Evaluating the Numerical Accuracy of Analyseit.
This document describes the performance of Analyseit for Microsoft Excel version 4.00 against the NIST StRD.
In response to industry concerns about the numerical accuracy of statistical software, the Statistical Engineering and Mathematical and Computational Sciences Divisions of NIST’s Information Technology Laboratory developed datasets with certified values for a variety of statistical methods.
For more information about the datasets see:
http://www.itl.nist.gov/div898/strd/
The results obtained from statistical software packages can be compared against the certified values. The certified values are accurate to 15 significant digits and computed using ultrahigh precision floating point arithmetic.
Most statistical packages use IEEE754 double precision (64bit) floating point arithmetic and due to finite precision, roundoff and truncation errors involved in numerical operations, will be unable to obtain the exact certified value. Therefore, a good measure of the accuracy of a result x against the certified value c, can be calculated as the base10 logarithm of the absolute value of the relative error:
LRE = log10 (x  c / c)
if c ≠ 0, otherwise,
LRE = log10 x
LRE is the number of significant digits in common with the certified value. Higher LRE values are better, and the maximum LRE obtainable is 15.
Performance benchmarks against the NIST StRD
We tested version 4.00 of Analyseit using the NIST StRD on an Intel Xeon dual processor PC. No statistical package achieves perfect accuracy for all the tests and no one package performs best for every test. In the tests:
 Analyseit performed consistently and amongst the best on all tests.
 Analyseit performed better than some of the more popular wellknown statistical packages.
Some developers of popular statistical software packages have published their own benchmarks against the NIST StRD, and some independent authors have also published benchmarks, see:
 A comparative study of the reliability of nine statistical software packages,
Computational Statistics & Data Analysis, 51(8), 38113831,
Keeling, K. B., & Pavur, R. J.
 On the Accuracy of Statistical Procedures in Microsoft Excel 97,
Computational Statistics and Data Analysis, July 1999, Volume 31, Number 1, pp 2737,
McCullough, B.D. and Wilson, B.
 Assessing the Reliability of Statistical Software: Part I,
The American Statistician, Volume 52, Number 4, pp 358366,
McCullough, B.D.
 Assessing the Reliability of Statistical Software: Part II,
The American Statistician, May 1999, Volume 53, Number 2, pp 149159,
McCullough, B.D.
To download the Excel worksheets containing the Analyseit analyses to perform the NIST StRD, and see comparisons against the published results for other packages, see:
http://analyseit.com/support/AnalyseitNISTStRDValidation.zip
Summary of results
The LRE obtained testing Analyseit against the NIST StRD are summarized below.
Univariate summary statistics
The univariate tests consist of nine datasets classified by difficulty.
The mean and standard deviation were computed using the Distribution analysis and compared to the certified values.
The lag1 autocorrelation is not computed by Analyseit.


LRE

Test

Difficulty

Mean

SD

PiDigits

Lower

15.0

15.0

Lottery

Lower

15.0

15.0

Lew

Lower

15.0

15.0

Marvo

Lower

15.0

13.1

Michelson

Lower

15.0

13.8

NumAcc1

Lower

15.0

15.0

NumAcc2

Average

15.0

15.0

NumAcc3

Average

15.0

9.5

NumAcc4

Higher

15.0

8.3

Average


15.0

13.3

Minimum


15.0

8.3

Maximum


15.0

15.0

Analysis of variance
The analysis of variance tests consist of eleven datasets classified by difficulty.
The F statistic was computed using the Compare Groups – ANOVA analysis and compared to the certified value.


LRE

Test

Difficulty

F

SiRstv

Lower

13.1

SmLs01

Lower

15.0

SmLs02

Lower

15.0

SmLs03

Lower

15.0

AtmWtAg

Average

10.2

SmLs04

Average

10.4

SmLs05

Average

10.2

SmLs06

Average

10.2

SmLs07*

Higher

4.4

SmLs08*

Higher

4.2

SmLs09*

Higher

4.2

Average


10.2 (12.4*)

Minimum


4.2 (10.2*)

Maximum


15.0

*Average/Minimum calculated excluding the marked tests.
NOTE: No statistical package has performed the SimonLesage tests 7, 8, and 9 (marked *) with more than 4.6 digits of accuracy, and although these programs reported 4.6 digits of accuracy for test 7 they performed markedly worse for tests 8 and 9. This is due to a flaw in the tests themselves rather than in the software packages, as the number 1,000,000,000,000.4 cannot be represented precisely using binary IEEE754 64bit double floating point representation. Instead it is represented as 1,000,000,000,000.4000244140625. Even simple summation of such a series of numbers leads to inaccuracy.
Linear regression
The linear regression tests consist of eleven datasets classified by difficulty.
The beta coefficients and standard error of the coefficients were computed using the Fit Model analysis and compared to the certified values. The minimum LRE value is reported for each analysis.
The R^{2} statistic was computed and compared to the certified value. No R^{2} value is computed for the no intercept models due to issues interpreting such a statistic.
The residual sum of square was computed and compared to the certified value.


LRE

Test

Difficulty

Beta

Beta SE

Residual SS

R^{2}

Norris

Lower

12.6

13.5

13.3

15.0

Pontius

Lower

12.4

12.7

12.4

15.0

NoInt1

Average

14.7

14.4

14.1



NoInt2

Average

15.0

15.0

14.6



Fillip

Higher

7.7

7.8

9.1

11.6

Longley

Higher

10.9

11.9

11.7

14.0

Wampler1

Higher

9.9

10.0

15.0

15.0

Wampler2

Higher

13.4

14.7

15.0

15.0

Wampler3

Higher

9.1

13.7

14.1

15.0

Wampler4

Higher

9.9

13.6

15.0

15.0

Wampler5

Higher

5.7

13.6

15.0

13.7

Average


11.0

12.8

13.6

14.4

Minimum


5.7

7.8

9.1

11.6

Maximum


15.0

15.0

15.0

15.0

NOTE: There may be slight variation in the LRE on repeated runs due to the use of available multiple processor cores when computing the QR decomposition of the matrix.