4.
Process Modeling
4.6. Case Studies in Process Modeling 4.6.3. Ultrasonic Reference Block Study


Weighting  Another approach when the assumption of constant variance of the errors is violated is to perform a weighted fit. In a weighted fit, we give less weight to the less precise measurements and more weight to more precise measurements when estimating the unknown parameters in the model.  
Finding an Appropriate Weight Function 
Techniques for determining an appropriate weight
function were discussed in detail in
Section 4.4.5.2.
In this case, we have replication in the data, so we can fit the power model $$ \begin{eqnarray} \ln{(\hat{\sigma}_i^2)} & = & \ln{(\gamma_1x_i^{\gamma_2})} \\ & & \\ & = & \ln{(\gamma_1)} + \gamma_2\ln{(x_i}) \end{eqnarray} $$ to the variances from each set of replicates in the data and use $$ w_i = \frac{1}{x^{\hat{\gamma}_2}_i} $$ for the weights. 

Fit for Estimating Weights 
The following results were obtained for the fit
of ln(variances) against ln(means) for the replicate
groups.
Parameter Estimate Stan. Dev t Value γ_{0} 2.5369 0.1919 13.1 γ_{1} 1.1128 0.1741 6.4 Residual standard deviation = 0.6099 Residual degrees of freedom = 20 The fit output and plot from the replicate variances against the replicate means shows that the linear fit provides a reasonable fit, with an estimated slope of 1.1128. Based on this fit, we used an estimate of 1.0 for the exponent in the weighting function. 

Residual Plot for Weight Function 
The residual plot from the fit to determine an appropriate weighting function reveals no obvious problems. 

Numerical Results from Weighted Fit 
The results of the weighted fit are shown below.
Parameter Estimate Stan. Dev t Value b1 0.146999 0.1505E01 9.8 b2 0.005280 0.4021E03 13.1 b3 0.012388 0.7362E03 16.8 Residual standard deviation = 4.11 Residual degrees of freedom = 211 

Plot of Predicted Values 
To assess the quality of the weighted fit, we first generate
a plot of the predicted line with the original data.
The plot of the predicted values with the data indicates a good fit. The model for the weighted fit is $$ \hat{y} = \frac{\exp(0.147x)}{0.00528 + 0.0124x} $$ 

6Plot of Fit 
We need to verify that the weighted fit does not violate the regression assumptions. The 6plot indicates that the regression assumptions are satisfied. 

Plot of Residuals 
In order to check the assumption of equal error variances in more detail, we generate a fullsized version of the residuals versus the predictor variable. This plot suggests that the residuals now have approximately equal variability. 