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:
Micromath Scientist Statistics Report |
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Input Information |
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Model: |
pK Determination.eqn |
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Data Set: |
pK Determination |
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Parameter Set: |
Parameter Set 3 |
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Report Options |
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Descriptive Statistics |
N |
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Goodness-of-Fit Statistics |
Y |
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Confidence Intervals |
Y |
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Variance-Covariance Matrix |
Y |
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Correlation Matrix |
Y |
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Rigorous Limits |
N |
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Residual Analysis |
Y |
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Confidence Interval |
95% |
50 |
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Goodness-of-Fit Statistics |
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Data Column Name: |
OHADD |
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Weighted |
Unweighted |
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Sum of squared observations: |
1.8453 |
1.8453 |
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Sum of squared deviations: |
3.2736E-006 |
3.2736E-006 |
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Standard deviation of data: |
0.00023965 |
0.00023965 |
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R-squared: |
0.99999 |
0.99999 |
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Coefficient of determination: |
0.99999 |
0.99999 |
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Correlation: |
1 |
1 |
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Data Set Name: |
pK Determination |
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Weighted |
Unweighted |
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Sum of squared observations: |
1.8453 |
1.8453 |
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Sum of squared deviations: |
3.2736E-006 |
3.2736E-006 |
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Standard deviation of data: |
0.00023965 |
0.00023965 |
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R-squared: |
0.99999 |
0.99999 |
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Coefficient of determination: |
0.99999 |
0.99999 |
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Correlation: |
1 |
1 |
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Model Selection Criterion: |
11.749 |
11.749 |
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Confidence Intervals |
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Parameter Name: |
PKH3A |
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Estimated Value: |
2.1459 |
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Standard Deviation: |
0.0013264 |
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95% Range (Univariate): |
2.1432 |
2.1485 |
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95% Range (Support Plane): |
2.1417 |
2.1501 |
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Parameter Name: |
PKH2A |
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Estimated Value: |
7.197 |
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Standard Deviation: |
0.0022613 |
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95% Range (Univariate): |
7.1925 |
7.2016 |
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95% Range (Support Plane): |
7.1898 |
7.2042 |
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Parameter Name: |
PKHA |
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Estimated Value: |
12.344 |
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Standard Deviation: |
0.0029078 |
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95% Range (Univariate): |
12.338 |
12.35 |
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95% Range (Support Plane): |
12.335 |
12.353 |
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Parameter Name: |
ATOT |
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Estimated Value: |
0.097024 |
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Standard Deviation: |
5.426E-005 |
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95% Range (Univariate): |
0.096915 |
0.097132 |
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95% Range (Support Plane): |
0.096851 |
0.097196 |
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Variance-Covariance Matrix |
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1.7592E-006 |
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1.0788E-006 |
5.1134E-006 |
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1.5902E-006 |
4.9092E-006 |
8.4551E-006 |
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3.2078E-008 |
9.9037E-008 |
1.4594E-007 |
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Correlation Matrix |
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0.99999 |
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0.3597 |
0.99999 |
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0.4123 |
0.74661 |
0.99999 |
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0.44573 |
0.80717 |
0.92501 |
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Residual Analysis |
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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. |
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Serial Correlation: |
-0.87966 |
is probably not significant. |
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Skewness |
-13.231 |
indicates the likelihood of a few large negative residuals having an unduly large effect on the fit. |
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Kurtosis |
6.8116 |
is probably not significant. |
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Weighting Factor: |
0 |
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Heteroscedacticity: |
1.3441 |
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Optimal Weighting Factor: |
1.3441 |
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