# coefCI

Confidence intervals of coefficient estimates of generalized linear regression model

## Description

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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Find the confidence intervals for the coefficients of a fitted generalized linear regression model.

Generate sample data using Poisson random numbers with two underlying predictors X(:,1) and X(:,2).

rng('default') % For reproducibility
rndvars = randn(100,2);
X = [2 + rndvars(:,1),rndvars(:,2)];
mu = exp(1 + X*[1;2]);
y = poissrnd(mu);

Create a generalized linear regression model of Poisson data.

mdl = fitglm(X,y,'y ~ x1 + x2','Distribution','poisson')
mdl =
Generalized linear regression model:
log(y) ~ 1 + x1 + x2
Distribution = Poisson

Estimated Coefficients:
Estimate       SE        tStat     pValue
________    _________    ______    ______

(Intercept)     1.0405      0.022122    47.034      0
x1              0.9968      0.003362    296.49      0
x2               1.987     0.0063433    313.24      0

100 observations, 97 error degrees of freedom
Dispersion: 1
Chi^2-statistic vs. constant model: 2.95e+05, p-value = 0

Find 95% (default) confidence intervals for the coefficients of the model.

ci = coefCI(mdl)
ci = 3×2

0.9966    1.0844
0.9901    1.0035
1.9744    1.9996

Find 99% confidence intervals for the coefficients.

alpha = 0.01;
ci = coefCI(mdl,alpha)
ci = 3×2

0.9824    1.0986
0.9880    1.0056
1.9703    2.0036

## Input Arguments

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Generalized linear regression model, specified as a GeneralizedLinearModel object created using fitglm or stepwiseglm, or a CompactGeneralizedLinearModel object created 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.

## Version History

Introduced in R2012a