Load the sample data.

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 a tall 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`

.

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 Indicator Variables.

The corresponding model is

$$\begin{array}{l}{y}_{im}={\beta}_{0}+{\beta}_{1}I{\left[MidAtl\right]}_{i}+{\beta}_{2}I{\left[ENCentral\right]}_{i}+{\beta}_{3}I{\left[WNCentral\right]}_{i}+{\beta}_{4}I{\left[SAtl\right]}_{i}\\ \text{\hspace{1em}}\text{\hspace{1em}}+{\beta}_{5}I{\left[ESCentral\right]}_{i}+{\beta}_{6}I{\left[WSCentral\right]}_{i}+{\beta}_{7}I{\left[Mtn\right]}_{i}+{\beta}_{8}I{\left[Pac\right]}_{i}+{b}_{0m}+{\epsilon}_{im},\text{\hspace{1em}}m=1,2,\mathrm{...},52,\end{array}$$

where y_{im} is
the observation *i* for level *m* of
grouping variable `Date`

, *β*_{j}, *j* =
0, 1, ..., 8, are the fixed-effects coefficients, with *β*_{0} being
the coefficient for region `NE`

. *b*_{0m} is
the random effect for level *m* of the grouping variable `Date`

,
and *ε*_{im} is
the observation error for observation *i*. The random
effect has the prior distribution, *b*_{0m} ~
N(0,σ_{b}^{2})
and the error term has the distribution, *ε*_{im} ~
N(0,σ^{2}).

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, σ^{2}_{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

$$\begin{array}{l}{\widehat{y}}_{WNCentral,10/9/2005}={\widehat{\beta}}_{0}+{\widehat{\beta}}_{3}I\left[WNCentral\right]+{\widehat{b}}_{10/9/2005}\\ \text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}=1.2233+0.23687-0.1718\\ \text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}=\mathrm{1.28837.}\end{array}$$

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.

In the previous calculation, `beta(1)`

corresponds
to the estimate for *β*_{0} and `beta(4)`

corresponds
to the estimate for *β*_{3} You
can simply display the fitted value using the `fitted`

method.

The estimated marginal response for region `WNCentral`

in
week 10/9/2005 is

$$\begin{array}{l}{\widehat{y}}_{WNCentral,10/9/2005}^{(\text{marginal})}={\widehat{\beta}}_{0}+{\widehat{\beta}}_{3}I\left[WNCentral\right]\\ \text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}=1.2233+0.23687\\ \text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}=\mathrm{1.46017.}\end{array}$$

Compute the fitted marginal response.