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# coefCI

Class: LinearMixedModel

Confidence intervals for coefficients of linear mixed-effects model

## 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.

## 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.

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.

## 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.