profileLikelihood

Profile loglikelihood for nonlinear regression model

Since R2025a

Syntax

[LV,PV] = profileLikelihood(mdl,coef)

[LV,PV,CI] = profileLikelihood(mdl,coef)

[___] = profileLikelihood(mdl,coef,Name=Value)

Description

[LV,PV] = profileLikelihood(mdl,coef) returns the profile loglikelihood for the model mdl and coefficient of interest coef.

[LV,PV,CI] = profileLikelihood(mdl,coef) also returns the likelihood-ratio confidence intervals for coef.

[___] = profileLikelihood(mdl,coef,Name=Value) specifies additional options using one or more name-value arguments. For example, you can specify the significance level for the confidence interval and the values for the coefficient of interest.

Examples

Calculate Profile Loglikelihood and Confidence Interval

Load a table of standardized variables generated from the carbig data set.

load standardizedcar.mat

The table tbl contains the variables Horsepower, Weight, and MPG, which represent car horsepower, weight, and miles per gallon, respectively.

Fit a nonlinear model to the data using Horsepower and Weight as predictors, and MPG as the response.

modelfun = @(b,x) exp(b(1)*x(:,1))+b(2)*x(:,2)+b(3);
beta0 = [1 1 1];
mdl = fitnlm(tbl,modelfun,beta0)

mdl = 
Nonlinear regression model:
    MPG ~ exp(b1*Horsepower) + b2*Weight + b3

Estimated Coefficients:
          Estimate       SE        tStat       pValue   
          ________    ________    _______    ___________

    b1    -0.57016    0.045819    -12.444     3.7325e-30
    b2    -0.39274    0.043737    -8.9797     1.1804e-17
    b3     -1.1417    0.034104    -33.476    1.3291e-116


Number of observations: 392, Error degrees of freedom: 389
Root Mean Squared Error: 0.516
R-Squared: 0.735,  Adjusted R-Squared 0.733
F-statistic vs. constant model: 539, p-value = 8.27e-113

mdl contains a fitted nonlinear regression model. The coefficient b1 is a nonlinear coefficient because it is inside the exponential term in the model function.

Calculate the profile loglikelihood and confidence interval for b1.

[LV,PV,CI] = profileLikelihood(mdl,"b1");
CI

CI = 1×2

   -0.6597   -0.4660

The output shows the 95% likelihood-ratio confidence interval for b1.

Plot the profile loglikelihood values for b1 using the plotProfileLikelihood function.

plotProfileLikelihood(mdl,"b1")

The plot shows the loglikelihood values together with the estimated value for b1, the Wald approximation, and the Wald and likelihood-ratio confidence intervals. The calculated values for b1 cover the confidence intervals, and the maximum likelihood estimate for b1 appears at the peak of the profile loglikelihood, confirming it is the maximum likelihood estimate. The likelihood-ratio confidence interval is slightly wider than the Wald interval, and is also asymmetric. However, the closeness of the two intervals suggests that the assumptions of the Wald approximation hold true for this model.

Calculate Profile Loglikelihood over Interval

Load the reaction data set.

load reaction

The variables reactants and rate contain data for the partial pressures of three chemicals and their reactant rates. The vector beta contains initial values for the Hougen-Watson model coefficients.

Fit the Hougen-Watson model to the data using the hougen function. Use reactants as the predictor data and rate as the response.

mdl = fitnlm(reactants,rate,@hougen,beta)

mdl = 
Nonlinear regression model:
    y ~ hougen(b,X)

Estimated Coefficients:
          Estimate       SE       tStat     pValue 
          ________    ________    ______    _______

    b1      1.2526     0.86701    1.4447    0.18654
    b2    0.062776    0.043561    1.4411    0.18753
    b3    0.040048    0.030885    1.2967    0.23089
    b4     0.11242    0.075157    1.4957    0.17309
    b5      1.1914     0.83671    1.4239     0.1923


Number of observations: 13, Error degrees of freedom: 8
Root Mean Squared Error: 0.193
R-Squared: 0.999,  Adjusted R-Squared 0.998
F-statistic vs. zero model: 3.91e+03, p-value = 2.54e-13

mdl contains the fitted nonlinear regression model. The estimate for b2 is near 0.06 and has a large p-value.

Calculate the profile loglikelihood for b2 in an interval around its estimated value. Plot the loglikelihood values against the specified values for b2.

[LV2,PV2] = profileLikelihood(mdl,"b2",CoefficientValues=[0.01:0.01:1]);
plot(PV2,LV2)
xlabel("b2")
ylabel("loglikelihood")

The profile loglikelihood has a nonlinear elbow shape and does not change significantly for values of b2 larger than 0.1. This result is consistent with the large p-value, which suggests that b2 does not have a statistically significant effect on the response variable.

Input Arguments

mdl — Nonlinear regression model

NonLinearModel object

Nonlinear regression model, specified as a NonLinearModel object created using fitnlm.

coef — Coefficient of interest

string | character array | index

Coefficient of interest, specified as a string, character array, or index. Coefficients not specified in coef are called nuisance coefficients. For each value of the coefficient of interest, profileLikelihood calculates values for the nuisance coefficients. For more information, see Profile Loglikelihood.

Example: "b1"

Example: 3

Data Types: single | double | char | string

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Example: [LV,PV,CI]=profileLikelihood(mdl,"b2",Alpha=0.01,Scale="log") calculates the 99% confidence intervals and values for the coefficient of interest on the log scale.

Alpha — Significance level

0.05 (default) | numeric value in the range [0,1]

Significance level for the confidence intervals, specified as a numeric value in the range [0,1]. The confidence level of CI is equal to 100(1 – alpha)%. Alpha is the probability that the confidence interval does not contain the true value.

Example: Alpha=0.01

Data Types: single | double

CoefficientValues — Values for coefficient of interest

numeric scalar | numeric vector

Values for the coefficient of interest, specified as a numeric scalar or vector. When you do not specify CoefficientValues, profileLikelihood selects values for coef that cover a typical confidence interval.

Example: CoefficientValues=[0:0.02:1]

Data Types: single | double

Scale — Scale for plotting

"linear" (default) | "log"

Scale for plotting the loglikelihood values, specified as "linear" or "log".

• When Scale is "linear", the function selects values for the coefficient of interest that lie on a Euclidean grid.

• When Scale is "log", the function selects values for the coefficient of interest that lie on a logarithmic grid. You can specify Scale as "log" only for positive coefficients.

Example: Scale="log"

Data Types: string | char

Output Arguments

LV — Loglikelihood values

numeric scalar | numeric vector

Loglikelihood values, returned as a numeric scalar or vector.

PV — Values for coefficient of interest

numeric scalar | numeric vector

Values for the coefficient of interest, returned as a numeric scalar or vector. You can specify PV using the CoefficientValues name-value argument. When you do not specify CoefficientValues, profileLikelihood returns values for PV that cover a typical confidence interval.

CI — Confidence interval

1-by-2 numeric vector

Confidence interval for the coefficient of interest, returned as a 1-by-2 numeric vector. CI is a likelihood-ratio confidence interval, which profileLikelihood calculates by determining the coefficient values for which the profile likelihood drops below a threshold.

Data Types: single | double

More About

Profile Loglikelihood

The profile loglikelihood describes the maximum possible likelihood for a set of nuisance coefficients, given a fixed value of the coefficient of interest. The coefficient of interest is a coefficient that you want to vary, and the nuisance coefficients are the other coefficients in the model formula.

The profile loglikelihood is described by the equation

$$L(\mu )=\underset{\theta }{\max }L(\theta |\mu ,X,Y),$$

which contains these variables and terms:

• $$\mu$$ — Coefficient of interest

• $$L(\mu )$$ — Profile loglikelihood function

• $$\theta$$ — Nuisance coefficients

• $$X$$ — Predictor input data

• $$Y$$ — Response input data

• $$L(\theta |\mu ,X,Y)$$ — Loglikelihood function for theta given $$\mu$$ and $$X$$

Alternative Functionality

You can calculate both Wald and likelihood-ratio confidence intervals for several coefficients using the coefCI function.

Version History

Introduced in R2025a