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# coefCI

Class: CompactLinearModel

Confidence intervals of coefficient estimates of linear model

## Syntax

`ci = coefCI(mdl)ci = coefCI(mdl,alpha)`

## Description

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

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

## Input Arguments

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Linear model object, specified as a full `LinearModel` object constructed using `fitlm` or `stepwiselm`, or a compacted `CompactLinearModel` object constructed using `compact`.

Confidence interval, specified as a numeric value in the range [0,1]. `alpha` is the probability that the confidence interval does not contain the true value.

## Output Arguments

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Confidence intervals, returned as a k-by-2 numeric matrix. The jth row of `ci` is the confidence interval of coefficient j of `mdl`. The name of coefficient j is stored in the `mdl` property `CoefficientNames`.

## Definitions

### Confidence Interval

Assume that model assumptions hold (data comes from a generalized linear model represented by the formula `mdl``.Formula`, and with observations that are independent conditional on the predictor values). Then row `j` of the confidence interval matrix `ci` gives a confidence interval `[ci(j,1),ci(j,2)]` computed such that, with repeated experimentation, a proportion `1 - alpha` of the intervals will contain the true value of the coefficient.

## Examples

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Fit a linear regression model for auto mileage based on the `carbig` data. Then obtain the default 95% confidence intervals for the resulting model coefficients.

Load the data and create a table.

```load carbig Origin = nominal(Origin); tbl = table(Horsepower,Weight,MPG,Origin); ```

Fit a linear regression model. Use horsepower, weight, and origin as predictor variables, and miles per gallon as the response variable.

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

View the names of the coefficients.

```mdl.CoefficientNames ```
```ans = 1×9 cell array Columns 1 through 4 '(Intercept)' 'Horsepower' 'Weight' 'Origin_France' Columns 5 through 8 'Origin_Germany' 'Origin_Italy' 'Origin_Japan' 'Origin_Sweden' Column 9 'Origin_USA' ```

Find confidence intervals for the coefficients of the model.

```ci = coefCI(mdl) ```
```ci = 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 for auto mileage based on the `carbig` data. Then obtain confidence intervals for the resulting model coefficients at the 99% level.

Load the data and create a table.

```load carbig Origin = nominal(Origin); tbl = table(Horsepower,Weight,MPG,Origin); ```

Fit a linear regression model using horsepower, weight, and origin as the predictor variables, and miles per gallon 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 = 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 5% confidence intervals.

## Alternatives

You can create the intervals from the model coefficients in `mdl``.Coefficients.Estimate` and an appropriate multiplier of the standard errors `sqrt(diag(mdl.CoefficientCovariance))`. The multiplier is `tinv(1-alpha/2,dof)`, where `level` is the confidence level, and `dof` is the degrees of freedom (number of data points minus the number of coefficients).