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

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).

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.

Admittedly, my machine example was a hack. Obviously when the two machines agree anywhere near as close as this, there is no practical difference between them. Your suggestion for an unpaired analysis is a good one. I’ll check about altering the algorithm.

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 . . .

When I “made up” the data, I used Excel’s NORMINV to compute the errors, and I think I used sigma values that differed by about 3:1, with higher variance in X. I was playing around to find points that looked good on screen.

Do you have any real world data that would satisfy your last sentence? It would be interesting. I presume you want more deviation than in my last chart in my first post?

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).