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:

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:

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: