# coefCI

Confidence intervals of coefficient estimates of linear regression model

## Description

example

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

example

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

## Examples

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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.

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
Columns 1 through 4

{'(Intercept)'}    {'Horsepower'}    {'Weight'}    {'Origin_France'}

Columns 5 through 7

{'Origin_Germany'}    {'Origin_Italy'}    {'Origin_Japan'}

Columns 8 through 9

{'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

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.

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

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

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

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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

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### Confidence Interval

The coefficient confidence intervals provide a measure of precision for regression coefficient estimates.

A 100(1 – α)% confidence interval gives the range that the corresponding regression coefficient will be in 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

${b}_{i}±{t}_{\left(1-\alpha /2,n-p\right)}SE\left({b}_{i}\right),$

where bi is the coefficient estimate, SE(bi) is the standard error of the coefficient estimate, and t(1–α/2,np) is the 100(1 – α/2) percentile of t-distribution with n – p degrees of freedom. n is the number of observations and p is the number of regression coefficients.