# coefCI

Confidence intervals for coefficients of linear mixed-effects model

## Syntax

``feCI = coefCI(lme)``
``feCI = coefCI(lme,Name,Value)``
``[feCI,reCI] = coefCI(___)``

## Description

example

````feCI = coefCI(lme)` returns the 95% confidence intervals for the fixed-effects coefficients in the linear mixed-effects model `lme`.```

example

````feCI = coefCI(lme,Name,Value)` returns the 95% confidence intervals for the fixed-effects coefficients in the linear mixed-effects model `lme` with additional options specified by one or more `Name,Value` pair arguments.For example, you can specify the confidence level or method to compute the degrees of freedom.```

example

````[feCI,reCI] = coefCI(___)` also returns the 95% confidence intervals for the random-effects coefficients in the linear mixed-effects model `lme`.```

## Examples

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Load the sample data.

`load('weight.mat')`

`weight` contains data from a longitudinal study, where 20 subjects are randomly assigned to 4 exercise programs, and their weight loss is recorded over six 2-week time periods. This is simulated data.

Store the data in a table. Define `Subject` and `Program` as categorical variables.

```tbl = table(InitialWeight, Program, Subject,Week, y); tbl.Subject = nominal(tbl.Subject); tbl.Program = nominal(tbl.Program);```

Fit a linear mixed-effects model where the initial weight, type of program, week, and the interaction between the week and type of program are the fixed effects. The intercept and week vary by subject.

`lme = fitlme(tbl,'y ~ InitialWeight + Program*Week + (Week|Subject)');`

Compute the fixed-effects coefficient estimates.

`fe = fixedEffects(lme)`
```fe = 9×1 0.6610 0.0032 0.3608 -0.0333 0.1132 0.1732 0.0388 0.0305 0.0331 ```

The first estimate, 0.6610, corresponds to the constant term. The second row, 0.0032, and the third row, 0.3608, are estimates for the coefficient of initial weight and week, respectively. Rows four to six correspond to the indicator variables for programs B-D, and the last three rows correspond to the interaction of programs B-D and week.

Compute the 95% confidence intervals for the fixed-effects coefficients.

`fecI = coefCI(lme)`
```fecI = 9×2 0.1480 1.1741 0.0005 0.0059 0.1004 0.6211 -0.2932 0.2267 -0.1471 0.3734 0.0395 0.3069 -0.1503 0.2278 -0.1585 0.2196 -0.1559 0.2221 ```

Some confidence intervals include 0. To obtain specific $p$-values for each fixed-effects term, use the `fixedEffects` method. To test for entire terms use the `anova` method.

Load the sample data.

`load carbig`

Fit a linear mixed-effects model for miles per gallon (MPG), with fixed effects for acceleration and horsepower, and a potentially correlated random effect for intercept and acceleration grouped by model year. First, store the data in a table.

`tbl = table(Acceleration,Horsepower,Model_Year,MPG);`

Fit the model.

`lme = fitlme(tbl, 'MPG ~ Acceleration + Horsepower + (Acceleration|Model_Year)');`

Compute the fixed-effects coefficient estimates.

`fe = fixedEffects(lme)`
```fe = 3×1 50.1325 -0.5833 -0.1695 ```

Compute the 99% confidence intervals for fixed-effects coefficients using the residuals method to determine the degrees of freedom. This is the default method.

`feCI = coefCI(lme,'Alpha',0.01)`
```feCI = 3×2 44.2690 55.9961 -0.9300 -0.2365 -0.1883 -0.1507 ```

Compute the 99% confidence intervals for fixed-effects coefficients using the Satterthwaite approximation to compute the degrees of freedom.

`feCI = coefCI(lme,'Alpha',0.01,'DFMethod','satterthwaite')`
```feCI = 3×2 44.0949 56.1701 -0.9640 -0.2025 -0.1884 -0.1507 ```

The Satterthwaite approximation produces similar confidence intervals than the residual method.

Load the sample data.

`load('shift.mat')`

The data shows the deviations from the target quality characteristic measured from the products that five operators manufacture during three shifts: morning, evening, and night. This is a randomized block design, where the operators are the blocks. The experiment is designed to study the impact of the time of shift on the performance. The performance measure is the deviation of the quality characteristics from the target value. This is simulated data.

`Shift` and `Operator` are nominal variables.

```shift.Shift = nominal(shift.Shift); shift.Operator = nominal(shift.Operator);```

Fit a linear mixed-effects model with a random intercept grouped by operator to assess if there is significant difference in the performance according to the time of the shift.

`lme = fitlme(shift,'QCDev ~ Shift + (1|Operator)');`

Compute the estimate of the BLUPs for random effects.

`randomEffects(lme)`
```ans = 5×1 0.5775 1.1757 -2.1715 2.3655 -1.9472 ```

Compute the 95% confidence intervals for random effects.

`[~,reCI] = coefCI(lme)`
```reCI = 5×2 -1.3916 2.5467 -0.7934 3.1449 -4.1407 -0.2024 0.3964 4.3347 -3.9164 0.0219 ```

Compute the 99% confidence intervals for random effects using the residuals method to determine the degrees of freedom. This is the default method.

`[~,reCI] = coefCI(lme,'Alpha',0.01)`
```reCI = 5×2 -2.1831 3.3382 -1.5849 3.9364 -4.9322 0.5891 -0.3951 5.1261 -4.7079 0.8134 ```

Compute the 99% confidence intervals for random effects using the Satterthwaite approximation to determine the degrees of freedom.

`[~,reCI] = coefCI(lme,'Alpha',0.01,'DFMethod','satterthwaite')`
```reCI = 5×2 -2.6840 3.8390 -2.0858 4.4372 -5.4330 1.0900 -0.8960 5.6270 -5.2087 1.3142 ```

The Satterthwaite approximation might produce smaller `DF` values than the residual method. That is why these confidence intervals are larger than the previous ones computed using the residual method.

## Input Arguments

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Linear mixed-effects model, specified as a `LinearMixedModel` object constructed using `fitlme` or `fitlmematrix`.

### Name-Value Arguments

Specify optional pairs of arguments as `Name1=Value1,...,NameN=ValueN`, where `Name` is the argument name and `Value` is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose `Name` in quotes.

Example: ` [feCI,reCI] = coefCI(lme,'Alpha',0.01) `

Significance level, specified as the comma-separated pair consisting of `'Alpha'` and a scalar value in the range 0 to 1. For a value α, the confidence level is 100*(1–α)%.

For example, for 99% confidence intervals, you can specify the confidence level as follows.

Example: `'Alpha',0.01`

Data Types: `single` | `double`

Method for computing approximate degrees of freedom for confidence interval computation, specified as the comma-separated pair consisting of `'DFMethod'` and one of the following.

 `'residual'` Default. The degrees of freedom are assumed to be constant and equal to n – p, where n is the number of observations and p is the number of fixed effects. `'satterthwaite'` Satterthwaite approximation. `'none'` All degrees of freedom are set to infinity.

For example, you can specify the Satterthwaite approximation as follows.

Example: `'DFMethod','satterthwaite'`

## Output Arguments

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Fixed-effects confidence intervals, returned as a p-by-2 matrix. `feCI` contains the confidence limits that correspond to the p fixed-effects estimates in the vector `beta` returned by the `fixedEffects` method. The first column of `feCI` has the lower confidence limits and the second column has the upper confidence limits.

Random-effects confidence intervals, returned as a q-by-2 matrix. `reCI` contains the confidence limits corresponding to the q random-effects estimates in the vector `B` returned by the `randomEffects` method. The first column of `reCI` has the lower confidence limits and the second column has the upper confidence limits.

## Version History

Introduced in R2013b