Comment from DaleW
Time: Tuesday, October 6, 2009, 12:49 pm

Hi Jon,

If I follow correctly, your Deming regression utility is designed for analyzing pairs of data where each point is immediately replicated, hence the requirement that both X and Y ranges have an even number of points?

Those replicates allow the algorithm to have a good estimate of the variance ratio for the errors (simple measurement repeatability, ignoring any bias to the nominal value). This Y to X variance ratio — that is assumed infinite for ordinary least squares where all the assumed error is Y — seems to be about 0.306 in your example dataset. (This might be another useful output for your utility?)

If I round to four decimal places, your Deming regression slope was given as 0.9940, your Demining utility gave 0.9942 for the rounded data on your screenshot, and I got an estimate of 0.9939 inputting my quick estimate of the variance ratio and your screenshot data into a commercial statistics package that supports Orthogonal Regression. The statistical uncertainty at 95% confidence level for the orthogonal slope is a couple orders of magnitude greater than these order 0.0001 differences, so I’m going to say our results agree.

Probably a better test case would have lower correlation between the two variables, so some meaningful differences might emerge by comparison with Ordinary Least Squares (both Y v. X and X v. Y).

Comment from DaleW
Time: Tuesday, October 6, 2009, 9:54 pm

Jon -

Unfortunately, the example from your prior post on Deming regression doesn’t show us that Deming regression will give a noticeably different result than ordinary least squares (OLS) regression.

The slope (Y to X) of a best fit line is about 0.993 by OLS and 0.994 by reverse OLS, with a standard error of 0.014 for either, so we can’t say whether the true slope is slightly more or less than one. Orthogonal regression can try to more precisely place the line between those limiting cases (no error in X or no error in Y), but it is such a small tweak here that it makes no practical difference.

To make a real difference, r-squared, which also equals the ratio between those two OLS slopes(!), generally has to be a lot less than 100%. It was 99.84% for your example. On the other hand, for a good measurement system, the vast majority of the total variance is explained by the sample differences, not the machine differences, so r-squared between paired machine measurements across the same samples tends to be very high.

Although I would prefer not to locate an example in my files of a bad enough measurement system with replicated measurements so that Deming regression could have been helpful, I’ll have to check . . .

Comment from Matt Healy
Time: Thursday, October 8, 2009, 10:30 pm

To follow up on previous comments, most variants of linear regression have the line pass through the point (mean_x, mean_y), so the difference is in the slope. If we call m1 the slope from regression of Y against X, and m2 the slope from regression of X against Y (swapping variables), then m1 times m2 will equal r-squared (try it yourself!). Of course r-squared will be the same either way. If one assumes the error in each variable is proportional to its value, then the best estimate for the slope will be simply the ratio of the standard deviations of X and Y.

Of course robust regression techniques use shrinkage estimators, order statistics, and the like to reduce the impact of outliers on the slope, which is another whole ball game. OLS and its variants assume an underlying Gaussian distribution but if the data come from a heavy-tailed distribution then OLS is too sensitive to outliers.

Comment from DaleW
Time: Friday, October 9, 2009, 10:31 am

To follow up on Matt’s comments: as an extension of OLS, Deming regression fits an XY slope m through the center of one’s data somewhere between m1 by OLS and 1/m2 by reverse OLS, depending on one’s estimate of the ratio of residual variances in the two variables.

Jon’s utility currently covers a very special case where the experimenter replicated every measurement to allow a good estimate of this Y:X variance ratio.

At the other extreme, in the absence of information, a data miner might guess that the error in each variable is proportional to its scatter (standard deviation), and Deming regression splits the difference, bisecting our data ellipse along its primary axis, with a slope that simplifies to sqrt(m1/m2).

Jon, the VBA code that you cited by E. Joseph Billo for that special case could be very easily modified to use a user selected estimate of the variance ratio “lambda” for the general XY regression problem, wouldn’t you say? Add radio buttons to the dialog for three or four limiting special cases, and you’d have a very cool Excel utility for general Deming regression (linear orthogonal regression for two variables).

Comment from Mike
Time: Thursday, November 5, 2009, 11:04 pm

Hi

When I use PTS Deming Regression Utility in Excel 2007 I get the following message:

“Run-time error ‘13′: Type mismatch”

Can you help me sort this out to get the utility working for me?

Thanks