Quadratic Models

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In statistics, linear regression is a linear approach to model the relationship between a response (or dependent variable) and one or more explanatory variables (or independent variables). The case of one explanatory variable is called simple linear regression. For more than one explanatory variable, the process is called multiple linear regression.

Linear regression models should have co-efficient of order one; but independent variables could have order more than one. So the models defined by below equations are referred as linear models.and

In fact, they are the second order polynomials in one and two variables respectively.

The polynomial models can be used in those situations where the relationship between study and explanatory variables is curvilinear. Sometimes a nonlinear relationship in a small range of explanatory variable can also be modeled by polynomials.

In the first equation given above; the coefficients β1 and β 2 are called the linear effect parameter and quadratic effect parameter respectively.

Considerations while fitting polynomial model are as follows:

  1. Order of the model: The order of the polynomial model is kept as low as possible. Some transformations can be used to keep the model to be of first order. If this is not satisfactory, then second order polynomial is tried. Arbitrary fitting of higher order polynomials can be a serious abuse of regression analysis.
  2. Model building strategy: One possible approach is to successively fit the models in increasing order and test the significance of regression coefficients at each step of model fitting.
  3. Extrapolation: One has to be very cautions in extrapolation with polynomial models. The curvatures in the region of data and region of extrapolation can be different.
  4. Ill-conditioning: A basic assumption in linear regression analysis is that rank of X-matrix is full column rank. In polynomial regression models, as the order increases, the X^2 and X matrix becomes ill-conditioned. As a result, the X^2 and X may not be accurate and parameters will be estimated with considerable error.

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