GLMMcosinor

An brief introduction to the cosinor model

A cosinor model aims to model the amplitude (\(A\)), acrophase (\(\phi\)), and MESOR (\(M\)) of a rhythmic dataset.

• MESOR (\(M\)) is the Midline Estimating Statistic of Rhythm, and may also be referred to as the equilibrium point.

• Amplitude (\(A\)) is the difference between the MESOR and the maximum height of the rhythm.

• Acrophase (\(\phi\)) is the phase at which the maximal response occurs.

These could be modelled using a cosine function:

\[Y(t) = M + Acos(\frac{2\pi t}{\tau} - \phi) + e(t)\] where \(e(t)\) is the error term.

However, these cannot be estimated using a linear modelling framework! Other packages, including {circacompare} (Parsons et al. 2020), fit this exact nonlinear model but most packages (including this one) decomposes this into linear terms, creating the cosinor model:

\[Y(t) = M + \beta x + \gamma z + e(t)\]

Where \(x =cos(\frac{2\pi t}{τ})\), \(z =sin(\frac{2\pi t}{τ})\), \(\beta = A cos(\phi)\), \(\gamma = A sin(\phi)\)

This linear model is passed to the package {glmmTMB} (Brooks et al. 2017) in lme4 syntax. If the model has no random effects, glmmTMB uses maximum likelihood estimation to estimate the linear coefficients of the model. For models with random effects, a Laplace approximation is used to integrate over the random effects. This approximation is handled by the {TMB}(Kristensen et al. 2016) package which uses automatic differentiation of the joint likelihood function to provide fast computations of parameter estimates. A detailed explanation of this process is described here (Kristensen et al. 2016)

glmmTMB returns the estimates of the linear coefficients from the linear model. To recover the estimates of the original parameters for amplitude (\(A\)) and acrophase (\(\phi\)), the estimates for \(\hat\beta\) and \(\hat\gamma\) must be transformed as per the following equations:

\[\hat\phi = \arctan(\frac{\hat\gamma}{\hat\beta}) \]

\[\hat A = \sqrt{\hat\beta ^2 + \hat\gamma ^ 2}\] These transformed parameters for acrophase and amplitude, along with MESOR are returned as part of the cglmm output. For a more thorough introduction to cosinor modelling, see here (Cornelissen 2014).

Introduction

{GLMMcosinor} allows the user to fit generalised linear models based on rhythmic data with a cosinor model. It allows users to summarise, predict, and plot these models too. Existing packages have focused primarily on Gaussian data. Some circadian regression modelling packages have allowed users to specify generalised linear models, but with limited flexibility. {GLMMcosinor} takes a comprehensive approach to modelling by harnessing the {glmmTMB} package, that has a wide range of available link functions, allowing users to model rhythmic data from a wide range of distributions (for full list - see ?family and ?glmmTMB::family_glmmTMB) including:

• Binomial
• Guassian
• Inverse Gaussian
• Gamma
• Poisson
• Negative Binomial

The table below shows what features are available within {GLMMcosinor} and other methods.

cglmm()

cglmm() wrangles the data appropriately to fit the cosinor model given the formula specified by the user. It returns a model, providing estimates of amplitude, acrophase, and MESOR (Midline Statistic Of Rhythm).

The formula argument for cglmm() is specified using the {lme4} style (for details see vignette("lmer", package = "lme4")). The only difference is that it allows for use of amp_acro() within the formula that is used to identify the cosinor (rhythmic) components and relevant variables in the provided data. Any other combination of covariates can also be included in the formula as well as random effects. Additionally, zero-inflation (ziformula) and dispersion (dispformula) formulae can be incorporated if required. For detailed examples of how to specify these types of models, see the mixed-models, model-specification and multiple-components vignettes.

For example, consider the following model and its output:

library(GLMMcosinor)

cosinor_model <- cglmm(
  vit_d ~ X + amp_acro(time, period = 12, group = "X"),
  data = vitamind
)

Notice how both the raw and transformed coefficients are provided as output. The adapted data.frame that was used to fit the raw model can be accessed from the model and includes main_rrr1 and main_sss1 columns of data:

head(cosinor_model$newdata)
#>      vit_d      time X  main_rrr1  main_sss1
#> 1 16.12091 11.439525 0  0.9572476 -0.2892699
#> 2 29.90624  5.807104 0 -0.9949038  0.1008285
#> 3 39.17572  1.045492 1  0.8538711  0.5204846
#> 4 35.15403  4.082983 1 -0.5371451  0.8434899
#> 5 43.67065 10.606247 1  0.7453295 -0.6666963
#> 6 31.20360  5.126054 0 -0.8971168  0.4417935

In this example, the main prefix indicates that this is the data for the conditional model, as opposed to (potential) dispersion or zero-inflation models, which have the prefixes disp and zi, respectively. The numeric suffix, indicates that this is the data for the first (and only) cosinor component. If there are multiple components, the columns of data will be named accordingly.

A basic overview of cglmm()

The cglmm() function is used to fit cosinor models to a variety of distributions using the glmmTMB() function.

cglmm(
  formula = vit_d ~ amp_acro(time, period = 12),
  data = vitamind,
  family = gaussian
)
#> 
#>  Conditional Model 
#> 
#>  Raw formula: 
#> vit_d ~ main_rrr1 + main_sss1 
#> 
#>  Raw Coefficients: 
#>             Estimate
#> (Intercept) 30.25467
#> main_rrr1    2.59418
#> main_sss1    5.75079
#> 
#>  Transformed Coefficients: 
#>             Estimate
#> (Intercept) 30.25467
#> amp          6.30883
#> acr          1.14703

• formula: A formula specifying the model structure, including the response variable and the cosinor components (using amp_acro()).
• data: The data.frame containing the variables used in the formula.
• family: The family of the distribution for the response variable (e.g., poisson, gaussian, or any family found in?family and ?glmmTMB::family_glmmTMB)

The amp_acro() function is used within the formula to specify the cosinor components. It allows you to specify the period of the rhythm and, if necessary, the grouping structure and the number of components. The arguments of amp_acro() are:

• group: The name of the grouping variable in the dataset.
• time_col: The name of the time column.
• n_components: The number of components in the cosinor model.
• period: The period(s) of the rhythm.

cglmm(
  vit_d ~ X + amp_acro(time, period = 12, group = "X"),
  data = vitamind
)
#> 
#>  Conditional Model 
#> 
#>  Raw formula: 
#> vit_d ~ X + X:main_rrr1 + X:main_sss1 
#> 
#>  Raw Coefficients: 
#>              Estimate
#> (Intercept)  29.68980
#> X1            1.90186
#> X0:main_rrr1  0.93079
#> X1:main_rrr1  6.51029
#> X0:main_sss1  6.20099
#> X1:main_sss1  4.81846
#> 
#>  Transformed Coefficients: 
#>             Estimate
#> (Intercept) 29.68980
#> [X=1]        1.90186
#> [X=0]:amp    6.27046
#> [X=1]:amp    8.09947
#> [X=0]:acr    1.42181
#> [X=1]:acr    0.63715

cglmm(
  vit_d ~ 0 + X + amp_acro(time,
    period = 12,
    group = "X"
  ),
  data = vitamind
)
#> 
#>  Conditional Model 
#> 
#>  Raw formula: 
#> vit_d ~ X + X:main_rrr1 + X:main_sss1 - 1 
#> 
#>  Raw Coefficients: 
#>              Estimate
#> X0           29.68980
#> X1           31.59165
#> X0:main_rrr1  0.93079
#> X1:main_rrr1  6.51029
#> X0:main_sss1  6.20101
#> X1:main_sss1  4.81847
#> 
#>  Transformed Coefficients: 
#>           Estimate
#> [X=0]     29.68980
#> [X=1]     31.59165
#> [X=0]:amp  6.27048
#> [X=1]:amp  8.09948
#> [X=0]:acr  1.42181
#> [X=1]:acr  0.63716

cosinor_model <- cglmm(
  vit_d ~ 0 + X + amp_acro(time,
    period = 12,
    group = "X"
  ),
  data = vitamind
)
autoplot(cosinor_model, predict.ribbon = FALSE)

More advanced cglmm() model specification

The cglmm() function allows you to specify different types of cosinor models with or without grouping variables. The function can also generate dispersion models and zero-inflation models. For more detailed explanations and examples, see the model-specification article.

Additionally, the cglmm() function provides more advanced functionality for multi-component models, and detailed explanations can be found in the multiple-components article.

The cglmm() function also allows mixed model specification. See the mixed-models article for more details.

Using summary() and testing for differences between estimates

The summary() method for the outputs from cglmm() provides a more detailed summary of the model and its parameter estimates and uncertainty. It outputs the estimates, standard errors, confidence intervals, and p-values for both the raw model parameters and the transformed parameters. The summary statistics do not represent a comparison between any groups for the cosinor components - that is the role of the test_cosinor_components() and test_cosinor_levels() functions.

Here is an example of how to use summary() with some simulated data:

testdata_simple <- simulate_cosinor(
  1000,
  n_period = 2,
  mesor = 5,
  amp = 2,
  acro = 1,
  beta.mesor = 4,
  beta.amp = 1,
  beta.acro = 0.5,
  family = "poisson",
  period = 12,
  n_components = 1,
  beta.group = TRUE
)

object <- cglmm(
  Y ~ group + amp_acro(times, period = 12, group = "group"),
  data = testdata_simple, family = poisson()
)
summary(object)
#> 
#>  Conditional Model 
#> Raw model coefficients:
#>                      estimate standard.error     lower.CI upper.CI    p.value
#> (Intercept)       4.998454142    0.003463730  4.991665356  5.00524 < 2.22e-16
#> group1           -1.002150001    0.005937109 -1.013786521 -0.99051 < 2.22e-16
#> group0:main_rrr1  1.082281784    0.003347565  1.075720677  1.08884 < 2.22e-16
#> group1:main_rrr1  0.876651963    0.006198710  0.864502714  0.88880 < 2.22e-16
#> group0:main_sss1  1.682350718    0.003919418  1.674668800  1.69003 < 2.22e-16
#> group1:main_sss1  0.481951763    0.005936670  0.470316104  0.49359 < 2.22e-16
#>                     
#> (Intercept)      ***
#> group1           ***
#> group0:main_rrr1 ***
#> group1:main_rrr1 ***
#> group0:main_sss1 ***
#> group1:main_sss1 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Transformed coefficients:
#>                    estimate standard.error     lower.CI upper.CI    p.value    
#> (Intercept)     4.998454142    0.003463730  4.991665356  5.00524 < 2.22e-16 ***
#> [group=1]      -1.002150001    0.005937109 -1.013786521 -0.99051 < 2.22e-16 ***
#> [group=0]:amp1  2.000409408    0.004275553  1.992029478  2.00879 < 2.22e-16 ***
#> [group=1]:amp1  1.000398004    0.006379631  0.987894156  1.01290 < 2.22e-16 ***
#> [group=0]:acr1  0.999134804    0.001439122  0.996314177  1.00196 < 2.22e-16 ***
#> [group=1]:acr1  0.502662067    0.005739524  0.491412807  0.51391 < 2.22e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

If we wanted to test the difference between the amplitude estimate for component 1 between group 1 and group 2, we can use the test_cosinor_levels() function:

test_cosinor_levels(object, x_str = "group", param = "amp")
#> Test Details: 
#> Parameter being tested:
#> Amplitude
#> 
#> Comparison type:
#> levels
#> 
#> Grouping variable used for comparison between groups: group
#> Reference group: 0
#> Comparator group: 1
#> 
#> cglmm model only has a single component and to compare
#>           between groups.
#> 
#> 
#> 
#> Global test: 
#> Statistic: 
#> 16955.27
#> 
#> P-value: 
#> 0
#> 
#> 
#> Individual tests:
#> Statistic: 
#> -130.21
#> 
#> P-value: 
#> 0
#> 
#> Estimate and 95% confidence interval:
#> -1 (-1.02 to -0.98)

The estimate here is the estimate of the difference between the inputted values, along with its confidence interval. The real parameters for amp in the first component were 2 and 1 for groups 0 and 1 respectively, and so the difference is approximately -1.

Now, consider an example where the difference is not so clear.

testdata_poisson <- simulate_cosinor(100,
  n_period = 2,
  mesor = 7,
  amp = c(0.1, 0.5),
  acro = c(1, 1),
  beta.mesor = 4.4,
  beta.amp = c(0.1, 0.46),
  beta.acro = c(0.5, -1.5),
  family = "poisson",
  period = c(12, 6),
  n_components = 2,
  beta.group = TRUE
)

cosinor_model <- cglmm(
  Y ~ group + amp_acro(times,
    period = c(12, 6),
    n_components = 2,
    group = "group"
  ),
  data = testdata_poisson,
  family = poisson()
)
test_cosinor_levels(
  cosinor_model,
  x_str = "group",
  param = "amp",
  component_index = 1
)
#> Test Details: 
#> Parameter being tested:
#> Amplitude
#> 
#> Comparison type:
#> levels
#> 
#> Grouping variable used for comparison between groups: group
#> Reference group: 0
#> Comparator group: 1
#> 
#> cglmm model has2 components. Component 1 is being used for comparison between groups.
#> 
#> 
#> 
#> Global test: 
#> Statistic: 
#> 0.05
#> 
#> P-value: 
#> 0.8245
#> 
#> 
#> Individual tests:
#> Statistic: 
#> -0.22
#> 
#> P-value: 
#> 0.8245
#> 
#> Estimate and 95% confidence interval:
#> 0 (-0.04 to 0.03)

In this example, there is no significant difference in the estimate of amp for the first component between the reference group and the comparator group. Also notice how if we are comparing between levels, we should keep the component the same, and that is what component_index sets. Likewise, when we test between components using test_cosinor_components(), we can indicate which level this comparison occurs using level_index. There may be multiple groups, in which case we can fix the group using the x_str argument.

As an example of testing the difference between components for the same level:

test_cosinor_components(
  cosinor_model,
  x_str = "group",
  param = "acr",
  level_index = 1
)
#> Test Details: 
#> Parameter being tested:
#> Acrophase
#> 
#> Comparison type:
#> components
#> 
#> Component indices used for comparison between groups: group
#> Reference component: 1
#> Comparator component: 2
#> 
#> 
#> Global test: 
#> Statistic: 
#> 154.29
#> 
#> P-value: 
#> 0
#> 
#> 
#> Individual tests:
#> Statistic: 
#> -12.42
#> 
#> P-value: 
#> 0
#> 
#> Estimate and 95% confidence interval:
#> -1.89 (-2.19 to -1.59)

In this situation, there is a significant difference between the acrophase for the comparator group between its two components.

Using predict()

The predict() method allows users to get predicted values from the model on either the existing or new data.

cbind(predictions = predict(cosinor_model, type = "response"), testdata_poisson)

#>   predictions    Y     times group
#> 1    865.8332  871 17.009450     0
#> 2    701.2750  714 10.503837     0
#> 3    798.4445  861  4.800118     0
#> 4   1551.0482 1541 18.409584     0
#> 5   1733.9702 1699 12.315885     0
#> 6   1972.1517 1951  1.072893     0

Plotting cglmm objects

The {GLMMcosinor} package includes two ways to visualise cglmm() objects. Firstly, the autoplot() method creates a time-response plot of the fitted model for all groups:

autoplot(cosinor_model, superimpose.data = TRUE)

This function also allows users to superimpose the data (that was used to fit the model) over the fitted model, using the superimpose.data = TRUE, as demonstrated above. By default, the generated plot will have x-limits corresponding to the minimum and maximum values of the time-vector in the original dataframe, although the x-limits can be manually defined by the user using the xlims argument. The details of using the autoplot function are found in the model-visualisations vignette.