fitted

Load the sample data.

load flu

The flu dataset array has a Date variable, and 10 variables containing estimated influenza rates (in 9 different regions, estimated from Google® searches, plus a nationwide estimate from the Center for Disease Control and Prevention, CDC).

To fit a linear-mixed effects model, your data must be in a properly formatted dataset array. To fit a linear mixed-effects model with the influenza rates as the responses and region as the predictor variable, combine the nine columns corresponding to the regions into an array. The new dataset array, flu2, must have the response variable, FluRate, the nominal variable, Region, that shows which region each estimate is from, and the grouping variable Date.

flu2 = stack(flu,2:10,'NewDataVarName','FluRate','IndVarName','Region');
flu2.Date = nominal(flu2.Date);

Fit a linear mixed-effects model with fixed effects for region and a random intercept that varies by Date.

Region is a categorical variable. You can specify the contrasts for categorical variables using the DummyVarCoding name-value pair argument when fitting the model. When you do not specify the contrasts, fitlme uses the 'reference' contrast by default. Because the model has an intercept, fitlme takes the first region, NE, as the reference and creates eight dummy variables representing the other eight regions. For example, $$I[MidAtl]$$ is the dummy variable representing the region MidAtl. For details, see Dummy Variables.

The corresponding model is

where $$y_{im}$$ is the observation $$i$$ for level $$m$$ of grouping variable Date, $${\beta }_j$$, $$j$$ = 0, 1, ..., 8, are the fixed-effects coefficients, with $${\beta }_0$$ being the coefficient for region NE. $$b_{0m}$$ is the random effect for level $$m$$ of the grouping variable Date, and $${\epsilon }_{im}$$ is the observation error for observation $$i$$. The random effect has the prior distribution, $$b_{0m}\sim N(0,{\sigma }_b^2)$$ and the error term has the distribution, $${\epsilon }_{im}\sim N(0,{\sigma }^2)$$.

lme = fitlme(flu2,'FluRate ~ 1 + Region + (1|Date)')

lme = 
Linear mixed-effects model fit by ML

Model information:
    Number of observations             468
    Fixed effects coefficients           9
    Random effects coefficients         52
    Covariance parameters                2

Formula:
    FluRate ~ 1 + Region + (1 | Date)

Model fit statistics:
    AIC       BIC       LogLikelihood    Deviance
    318.71    364.35    -148.36          296.71  

Fixed effects coefficients (95% CIs):
    Name                        Estimate    SE          tStat      DF     pValue        Lower        Upper    
    {'(Intercept)'     }          1.2233    0.096678     12.654    459     1.085e-31       1.0334       1.4133
    {'Region_MidAtl'   }        0.010192    0.052221    0.19518    459       0.84534    -0.092429      0.11281
    {'Region_ENCentral'}        0.051923    0.052221     0.9943    459        0.3206    -0.050698      0.15454
    {'Region_WNCentral'}         0.23687    0.052221     4.5359    459    7.3324e-06      0.13424      0.33949
    {'Region_SAtl'     }        0.075481    0.052221     1.4454    459       0.14902     -0.02714       0.1781
    {'Region_ESCentral'}         0.33917    0.052221      6.495    459    2.1623e-10      0.23655      0.44179
    {'Region_WSCentral'}           0.069    0.052221     1.3213    459       0.18705    -0.033621      0.17162
    {'Region_Mtn'      }        0.046673    0.052221    0.89377    459       0.37191    -0.055948      0.14929
    {'Region_Pac'      }        -0.16013    0.052221    -3.0665    459     0.0022936     -0.26276    -0.057514

Random effects covariance parameters (95% CIs):
Group: Date (52 Levels)
    Name1                  Name2                  Type           Estimate    Lower     Upper  
    {'(Intercept)'}        {'(Intercept)'}        {'std'}        0.6443      0.5297    0.78368

Group: Error
    Name               Estimate    Lower      Upper
    {'Res Std'}        0.26627     0.24878    0.285

The $$p$$-values 7.3324e-06 and 2.1623e-10 respectively show that the fixed effects of the flu rates in regions WNCentral and ESCentral are significantly different relative to the flu rates in region NE.

The confidence limits for the standard deviation of the random-effects term, $${\sigma }_b$$, do not include 0 (0.5297, 0.78368), which indicates that the random-effects term is significant. You can also test the significance of the random-effects terms using the compare method.

The conditional fitted response from the model at a given observation includes contributions from fixed and random effects. For example, the estimated best linear unbiased predictor (BLUP) of the flu rate for region WNCentral in week 10/9/2005 is

This is the fitted conditional response, since it includes contributions to the estimate from both the fixed and random effects. You can compute this value as follows.

beta = fixedEffects(lme);
[~,~,STATS] = randomEffects(lme); % Compute the random-effects statistics (STATS)
STATS.Level = nominal(STATS.Level);
y_hat = beta(1) + beta(4) + STATS.Estimate(STATS.Level=='10/9/2005')

y_hat = 
1.2884

In the previous calculation, beta(1) corresponds to the estimate for $${\beta }_0$$ and beta(4) corresponds to the estimate for $${\beta }_3$$. You can simply display the fitted value using the fitted method.

F = fitted(lme);
F(flu2.Date == '10/9/2005' & flu2.Region == 'WNCentral')

ans = 
1.2884

The estimated marginal response for region WNCentral in week 10/9/2005 is

Compute the fitted marginal response.

F = fitted(lme,'Conditional',false);
F(flu2.Date == '10/9/2005' & flu2.Region == 'WNCentral')

ans = 
1.4602

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 fitted values and raw residuals.

F = fitted(lme);
R = residuals(lme);

Plot the residuals versus the fitted values.

plot(F,R,'bx')
xlabel('Fitted Values')
ylabel('Residuals')

Now, plot the residuals versus the fitted values, grouped by program.

figure()
gscatter(F,R,Program)