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Fit Analyses

Weighted Analyses

If the errors \varepsilon_{i} do not have a common variance in the regression model

y_{i} = f( x_{i}) + \varepsilon_{i}
a weighted analysis may be appropriate. The observation weights are the values of the Weight variable you specified.

In parametric regression, the linear model is given by

y = X {\beta} + {{\epsilon}}

Let W be an n×n diagonal matrix consisting of weights w1>0, w2>0, ..., and wn>0 for the observations, and let W1/2 be an n×n diagonal matrix with diagonal elements w11/2, w21/2, ..., and wn1/2.

The weighted fit analysis is equivalent to the usual (unweighted) fit analysis of the transformed model

y^{{\ast}} = X^{{\ast}} {\beta} + {\epsilon}^{{\ast}}
where { y^{{\ast}} = W^{1/2} y}, { X^{{\ast}} = W^{1/2} X}, and { {\epsilon}^{{\ast}} = W^{1/2} {\epsilon}}.

The estimate of \beta is then given by

bw = (X'WX)-1 X'Wy

For nonparametric weighted regression, the minimizing criterion in spline estimation is given by

S(\lambda) = \frac{1}{\sum_{i=1}^n{w_{i}}} \sum_{i=1}^n{w_{i} \{ y_{i} - \hat... ... \}^2} + \lambda \int_{-{\infty}}^{{\infty}}{\{ {\hat{f_\lambda}"}(x) \}^2 dx}

In kernel estimation, individual weights are

W(x, x_{i};\lambda) = \frac{w_{i} K_{0}(\frac{x- x_{i}}{\lambda})}{\sum_{j=1}^n{w_{j} K_{0}(\frac{x- x_{j}}{\lambda})} }

For generalized linear models, the function { a_{i}(\phi) = \phi / ( m_{i} w_{i})} for binomial distribution with mi trials in the ith observation, { a_{i}(\phi) = \phi / w_{i}} for other distributions. The function { a_{i}(\phi)} is used to compute the likelihood function and the diagonal matrices Wo and We.

The individual deviance contribution di is obtained by multiplying the weight wi by the unweighted deviance contribution. The deviance is the sum of these weighted deviance contributions.

The Pearson \chi^2 statistic is

\chi^2 = \sum_{i=1}^n{w_{i} m_{i} ( y_{i}- \mu_{i})^2 / V( \mu_{i}) }
for binomial distribution with mi trials in the ith observation,
\chi^2 = \sum_{i=1}^n{w_{i} ( y_{i}- \mu_{i})^2 / V( \mu_{i}) }
for other distributions.

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