fixedEffects

Load the sample data.

load('weight.mat');

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'}

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 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     pValue        Lower       Upper   
    {'(Intercept)' }          50.133       2.2652     22.132    389    7.7727e-71      45.679      54.586
    {'Acceleration'}        -0.58327      0.13394    -4.3545    389    1.7075e-05    -0.84661    -0.31992
    {'Horsepower'  }        -0.16954    0.0072609     -23.35    389     5.188e-76    -0.18382    -0.15527

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

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 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       Lower      Upper 
    {'(Intercept)'  }         3.1196     0.88681      3.5178    12    0.0042407    0.41081    5.8284
    {'Shift_Morning'}        -0.3868     0.48344    -0.80009    12      0.43921    -1.8635    1.0899
    {'Shift_Night'  }         1.9856     0.48344      4.1072    12    0.0014535     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     Lower       Upper 
    {'(Intercept)'  }         3.1196     0.88681      3.5178    6.123    0.01214    -0.14122    6.3804
    {'Shift_Morning'}        -0.3868     0.48344    -0.80009       10    0.44225      -1.919    1.1454
    {'Shift_Night'  }         1.9856     0.48344      4.1072       10    0.00212     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).