Statistics Output
Scientist provides a broad range of statistical output, including
parameter estimates, confidence limits, various measures of goodnesstofit,
variancecovariance 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 variancecovariance 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 nonlinear 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:
Micromath Scientist Statistics Report 





Input Information 

Model: 
pK Determination.eqn 



Data Set: 
pK Determination 



Parameter Set: 
Parameter Set 3 







Report Options 

Descriptive Statistics 
N 



GoodnessofFit Statistics 
Y 



Confidence Intervals 
Y 



VarianceCovariance Matrix 
Y 



Correlation Matrix 
Y 



Rigorous Limits 
N 



Residual Analysis 
Y 



Confidence Interval 
95% 
50 






GoodnessofFit Statistics 

Data Column Name: 
OHADD 




Weighted 
Unweighted 


Sum of squared observations: 
1.8453 
1.8453 


Sum of squared deviations: 
3.2736E006 
3.2736E006 


Standard deviation of data: 
0.00023965 
0.00023965 


Rsquared: 
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: 
3.2736E006 
3.2736E006 


Standard deviation of data: 
0.00023965 
0.00023965 


Rsquared: 
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: 
5.426E005 



95% Range (Univariate): 
0.096915 
0.097132 


95% Range (Support Plane): 
0.096851 
0.097196 






VarianceCovariance Matrix 


1.7592E006 




1.0788E006 
5.1134E006 



1.5902E006 
4.9092E006 
8.4551E006 


3.2078E008 
9.9037E008 
1.4594E007 





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 


