coefCI

Confidence intervals of coefficient estimates of linear regression model

Syntax

ci = coefCI(mdl)

ci = coefCI(mdl,alpha)

Description

ci = coefCI(mdl) returns 95% confidence intervals for the coefficients in mdl.

ci = coefCI(mdl,alpha) returns confidence intervals using the confidence level 1 – alpha.

Examples

Find Confidence Intervals for Model Coefficients

Fit a linear regression model and obtain the default 95% confidence intervals for the resulting model coefficients.

Load the carbig data set and create a table in which the Origin predictor is categorical.

load carbig
Origin = categorical(cellstr(Origin));
tbl = table(Horsepower,Weight,MPG,Origin);

Fit a linear regression model. Specify Horsepower, Weight, and Origin as predictor variables, and specify MPG as the response variable.

modelspec = 'MPG ~ 1 + Horsepower + Weight + Origin';
mdl = fitlm(tbl,modelspec);

View the names of the coefficients.

mdl.CoefficientNames

ans = 1x9 cell
    {'(Intercept)'}    {'Horsepower'}    {'Weight'}    {'Origin_France'}    {'Origin_Germany'}    {'Origin_Italy'}    {'Origin_Japan'}    {'Origin_Sweden'}    {'Origin_USA'}

Find confidence intervals for the coefficients of the model.

ci = coefCI(mdl)

ci = 9×2

   43.3611   59.9390
   -0.0748   -0.0315
   -0.0059   -0.0037
  -17.3623   -0.3477
  -15.7503    0.7434
  -17.2091    0.0613
  -14.5106    1.8738
  -18.5820   -1.5036
  -17.3114   -0.9642

Specify Confidence Level

Fit a linear regression model and obtain the confidence intervals for the resulting model coefficients using a specified confidence level.

Load the carbig data set and create a table in which the Origin predictor is categorical.

load carbig
Origin = categorical(cellstr(Origin));
tbl = table(Horsepower,Weight,MPG,Origin);

Fit a linear regression model. Specify Horsepower, Weight, and Origin as predictor variables, and specify MPG as the response variable.

modelspec = 'MPG ~ 1 + Horsepower + Weight + Origin';
mdl = fitlm(tbl,modelspec);

Find 99% confidence intervals for the coefficients.

ci = coefCI(mdl,.01)

ci = 9×2

   40.7365   62.5635
   -0.0816   -0.0246
   -0.0062   -0.0034
  -20.0560    2.3459
  -18.3615    3.3546
  -19.9433    2.7955
  -17.1045    4.4676
  -21.2858    1.2002
  -19.8995    1.6238

The confidence intervals are wider than the default 95% confidence intervals in Find Confidence Intervals for Model Coefficients.

Input Arguments

mdl — Linear regression model object

LinearModel object | CompactLinearModel object

Linear regression model object, specified as a LinearModel object created by using fitlm or stepwiselm, or a CompactLinearModel object created by using compact.

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

$$b_i\pm t_{\left(1-\alpha /2,n-p\right)}SE\left(b_i\right),$$

where b_i is the coefficient estimate, SE(b_i) 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.

Extended Capabilities

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

This function fully supports GPU arrays. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Version History

Introduced in R2012a