Investigating the Significance of a Correlation Coefficient using Jackknife Estimates


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1 Iteratioal Joural of Scieces: Basic ad Applied Research (IJSBAR) ISSN (Prit & Olie) Ivestigatig the Sigificace of a Correlatio Coefficiet usig Jackkife Estimates Athoy Akpata a, Idika Okorie b a,b Departmet of Statistics, Abia State Uiversity Uturu,Nigeria a b Abstract Ofte i Applied statistics, populatio parameters are ot kow ad could be iferred usig the available sample data ad this is the uderpiig of statistical iferece. Resamplig techique such as jackkife offers effective estimates of parameters ad its asymptotic distributio. I this paper, we preset the jackkife estimate of the parameters of a simple liear regressio model with particular iterest o the correlatio coefficiet. This procedure provides a effective alterative test statistic for testig the ull hypothesis of o associatio betwee the explaatory variables ad a respose variable. Keywords: Jackkife; simple liear regressio; correlatio coefficiet; ols estimates; bias. 1. Itroductio After estimatio of parameters i applied statistics it is always crucial to assess the accuracy of the estimator by its stadard error ad costructio of cofidece itervals for the parameter [1]. Queouille i 1956 developed a cross validatio procedure kow as jackkife (leaveoeout procedure) for estimatig the bias of a estimator [2]. Two years later this method was further exteded by Joh Tukey to estimate the variace of a estimator ad the ame Jackkife was coied for this cross validatio method [3] * Correspodig author. address: 441
2 Iteratioal Joural of Scieces: Basic ad Applied Research (IJSBAR)(2015) Volume 22, No 2, pp The jackkife algorithm is a iterative procedure. The iitial step is to estimate the parameter(s) from the etire sample. The the ith elemet (datum) is sequetially dropped from the sample ad the model parameters estimated from the reduced sample data. The resultat estimates are called the partial estimate (pseudo estimates) [4]. The mea of the pseudo estimates is referred to as the jackkife estimate used i place of the mai parameter value [5]. Also, from the pseudo estimates the stadard errors of the parameters could be estimated usig the stadard deviatio i order to eable a statistically sigificat test of the parameters ad the costructio of the cofidece iterval [6]. Regressio aalysis has bee widely used to explai the relatioship betwee the explaatory variables ad a respose variable. However, jackkife was foud viable i estimatig the samplig distributio of the regressio coefficiets i the work of Efro [7], ad further exteded by Freedma [8] ad Wu [9]. With a special case of the simple liear regressio model, this article is aimed at illustratig a alterative to the classic test statistic for assessig the sigificace of the correlatio coefficiet usig jackkife estimates. 2. Methods The liear regressio model could be give i matrix form Y = Xθ + ε (1) Where x 11 x 12 x 1p x 21 x 22 x 2p X = x 1 x 2 x p p, is the p desig matrix (matrix of the explaatory variables) ad the remaiig quatities are vectors correspodig to p 1 regressio parameters, 1 respose variable ad 1 ormally distributed error term with zero mea ad costat variace, defied by θ 1 θ = θ p p 1 Y 1, Y = Y 1 ad ε = ε 1 ε 1. The simple liear regressio model with oe explaatory variable (x i, i = 1,2,3,, ) ad two parameters θ (0) ad θ (1) correspodig to the itercept ad slope parameter is a special case of (1). Hece, the ordiary least square (ols) estimator of this model is θ ols (0) ols θ = (X X) 1 X Y (2) (1) 442
3 Iteratioal Joural of Scieces: Basic ad Applied Research (IJSBAR)(2015) Volume 22, No 2, pp Where 1 x 1 1 x X = 2 1 x 2 With variace covariace matrix of θ ols (0) ad θ ols (1) give by var/cov θ ols (0) ols θ = σ 2 (X X) (3) (1) Where the diagoal elemets of (3) are the variaces of θ ols (0) ad θ ols (1) respectively, ad the offdiagoals are their covariaces. Also, the least squares estimate of the correlatio coefficiet which measures the stregth of a liear relatioship is give by the Pearso product momet estimate x i y i x i y i ρ x,y = ( x 2 i [ x i )( y 2 i [ y i ). (4) ] 2 ] 2 This measure of stregth lies withi 1 ρ x,y 1 where the closer it is to 1 the stroger the positive, if closer to 1 the the stroger the egative relatioship, ad the closer it is to 0, the weaker the relatioship. Iterestigly, 1, 0 ad 1 estimates of this measure imply perfect egative, o ad perfect positive relatioships, respectively. Also, it is ofte ecessary to test the sigificace of this parameter with the followig hypothesis ad test statistic Hypothesis: H 0 : ρ x,y = 0 H 1 : ρ x,y 0 Test Statistic ρ x,y 2 1 ρ x,y 2 ~t α,( 2). However, the test statistic above is classical, ad i this article we propose a jackkife based statistic ρ x,y(j) var ρ x,y(j) ~t α,( 2) for testig the above hypothesis. The jackkife estimates of (2) ad (4) is obtaied by leavigout the ith observatio of the pair y i, x i ; i = 443
4 Iteratioal Joural of Scieces: Basic ad Applied Research (IJSBAR)(2015) Volume 22, No 2, pp ,2,3,, ad evaluatig θ ols (J) ad ρ x,y (J) the least squares estimates based o the remaiig observatios [10]. The estimates of θ J ad ρ J, bias ad variace usig the pseudo values θ Ji ad ρ x,y(ji) are θ J = θ Ji (5) With bias bias = θ ols θ Ji (6) Or more succictly bias = θ ols θ J (7) Ad the variace var θ J = θ Ji θ J 2 ( 1) (8) Also, ρ x,y(j) = ρ x,y(ji) (9) With bias bias = ρ x,y ρ x,y(ji) (10) Or bias = ρ x,y ρ x,y(j) (11) Ad variace var ρ x,y(j) = ρ x,y(ji) ρ x,y(j) 2 ( 1) (12) 2.1 Algorithm for Jackkifig Simple Liear Regressio Model Steps: 444
5 Iteratioal Joural of Scieces: Basic ad Applied Research (IJSBAR)(2015) Volume 22, No 2, pp Usig a pair of idepedet sample of size () of explaatory ad respose variables (x i, y i ), i = 1, 2, 3,,. Drop the first datum i both variable ad estimate the ordiary least squares (ols) regressio coefficiets θ (0)J1 ad θ (1)J1 ad the correlatio coefficiet ρ x,y(j1) usig 1 observatios. Drop the secod datum ad replace the iitially dropped datum i (ii) ad compute the ordiary least squares (ols) regressio coefficiets θ (0)J2 ad θ (1)J2 ad the correlatio coefficiet ρ x,y(j2) usig 1 observatios. Repeat steps (ii) ad (iii) by replacig the (i 1)th previously dropped observatio ad droppig the ith observatio ad the computig the ordiary least squares (ols) regressio coefficiets θ (0)Ji ad θ (1)Ji, i = 3, 4, 5,, ad the correlatio coefficiet ρ x,y(ji), i = 3, 4, 5,, usig 1 observatios at each iteratio util all the observatios i the pair (x i, y i ), i = 1, 2, 3,, has bee sequetially dropped ad replaced i turs. Steps (ii) to (iv) results to a dimesioal vectors of pseudo values correspodig to θ (0)Ji, θ (1)Ji ad ρ x,y(ji). Compute the jackkife regressio parameters, correlatio coefficiets ad their correspodig bias ad stadard errors usig (5), (7), (8), (9), (11), ad (12). 3. Data ad Simulatio We have used the total demad ad supply of FOREX (USD millio) data from Jauary 2008 to May (77 data poits) available o the Cetral Bak of Nigeria official website [11]. All computatios are doe usig R programs for widows. 3.1 Simulatio Results Usig the data i 2.0 we fit a simple liear regressio model ad the result is show i Table 1. Table 1: Parameter Estimates for the Fitted Simple Liear Regressio Model Parameters θ (0) θ (1) ρ x,y Estimate Stadard Error Jackkifig the Simple Liear Regressio Model Table 2 shows the ols estimates of the pseudo values, jackkife estimates ad their correspodig stadard errors obtaied from the leaveoeout procedure. 445
6 Iteratioal Joural of Scieces: Basic ad Applied Research (IJSBAR)(2015) Volume 22, No 2, pp Table 2: ols Estimates S/N θ (0)Ji θ (1)Ji ρ x,y(ji) θ (0)J θ (1)J ρ x,y(j) SE θ (0)J SE θ (1)J SE ρ x,y(j) Table 3: Compariso betwee ols ad Jackkife ols Estimates Estimates ols Jackkife Bias θ (0) SE θ (0) θ (1) SE θ (1) ρ x,y SE ρ x,y Testig the sigificace of the correlatio coefficiet We shall proceed to test the sigificace of the correlatio coefficiet at 5% level of sigificace as follows: H 0 : ρ x,y = 0 H 1 : ρ x,y 0 446
7 Iteratioal Joural of Scieces: Basic ad Applied Research (IJSBAR)(2015) Volume 22, No 2, pp classic = ρ x,y 2 1 ρ x,y 2 = ( ) ( ) 2 = jackkife = ρ x,y(j) SE ρ x,y(j) = = with critical value t α,( 2) = t 0.05,(77 2) = Decisio: Sice both test statistics are larger tha the critical value, we therefore coclude that there is eough evidece agaist the ull hypothesis; hece, the correlatio coefficiet is sigificatly differet from 0 at 5% level of sigificace Discussios The jackkife (leaveoeout) ols estimator provides better estimates of the regressio parameters tha the ols method. From Table 3 above it could be see that the Jackkife estimates of both the regressio coefficiets θ (0) ad θ (1) ad the correlatio coefficiet ρ x,y are approximately the ols estimates with very small bias, it is iterestig to observe that the Jackkife estimates has smaller stadard errors (Efficiecy property), a uique feature of a good estimator i compariso to their ols couterpart. The classic test statistic value for testig the sigificace of the correlatio coefficiet is smaller tha the value obtaied from the proposed jackkife test statistic; this is a cosequece of a large variace of the ols estimates. 4. Coclusio Jackkife results are misleadig whe the sample size is ot large eough ( < 50), [12]. Factually, the 77 observatios used i this study reveals that the Jackkife estimators are more efficiet tha their ols couterpart i estimatig the coefficiets of a liear regressio model ad the correlatio coefficiet. It also provides the asymptotic distributio of the above metioed parameters, e.g., Table 2. The classic test statistic for testig the sigificace of the correlatio coefficiet is uderestimated, a effect of large stadard error of the ols estimators ad cosequetly, could lead to erroeously acceptig the ull hypothesis (Type II error). Without loss of geerality, the jackkife based test statistic is better tha its classic couterpart. Refereces [1] M. R. Cherick. Bootstrap Methods a Guide for Practitioers ad Researchers. 2d ed; Joh Wiley & Sos Ic., New Jersey, [2] M. H. Queouille. "Notes o Bias i Estimatio", Biometrika, 61, pp. 117, [3] J. W.Tukey."Bias, ad Cofidece i ot Quite Large Samples (Abstract)" Aals of Mathematical 447
8 Iteratioal Joural of Scieces: Basic ad Applied Research (IJSBAR)(2015) Volume 22, No 2, pp Statistics, 29, pp. 614, [4] H. Friedl ad E. Stampfer. "Jackkife Resamplig", Ecyclopaedia of Ecoometrics, 2, pp , [5] S. Sahiler ad D.Topuz. "Bootstrap ad Jackkife Resamplig Algorithms for Estimatio of Regressio Parameters", Joural of Applied Quatitative Methods, Vol. 2. No. 2. pp , [6] H. Abdi ad J. L. Williams. "Jackkife", I Neil Salkid (Ed.), Ecyclopaedia of Research Desig. Thousad Oaks, CA: Sage,2010. [7] B. Efro. "Bootstrap Method; aother Look at Jackkife". Aals of Statistics, Vol. 7, pp. 126, [8] D.A. Freedma. "Bootstrappig Regressio Models", Aals of Statistics. Vol.1, No. 6, pp , 1981 [9] C. F. J. Wu. "Jackkife, Bootstrap ad other Resamplig Methods i Regressio Aalysis", Aals of Statistics, Vol. 14, No. 4, pp ,1986. [10] J. Shao ad D. Tu. The Jackkife ad Bootstrap, Spriger Verlag, New York, [11] http// date accessed 1\5\2015. [12] Zakariya, Y. A. ad Khairy, B. R., (2010), Resamplig i Liear Regressio Model Usig Jackkife ad Bootstrap, Iraqi Joural of Statistical Sciece. Vol. 18, pp
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