be shown also from Table 3 that, among the proposed robust methods, only three estmators, that s the DRGP-MM, GM-DRGP-LTS and RLS-LMS can solve the multcollnearty problems. It s nterestng to note that the DRGP- MM has the least standard devaton error, followed by the GM-DRGP- LTS and RLS-LMS.Thus the new proposed estmators namely the DRGP-MM and the GM-DRGP-LTS, outperforms all other

7 DISCUSSION CONCLUSION Let us frst focus our attenton to the result of modfed chld mortalty data set whch s dsplayed n Table 1. The classcal dagnostcs measures of the orgnal data clearly ndcate that the data set does not have collnear explanatory varables. The T and F- tests confrm that there exsts relatonshp between the explanatory and response varable. Ths data set has two multple hgh leverage ponts based on the hat matrx by twce the mean-rule cutoff pont, whle DRGP (MVE) can detect 11 observatons as multple hgh leverage ponts. The hgh leverage ponts are not collnearty-enhansng observatons evdent by the small value of correlaton matrx and VIF (Table 1). The results of Table sgnfy that all the T of these multple hgh leverage ponts for the orgnal data exceeds the cutoff pont of 3 whch can be consdered as hgh leverage ponts n x, except for observaton 1,5,38and 54. It s nterestng to pont out that after the modfcaton (values for varable x 1 are modfed to become hgh leverage collnearty-enhancng observatons ), the hat matrx can not detect all of these modfed observatons as multple hgh leverage ponts, whle the DRGP (MVE) statstcs dentfed them as hgh leverage ponts. The result of Table 1 suggests that there s a strong multcollnearty n the modfed data set. Moreover, the non-sgnfcant of the t- statstcs and the sgnfcant of the F-statstcs of the two coeffcent estmatons confrmed the presence of multcollnearty n the modfed data. The presence of multcollnearty has produced larger standard devaton of the errors for the modfed data as well. It s mportant to pont out that the F-statstcs for the DRGP-MM estmator as shown n Table 3 can not be obtaned because t s not a one step reweghted estmator. It can be observed from Table 3 that, among the proposed robust methods, only three estmators, that s the DRGP-MM, GM-DRGP-LTS and RLT-LMS can solve the multcollnearty problems. Ths result also suggests that the other methods can hardly rectfy the multcollnearty problem evdent by the larger p values and hgher VIF values. It s nterestng to note that the DRGP-MM has the least standard devaton error, followed by the GM-DRGP-LTS and RLS-LMS. We have not pursued the analyss of ths example to the fnal concluson, but a reasonable nterpretaton up to ths stage s that the proposed Mult-stage GMncorporated the DRGP are able to solve the problem of multcollnearty whch s caused by hgh leverage ponts. 135 Outlers n the X-drecton whch are refer as multple hgh leverage ponts can render least squares estmaton meanngless and cause multcollnearty problems. Many robust methods have been developed to reduce the effect of outlers n the X-drecton. Nonetheless, the development of robust methods that deal wth the multcollnearty problems whch are manly due to multple hgh leverage ponts has not been publshed extensvely n the lterature. The man focus of ths study s to develop a relable method for correctng the problem of hgh leverage ponts enhancng multcollnearty. In ths study, we ncorporate the DRGP (MVE), one of the latest multple hgh leverage dagnostcs method wth dfferent types of robust estmators. The emprcal study ndcates that the DRGP-MM emerge to be more effcent and more relable than other methods, followed by the GM-DRGP-LTS as they are able to reduce the most effect of multcollnearty. The results seem to suggest that the DRGP-MM offers a substantal mprovement over other methods for correctng the problems of hgh leverage ponts enhancng multcollnearty. REFERENCES Andersen, R., 008. Modern Methods for Robust Regresson. 1st Edn., Stage Publcaton, The Unted States of Amerca, ISBN: , pp: 107. Armstrong, R.D. and M.T. Kung, Least absolute values estmates for a smple lnear regresson problem. J. R. Sc. Soc., 7: Beaton, A.E. and J.W. Tukey, The fttng of power seres, meanng polynomals, llustrated on band-spectroscopc data. Technometrcs, 16: Belsley, D.A., 1991.Condtonng Dagnostcs: Collnearty and Weak DAT n Regresson. 1st Edn., Wley, New York, ISBN: 10: , pp: 396. Belsley, D.A., E. Kuh and R.E. Welsch, Regresson Dagnostcs: Identng Influental Data and Sources of Collnearty. 1st Edn., Wley. New York, ISBN: , pp: 9. Ghaz, F.M., A. Majed, Alsharayr and M.D. Mwafaq, 010. Impact of frm s characterstcs on determnng the fnancal structure on the nsurance sector frms n jordan. J. Soc. Sc., 6: DOI: /jssp

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