yfit = fitted(lme)
yfit = fitted(lme,Name,Value)
lme— Linear mixed-effects model
Linear mixed-effects model, returned as a
For properties and methods of this object, see
Specify optional comma-separated pairs of
Name is the argument
Value is the corresponding
Name must appear
inside single quotes (
You can specify several name and value pair
arguments in any order as
'Conditional'— Indicator for conditional responseTrue (default) | False
Indicator for conditional response, specified as the comma-separated
pair consisting of
'Conditional' and either of
|Contribution from both fixed effects and random effects (conditional)|
|Contribution from only fixed effects (marginal)|
Load the sample data.
flu dataset array has a
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 a tall array. The new dataset array,
must have the response variable,
FluRate, the nominal
Region, that shows which region each
estimate is from, and the grouping variable
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
Region is a categorical variable. You can specify the contrasts
for categorical variables using the
pair argument when fitting the model. When you do not specify the
fitlme uses the
by default. Because the model has an intercept,
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
For details, see Dummy Indicator Variables.
The corresponding model is
where yim is
the observation i for level m of
Date, βj, j =
0, 1, ..., 8, are the fixed-effects coefficients, with β0 being
the coefficient for region
NE. b0m is
the random effect for level m of the grouping variable
and εim is
the observation error for observation i. The random
effect has the prior distribution, b0m ~
and the error term has the distribution, εim ~
lme = fitlme(flu2,'FluRate ~ 1 + Region + (1|Date)')
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
significantly different relative to the flu rates in region
The confidence limits for the standard deviation of the random-effects
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
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
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,
to the estimate for β0 and
to the estimate for β3 You
can simply display the fitted value using the
F = fitted(lme); F(flu2.Date == '10/9/2005' & flu2.Region == 'WNCentral')
ans = 1.2884
The estimated marginal response for region
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
Navigate to a folder containing sample data.
Load the sample data.
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
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.
A conditional response includes contributions from both fixed and random effects, whereas a marginal response includes contribution from only fixed effects.
Suppose the linear mixed-effects model,
has an n-by-p fixed-effects
design matrix X and an n-by-q random-effects
design matrix Z. Also, suppose the p-by-1
estimated fixed-effects vector is , and the q-by-1
estimated best linear unbiased predictor (BLUP) vector of random effects
is . The fitted conditional response
and the fitted marginal response is