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Statistics Output

Scientist provides a broad range of statistical output, including parameter estimates, confidence limits, various measures of goodness-to-fit, variance-covariance and correlation information, and analysis of residuals. Confidence limits for parameter estimates are calculated using the customary approach involving a local linearization of the model or a more rigorous approach that locates various points on constant sum of squares contours.

Data Statistics

Statistical information based solely on the observed data include, minimum, maximum, the algebraic, absolute, geometric, harmonic, quadratic and weighted means, the median, sum of squares, deviations, variance, skewness and kurtosis.

Parameter Statistics

The statistical information on parameters provided by Scientist include:

  • The best fit estimate for each parameter
  • The standard deviation for each estimate
  • The univariate and support plane 95% confidence ranges for each parameter.
  • The variance-covariance matrix and the correlation matrix

The 95% confidence ranges can be calculated making the assumption of linearity near the minima or using a more rigorous iterative method that assumes a non-linear response in the region of the parameter estimates.

Model Selection Criterion

The Akaike Information Criterion (AIC) is defined by the formula:

Akaike Information Criterion Equation

The AIC attempts to represent the “information content” of a given set of parameter estimates by relating the coefficient of determination to the number of parameters (or equivalently, the number of degrees of freedom) that were required to obtain the fit. When comparing two models with different numbers of parameters, this criterion places a burden on the model with more parameters to not only have a better coefficient of determination, but quantifies how much better it must be for the model to be deemed more appropriate.

The AIC, as defined above, is dependent on the magnitude of the data points as well as the number of observations. In addition, the most appropriate model is the one with the smallest value of the AIC.

Scientist uses a modified AIC which we call the Model Selection Criterion (MSC) defined by the formula:

Model Selection Criterion Formula

The MSC will give the same rankings between models as the AIC and has been normalized so that it is independent of the scaling of the data points. Furthermore, the most appropriate model will be that with the largest MSC (because we want to maximize the “information content” of the model). It is therefore quite easy, with experience, to develop a feeling for what the MSC means in terms of how well the model fits the data.

The following is a typical Scientist Statistics Report:

Micromath Scientist Statistics Report





Input Information

Model: pK Determination.eqn


Data Set: pK Determination


Parameter Set: Parameter Set 3






Report Options

Descriptive Statistics N


Goodness-of-Fit Statistics Y


Confidence Intervals Y


Variance-Covariance Matrix Y


Correlation Matrix Y


Rigorous Limits N


Residual Analysis Y


Confidence Interval 95% 50





Goodness-of-Fit Statistics

Data Column Name: OHADD



Weighted Unweighted

Sum of squared observations: 1.8453 1.8453

Sum of squared deviations: 3.2736E-006 3.2736E-006

Standard deviation of data: 0.00023965 0.00023965

R-squared: 0.99999 0.99999

Coefficient of determination: 0.99999 0.99999

Correlation: 1 1






Data Set Name: pK Determination



Weighted Unweighted

Sum of squared observations: 1.8453 1.8453

Sum of squared deviations: 3.2736E-006 3.2736E-006

Standard deviation of data: 0.00023965 0.00023965

R-squared: 0.99999 0.99999

Coefficient of determination: 0.99999 0.99999

Correlation: 1 1

Model Selection Criterion: 11.749 11.749





Confidence Intervals

Parameter Name: PKH3A


Estimated Value: 2.1459


Standard Deviation: 0.0013264


95% Range (Univariate): 2.1432 2.1485

95% Range (Support Plane): 2.1417 2.1501






Parameter Name: PKH2A


Estimated Value: 7.197


Standard Deviation: 0.0022613


95% Range (Univariate): 7.1925 7.2016

95% Range (Support Plane): 7.1898 7.2042






Parameter Name: PKHA


Estimated Value: 12.344


Standard Deviation: 0.0029078


95% Range (Univariate): 12.338 12.35

95% Range (Support Plane): 12.335 12.353






Parameter Name: ATOT


Estimated Value: 0.097024


Standard Deviation: 5.426E-005


95% Range (Univariate): 0.096915 0.097132

95% Range (Support Plane): 0.096851 0.097196





Variance-Covariance Matrix


1.7592E-006



1.0788E-006 5.1134E-006


1.5902E-006 4.9092E-006 8.4551E-006


3.2078E-008 9.9037E-008 1.4594E-007





Correlation Matrix


0.99999



0.3597 0.99999


0.4123 0.74661 0.99999


0.44573 0.80717 0.92501





Residual Analysis

Expected Value: The following are normalized parameters with an expected value of 0.0. Values are in units of standard deviations from the expected value.

Serial Correlation: -0.87966 is probably not significant.

Skewness -13.231 indicates the likelihood of a few large negative residuals having an unduly large effect on the fit.

Kurtosis 6.8116 is probably not significant.

Weighting Factor: 0


Heteroscedacticity: 1.3441


Optimal Weighting Factor: 1.3441