Okolo, A

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  GLOBAL JOURNAL OF MATHEMATICAL SCIENNCES VOL. 8, N0. 1, 2009: 19 - 22 COPYRIGHT (C) BACHUDO SCIENCE CO. LTD. PRINTED IN NIGERIA. ISSN 1596-6208 TRANSFORMATION OF INDEPENDENT VARIABLES IN POLYNOMIAL REGRESSION MODELS                       A. OKOLO                  (Received 24 July 2008; Revision Accepted 5 December 2008)  ABSTRACT  In representing a relationship between a response and a number of independent variables, it is preferable when possible to work with a simple functional form in transformed variables rather than with a more complicated form in the original variables. In this paper, it is shown that linear transformations applied to independent variables in polynomial regression models affect the t ratio and hence the statistical significance for certain parameters of the polynomial regression models.  KEY WORDS:  Transformation, Polynomial regression, Independent variables.  INTRODUCTION  Transformations in experimental, mathematical and statistical work have found use in two major areas: i. that of providing theoretical approximations, and  ii. that of bending the data to conform to assumptions underlying conventional analysis.  The usual techniques, according to Draper and Smith (1998), for the analysis of linear models as exemplified by regression analysis are usually justified by assuming: 1. Simplicity of structure for average values  for the dependent variable, y, 2. constancy of error variance about these average values, 3. normality of distributions, and 4. independence of observations.  Bates and Watts (2007) reported that if assumptions 1 and 3 are not satisfied in terms of the original observations, y, a linear or non-linear transformation of variables might be desirable to produce the simplest possible regression model in the transformed variables.  In regression problems, assumption 1 might be that E(y) is adequately represented by a rather simple empirical function of the independent variables x1, x2, …, xn and we would want to transform so that this assumption together with assumptions 2 and 3, is approximately satisfied. Each of the assumptions 1 and 3 can and has been used separately to select a suitable candidate from a parametric family of transformations (see Neter, Wasserman and Kutner, 1989). The majority of the literature on transformation is concerned with transforming the y’s to achieve simplicity when the necessary assumptions could not otherwise be realistically made (Hocking, 1983; Bates and Watts, 2007).  Most emphasis has been placed therefore on transformations of E(y) which may be expected to stabilize the variance or reduce the regression function to linearity in the parameters.  In regression problems, one can transform both the dependent and the independent variables (Bernhardt and Jung, 1979).  It has been pointed out in Weisberg (2005) that replacing either the dependent, the independent, or both by nonlinear transformations of them is an important tool that the analyst can use to extend the number of problems for which linear regression methodology is appropriate. However, when the independent variables are transformed, it is shown in this paper that additivity is a concern in tests of statistical significance for individual parameters.  TRANSFORMATIONS ON THE X’S Suppose we have a model under consideration, which can be written in the form  19 A. Okolo, Department of Statistics and Operations Research, Federal University of Technology, Yola, Nigeria     Y = xβ ε+ ,          (1) where Y is the nx1 vector of dependent variables, x is the nxk matrix of independent variables, β is the kx1 vector of parameters to be estimated and ε is an nx1 vector of errors.  In the present investigation we suppose that the errors in the Y’s are at least approximately normally and independently distributed with constant variance σ2, where E(ε) = 0 and Var (ε) = Iσ2.  The least squares estimators of β and σ2 are             (2) $ ( )β = ′ ′−x x x Y1and          (3) ( ) ( ) ($ $ $ /σ β β2 = − ′ − −y x y x n k )β β respectively (Draper and Smith, 1998).  Consider a transformation Z = xT of the matrix x.  The transformed model becomes  Y= Zγ + ε = xTγ + ε   = x(Tγ) +ε  =    xβ  +  ε, so γ is a k x 1  vector of coefficient parameters, equal to T-1β.  It then follows that the least squares estimator of γ is .  Since , the two models have the same predicted  $ $γ = −T 1 Ζ $ $γ = xvalues and residuals; values of R2 and  also are identical.  Since  is the same, for simplicity sake, let it be unity.  The respective variance – covariance matrices of the estimators   and $σ 2 $σ 2$β $γ  are  and respectively. ( )′ −x x 1 T x x T− − −′1 1 1( ) ( )  1 TRANSFORMATIONS IN POLYNOMIAL REGRESSION MODELS  ILLUSTRATIVE EXAMPLE   Consider quadratic models in (4) and (5) with the transformationΖ i ia bx= + :  Y xii = +β β0 1 β22xi +    +  εi         (4)  Y zi i= +γ γ0 1 γ 22zi +    +  εi          (5)      = γ γ0 1+ +( )a bxi   +     +  γ 22( )a bxi+ εi        =  +  ( )( )γ γ γ0 1 2 2+ +a a b ab ixγ γ1 22+  +   b   +  xi2 2 2γ εi   From above, β = Tγ  when       1 a a2     T  = 0 b 2ab    ,      0 0 b2  so that    1 −ab   ab22   T-1 = 0 1b   −22ab     .    0 0  1 2b   Using the previous result that $γ   =  , it follows that T −1 $β    $γ 0    1 −ab      a   b22$β0 $γ   = $γ 1   =   T  = 0 −1 $β 1b    −2     $β1ab2  $γ 2    0 0    1b    2$β2 20 A. OKOLO   TRANSFORMATION OF INDEPENDENT VARIABLES IN POLYNOMIAL REGRESSION MODELS               21         -    $β0ab$β1    +  ab22 2$β   =              11b$β     -   ab22 2$β         (6)              12 2b$β   Denote the elements of   by ( ,  i, j = 0, 1, 2. ( )′ −x x 1 )Cij Then                C00     C01 C02  (   =                  C11 C12        (7) )′ −x x 1               C22  where (Cij)   =  (Cji),  so that  Var ( $)γ   =   T-1    (T-1)1 ( )′ −x x 1  1         −ab    ab22         C00      C01 C02   1    0     0   =    0 1b    −22ab      C10      C11 C12  −ab    1b          0  0 0    1 2b         C20      C21 C22  ab22  −2 2ab    12b            CabCabC00 1022 20− +      CabCabC01 1122 21− +       CabCabC02 1222 22− +      1          0      0 =         1 210 2 20bCabC−              1 211 2 21bCabC−    1 212 2 22bCabC−               −ab     1b        0             12 20bC    12 21bC    12 22bC       a b22  −2 2ab  12b      C002abC012a2b2C02a2b2C112a3b3C12a4b4C22          1bC01ab2C113a2b3C122ab2C022a3b4C22            1b2C02ab3C12a2b4C22− + + − + − + − − − +       .8             12 1143 124 24 22bCabCabC− +            13 12bC -  24 22abC                   14 22bC        22 A. OKOLO Let t (βi) and t (γi) denote the t ratio associated with βi and γi respectively.  Using (7),    t (β1)  =  $β111C and t (β2)  =  $β222C.       (9)  Furthermore, using (6) and (8), we obtain  t (γ1)   =   1 21 4 41 211 1222 22babbCabCabC$ $β β−⎛⎝⎜⎞⎠⎟− +  and   t (γ2)   =  112 22 22bbC$β.           (10)  We observe that t (β1)  t (γ1), while t (β2) = t (γ2).  Notice that  t (γ1) will equal  ≠t (β1) when a is zero.  That is, the additive quantity a, in the transformation Z = a + bx is responsible for the non-equality of the t ratio for the lower order coefficient.   CONCLUSION   The most frequent purpose of transformations is to achieve a mean function that is linear in the transformed scale.  Over the years, emphasis for transformation has tended to be on obtaining a constant error variance.  In all cases, we are concerned not merely to find a transformation which will justify assumptions but rather to find, where possible, a metric in terms of which the findings may be succinctly expressed.  Simplicity and ease commend transformations because it is much simpler to transform the variables, estimate the coefficients, and merely inspect the t ratios for possible differences.  It has been shown that additivity is a problem when using linear transformation in  polynomial regression models.  REFERENCES  Bates, D. M. and Watts, D. G., 2007. Nonlinear Regression Analysis and Its Applications.John  Wiley and Sons, Inc., New York, 32pp.  Bernhardt, I. and Jung, B. S., 1979. The Interpretation of Least Squares Regression with Interaction or Polynomial Terms. The Review of Economics and Statistics, 61: 481 – 483.  Draper, N. R. and Smith, H., 1998. Applied Regression Analysis. 3rd Edition, John Wiley and Sons,  Inc., New York, 253pp.  Hocking, R. R., 1983. Developments in Linear Regression Methodology (with discussion).   Technometrics, 25 (3): 219 – 249.  Neter, J., Wasserman, W. and Kutner, M. H., 1989. Applied Linear Regression Models. 2nd Edition,  Homewood, NJ, Irwin, 506pp.  Weisherg, S., 2005. Applied Linear Regression.  3rd Edition, John Wiley and Sons, Inc., New York,  146pp.      

Transformation of independent variables in polynominal regression models

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Transformation of independent variables in polynominal regression models

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