coefCI
Confidence intervals of coefficient estimates of nonlinear regression model
Description
Examples
Default Confidence Intervals
Create a nonlinear model for auto mileage based on the carbig
data. Then obtain confidence intervals for the resulting model coefficients.
Load the data and create a nonlinear model.
load carbig ds = dataset(Horsepower,Weight,MPG); modelfun = @(b,x)b(1) + b(2)*x(:,1) + ... b(3)*x(:,2) + b(4)*x(:,1).*x(:,2); beta0 = [1 1 1 1]; mdl = fitnlm(ds,modelfun,beta0)
mdl = Nonlinear regression model: MPG ~ b1 + b2*Horsepower + b3*Weight + b4*Horsepower*Weight Estimated Coefficients: Estimate SE tStat pValue __________ __________ _______ __________ b1 63.558 2.3429 27.127 1.2343e-91 b2 -0.25084 0.027279 -9.1952 2.3226e-18 b3 -0.010772 0.00077381 -13.921 5.1372e-36 b4 5.3554e-05 6.6491e-06 8.0542 9.9336e-15 Number of observations: 392, Error degrees of freedom: 388 Root Mean Squared Error: 3.93 R-Squared: 0.748, Adjusted R-Squared 0.746 F-statistic vs. constant model: 385, p-value = 7.26e-116
All the coefficients have extremely small -values. This means a confidence interval around the coefficients will not contain the point 0
, unless the confidence level is very high.
Find 95% confidence intervals for the coefficients of the model.
ci = coefCI(mdl)
ci = 4×2
58.9515 68.1644
-0.3045 -0.1972
-0.0123 -0.0093
0.0000 0.0001
The confidence interval for b4
seems to contain 0
. Examine it in more detail.
ci(4,:)
ans = 1×2
10-4 ×
0.4048 0.6663
As expected, the confidence interval does not contain the point 0
.
Input Arguments
mdl
— Nonlinear regression model object
NonLinearModel
object
Nonlinear regression model object, specified as a NonLinearModel
object created by using fitnlm
.
alpha
— Significance level
0.05 (default) | numeric value in the range [0,1]
Significance level for the confidence interval, 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: 0.01
Data Types: single
| double
Output Arguments
ci
— Confidence intervals
numeric matrix
Confidence intervals, returned as a k-by-2 numeric matrix, where
k is the number of coefficients. The jth row
of ci
is the confidence interval of the jth
coefficient of mdl
. The name of coefficient j is
stored in the CoefficientNames
property of
mdl
.
Data Types: single
| double
More About
Confidence Interval
The coefficient confidence intervals provide a measure of precision for regression coefficient estimates.
A 100(1 – α)% confidence interval gives the range for the corresponding regression coefficient with 100(1 – α)% confidence, meaning that 100(1 – α)% of the intervals resulting from repeated experimentation will contain the true value of the coefficient.
The software finds confidence intervals using the Wald method. The 100(1 – α)% confidence intervals for regression coefficients are
where bi is the coefficient estimate, SE(bi) is the standard error of the coefficient estimate, and t(1–α/2,n–p) is the 100(1 – α/2) percentile of the t-distribution with n – p degrees of freedom. n is the number of observations and p is the number of regression coefficients.
Version History
Introduced in R2012a
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