# Orthogonal Regression: Testing the Equivalence of Instruments

I recently got a request from one of our Facebook fans to do a post about orthogonal regression, which I admit is not a subject I’m very familiar with. However, with a little help from Minitab’s help resources and by consulting a few Minitab experts, I think I came up with a post that will be useful. I thought it would help to discuss orthogonal regression with an example, but first...

## What the Heck Is Orthogonal Regression?

Orthogonal regression is also known as “Deming regression” and examines the linear relationship between two continuous variables. It’s often used to test whether two instruments or methods are measuring the same thing, and is most commonly used in clinical chemistry to test the equivalence of instruments.

## How Is Orthogonal Regression Different from Simple Regression?

Unlike simple linear regression, both the response and predictor in orthogonal regression contain measurement error. In simple regression, only the response variable contains measurement error.

## An Example Using Orthogonal Regression

Suppose there is a medical equipment company that wants to determine if their blood pressure monitor is equivalent to a similar model on the market. The company obtains systolic blood pressure readings on a random sample of 60 people using the two instruments, and records the data in a Minitab worksheet:

Based on a previously conducted study, the company knows that the error variance ratio is 0.90.

However, note that the error variance ratio could be calculated by performing an independent Gage R&R study for each measurement device to produce a variance component (VarComp) for Repeatability for each device. The ratio of the two VarComp for Repeatability can be used as the input for the error variance ratio field.

To perform the analysis, a quality practitioner at the company chooses Stat > Regression > Orthogonal Regression in Minitab:

At the first dialogue box, he fills in the Response, Predictor, and error variance ratio. Then he clicks OK. :

The quality practitioner is presented with the following session window output and line plot:

## What Does this Orthogonal Regression Output Mean?

Because 0 is contained in the confidence interval for the intercept (-2.77513, 4.06395) and 1 is contained in the confidence interval for the slope (0.96769, 1.02315), there is no evidence that the two instruments measure different things. Essentially, the analysis is saying that the company’s blood pressure monitor readings are equivalent to readings from the similar model.

And remember that in orthogonal regression, the following assumptions must be met:

• Both the predictor and the response contain a fixed unknown quantity denoted as X and Y,  respectively, and an error component.
• The error terms are independent.
• The error terms have means of zero and constant variances.
• The predictor and response are linearly related.

Do you have any tips for performing orthogonal regression?

Name: Nail • Tuesday, January 29, 2013

Why can't we use Two sample t-test in MINITAB for this case?
T-Test of difference = 0 (vs not =): T-Value = 0,03 P-Value = 0,974 DF = 117

Name: Carly Barry • Tuesday, January 29, 2013

Thanks for your comment! One reason why we can't use 2-sample t-tests here is that 2-sample t-tests assume the data are independent. However, with orthogonal regression, they are not independent.

In other words, if you randomized the 2-sample t data, you’d still get the same results, but this is not the case with orthogonal regression.

Name: Paul Sheldon • Tuesday, January 29, 2013

Other methods for comparison of two devices or laboratories: (1) the PAIRED t-test; (2) the Bland-Altman macro; (3) Minitab does not have a Youden style plot macro, but that is another useful approach.

Name: John • Tuesday, January 29, 2013

What do you mean by "error variance ratio"? I am sure I've used this in my MSE work and just not realized it. Thanks

Name: Carly Barry • Wednesday, January 30, 2013

Name: Carly Barry • Wednesday, January 30, 2013

Hi John – Thanks for your question. The error variance ratio is the ratio of the measurement error variances in Y and X. One way to obtain estimates of the error variances is to perform separate Gage R&R studies for X and Y.

For example, in this case, the error variance ratio could be calculated by performing an independent Gage R&R study for each measurement device to produce a variance component (VarComp) for Repeatability for each device. The ratio (Y/X) of the two VarComp for Repeatability values can be used as the input for the error variance ratio field. Hope this helps!

Carly

Name: John • Wednesday, January 30, 2013

Thanks, Carly. I love this blog. It is very helpful.

Name: Carly Barry • Wednesday, January 30, 2013

You're welcome, John!

Name: Rodolfo • Tuesday, February 26, 2013

Carly, you mentioned you can get the error variance ratio from doing two independent Gage R&R for each instrument. Would the two independent Gage R&R tell you if the two instruments read the same? why would want to run an orthogonal regression afterwards? could you estimate the error variance ratio another way?

Name: Carly Barry • Wednesday, February 27, 2013

Hi Rodolfo, I checked with a couple experts here at Minitab to answer some of your questions. Here's a recap of what they told me:

Two independent Gage R&R studies would allow you to assess each measurement system separately – with respect to the variation – but would not tell you if the two measurement systems provided the same readings. (That's why you'd still want to use orthogonal regression.)

One expert I consulted said he was virtually certain that there are other ways to estimate the error variance ratio. He said that Gage R&R is essentially a specialized ANOVA analysis. So you could presumably get the same info from just doing an ANOVA or even a regular regression analysis. You’d conduct separate analyses using measurement data for each instrument. It might even be a bit simpler to get the information that way because you wouldn’t need to conduct a full Gage R&R analysis, which adds extra things. We'd have to take some time to figure out exactly what type of design you'd need and the numbers to use.

However, if you are interested in documenting whether two instruments produce the same readings, Gage R&R analysis and Orthogonal Regression results are a good place to start.

Even though there might be a simpler way to get the error variance ratio rather than performing a full Gage R&R, it's still probably a good thing to perform a Gage R&R analysis anyways to gain some insight into your measurement systems.

Hope this helps!
Carly

Name: Stacy • Wednesday, May 8, 2013

Carly you did a great job explaining why a two sample t-test can't be used instead of orthogonal regression. Why wouldn't a paired t-test be sufficient since it assume the data is dependent.

Name: Carly Barry • Thursday, May 9, 2013

Hi Stacy - Thanks for reading! I checked with a member of our statistical Tech Support Team (-thank you Cathy!) in order to answer your question. Here's what she said:

Under the assumption that there is no linearity (bias is not changing with the magnitude of measurements), a paired t test would tell you the average difference between the readings. However, linearity may be present and the paired t-test would not account for this.

Also, check out this blog post by Rick Haynes, which goes into more detail about this topic: https://www.smartersolutions.com/blog/wordpress/2011/06/09/orthogonal-regression-a-new-minitab-function/

Hope this helps!
Carly

Name: Stacy • Monday, May 13, 2013

Thanks for the explanation. This makes sense.

Name: junhoe • Wednesday, February 26, 2014

Thanks, this was useful!

Name: Carly Barry • Thursday, February 27, 2014