Have you ever wondered whether the outgoing inspection values from your vendor are equivalent to the values of your incoming inspection? Maybe it’s time to use orthogonal regression to see if one of you can stop inspecting.

## If your vendor is doing final outgoing inspection, do you need to also do incoming inspection on the same materials?

This beverage company purchases PET (plastic) blow molded bottles from one main vendor for its carbonated liquid refreshment beverages. Consistency and correctness of the bottle height is critical. Bottles are filled on a very high-speed sensitive piece of equipment. Variations in dimension can cause work stoppage and waste if there are jam-ups and the machine goes down.

The vendor selects sample bottles for final inspection. When the vendor delivery trucks arrive at the shipping dock, the company selects samples and does incoming inspection on the bottles from the vendor.

Unfortunately, there are often differences in measurements between the vendor and the company. Can the company be assured that the two processes for measurement are equivalent? If not, then there is a problem. If yes, possibly the company could stop doing incoming inspections, at least to the degree that they are currently doing. This would save time and money and a lot of discussion between the two Quality departments.

Linda, the newly hired Master Black Belt (MBB) was asked to statistically test whether the values provided by the vendor and the company’s own quality control incoming inspection numbers were statistically equivalent.

## Linda decided her first step was to check both measurement systems using a Crossed Gage R&R.

A Crossed Gage R&R will demonstrate whether the measurement systems at the vendor and the company are capable of discriminating variation to the degree needed to get reliable measurements. The following procedure was used:

• The company requested that the vendor do a Crossed Gage R&R
• A sample of 20 bottles was randomly selected
• 3 QC associates were selected to do the study
• The height of each bottle was measured 3 times
• Results were analyzed using a popular statistical software package
• The same 20 bottles were shipped to the company who performed the same study using their equipment and their QC associates
• The purpose of these two Gage R&R studies was to establish whether both of the measurement systems could be trusted
• Both studies came back with sufficient Part-to-Part % Contribution (VarComp) to have confidence that they could distinguish variation in the bottles

Below are the outputs for both studies with the vendor results first, followed by the company results:

## Linda was now ready to go on to the next step.

Linda concluded that both the vendor and her company’s measurement systems can each detect variation and precision in bottle size with the company being a bit better. But how do I compare the two with respect to accuracy?

To answer that question, Linda had both the vendor and her company QC department randomly select 50 bottles from their respective warehouses and measure them using their measurement systems. Her first thought was to do a 2-sample t test to determine if the measurements were different.

Unfortunately, this only tests for the difference in means and doesn’t take into consideration variation. It tells you that the population that the bottles came from were the same. They should be. It doesn’t tell you about the equivalency of the two measurement systems. Below are the results from the 2-sample t test.

## Linda then had a different thought.

Linda considered running a regression and regressing the vendor values with the company values. To do this, Linda did the following:

• Linda asked the vendor to randomly select 50 bottles
• The vendor measured them using the current standard procedure
• The bottles were then sent to the company who measured the same 50 bottles using their standard procedure
• The company values were chosen to be the response variable and the vendor values to be the predictor variable

The problem is error is measured only in the response variable (vertical distance of points to the prediction line) and not in the predictor error.

The results below show there was a good relationship and correlation based on the Rsq value but the residuals indicate something was unusual about the data.

## Linda was not satisfied with her analysis, so she made a decision to use Orthogonal Regression to answer her question about the equivalency of values between the company and the vendor.

Orthogonal regression is a powerful statistical tool although not commonly used. Linda met with the senior QC staff and explained what orthogonal regression is and how it can help her answer the question of equivalency of values between the company and the vendor.

Orthogonal regression, also known as total least squares regression, is a statistical method for fitting a linear relationship between two variables in which both variables are assumed to have measurement errors. This is true of Linda’s situation since the measurements of bottle height at both the vendor and company will have some inherent measurement error.

In traditional simple linear regression, only the dependent variable is assumed to have measurement error, while the independent variable is assumed to be measured without error. However, in some situations, such as in physical measurements, both variables can have measurement errors. Orthogonal regression considers the errors in both variables by minimizing the perpendicular distances between the observed data points and the fitted line, rather than just the vertical distances as in traditional linear regression. The result is a line that provides the best fit to the data, considering the measurement errors in both variables.

Orthogonal regression, like any statistical method, makes certain assumptions about the data being analyzed. The main assumptions of orthogonal regression are:

1. Linearity – The relationship between the two variables is assumed to be linear, meaning that a straight line is the best way to fit the data.
2. Homoscedasticity – The variances of the measurement errors in both variables are assumed to be constant across the range of the data.
3. Errors in both variables – Both the independent and dependent variables are assumed to have measurement errors. The errors are assumed to be uncorrelated with each other and with the true values of the variables.
4. Normality – The errors in both variables (residuals) are assumed to be normally distributed. This assumption is important because the regression analysis relies on the normality assumption to estimate the uncertainty of the regression coefficients.

When these assumptions are met, orthogonal regression can provide an accurate and robust estimate of the true relationship between the two variables. However, if the assumptions are violated, the results of the analysis may be biased or unreliable. It is important to check the assumptions of orthogonal regression before interpreting the results.

## 3 best practices when using Orthogonal Regression.

Here are some best practices for using orthogonal regression:

1. Use orthogonal regression when appropriate

Orthogonal regression is appropriate when both variables in the regression equation are subject to measurement error. If only one variable has measurement error, then traditional least squares regression is more appropriate.

1. Check the assumptions

As with any regression technique, it is important to check the assumptions of orthogonal regression. You should check for linearity of the relationship, homoscedasticity of the errors, and normality of the errors.

1. Be careful with interpretation

The interpretation of the slope and intercept in orthogonal regression is not always straightforward. In particular, the slope represents the correlation between the two variables, rather than the causal relationship. Therefore, you should be careful when interpreting the results.

1. Use appropriate software

Not all statistical software packages have built-in functions for orthogonal regression. Therefore, it is important to use software that can perform orthogonal regression, or to write your own code if necessary.

1. Consider the sample size

Orthogonal regression requires a larger sample size than traditional least squares regression. Therefore, it is important to ensure that you have enough data to perform the analysis.

## The final analysis was done with orthogonal regression shown below.

Unfortunately, the final results indicated that Linda and the company could not establish the equivalency of the vendor’s final inspection numbers and that of the company’s incoming QC values. A team was created, composed of QC specialists from the vendor and the company, who did a deep dive study of the two measurement procedures. After considerable process mapping, they discovered that the sequence of testing was different and could account for the variation in measurements.

The vendor agreed to follow the revised procedures and do the appropriate final inspections. The company stopped full incoming inspection and relied on intermittent sampling to assure themselves that the incoming bottles are as they should be. This reduced level of incoming inspection saved the company more than \$175,000 a year in reduced labor and testing materials.