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Growth Curve Modeling: Theory and Applications by Michael J. Panik

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6

NONLINEAR REGRESSION

6.1 INTRINSIC LINEARITY/NONLINEARITY

In the classical linear regression model, the multivariate regression hyperplane has the form

images

where we have m predetermined explanatory variables or regressors X1, ..., Xm and ε is a random error term. Here Equation 6.1 depicts a linear regression model since it is linear in the parameters β0, β1, ..., βm. It is assumed that n > p = m + 1 and no exact linear relationship exists between the Xj's, j = 1,..., m.

For fixed Xj's, j = 1,..., m, the population regression hyperplane is specified as the conditional mean of Y given the Xj's or

images

given that E(ε) = 0. Given Equation 6.2,

images

that is, the population intercept is the mean of Y given that all of the Xj's are set equal to zero. Also,

images

is termed the jth partial regression coefficient; that is, as Xj increases by one unit, the average value of Y changes by βj units, given that all remaining explanatory variables are held constant. Once the βk's, k = 0, 1,..., m, are estimated from the sample information, we obtain the sample regression hyperplane

where Ŷ is the estimated value of ...

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