Applications. The General Linear Model. Transformations. Transformations to Linearity. Many non-linear curves can be put into a linear form by appropriate transformations of the either the dependent variable Y or some (or all) of the independent variables X 1 , X 2 , ... , X p.

Download Presentation

Applications

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

Transformations to Linearity • Many non-linear curves can be put into a linear form by appropriate transformations of the either • the dependent variable Y or • some (or all) of the independent variables X1, X2, ... , Xp . • This leads to the wide utility of the Linear model. • We have seen that through the use of dummy variables, categorical independent variables can be incorporated into a Linear Model. • We will now see that through the technique of variable transformation that many examples of non-linear behaviour can also be converted to linear behaviour.

The ln-transformation is a member of the Box-Cox family of transformations with l= 0 • If you decrease the value of lthe effect of the transformation will be greater. • If you increase the value of lthe effect of the transformation will be less.

Non-Linear Growth models The Mechanistic Growth Model • many models cannot be transformed into a linear model Equation: or (ignoring e)“rate of increase in Y”=

Comment: A cubic polynomial in x can be fitted to y by defining the variables X1 = x, X2 = x2, and X3 = x3 Then fitting the linear model The Model – Cubic polynomial (degree 3)

Dependent variable Y and two independent variables x1 and x2. (These ideas are easily extended to more the two independent variables) Response Surface models (2 independent vars.) The Model (A cubic response surface model) Compare with a linear model:

The response surface model can be put into the form of a linear model : Y = b0 + b1X1 +b2X2 + b3X3 +b4X4 + b5X5 + b6X6 + b7X7 + b8X8 + b9X9+ e by defining

More Generally, consider the random variable Y with 1. E[Y] = g(U1 ,U2 , ... , Uk) = b1f1(U1 ,U2 , ... , Uk) + b2f2(U1 ,U2 , ... , Uk) + ... + bpfp(U1 ,U2 , ... , Uk) = and 2. var(Y) = s2 • where b1, b2 , ... ,bp are unknown parameters • and f1 ,f2 , ... , fp are known functions of the nonrandom variables U1 ,U2 , ... , Uk. • Assume further that Y is normally distributed.

Now suppose that n independent observations of Y, (y1, y2, ..., yn) are made corresponding to n sets of values of (U1 ,U2 , ... , Uk) : (u11 ,u12 , ... , u1k), (u21 ,u22 , ... , u2k), ... (un1 ,un2 , ... , unk). Let xij = fj(ui1 ,ui2 , ... , uik) j =1, 2, ..., p; i =1, 2, ..., n. Then or

Definition Consider the values x0, x1, … , xnand the polynomials are orthogonal relative to x0, x1, … , xnif: If in addition , they are called orthonormal