Chapter 6 – Estimating Variance Components of Tack Measurements

1. Chapter 6 – Estimating Variance Components of Tack Measurements An important consideration in the manufacture of type of item is consistency; all items of a particular type coming off an assembly line should have the same measured dimensions. Some variation in these dimensions is unavoidable, but the manufacturer wants to minimize such variation. Therefore the manufacturer employs a a staff of people to do quality control; at regular times, the quality control staff will select a sample of manufactured items and make measurements. However, the measurements themselves are also subject to variation, called measurement error. It is important to distinguish between measurement error and variation due to actual differences among the manufactured items.

2. Hence the quality control staff will also do calibration studies, to estimate the relative importance of measurement error in their data. You are a statistician working in the Quality Department of the Sharp Point Tack Company. The company has recently set up a new factory. Plant manager William Bossman wants an evalution of the process for measuring the lengths of nominal ¾ inch carpet tacks. Operators routinely take these measurements as part of the plant’s quality control effort. The measurement device is a micrometer.

3. A tack length measurement equals the true length plus measurement error. We want variation among the actual true tacks lengths to be small, to assure the quality of the manufactured product. But we want measurement error to be even smaller, to enhance the quality control process. There are three sources of measurement error: a) operator-to-operator variability, b) micrometer-to-micrometer variability, and c) inherent variability (lack of repeatability) in the measurement process.

4. Operator-to-operator differences can be reduced by training the operators to follow a measurement protocol. Micrometer-to -micrometer differences may be reduced by frequently calibrating the micrometers. Inherent variability is independent of operator-to -operator differences and micrometer-to-micrometer differences. A reduction in inherent variability would require a fundamental change in the measurement process, which could be expensive to implement. We will perform an experiment, called a calibration study, to allow us to estimate the amount of variation due to each source.

5. The model for our calibration study is as follows: where Yijkm is the mth measurement of the ith tack by the jth operator using the kth micrometer;  is an overall grand mean; Ti is a tack effect random variable, having variance ; Oj is an operator effect random variable, with variance ; Mk is a micrometer effect random variable, with variance ; and ijkm is a random error term.