Introduction to the Nonlinear Fit Curve Personality
Some models are linear in the parameters (for example, a quadratic or other polynomial); others can be transformed to be such (for example, when you use a log transformation of x). The Fit Model or Fit Y by X platforms are more appropriate in these situations. An example in the Model Specification chapter in the Fitting Linear Models book shows a significant linear relationship between oxygen uptake and time spent running. For more information about Fit Model, see the Model Specification chapter in the Fitting Linear Models book. For more information about Fit Y by X, see the Introduction to Fit Y by X chapter in the Basic Analysis book.
The Nonlinear platform enables you to fit models that are nonlinear in the parameters. The initial example in this chapter shows the analysis of a nonlinear relationship: drug toxicity as a function of concentration. The effect of concentration on toxicity changes from low to high doses, so this relationship is nonlinear.
The following are examples of equations for linear and nonlinear functions.
• Linear function: Y=β0+β1ex
• Nonlinear function: Y=β0+β1eβ2x
The Nonlinear platform’s Fit Curve personality provides predefined models, such as polynomial, logistic, Gompertz, exponential, peak, and pharmacokinetic models. Specifying a grouping variable lets you estimate separate model parameters for each level of the grouping variable. The fitted models and estimated parameters can be compared across the levels of the grouping variable.
Fit Curve also enables you to build a model to create the prediction formula. Then you set upper and lower parameter limits in Nonlinear. For details, see “Example of Setting Parameter Limits” in the “Nonlinear Regression” chapter.
Example Using the Fit Curve Personality
This example shows how to build a model for toxicity as a function of the concentration of a drug. You have a standard formulation of the drug and want to compare it to three new formulations.
You are interested in a toxicity ratio of surviving to non-surviving cells at a specific concentration of each drug. From prior research, you know the toxicity ratios for 16 different concentrations of each drug formulation. A lower ratio indicates more toxicity, which could be detrimental to development of the drug. Log concentration was calculated to decrease the range of concentration values and make it easier to detect differences in the curves.
Follow these steps to build the model:
1. Select Help > Sample Data Library and open Nonlinear Examples/Bioassay.jmp.
2. Select Analyze > Specialized Modeling > Nonlinear.
3. Assign Toxicity to the Y, Response role.
4. Assign log Conc to the X, Predictor Formula role.
5. Assign Formulation to the Group role.
6. Click OK.
The Fit Curve Report appears as shown in Figure 12.2. The Plot report contains an overlaid plot of the fitted model of each formulation.
Figure 12.2 Initial Fit Curve Report
7. To see a legend identifying each drug formulation, right-click one of the graphs and select Row Legend. Select Formulation for the column and click OK. The plot shown in Figure 12.3 appears.
Figure 12.3 Fit Curve Report with Plot Legend
The curves appear S-shaped, so a sigmoid curve would be an appropriate fit. Table 12.1 shows formulas and graphical depictions of the different types of models that the Fit Curve personality offers.
8. Select Sigmoid Curves > Logistic Curves > Fit Logistic 4P from the Fit Curve red triangle menu.
Figure 12.4 Logistic 4P Report
The Logistic 4P report appears (Figure 12.4). There is also a separate plot for each drug formulation. The plot of the fitted curves suggests that formulation B might be different, because the test B curve starts to rise sooner than the others. Inflection point parameters cause this rise.
9. Select Compare Parameter Estimates from the Logistic 4P red triangle menu.
A portion of the Parameter Comparison report is shown in Figure 12.5.
Figure 12.5 Parameter Comparison Report
Notice that the Inflection Point parameter for the test B formulation is significantly lower than the average inflection point. This agrees with the plots shown in Figure 12.4. Drug formulation B has a lower toxicity ratio than the other formulations.
Launch the Nonlinear Platform
To launch the Nonlinear platform, select Analyze > Specialized Modeling > Nonlinear. The launch window is shown in Figure 12.6.
Figure 12.6 Nonlinear Platform Launch Window
The Nonlinear platform launch window has the following features:
Y, Response
Select the Y variable.
X, Predictor Formula
Select the X variable (selecting a column containing a model formula launches the Nonlinear platform, as explained in the “Nonlinear Regression” chapter).
Group
Specify a grouping variable. The fitted model has separate parameters for each level of the grouping variable. This enables you to compare fitted models and estimated parameters across the levels of the grouping variable.
Weight
Specify a variable containing the weights of observations.
Freq
Specify a variable representing the frequency of an observation.
Loss
Specify a formula column giving a loss function (only for custom models).
By
Specify a variable to perform a separate analysis for every level of the variable.
The Model Library, formula, and numerical options are only for classic nonlinear analysis. For more details, see the “Nonlinear Regression” chapter.
The Fit Curve Report
The Fit Curve report initially contains only a plot of Y versus X (Figure 12.7). If you specify a Group variable, the report includes overlaid and individual plots for each group of the fitted model (shown in Figure 12.7 on the right).
Figure 12.7 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.
See the JMP Reports chapter in the Using JMP book for more information about the following options:
Local Data Filter
Shows or hides the local data filter that enables you to filter the data used in a specific report.
Redo
Contains options that enable you to repeat or relaunch the analysis. In platforms that support the feature, the Automatic Recalc option immediately reflects the changes that you make to the data table in the corresponding report window.
Save Script
Contains options that enable you to save a script that reproduces the report to several destinations.
Save By-Group Script
Contains options that enable you to save a script that reproduces the platform report for all levels of a By variable to several destinations. Available only when a By variable is specified in the launch window.
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 Figure 12.8, select Sigmoid Curves > Logistic Curves > Fit Logistic 4P and Sigmoid Curves > Fit Gompertz 4P from the Fit Curve red triangle menu.
Figure 12.8 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 the Statistical Details appendix in the Fitting Linear Models book.
AICc Weight
Gives normalized AICc values that sum to one. The AICc weight can be interpreted as the probability that a particular model is the true model given that one of the fitted models is the truth. 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, as follows:
AICcWeight = exp[-0.5(AICc-min(AICc))] / sum(exp[-0.5(AICc-min(AICc))])
where min(AICc) is the smallest AICc value among the fitted models. The AICc Weight column is then sorted in decreasing order.
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 the Statistical Details appendix in the Fitting Linear Models book.
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” chapter 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 Figure 12.9. The plots are shown only when you select a Grouping variable on the platform launch window.
Figure 12.9 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.
Plot Actual by Predicted
Plots actual Y values on the vertical axis and predicted Y values on the horizontal axis.
Plot Residual by Predicted
Plots the residuals on the vertical axis and the predicted Y values on the horizontal axis.
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 Standard Least Squares chapter in the Fitting Linear Models book.
Remove Fit
Removes the model report, the entry from the Model Comparison report, and the fitted line from the plot.
Fit Curve Options
Model Formulas
Table 12.1 provides the formulas for the models on the Fit Curve red triangle menu.
|
Model
|
Formula
|
|
Polynomials
 |
 where k is the order of the polynomial. These models can also be fit using the Fit Model and Fit Y by X platforms.
|
|
Logistic 2P
 |
 a = Growth Rate
b = Inflection Point
|
|
Logistic 3P
 |
 a = Growth Rate
b = Inflection Point
c = Asymptote
|
|
Logistic 4P
 |
 a = Growth Rate
b = Inflection Point
c = Lower Asymptote
d = Upper Asymptote
|
|
Logistic 5P
 |
 a = Growth Rate
b = Inflection Point
c = Asymptote 1
d = Asymptote 2
f = Power
|
|
Gompertz 3P
 |
 a = Asymptote
b = Growth Rate
c = Inflection Point
|
|
Gompertz 4P
 |
 a = Lower Asymptote
b = Upper Asymptote
c = Growth Rate
d = Inflection Point
|
|
Exponential 2P
 |
 a = Scale
b = Growth Rate
|
|
Exponential 3P
 |
 a = Asymptote
b = Scale
c = Growth Rate
|
|
Biexponential 4P
 |
 a = Scale 1
b = Decay Rate 1
c = Scale 2
d = Decay Rate 2
|
|
Biexponential 5P
 |
 a = Asymptote
b = Scale 1
c = Decay Rate 1
d = Scale 2
f = Decay Rate 2
|
|
Mechanistic Growth
 |
 a = Asymptote
b = Scale
c = Growth Rate
|
|
Gaussian Peak
 |
 a = Peak Value
b = Critical Point
c = Growth Rate
|
|
Lorentzian Peak
 |
 a = Peak Value
b = Growth Rate
c = Critical Point
|
|
One Compartment Oral Dose
  |
 a = Area Under Curve
b = Elimination Rate
c = Absorption Rate
|
|
Two Compartment IV Bolus Dose
  |
 α =
 β =
 a = Initial Concentration
b = Transfer Rate In
c = Transfer Rate Out
d = Elimination Rate
|
|
Michaelis-Menten
 |
 a = Max Reaction Rate
b = Inverse Affinity
|
Test Parallelism
The Test Parallelism option provides an analysis for testing if the fitted models between groups have the same shape, but are shifted along the X-axis (Figure 12.10). In the Bioassay example, the curve for drug formulation B is shifted to the left of the other three curves. However, you do not know whether the curves still have the same shape (are parallel), or if formulation B is different. The Parallelism Test tells us if the shapes for the different drug formulations have similar shapes and are shifted along the horizontal axis. Select Test Parallelism from the fitted model’s red triangle menu to add the report.
Figure 12.10 Parallelism Test
The report gives the following results:
Test Results
Gives the results of an F Test and a Chi-Square Test for parallelism. The F Test compares the error sums-of-squares for a full and a reduced model. The full model gives each group different parameters. The reduced model forces the groups to share every parameter except for the inflection point. In this example, the p-value is greater than 0.05, indicating that there is not enough evidence to conclude that differences exist between the curves.
Parallel Fit Parameter Estimates
Gives the parameter estimates under the reduced model (same parameters, except for inflection point). A plot of the fitted curves under the reduced model is provided. The inflection point for drug formulation B is much lower than that of the other three drug formulations.
Relative Potencies
Gives the relative potency for each level of the grouping variable. The relative potency is 10^(EC50), where EC50 is the concentration at which the response half way between baseline and maximum is obtained. For the Logistic 2P, 3P, and 4P, the relative potency is 10^(inflection point parameter).
Figure 12.11 Relative Potencies by group
In the Relative Potency versus standard panel from Figure 12.11, note that the relative potencies for drug formulations A and C are nearly one. This indicates that their potencies are similar to that of the standard formulation. The potency for drug formulation B is lower than that of the standard. This means that drug formulation B increases in toxicity as a function of concentration faster than the standard.
In the parallelism test, the curves are parallel, which enables you to calculate relative potencies. Based on the relative potencies, you conclude that formulation B is more potent than the other drug formulations. Taken with the prior findings, drug formulation B appears to be more toxic.
Compare Parameter Estimates
The Compare Parameter Estimates report gives results for testing the equality of parameters across the levels of the grouping variable. There is an Analysis of Means (ANOM) report for each parameter, which tests whether the parameters are equal to an overall mean. If the result for a parameter exceeds the decision limits, then the parameter is different from the overall mean. Figure 12.12 shows the ANOM report for growth rate estimates. Select Compare Parameter Estimates from the fitted model’s red triangle menu to add the report.
Figure 12.12 Parameter Comparison for Growth Rate Estimates
The Analysis of Means red triangle menu has the following options:
Set Alpha Level
Sets the alpha level for the test.
Show Summary Report
Shows or hides a report containing the parameter estimates, the decision limits, and whether the parameter exceeded the limits.
Display Options
Contains options for showing or hiding decision limits, shading, and the center line. Also contains options for changing the appearance of the points.
For more information about the Analysis of Means report, see the Oneway chapter in the Basic Analysis book.
Equivalence Test
The Equivalence Test report gives an analysis for testing the equivalence of models across levels of the grouping variable (Figure 12.13). After selecting the option, you specify the level of the grouping variable that you want to test against every other level. There is a report for every level versus the chosen level. Select Equivalence Test from the fitted model’s red triangle menu to add the report.
The equality of the parameters is tested by analyzing the ratio of the parameters. The default decision lines are placed at ratio values of 0.8 and 1.25, representing a 25% difference.
If all of the confidence intervals are inside the decision lines, then the two groups are practically equal. If a single interval falls outside the lines (as shown in Figure 12.13), then you cannot conclude that the groups are equal. The inflection point for drug formulation B is lower than the standard, which agrees with the previous findings.
Figure 12.13 Equivalence Test
The Equivalence red triangle menu has the following options:
Set Alpha Level
Sets the alpha level for the test. The default value is 0.05.
Set Decision Lines
Changes the decision limits for the ratio. The default values are set at 0.8 and 1.25, representing a 25% difference.
Show Summary Report
Shows or hides a report containing the parameter estimates, the decision limits, and whether the parameter exceeded the limits.
Display Options
Contains options for showing or hiding decision limits, shading, and the center line. Also contains options for changing the appearance of the points. For additional formatting options, right-click the graph and select Customize.
..................Content has been hidden....................
You can't read the all page of ebook, please click here login for view all page.