Specialized Models • Nonlinear Regression with Built-In Models • The Fit Curve Report
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The Fit Curve Report
The Fit Curve report initially contains only a plot of Y versus X (Fit Curve Reports: No Grouping Variable (left) and with Group Variable (right)). If you specify a Group variable, the report includes overlaid and individual plots for each group of the fitted model (shown in Fit Curve Reports: No Grouping Variable (left) and with Group Variable (right) on the right).
Fit Curve Reports: No Grouping Variable (left) and with Group Variable (right)
Select any of the following built-in models from the Fit Curve red triangle menu:
Polynomials
Fits first degree to fifth degree polynomials.
Sigmoid Curves
Fits Logistic and Gompertz models. These models are S-shaped and have both upper and lower asymptotes. The Logistic 2P, 3P, and 4P models are symmetric. The Logistic 5P and both Gompertz models are not symmetric. The Logistic 2P is available only when the response is between 0 and 1. Examples of Sigmoid curves include learning curves and modeling tumor growth, both of which increase initially and then taper off.
Exponential Growth and Decay
Fits Exponential, Biexponential, and Mechanistic Growth models. The Exponential 2P and 3P are similar, but the 3P model has an asymptote. The Biexponential models assume there are two separate growth or decay processes. The Mechanistic Growth and Exponential 3P models always increase, but the rate of growth slows so that the model has an asymptote. Examples of exponential growth and decay functions are virus spread and drug half-life, respectively.
Peak Models
Fits Gaussian Peak and Lorentzian Peak models. These models increase up to a peak and then decrease. The Gaussian Peak model is a scaled version of the Gaussian probability density function (PDF). The Lorentzian Peak model is a scaled version of the Cauchy distribution, a continuous probability distribution. These models can be used for some chemical concentration assays and artificial neural networks.
Pharmacokinetic Models
Fits the One Compartment Oral Dose model, the Two Compartment IV Bolus Dose model, and the Biexponential 4P model. This option is used to model the concentration of drugs in the body.
Fit Michaelis-Menten
Fits the Michaelis-Menten biochemical kinetics model, which relates the rate of enzymatic reactions to substrate concentration.
Script
Contains options that are available to all platforms. See Using JMP.
Initial Fit Curve Reports
Before fitting a model, the Fit Curve report contains only a plot of Y versus X. After fitting a model, the fitted model is added to the plot (when no grouping variable is specified on the platform launch window). The report contains the following results:
Model Comparison Report
To create the report shown in Model Comparison Report, select Sigmoid Curves > Logistic Curves > Fit Logistic 4P and Sigmoid Curves > Fit Gompertz 4P from the Fit Curve red triangle menu.
Model Comparison Report
The Model Comparison report shows fit statistics used for comparing multiple models. The statistics are AICc, AICc Weight, BIC, SSE, MSE, RMSE, and R-Square, and are defined below.
AICc
Gives a measure of the goodness of fit of an estimated statistical model that can be used to compare two or more models. AICc is a modification of the AIC adjusted for small samples. AICc can only be computed when the number of data points is at least two greater than the number of parameters. The model with the lowest AICc value is the best, which is the Logistic 4P in our example. See Fitting Linear Models.
AICc Weight
A convenient way of normalizing AIC values so that they sum to one. The AICc weight can be interpreted as the probability that model is the best model among the fitted models. Therefore, the model with the AICc weight closest to one is the better fit. In our example, the Logistic 4P model is clearly the better fit. The AICc weights are calculated using only nonmissing AICc values.
BIC
Gives a measure based on the likelihood function of model fit that is helpful when comparing different models. The model with the lower BIC value is the better fit. See Fitting Linear Models.
SSE
The sum of the squared differences between each observation and its predicted value.
MSE
Gives the average of the squares of the errors of each value.
RMSE
The square root of the MSE that estimates the standard deviation of the random error.
R-Square
Estimates the proportion of variation in the response that can be attributed to the model rather than to random error. The model with the R-Square value closest to one is the better fit.
The Model Comparison platform provides additional options, such as plotting residual and actual values. See the Model Comparison section for more information.
Model Reports
A report is created for each fitted model. The red triangle menu for each model report provides the following options.
Prediction Model
Gives the algebraic form of the prediction formula and the parameters.
Summary of Fit
Gives the same fit statistics as the Model Comparison report.
Parameter Estimates
Gives the estimates of the parameters, standard errors, and confidence intervals. The correlations and covariances of the estimates are given also.
Plot
Gives plots of the data with the fitted model. See Initial Model Reports for Logistic 4P Model. The plots are shown only when you select a Grouping variable on the platform launch window.
Initial Model Reports for Logistic 4P Model
Each model report contains a red triangle menu with some or all of the following options:
Test Parallelism
Helps determine whether the curves are similar in shape when they are shifted along the x-axis. In certain situations, it is important to establish parallelism before making further comparisons between groups. This option is available only when a Group variable is specified on the platform launch window. This option is available for the Sigmoid models (Logistic and Gompertz), as well as the Linear Regression model, with the exception of higher-order polynomials. For details, see Test Parallelism.
Area Under Curve
Gives the area under the fitted curve. This option is available only for the following models: One Compartment, Two Compartment, Gaussian Peak, and Lorentzian Peak. The range of integration depends on the type of model and is specified in the report.
If a Grouping variable is specified on the platform launch window, an Analysis of Means is performed for comparing the estimates across groups. If the result for a group exceeds a decision limit, the result is considered different from the overall mean of AUC.
Compare Parameter Estimates
Gives an analysis for testing the equality of parameters across levels of the grouping variable. This option is available only when a Group variable is specified on the platform launch window. For details, see Compare Parameter Estimates.
Equivalence Test
Gives an analysis for testing the equivalence of models across levels of the grouping variable. This option is available only when a Group variable is specified on the platform launch window. For details, see Equivalence Test.
Make Parameter Table
Saves the parameter estimates, standard errors, and t-ratios in a data table. This option is available only when a Group variable is specified on the platform launch window.
Profiler
Shows or hides a profiler of the fitted prediction function. The derivatives are derivatives of the prediction function with respect to the X variable. For more information about profilers, see the Profiler chapter in the Profilers book.
Save Formulas
Contains options for saving a variety of formula columns in the data table.
Save Prediction Formula saves the prediction equation.
Save Std Error of Predicted saves the standard error of the predicted values.
Save Parametric Prediction Formula saves the prediction equation in parametric form. This is helpful if you want to use the fitted model in the full personality of the Nonlinear platform.
Save Residual Formula saves the residuals.
Save Studentized Residual Formula saves the studentized residual formula, a standard residual that is divided by its estimated standard deviation.
Save First Derivative saves the derivative of the prediction function with respect to the X variable.
Save Std Error of First Derivative saves the equation of the standard error of the first derivative.
Save Inverse Prediction Formula saves the equation for predicting X from Y.
Custom Inverse Prediction
Predicts an X value for a specific Y value. For more information about inverse prediction, see the Multivariate Methods book.
Remove Fit
Removes the model report, the entry from the Model Comparison report, and the fitted line from the plot.