Chapter 6 – Estimating Variance Components of Tack Measurements

An important consideration in the manufacture of type of item isconsistency; all items of a particular type coming off an assemblyline should have the same measured dimensions. Some variationin these dimensions is unavoidable, but the manufacturer wants tominimize such variation. Therefore the manufacturer employs aa staff of people to do quality control; at regular times, the qualitycontrol staff will select a sample of manufactured items and makemeasurements. However, the measurements themselves are alsosubject to variation, called measurement error. It is important todistinguish between measurement error and variation due toactual differences among the manufactured items.

Hence the quality control staff will also do calibration studies, toestimate the relative importance of measurement error in theirdata.You are a statistician working in the Quality Department of theSharp Point Tack Company. The company has recently set up anew factory. Plant manager William Bossman wants an evalutionof the process for measuring the lengths of nominal ¾ inch carpettacks. Operators routinely take these measurements as part of theplant’s quality control effort. The measurement device is amicrometer.

A tack length measurement equals the true length plusmeasurement error. We want variation among the actual truetacks lengths to be small, to assure the quality of themanufactured product. But we want measurement error to beeven smaller, to enhance the quality control process. There arethree sources of measurement error: a) operator-to-operatorvariability, b) micrometer-to-micrometer variability, and c)

inherentvariability (lack of repeatability) in the measurement process.

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

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.

ijkmkjiijkm MOTY

2T

2O

2M