Durban Watson for Testing the Lack-of-Fit of Polynomial Regression Models without Replications
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1 Durban Watson for Testng the Lack-of-Ft of Polynomal Regresson Models wthout Replcatons Ruba A. Alyaf, Maha A. Omar, Abdullah A. Al-Shha Department of Statstcs and Operatons Research, College of Scences Kng Saud Unversty, Saud Araba Abstract: A parametrc approach usng Durban Watson test s developed to test the lack of ft of the followng regresson model when replcaton s not avalable. p Y = β 0 + β j=1 j X j + ε ; = 1, 2,, n. Ths paper s based on the dea of testng the lack of ft dependng on the randomness of the sample v 1, v 2,, v n. We assumed that the errors e 1, e 2,, e n are a random sample from a normal dstrbuton wth mean equal to zero and varance σ 2. Smulaton results are presented to show the power and the sze of the modfed Durban Watson test (1951). Key words and phrases: Parametrc test, Durban Watson, lack of ft, model adequacy, regresson model, power of the test, sze of the test and replcaton. 1. Introducton A very well-known Technque, the classcal lack of ft F-test Fsher (1922), s used to test the lack of ft for regresson models when replcaton s avalable. On the other hand, when replcaton s not avalable most of the studes tend to partton the sample space of the contnuous covarant. Ths study suggests a smple approach to test the adequacy of the regresson model usng Durban Watson test technque. We wll test the lack of ft of the followng regresson model (1985) when replcaton s not avalable. p Y = β 0 + β j=1 j X j + ε ; = 1, 2,, n. (1.1) If the regresson model s approprate then the errors ε 1, ε 2,, ε n s a random sample from a normal dstrbuton wth mean equal to zero and varance σ 2. Approxmately, t can be consdered JGRMA 2017, All Rghts Reserved 165
2 that the resduals e s ( = 1,2,, n) s a random sample from a normal dstrbuton wth mean equal to zero and varance σ 2. The value X s the value of the ndependent varable for the th tral. Let X (1) < X (2) < X (n) be the ordered values for the ndependent varable and v = e () s the resduals assocated wth the ordered value X (). In ths test, we bult a lack of ft test dependng on the randomness of the sample v 1,v 2,, v n. To test the adequacy of the model based on the proposed lack of ft tests wth the hypotheses H 0 : the sample v 1, v 2,, v n s random vs. H 1 : the sample v 1,v 2,, v n s not random, we run a smulaton study and calculated the emprcal sze and power of the tests. In ths smulaton, we consdered testng the lack of ft of the followng three regresson models,.e. when p = 1, 2 and 3 n (1.1): Model (1): y 1 X ; = 1, 2,, n. (1.2) o Model (2): Model (3): y y X o X o X ; = 1, 2,, n. (1.3) X X ; = 1, 2,, n. (1.4) Note that Model (0) refers to the case y ; = 1, 2,, n. o The smulaton s done usng the software Mathematca 8 and the tests are repeated tmes. We generated the ndependent varable X form random real numbers wth mnmum value 0 and a maxmum value c (where c = 1, 5, 20 and 50). The dependent varable Y s calculated from the specfed model (true model (1) or (2) or (3)) then we ft Y to the true model and the false model (the false model s a model wth a lower degree). In the smulaton, we consdered the followng values: n s the sample sze that takes the values 10, 30 and 100. σ 2 s the varance that takes the values 0.1, 0.5, 1 and 5. α s the sgnfcant level that takes the values 0.05 and 0.1. We assumed β 0 = 1 and β 1 = 1 (when dealng wth model (3)). j 0.5,1,5 for j 1,2,..., p. For each run, the emprcal sze of the test α (the test s probablty to reject H 0 when H 0 s true ) s calculated as follows: JGRMA 2017, All Rghts Reserved 166
3 the emprcal sze of the test = number of rejectng the true model when the y values are generated from that model number of runs The emprcal power of the test 1 β (the test s probablty to reject H 0 when H 0 s false) s calculated as follows: the emprcal power of the test = number of rejectng the false model when the y values are generated from the true model number of runs For both, the true and false models, the resduals and the standardzed resduals assocated wth the ordered X values are calculated. The Durbn Watson test statstc s calculated for both the true and the false model. Usng the two-sded rejecton rule the emprcal sze and power of the test s obtaned. A counter s ntated, whenever the test s nconclusve the counter adds up and the sample s omtted. Note: The Durbn-Watson test s not applcable when usng Model (0) as the false model and Model (1) as the true model,.e. when p = 1, snce Durbn-Watson tables are gven only when a constant term n the regresson model exst and at least one ndependent varable. 2. The Test In ths test, we propose testng the lack of ft of the regresson model (1.1).e. the randomness of the resduals v 1,v 2,, v n assocated wth the ordered x values by testng the seral correlaton of the ordered resduals usng the Durbn-Watson (1951) test. Ths test tests the null hypothess that the resduals are not auto-correlated H 0 : all ρ s = 0 aganst the alternatve H 1 : ρ s = ρ s whch s equvalent to H 0 = the proposed model s approprate aganst H 1 = the proposed model s not approprate. The alternatve takes place from the assumpton that the errors n the regresson model (1.1) are generated by frst-order autoregressve processes that are observed at equally spaced tme perods. such that: ε t = ρε t 1 + z t (2.2) Where ε t s the error component at tme t and z t are.. d. N(0,σ 2 ). ε t and z t are ndependent. The Durbn Watson test statstc d s used on model (1.1). Usng the two-sded rejecton rule, at level α, we reject H 0 f d < d L or 4 d < d L, not reject H 0 f d > d U and 4 d > d U, otherwse the test s sad to be nconclusve. JGRMA 2017, All Rghts Reserved 167
4 The fgure and the table below dsplay some cases of the emprcal sze and the power of the test to llustrate our method. Fgure 1 shows the power and sze of the test for testng the lack of ft for model (1.1) when p = 2, for α =.05, β 1 =.5 β 2 = 5 and dfferent values of n, c and σ 2. Table 1 shows the power and sze of the test for both resduals and standardzed resduals when testng the lack of ft for model (1.1) when p = 2, for σ 2 =.1, β 1 =.5 and dfferent values of n, c, β 2 and α. JGRMA 2017, All Rghts Reserved 168
5 Sze of the test power of the test Fgure(1) The sze and the power of the modfed Durban Watson test ( α =. 05, β 1 =. 5 β 2 = 5) JGRMA 2017, All Rghts Reserved 169
6 Table 1 The sze and the power of the modfed Durban Watson test ( σ 2 =. 5 β 1 =. 5) α β 2 n resduals Standardzed resduals c=1 c=5 c=20 c=50 c=1 c=5 c=20 c=50 α 1 β α 1 β α 1 β α 1 β α 1 β α 1 β α 1 β α 1 β JGRMA 2017, All Rghts Reserved 170
7 3. Conclusons: Our followng conclusons apples for p = 2 and Comparng the performance of the test usng standardzed resduals and regular resduals, we get almost the same results. 2- When n=10 sometmes the test s sad to be nconclusve and ths s due to the large range between the d l and d u value relatvely wth the Durbn Watson test statstc, as n ncreases the nconclusve cases decreases. Example: When n = 10, β 1 =.5, β 2 =.5,σ 2 =.1,c = 5. For α =.05 : Number of nconclusve for true model.e. model 2 (1.3) s Number of nconclusve for false model.e. model 1 (1.2) s For α =.1 : Number of nconclusve for true model.e. model 2 (1.3) s Number of nconclusve for false model.e. model 1 (1.2) s When n = 100, β 1 =.5, β 2 =.5, σ 2 =.1,c = 5. For α =.05 : Number of nconclusve for true model.e. model 2 (1.3) s 528. Number of nconclusve for false model.e. model 1 (1.2) s 0. For α =.1 : Number of nconclusve for true model.e. model 2 (1.3) s 900. Number of nconclusve for false model.e. model 1 (1.2) s The emprcal sze of the test almost decreases as n ncreases. 4- The emprcal sze of the test ncreases as α ncreases. 5- The emprcal sze of the test ncreases as c ncreases. 6- The emprcal power of the test ncreases when n ncreases. 7- The emprcal power of the test ncreases when c ncreases. 8- The emprcal power of the test slghtly ncreases when β 1 ncreases. 9- The emprcal power of the test ncreases when β 2 ncreases. 10- The emprcal power of the test ncreases when α ncreases. JGRMA 2017, All Rghts Reserved 171
8 11- As n ncreases the emprcal power of the test almost always ncreases especally for larger c values. 12- As σ 2 ncreases the emprcal power of the test decreases. References [1] Durbn, J., and Watson, G. S. (1950) "Testng for Seral Correlaton n Least Squares Regresson, I." Bometrka 37, [2] Durbn, J., and Watson, G. S. (1951) "Testng for Seral Correlaton n Least Squares Regresson, II." Bometrka 38, [3] Fsher, R. A. (1922), The Goodness of Ft of Regresson Formulae and the Dstrbuton of Regresson Coeffcents. J. Roy. Statst. Soc. 85, [4] John, N., Mchael, H. and Wllam, W.(1985) Appled Lnear Statstcal Models. JGRMA 2017, All Rghts Reserved 172
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