Least-Squares Regression Models Results

Note:This section is meant to provide guidance and references to aid in the interpretation of modeling output, and is not a replacement for a PK or statistics textbook.

After a Classic model is run, the output is displayed on the Results tab in Phoenix. The output is discussed in the following sections.

Core Output: text version of all model settings and output, including any errors that occurred during modeling. See “Core Output File” for a full description.

Settings: test version of all user-defined settings.

Worksheet output: worksheets listing input data, modeling iterations and output parameters, as well as several measures of fit.

Plot output: plots of observed and predicted data, residuals, and other quantities, depending on the model run.

Worksheet output

Worksheet output contains summary tables of the modeling data and a summary of the information in the Core Output. The worksheets generated depend on the analysis type and model settings. They present the output in a form that can be used for reporting and further analyses and are listed on the Results tab underneath Output Data.

Condition Numbers: Rank and condition number of the matrix of partial derivatives for each iteration.
The matrix is of full rank, since Rank is equal to the number of parameters. If the Rank were less than three, that would indicate that there was not enough information in the data to estimate all three parameters. The condition value is the square root of the ratio of the largest to the smallest eigenvalue and values should be less than 10^n where n is the number of parameters.

Correlation Matrix: A correlation matrix for the parameters, for each sort level. If any values get close to 1 or –1, there may be too many parameters in the model and a simpler model may work better.

Diagnostics: Diagnostics for each function in the model and for the total:

CSS: corrected sum of squared observations
WCSS: weighted corrected sum of squared observations
SSR: sum of squared residuals
WSSR: weighted sum of squared residuals
S: estimate of residual standard deviation
DF: degrees of freedom
CORR_(OBS,PRED): correlation between observed Y and predicted Y
WT_CORR_(OBS,PRED): weighted correlation
AIC: Akaike Information Criterion goodness of fit measurement
SBC: Schwarz Bayesian Criterion goodness of fit measurement^a

Differential Equations: The value of the partial derivatives for each parameter at each time point for each value of the sort variables.

Dosing Used: The dosing regimen specified for the modeling.

Eigenvalues: Eigenvalues for each level of the sort variables. (An eigenvalue of matrix A is a number l, such that Ax=lx for some vector x, where x is the eigenvector. Eigenvalues and their associated eigenvectors can be thought of as building blocks for matrices.)

Final Parameters and Final Parameters Pivoted: Parameter names, units, estimates, standard error of the estimates, CV% (values < 20% are generally considered to be very good), univariate intervals, and planar intervals for each level of the sort variables.

Fitted Values: (Dissolution models) Predicted data for each profile.

Initial Estimates: Parameter names, initial values, and lower and upper bounds for each level of the sort variables.

Minimization Process: Iteration number, weighted sum of squares, and value for each parameter, for each level of the sort variables. This worksheet shows how parameter values converged as the iterations were performed. If the number of iterations is approaching the specified limit, there may be some problems with the model.

Parameters: (Dissolution models) The smoothing parameter delta and absorption lag time for each profile.

Partial Derivatives and Stacked Partial Derivatives: Values of the differential equations at each time in the dataset.

Predicted Data: Time and predicted Y for multiple time points, for each sort level.

Secondary Parameters and Secondary Parameters Pivoted: Available for Michaelis-Menten, PK, PD, PK/PD Linked and ASCII models. Secondary parameter name, units, estimate, standard error of the estimate, and CV% for each sort level.

Summary Table^b: The sort variables, X, Y, transformed X, transformed Y, predicted Y, residual, weight, standard error of predicted Y, standardized residuals, for each sort level. For link models, also includes C_P and C_e. For indirect response models, also includes C_P.

Values: (Dissolution models) Time, input rate, cumulative amount (Cumul_Amt, using the dose units) and fraction input (Cumul_Amt/test dose or, if no test doses are given, then fraction input approaches one) for each profile.

Variance Covariance Matrix: A variance-covariance matrix for the parameters, for each sort level.

User Settings: Model number, minimization method, convergence criterion, maximum number of iterations allowed, and the weighting scheme.

^aAIC and SBC are only meaningful during comparison of models. A smaller value is better, negative is better than positive, and a more negative value is even better. AIC is computed as:
AIC=N log (WRSS)+2P, where N is the number of observations with positive weight, log is the natural logarithm, WRSS is the weighted residual sum of squares, P is the number of parameters.

^bIf there are no statements to transform the data, then X and Y will equal X(obs) and Y(obs).

Plot output

Analysis produces up to eight graphs that are divided by each level of the sort variable. Plot output is listed underneath Plots in the Results tab.

Cumulative Rates: (Dissolution models) Cumulative drug input vs. time.

Fitted Curves: (Dissolution models) Observed time-concentration data vs. the predicted curve.

Input Rates: (Dissolution models) Rate of drug input vs. time.

Observed Y and Predicted Y vs X: plots the predicted curves as a function of X, with the Observed Y overlaid on the plot. Used for assessing the model fit.

Partial Derivatives Plot: plots the partial derivative of the model with respect to each parameter as a function of the x variable. If f(x; a, b, c) is the model as a function of x, based on parameters a, b, and c, then df(x; a, b, c)/da, df(x; a, b, c)/db, df(x; a, b, c)/dc are plotted versus x. Each derivative is evaluated at the final parameter estimates. Data taken at larger partial derivatives are more influential than those taken at smaller partial derivatives for the parameter of interest.

Predicted Y vs Observed Y: plots weighted Y against observed Y. Scatter should lie close to the 45 degree line.

Residual Y vs Predicted Y: used to assess whether error distribution is appropriately modeled throughout the range of the data.

Residual Y vs X: used to assess whether error distribution is appropriately modeled across the range of the X variable.

Weighted Predicted Y vs Observed Y: plots weighted predicted Y against observed Y. Scatter should lie close to the 45 degree line; only produced if a non-Uniform weighting scheme is used.
Weighted Residual Y vs Observed Y: used to assess whether error distribution is appropriately modeled across the range of the observed Y variable; only produced if a non-Uniform weighting scheme is used.
Weighted Residual Y vs Predicted Y: used to assess whether error distribution is appropriately modeled across the range of the predicted variable; only produced if a non-Uniform weighting scheme is used.
Weighted Residual Y vs Weighted Predicted Y: used to assess whether error distribution is appropriately modeled throughout the range of the data; only produced if a non-Uniform weighting scheme is used.
Weighted Residual Y vs X: used to assess whether error distribution is appropriately modeled across the range of the X variable; only produced if a non-Uniform weighting scheme is used.

Users can double-click any plot in the Results tab to edit it. (See the “Plot Options tab” description for editing options.)