`{GLMMcosinor}`

allows specification of mixed models
accounting for fixed and/or random effects. Mixed model specification
follows the `{lme4}`

format. See their vignette, Fitting
Linear Mixed-Effects Models Using lme4, for details about how to
specify mixed models.

To illustrate an example of using a model with random effects on the
cosinor components, we will first simulate some data with
`id`

-level differences in amplitude and acrophase.

```
f_sample_id <- function(id_num,
n = 30,
mesor,
amp,
acro,
family = "gaussian",
sd = 0.2,
period,
n_components,
beta.group = TRUE) {
data <- simulate_cosinor(
n = n,
mesor = mesor,
amp = amp,
acro = acro,
family = family,
sd = sd,
period = period,
n_components = n_components
)
data$subject <- id_num
data
}
dat_mixed <- do.call(
"rbind",
lapply(1:30, function(x) {
f_sample_id(
id_num = x,
mesor = rnorm(1, mean = 0, sd = 1),
amp = rnorm(1, mean = 3, sd = 0.5),
acro = rnorm(1, mean = 1.5, sd = 0.2),
period = 24,
n_components = 1
)
})
)
dat_mixed$subject <- as.factor(dat_mixed$subject)
```

A quick graph shows how there are individual differences in terms of MESOR, amplitude and phase.

For the model, we should include a random effect for the MESOR, amplitude and acrophase as these are clustered within individuals.

In the model formula, we can use the special `amp_acro[n]`

which represents the n^{th} cosinor component. In this case, we
only have one component so we use `amp_acro1`

. Following the
`{lme4}`

-style mixed model formula, we add our random effect
for this component and the intercept term (MESOR) clustered within
subjects by using `(1 + amp_acro1 | subject)`

. The code below
fits this model

```
mixed_mod <- cglmm(
Y ~ amp_acro(times, n_components = 1, period = 24) +
(1 + amp_acro1 | subject),
data = dat_mixed
)
```

This works by replacing the amp_acro1 with the relevant cosinor
components when the data is rearranged and the formula created. The
formula created can be accessed using `.$formula`

, and shows
the `amp_acro1`

is replaced by the `main_rrr1`

and
`main_sss1`

(the cosine and sine components of time that also
appear in the fixed effects).

```
mixed_mod$formula
#> Y ~ main_rrr1 + main_sss1 + (1 + main_rrr1 + main_sss1 | subject)
#> <environment: 0x000001d112b85738>
```

The mixed model can also be plotted using `autoplot`

, but
some of the plotting features that are available for fixed-effects
models may not be available for mixed-effect models.

The summary of the model shows that the input means for MESOR, amplitude and acrophase are similar to what we specified in the simulation (0, 3, and 1.5, respectively).

```
summary(mixed_mod)
#>
#> Conditional Model
#> Raw model coefficients:
#> estimate standard.error lower.CI upper.CI p.value
#> (Intercept) -0.09323335 0.17907993 -0.44422356 0.25776 0.60263
#> main_rrr1 0.18030389 0.11089882 -0.03705381 0.39766 0.10398
#> main_sss1 2.85563791 0.09447410 2.67047207 3.04080 < 2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Transformed coefficients:
#> estimate standard.error lower.CI upper.CI p.value
#> (Intercept) -0.09323335 0.17907993 -0.44422356 0.25776 0.60263
#> amp1 2.86132441 0.09431188 2.67647651 3.04617 < 2e-16 ***
#> acr1 1.50774041 0.03880609 1.43168187 1.58380 < 2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```

We can see that the predicted values from the model closely resemble the patterns we see in the input data.

```
ggplot(cbind(dat_mixed, pred = predict(mixed_mod))) +
geom_point(aes(x = times, y = Y, col = subject)) +
geom_line(aes(x = times, y = pred, col = subject))
```

This looks like a good model fit for these data. We can highlight the
importance of using a mixed model in this situation rather than a fixed
effects only model by creating that (bad) model and comparing the two by
using the Akaike information criterion using `AIC()`

.

```
fixed_effects_mod <- cglmm(
Y ~ amp_acro(times, n_components = 1, period = 24),
data = dat_mixed
)
AIC(fixed_effects_mod$fit)
#> [1] 2834.918
AIC(mixed_mod$fit)
#> [1] 144.208
```

Aside from not being able to be useful to see the differences between subjects from the model, we end up with much worse model fit and likely biased and/or imprecise estimates of our fixed effects that we are interested in!