Experimental design in the presence of an uncontrollable variable: Model characteristics and design augmentation in a front wheel alignment experiment.

Abstract

An off-line experiment was performed to estimate a model for process control describing the empirical relationship between the design and production factors affecting the front wheel camber angle. During the experiment, an additional factor was identified which could not be manipulated. Labeled an uncontrollable variable, this factor had to be included in the experiment since it was shown to affect the camber angle. Motivated by this experiment, problems are identified in experimental designs which ignore an uncontrollable variable. It is shown how the uncontrollable variable affects the determinant of the information matrix and the variance of predictions generated from the model. Numerous examples are given. A methodology is developed for improving the original experimental design by augmenting the experiment with another observation of the uncontrollable variable modeled as a normally distributed random variable. Under the constraints of a cost model, four sampling options are evaluated for choosing a value of the uncontrollable variable for the augmented design. Augmented designs are evaluated using two criteria: the expected value of $\vert$X$\sp\prime$X$\vert$, and the expected center the information function (with respect to the uncontrollable variable). The methodology augments the experiment with either a 3$\sp3$, a 3 x 2 x 2, or a 2$\sp3$ factorial design. Expressions for the expected value of $\vert$X$\sp\prime$X$\vert$, the expected center of the information function, and the expected cost of the augmented design are derived. Depending upon the cost of measuring the uncontrollable variable, more than one sampling option results in an expected value of $\vert$X$\sp\prime$X$\vert$ which is greater than the value expected by merely selecting the next observation of the uncontrollable variable. More than one sampling option also results in an expected center of the information function closer to the mean of the uncontrollable variable distribution than the value expected by merely selecting the next observation. Strategies to center the information function can lead to a greater expected value of $\vert$X$\sp\prime$X$\vert$. Both desirable design criteria are expected depending upon the cost of measuring the uncontrollable variable. The methodology improves the process control model by allowing minimal variance predictions to be generated for vehicles representative of those produced in the assembly plant.