The Logistic w Loss.jmp data table in the Nonlinear Examples sample data folder has an example for fitting a logistic regression using a loss function. The Y column is the proportion of ones for equal-sized samples of x values. The Model Y column has the linear model, and the Loss column has the loss function. In this example, the loss function is the negative log-likelihood for each observation, or the negative log of the probability of getting the observed response.
1.
Open Logistic w Loss.jmp from the Nonlinear Examples sample data folder.
2.
Select Analyze > Modeling > Nonlinear.
3.
Assign Model Y to the X, Predictor Formula role.
4.
Assign Loss to the Loss role.
Nonlinear Launch Window
5.
Nonlinear Fit Control Panel
6.
Solution Report
1.
Display the Logistic w Loss.jmp sample data table again.
2.
Select Analyze >Modeling > Nonlinear.
3.
Assign Model2 Y to the X, Predictor Formula role.
4.
Assign Loss2 to the Loss role.
5.
Select the Second Derivatives option.
Nonlinear Launch Window for Second Derivatives
6.
Specify the Stop Limit
8.
The Standard Error is different
The Ingots2.jmp sample data table includes the numbers of ingots tested for readiness after different treatments of heating and soaking times. The response variable, NReady, is binomial, depending on the number of ingots tested (Ntotal) and the heating and soaking times. Maximum likelihood estimates for parameters from a probit model with binomial errors are obtained using:
Normal Distribution(b0+b1*Heat+b2*Soak)
The argument to the Normal Distribution function is a linear model of the treatments.
-(Nready*Log(p) + (Ntotal - Nready)*Log(1 - p))
1.
Select Analyze > Modeling > Nonlinear.
2.
Assign P to the X, Predictor Formula role,
3.
Assign Loss to the Loss role.
4.
Select the Numeric Derivatives Only option.
5.
6.
Solution for the Ingots2 Data
, n = 0, 1, 2, …
where μ can be a single parameter, or a linear model with many parameters. Many texts and papers show how the model can be transformed and fit with iteratively reweighted least squares (Nelder and Wedderburn 1972). However, in JMP it is more straightforward to fit the model directly. For example, McCullagh and Nelder (1989) show how to analyze the number of reported damage incidents caused by waves to cargo-carrying vessels.
The data are in the Ship Damage.jmp sample data table. The model formula is in the model column, and the loss function (or –log-likelihood) is in the Poisson column. To fit the model, follow the steps below:
1.
Select Analyze > Modeling > Nonlinear.
2.
Assign model to the X, Predictor Formula role.
3.
Assign Poisson to the Loss role.
4.
5.
Set the Current Value (initial value) for b0 to 1, and the other parameters to 0 (Enter New Parameters).
Enter New Parameters
6.
7.
Click the Confidence Limits button.
The Solution report is shown in Solution Table for the Poisson Loss Example. The results include the parameter estimates and confidence intervals, and other summary statistics.
Solution Table for the Poisson Loss Example