# fixedEffects

Class: LinearMixedModel

Estimates of fixed effects and related statistics

## Description

beta = fixedEffects(lme) returns the estimated fixed-effects coefficients, beta, of the linear mixed-effects model lme.

example

[beta,betanames] = fixedEffects(lme) also returns the names of estimated fixed-effects coefficients in betanames. Each name corresponds to a fixed-effects coefficient in beta.

example

[beta,betanames,stats] = fixedEffects(lme) also returns the estimated fixed-effects coefficients of the linear mixed-effects model lme and related statistics in stats.

example

[beta,betanames,stats] = fixedEffects(lme,Name,Value) also returns the estimated fixed-effects coefficients of the linear mixed-effects model lme and related statistics with additional options specified by one or more Name,Value pair arguments.

## Input Arguments

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

### Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

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 the t-statistic that tests the fixed-effects coefficients against 0, 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 coefficients estimates of the fitted linear mixed-effects model lme, returned as a vector.

Names of fixed-effects coefficients in beta, returned as a table.

Fixed-effects estimates and related statistics, returned as a dataset array that has one row for each of the fixed effects and one column for each of the following statistics.

 Name Name of the fixed effect coefficient Estimate Estimated coefficient value SE Standard error of the estimate tStat t-statistic for a test that the coefficient is zero DF Estimated degrees of freedom for the t-statistic pValue p-value for the t-statistic Lower Lower limit of a 95% confidence interval for the fixed-effect coefficient Upper Upper limit of a 95% confidence interval for the fixed-effect coefficient

## Examples

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The data set 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 week and program are the fixed effects. The intercept and week vary by subject.

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

Display the fixed-effects coefficient estimates and corresponding fixed-effects names.

[beta,betanames] = fixedEffects(lme)
beta = 9×1

0.6610
0.0032
0.3608
-0.0333
0.1132
0.1732
0.0388
0.0305
0.0331

betanames=9×1 table
Name
__________________

{'(Intercept)'   }
{'InitialWeight' }
{'Program_B'     }
{'Program_C'     }
{'Program_D'     }
{'Week'          }
{'Program_B:Week'}
{'Program_C:Week'}
{'Program_D:Week'}

Fit a linear mixed-effects model for miles per gallon (MPG), with fixed effects for acceleration and horsepower, and potentially correlated random effects 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 coefficients estimates and related statistics.

[~,~,stats] = fixedEffects(lme)
stats =
Fixed effect coefficients: DFMethod = 'Residual', Alpha = 0.05

Name                    Estimate    SE           tStat      DF
{'(Intercept)' }          50.133       2.2652     22.132    389
{'Acceleration'}        -0.58327      0.13394    -4.3545    389
{'Horsepower'  }        -0.16954    0.0072609     -23.35    389

pValue        Lower       Upper
7.7727e-71      45.679      54.586
1.7075e-05    -0.84661    -0.31992
5.188e-76    -0.18382    -0.15527

The small $p$-values (under pValue) indicate that all fixed-effects coefficients are significant.

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 performance significantly differs according to the time of the shift.

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

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

[~,~,stats] = fixedEffects(lme,'alpha',0.01)
stats =
Fixed effect coefficients: DFMethod = 'Residual', Alpha = 0.01

Name                     Estimate    SE         tStat       DF    pValue
{'(Intercept)'  }         3.1196     0.88681      3.5178    12    0.0042407
{'Shift_Morning'}        -0.3868     0.48344    -0.80009    12      0.43921
{'Shift_Night'  }         1.9856     0.48344      4.1072    12    0.0014535

Lower      Upper
0.41081    5.8284
-1.8635    1.0899
0.5089    3.4623

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

[~,~,stats] = fixedEffects(lme,'DFMethod','satterthwaite','alpha',0.01)
stats =
Fixed effect coefficients: DFMethod = 'Satterthwaite', Alpha = 0.01

Name                     Estimate    SE         tStat       DF       pValue
{'(Intercept)'  }         3.1196     0.88681      3.5178    6.123    0.01214
{'Shift_Morning'}        -0.3868     0.48344    -0.80009       10    0.44225
{'Shift_Night'  }         1.9856     0.48344      4.1072       10    0.00212

Lower       Upper
-0.14122    6.3804
-1.919    1.1454
0.45343    3.5178

The Satterthwaite approximation usually produces smaller DF values than the residual method. That is why it produces larger $p$-values (pValue) and larger confidence intervals (see Lower and Upper).