Linear Least Squares Regression Background Information
NIST StRD    Data Archive

Linear Regression

Background Information

Even with the availability of reliable code for linear least squares fitting, problems persist. Failure to use the best algorithms and to implement them most effectively is often the cause. Therefore, we provide datasets with certified values for key statistics for testing linear least squares code.

Both generated and "real-world" data are included. Generated datasets challenge specific computations and include the Wampler data developed at NIST (formerly NBS) in the early 1970's. Real-world data include the challenging Longley data, as well as more benign datasets from our statistical consulting work at NIST.

Datasets are ordered by level of difficulty (lower, average, and higher). Strictly speaking the level of difficulty of a dataset depends on the algorithm used. These levels are intended to provide rough guidance for the user. Datasets of lower level of difficulty should pose few problems for most code. Discrepancies here may indicate a failure to use correct options for the code. Two datasets are included for fitting a line through the origin. We have encountered codes that produce negative R-squared and incorrect F-statistics for these datasets. Therefore, we assign them an "average" level of difficulty. Finally, several datasets of higher level of difficulty are provided. These datasets are multicollinear. They include the Longley data and several NIST datasets developed by Wampler.

Producing correct results on all datasets of higher difficulty does not imply that your software will pass all datasets of average or even lower difficulty. Similarly, producing correct results for all datasets in this collection does not imply that your software will do the same for your particular dataset. It will, however, provide some degree of assurance, in the sense that your package provides correct results for datasets known to yield incorrect results for some software.

Certified values are provided for the parameter estimates, their standard deviations, the residual standard deviation, R-squared, and the standard ANOVA table for linear regression. Certified values are quoted to 16 significant digits and are accurate up to the last digit, due to possible truncation errors. For more information on certification methodology, see the description provided for each dataset.

If your code fails to produce correct results for a dataset of higher level of difficulty, one possible remedy is to center the data and rerun the code. Centering the data, i.e., subtracting the mean for each predictor variable, reduces the degree of multicollinearity. The code may produce correct results for the centered data. You can judge this by comparing predicted values from the fit of centered data with those from the certified fit.

We plan to update this collection of datasets, and welcome your feedback on specific datasets to include, and on other ways to improve this web service.

See also: General Background Information