Open challenges: Correlation of intermediate and final measurements

Problem statement

In the relentless pursuit of product quality, companies know that they must also stay financially viable. While quality improvement efforts can eliminate the need for quality testing and measurement, in regulated industries, measurement and testing of products is a necessary element to ensure high quality and to meet customer requirements. While measurement traditionally has been done on the final product prior to shipping to the customer, it is becoming easier to measure more frequently at intermediate stages of the manufacturing process. These upstream, intermediate measurements are often highly correlated with the final product measurements. The upstream and final measurements are often done using the same measurement methods and result in the same measurement units.

For example, consider a manufacturing process of a medical device where a final sterilization step is done prior to shipping the device to a customer. Measurement of the product typically occurs after the final process step. There are customer specifications associated with the final measurements which must be met to ensure product quality. For practical reasons, a manufacturer would like to be able to measure product prior to the sterilization step to avoid additional costs associated with sterilization of any bad devices.

The measurements of products can be destructive, in that once the product is tested, it is no longer usable or able to be shipped to a customer. Alternatively, measurements can be nondestructive where repeated measurements can be made on the same product without impacting its ability to function as intended.

When measurements are nondestructive, then intermediate measurements can be a valuable way to screen out lower quality product prior to continue processing. Of course, this assumes that intermediate measurements provide a cost savings over that of continued processing and final measurements. In addition, the correlation between intermediate measurements and the final measurements can be easily determined. An appropriate model, such as a simple linear regression, can be used to quantify the correlation and correctly account for the variability in the measurement process.

However, when the measurements are destructive, then it is more challenging to use intermediate measurements. Because you are no longer able to test the same product twice, you cannot quantify the correlation between the intermediate and final measurements. Doing intermediate and final measurements on different products reduces the amount of product available for sale at the end of the process.

Thus the manufacturer would like to be able to test at an intermediate stage of the process and take advantage of the presumably high (but unknown) correlation between the intermediate and final measurements to ensure high product quality.

Mathematically, we assume that the intermediate measurement data is denoted as a random variable X with some known mean and variance and that the final measurement result is represented by a random variable Y, also with some mean and variance. We also assume that we have an available dataset with n₁ samples that were measured at an intermediate stage and n₂ samples that were measured at the final stage. This intermediate measurement data is denoted as x₁, x₂ , …, x_n1 and the final measurement data is denoted as y₁, y₂ , …, y_n2. Because of the destructive measurements, the covariance of X and Y is not known.

So, the problem becomes the following: given a set of data for both the intermediate and final measurements, and given a specification on the final measurement, what requirement should be used on the intermediate measurement to ensure that the specification will be met on the final measurement?

There are many other similar applications where intermediate measurements are more easily obtainable than final measurements, such as for accelerated testing conditions which mimic real world aging effects. Another potential application is when measurement is done on a convenient location of a product which may have a bias relative to a randomly selected location.

Available data

This generated dataset is representative of a real data situation described previously about a sterilization process and is shown in Table 1. We have available n₁ = 20 intermediate measurements prior to sterilization and n₂ = 20 final measurements taken after sterilization. Graphically, we can represent the data in Figure 1. We assume that for the final measurement, Y, the specification is such that the values must be between 25 and 35. We have labeled these values respectively as the lower specification limit (LSL) and upper specification limit (USL) on Figure 1. We would like to determine the appropriate LSL and USL for the intermediate measurement based on this dataset. We also assume that these measurements include both the short-term and long-term variability of the process, even though from Figure 1 we see that the intermediate measurements have more variability than the final measurements. The process is generally stable and capable as evidenced by the fact that none of the final measurements fall outside the specifications. Because this is a single, nonrecurring calculation that needs to be done to establish some specification limits to be used in the future process, computationally intensive methods could reasonably be used.

Table 1. Raw data.

X	Y
32.49	26.56
34.07	27.43
35.17	28.17
35.04	28.65
37.58	28.77
37.72	29.62
38.77	29.9
40.07	30.07
40.43	30.13
41.92	30.36
42.36	30.51
42.4	30.55
44.19	30.97
45.07	31.08
45.73	31.53
46.54	31.65
46.71	31.9
48.21	32.13
49.82	32.31
51.27	32.61

Figure 1. Intermediate and final measurement data.