This function allows models to be defined using a flexible formula interface that supports fixed effects, random effects (using lme4 syntax), and random walks. The formula syntax builds on standard R formula notation and extends it with lme4 style random effects and custom random walk terms. Users familiar with mixed models in lme4 or brms will recognise the syntax. Note that the returned fixed effects design matrix is sparse and so the index supplied is required to link observations to the appropriate design matrix row.
Arguments
- formula
A model formula that may use standard fixed effects, random effects using lme4 syntax (see
re()), and random walks defined using therw()helper function. See the Details section below for a comprehensive explanation of the supported syntax.- data
A
data.frameof observations. It must include all variables used in the supplied formula.- sparse
Logical, defaults to
TRUE. Should the fixed effects design matrix be sparely defined.
Value
A list containing the following:
formula: The user supplied formulaparsed_formula: The formula as parsed byparse_formula()expanded_formula: The flattened version of the formula with both user supplied terms and terms added for the user supplied complex model components.fixed: A list containing the fixed effect formula, sparse design matrix, and the index linking the design matrix with observations.random: A list containing the random effect formula, sparse design matrix, and the index linking the design matrix with random effects.
Details
Formula syntax overview
The formula interface supports three types of model components:
Fixed effects: Standard R formula syntax as used in stats::lm() and
similar functions. For example:
~ 1: intercept only~ age_group: intercept plus categorical predictor~ age_group + location: multiple predictors~ 0 + age_group: no intercept (contrasts)
Random effects: Uses lme4 syntax with vertical bar notation.
Random effects allow parameters to vary by group whilst sharing information
across groups through partial pooling. Note that epinowcast assumes
independent standard deviations for random effects rather than correlated
random effects as supported by lme4. For example:
~ 1 + (1 | location): random intercepts by location~ 1 + age_group + (1 | location): fixed age effect with random location intercepts~ (age_group | location): random slopes for age within each location~ (1 + week | location:month): random intercepts and week effects for each location-month combination (using interaction to create independent random effects per strata)
Interactions (e.g., location:month) can be used on the right-hand side
of the vertical bar to specify independent random effects for each
combination of the interacting variables.
See the lme4 package documentation for more details on random effects syntax.
Random walks: Uses the rw() helper function to specify time-varying
effects that evolve smoothly over time. For example:
~ rw(week): a random walk over weeks~ rw(week, location): independent random walks for each location~ rw(week, location, type = "dependent"): random walks with shared variance across locations
These three types of effects can be combined in a single formula, for example:
~ 1 + age_group + (1 | location) + rw(week, location) specifies fixed
age effects, random location intercepts, and location-specific random
walks over time.
Turning off model components
In epinowcast model specification functions (such as enw_reference(),
enw_report(), enw_expectation()), formula arguments can be set to ~0
to disable that model component entirely. This is a package-specific
convention. Note that when a formula is specified as ~0, it is typically
converted internally to ~1 (intercept only) to ensure valid model
structure, but the component is flagged as inactive.
How formulas map to the model
The formula you specify controls which covariates and effects enter the
linear predictor of the model. For instance, in the reference date model
(enw_reference()), the formula determines how reporting delay parameters
vary by covariates and groups. The formula is converted to design matrices:
a fixed effects matrix (which may be sparse for computational efficiency)
and a random effects matrix that defines the hierarchical structure.
References
For users new to formula syntax in R:
Fixed effects: See
?formulaand the "Statistical Models in R" chapter of "An Introduction to R": https://cran.r-project.org/doc/manuals/r-release/R-intro.html#Statistical-models-in-RRandom effects: See the lme4 package documentation and vignettes.
Mixed models: Bates et al. (2015) "Fitting Linear Mixed-Effects Models Using lme4". Journal of Statistical Software, 67(1), 1-48. doi:10.18637/jss.v067.i01
See also
Functions used to help convert formulas into model designs
as_string_formula(),
construct_re(),
construct_rw(),
enw_manual_formula(),
parse_formula(),
re(),
remove_rw_terms(),
rw(),
rw_terms(),
split_formula_to_terms()
Examples
# Use meta data for references dates from the Germany COVID-19
# hospitalisation data.
obs <- enw_filter_report_dates(
germany_covid19_hosp[location == "DE"],
remove_days = 40
)
obs <- enw_filter_reference_dates(obs, include_days = 40)
pobs <- enw_preprocess_data(
obs, by = c("age_group", "location"), max_delay = 20
)
data <- pobs$metareference[[1]]
# Intercept only
enw_formula(~ 1, data)
#> $formula
#> [1] "~1"
#>
#> $parsed_formula
#> $parsed_formula$fixed
#> [1] "1"
#>
#> $parsed_formula$random
#> NULL
#>
#> $parsed_formula$rw
#> character(0)
#>
#>
#> $expanded_formula
#> [1] "~1"
#>
#> $fixed
#> $fixed$formula
#> [1] "~1"
#>
#> $fixed$design
#> (Intercept)
#> 1 1
#>
#> $fixed$index
#> [1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
#> [38] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
#> [75] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
#> [112] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
#> [149] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
#> [186] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
#> [223] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
#> [260] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
#>
#>
#> $random
#> $random$formula
#> [1] "~1"
#>
#> $random$design
#> (Intercept)
#> attr(,"assign")
#> [1] 0
#>
#> $random$index
#> integer(0)
#>
#>
#> attr(,"class")
#> [1] "enw_formula" "list"
# Fixed effect
enw_formula(~ 1 + age_group, data)
#> $formula
#> [1] "~1 + age_group"
#>
#> $parsed_formula
#> $parsed_formula$fixed
#> [1] "1" "age_group"
#>
#> $parsed_formula$random
#> NULL
#>
#> $parsed_formula$rw
#> character(0)
#>
#>
#> $expanded_formula
#> [1] "~1 + age_group"
#>
#> $fixed
#> $fixed$formula
#> [1] "~1 + age_group"
#>
#> $fixed$design
#> (Intercept) age_group00+ age_group05-14 age_group15-34 age_group35-59
#> 1 1 1 0 0 0
#> 41 1 0 0 0 0
#> 81 1 0 1 0 0
#> 121 1 0 0 1 0
#> 161 1 0 0 0 1
#> 201 1 0 0 0 0
#> 241 1 0 0 0 0
#> age_group60-79 age_group80+
#> 1 0 0
#> 41 0 0
#> 81 0 0
#> 121 0 0
#> 161 0 0
#> 201 1 0
#> 241 0 1
#>
#> $fixed$index
#> [1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
#> [38] 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
#> [75] 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
#> [112] 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
#> [149] 4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
#> [186] 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
#> [223] 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7
#> [260] 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7
#>
#>
#> $random
#> $random$formula
#> [1] "~1"
#>
#> $random$design
#> (Intercept)
#> 1 1
#> 2 1
#> 3 1
#> 4 1
#> 5 1
#> 6 1
#> attr(,"assign")
#> [1] 0
#>
#> $random$index
#> [1] 1 2 3 4 5 6
#>
#>
#> attr(,"class")
#> [1] "enw_formula" "list"
# Random intercepts
enw_formula(~ 1 + (1 | age_group), data)
#> $formula
#> [1] "~1 + (1 | age_group)"
#>
#> $parsed_formula
#> $parsed_formula$fixed
#> [1] "1"
#>
#> $parsed_formula$random
#> $parsed_formula$random[[1]]
#> 1 | age_group
#>
#>
#> $parsed_formula$rw
#> character(0)
#>
#>
#> $expanded_formula
#> [1] "~1 + age_group"
#>
#> $fixed
#> $fixed$formula
#> [1] "~1 + age_group"
#>
#> $fixed$design
#> (Intercept) age_group00-04 age_group00+ age_group05-14 age_group15-34
#> 1 1 0 1 0 0
#> 41 1 1 0 0 0
#> 81 1 0 0 1 0
#> 121 1 0 0 0 1
#> 161 1 0 0 0 0
#> 201 1 0 0 0 0
#> 241 1 0 0 0 0
#> age_group35-59 age_group60-79 age_group80+
#> 1 0 0 0
#> 41 0 0 0
#> 81 0 0 0
#> 121 0 0 0
#> 161 1 0 0
#> 201 0 1 0
#> 241 0 0 1
#>
#> $fixed$index
#> [1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
#> [38] 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
#> [75] 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
#> [112] 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
#> [149] 4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
#> [186] 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
#> [223] 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7
#> [260] 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7
#>
#>
#> $random
#> $random$formula
#> [1] "~0 + fixed + age_group"
#>
#> $random$design
#> fixed age_group
#> 1 0 1
#> 2 0 1
#> 3 0 1
#> 4 0 1
#> 5 0 1
#> 6 0 1
#> 7 0 1
#> attr(,"assign")
#> [1] 1 2
#>
#> $random$index
#> [1] 1 2 3 4 5 6 7
#>
#>
#> attr(,"class")
#> [1] "enw_formula" "list"
# Random walk
enw_formula(~ 1 + rw(week), data)
#> $formula
#> [1] "~1 + rw(week)"
#>
#> $parsed_formula
#> $parsed_formula$fixed
#> [1] "1"
#>
#> $parsed_formula$random
#> NULL
#>
#> $parsed_formula$rw
#> [1] "rw(week)"
#>
#>
#> $expanded_formula
#> [1] "~1 + cweek1 + cweek2 + cweek3 + cweek4 + cweek5"
#>
#> $fixed
#> $fixed$formula
#> [1] "~1 + cweek1 + cweek2 + cweek3 + cweek4 + cweek5"
#>
#> $fixed$design
#> (Intercept) cweek1 cweek2 cweek3 cweek4 cweek5
#> 1 1 0 0 0 0 0
#> 8 1 1 0 0 0 0
#> 15 1 1 1 0 0 0
#> 22 1 1 1 1 0 0
#> 29 1 1 1 1 1 0
#> 36 1 1 1 1 1 1
#>
#> $fixed$index
#> [1] 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 5 5 5 5 5 5 5 6 6
#> [38] 6 6 6 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 5 5 5 5 5 5
#> [75] 5 6 6 6 6 6 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 5 5 5
#> [112] 5 5 5 5 6 6 6 6 6 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4
#> [149] 5 5 5 5 5 5 5 6 6 6 6 6 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4
#> [186] 4 4 4 5 5 5 5 5 5 5 6 6 6 6 6 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 3 3 4
#> [223] 4 4 4 4 4 4 5 5 5 5 5 5 5 6 6 6 6 6 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3
#> [260] 3 3 4 4 4 4 4 4 4 5 5 5 5 5 5 5 6 6 6 6 6
#>
#>
#> $random
#> $random$formula
#> [1] "~0 + fixed + rw__week"
#>
#> $random$design
#> fixed rw__week
#> 1 0 1
#> 2 0 1
#> 3 0 1
#> 4 0 1
#> 5 0 1
#> attr(,"assign")
#> [1] 1 2
#>
#> $random$index
#> [1] 1 2 3 4 5
#>
#>
#> attr(,"class")
#> [1] "enw_formula" "list"
# Model with a random effect for age group and a random walk
enw_formula(~ 1 + (1 | age_group) + rw(week), data)
#> $formula
#> [1] "~1 + (1 | age_group) + rw(week)"
#>
#> $parsed_formula
#> $parsed_formula$fixed
#> [1] "1"
#>
#> $parsed_formula$random
#> $parsed_formula$random[[1]]
#> 1 | age_group
#>
#>
#> $parsed_formula$rw
#> [1] "rw(week)"
#>
#>
#> $expanded_formula
#> [1] "~1 + age_group + cweek1 + cweek2 + cweek3 + cweek4 + cweek5"
#>
#> $fixed
#> $fixed$formula
#> [1] "~1 + age_group + cweek1 + cweek2 + cweek3 + cweek4 + cweek5"
#>
#> $fixed$design
#> (Intercept) age_group00-04 age_group00+ age_group05-14 age_group15-34
#> 1 1 0 1 0 0
#> 8 1 0 1 0 0
#> 15 1 0 1 0 0
#> 22 1 0 1 0 0
#> 29 1 0 1 0 0
#> 36 1 0 1 0 0
#> 41 1 1 0 0 0
#> 48 1 1 0 0 0
#> 55 1 1 0 0 0
#> 62 1 1 0 0 0
#> 69 1 1 0 0 0
#> 76 1 1 0 0 0
#> 81 1 0 0 1 0
#> 88 1 0 0 1 0
#> 95 1 0 0 1 0
#> 102 1 0 0 1 0
#> 109 1 0 0 1 0
#> 116 1 0 0 1 0
#> 121 1 0 0 0 1
#> 128 1 0 0 0 1
#> 135 1 0 0 0 1
#> 142 1 0 0 0 1
#> 149 1 0 0 0 1
#> 156 1 0 0 0 1
#> 161 1 0 0 0 0
#> 168 1 0 0 0 0
#> 175 1 0 0 0 0
#> 182 1 0 0 0 0
#> 189 1 0 0 0 0
#> 196 1 0 0 0 0
#> 201 1 0 0 0 0
#> 208 1 0 0 0 0
#> 215 1 0 0 0 0
#> 222 1 0 0 0 0
#> 229 1 0 0 0 0
#> 236 1 0 0 0 0
#> 241 1 0 0 0 0
#> 248 1 0 0 0 0
#> 255 1 0 0 0 0
#> 262 1 0 0 0 0
#> 269 1 0 0 0 0
#> 276 1 0 0 0 0
#> age_group35-59 age_group60-79 age_group80+ cweek1 cweek2 cweek3 cweek4
#> 1 0 0 0 0 0 0 0
#> 8 0 0 0 1 0 0 0
#> 15 0 0 0 1 1 0 0
#> 22 0 0 0 1 1 1 0
#> 29 0 0 0 1 1 1 1
#> 36 0 0 0 1 1 1 1
#> 41 0 0 0 0 0 0 0
#> 48 0 0 0 1 0 0 0
#> 55 0 0 0 1 1 0 0
#> 62 0 0 0 1 1 1 0
#> 69 0 0 0 1 1 1 1
#> 76 0 0 0 1 1 1 1
#> 81 0 0 0 0 0 0 0
#> 88 0 0 0 1 0 0 0
#> 95 0 0 0 1 1 0 0
#> 102 0 0 0 1 1 1 0
#> 109 0 0 0 1 1 1 1
#> 116 0 0 0 1 1 1 1
#> 121 0 0 0 0 0 0 0
#> 128 0 0 0 1 0 0 0
#> 135 0 0 0 1 1 0 0
#> 142 0 0 0 1 1 1 0
#> 149 0 0 0 1 1 1 1
#> 156 0 0 0 1 1 1 1
#> 161 1 0 0 0 0 0 0
#> 168 1 0 0 1 0 0 0
#> 175 1 0 0 1 1 0 0
#> 182 1 0 0 1 1 1 0
#> 189 1 0 0 1 1 1 1
#> 196 1 0 0 1 1 1 1
#> 201 0 1 0 0 0 0 0
#> 208 0 1 0 1 0 0 0
#> 215 0 1 0 1 1 0 0
#> 222 0 1 0 1 1 1 0
#> 229 0 1 0 1 1 1 1
#> 236 0 1 0 1 1 1 1
#> 241 0 0 1 0 0 0 0
#> 248 0 0 1 1 0 0 0
#> 255 0 0 1 1 1 0 0
#> 262 0 0 1 1 1 1 0
#> 269 0 0 1 1 1 1 1
#> 276 0 0 1 1 1 1 1
#> cweek5
#> 1 0
#> 8 0
#> 15 0
#> 22 0
#> 29 0
#> 36 1
#> 41 0
#> 48 0
#> 55 0
#> 62 0
#> 69 0
#> 76 1
#> 81 0
#> 88 0
#> 95 0
#> 102 0
#> 109 0
#> 116 1
#> 121 0
#> 128 0
#> 135 0
#> 142 0
#> 149 0
#> 156 1
#> 161 0
#> 168 0
#> 175 0
#> 182 0
#> 189 0
#> 196 1
#> 201 0
#> 208 0
#> 215 0
#> 222 0
#> 229 0
#> 236 1
#> 241 0
#> 248 0
#> 255 0
#> 262 0
#> 269 0
#> 276 1
#>
#> $fixed$index
#> [1] 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4
#> [26] 4 4 4 5 5 5 5 5 5 5 6 6 6 6 6 7 7 7 7 7 7 7 8 8 8
#> [51] 8 8 8 8 9 9 9 9 9 9 9 10 10 10 10 10 10 10 11 11 11 11 11 11 11
#> [76] 12 12 12 12 12 13 13 13 13 13 13 13 14 14 14 14 14 14 14 15 15 15 15 15 15
#> [101] 15 16 16 16 16 16 16 16 17 17 17 17 17 17 17 18 18 18 18 18 19 19 19 19 19
#> [126] 19 19 20 20 20 20 20 20 20 21 21 21 21 21 21 21 22 22 22 22 22 22 22 23 23
#> [151] 23 23 23 23 23 24 24 24 24 24 25 25 25 25 25 25 25 26 26 26 26 26 26 26 27
#> [176] 27 27 27 27 27 27 28 28 28 28 28 28 28 29 29 29 29 29 29 29 30 30 30 30 30
#> [201] 31 31 31 31 31 31 31 32 32 32 32 32 32 32 33 33 33 33 33 33 33 34 34 34 34
#> [226] 34 34 34 35 35 35 35 35 35 35 36 36 36 36 36 37 37 37 37 37 37 37 38 38 38
#> [251] 38 38 38 38 39 39 39 39 39 39 39 40 40 40 40 40 40 40 41 41 41 41 41 41 41
#> [276] 42 42 42 42 42
#>
#>
#> $random
#> $random$formula
#> [1] "~0 + fixed + age_group + rw__week"
#>
#> $random$design
#> fixed age_group rw__week
#> 1 0 1 0
#> 2 0 1 0
#> 3 0 1 0
#> 4 0 1 0
#> 5 0 1 0
#> 6 0 1 0
#> 7 0 1 0
#> 8 0 0 1
#> 9 0 0 1
#> 10 0 0 1
#> 11 0 0 1
#> 12 0 0 1
#> attr(,"assign")
#> [1] 1 2 3
#>
#> $random$index
#> [1] 1 2 3 4 5 6 7 8 9 10 11 12
#>
#>
#> attr(,"class")
#> [1] "enw_formula" "list"
# Model defined without a sparse fixed effects design matrix
enw_formula(~1, data[1:20, ], sparse = FALSE)
#> $formula
#> [1] "~1"
#>
#> $parsed_formula
#> $parsed_formula$fixed
#> [1] "1"
#>
#> $parsed_formula$random
#> NULL
#>
#> $parsed_formula$rw
#> character(0)
#>
#>
#> $expanded_formula
#> [1] "~1"
#>
#> $fixed
#> $fixed$formula
#> [1] "~1"
#>
#> $fixed$design
#> (Intercept)
#> 1 1
#> 2 1
#> 3 1
#> 4 1
#> 5 1
#> 6 1
#> 7 1
#> 8 1
#> 9 1
#> 10 1
#> 11 1
#> 12 1
#> 13 1
#> 14 1
#> 15 1
#> 16 1
#> 17 1
#> 18 1
#> 19 1
#> 20 1
#> attr(,"assign")
#> [1] 0
#>
#> $fixed$index
#> [1] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
#>
#>
#> $random
#> $random$formula
#> [1] "~1"
#>
#> $random$design
#> (Intercept)
#> attr(,"assign")
#> [1] 0
#>
#> $random$index
#> integer(0)
#>
#>
#> attr(,"class")
#> [1] "enw_formula" "list"
# Model using an interaction in the right hand side of a random effect
# to specify an independent random effect per strata.
enw_formula(~ (1 + day | week:month), data = data)
#> $formula
#> [1] "~(1 + day | week:month)"
#>
#> $parsed_formula
#> $parsed_formula$fixed
#> [1] "1"
#>
#> $parsed_formula$random
#> $parsed_formula$random[[1]]
#> 1 + day | week:month
#>
#>
#> $parsed_formula$rw
#> character(0)
#>
#>
#> $expanded_formula
#> [1] "~1 + week:month + day:week:month"
#>
#> $fixed
#> $fixed$formula
#> [1] "~1 + week:month + day:week:month"
#>
#> $fixed$design
#> (Intercept) week0:month0 week1:month0 week2:month0 week3:month0 week4:month0
#> 1 1 1 0 0 0 0
#> 2 1 1 0 0 0 0
#> 3 1 1 0 0 0 0
#> 4 1 1 0 0 0 0
#> 5 1 1 0 0 0 0
#> 6 1 1 0 0 0 0
#> 7 1 1 0 0 0 0
#> 8 1 0 1 0 0 0
#> 9 1 0 1 0 0 0
#> 10 1 0 1 0 0 0
#> 11 1 0 1 0 0 0
#> 12 1 0 1 0 0 0
#> 13 1 0 1 0 0 0
#> 14 1 0 1 0 0 0
#> 15 1 0 0 1 0 0
#> 16 1 0 0 1 0 0
#> 17 1 0 0 1 0 0
#> 18 1 0 0 1 0 0
#> 19 1 0 0 1 0 0
#> 20 1 0 0 1 0 0
#> 21 1 0 0 1 0 0
#> 22 1 0 0 0 1 0
#> 23 1 0 0 0 1 0
#> 24 1 0 0 0 1 0
#> 25 1 0 0 0 1 0
#> 26 1 0 0 0 1 0
#> 27 1 0 0 0 1 0
#> 28 1 0 0 0 1 0
#> 29 1 0 0 0 0 1
#> 30 1 0 0 0 0 1
#> 31 1 0 0 0 0 0
#> 32 1 0 0 0 0 0
#> 33 1 0 0 0 0 0
#> 34 1 0 0 0 0 0
#> 35 1 0 0 0 0 0
#> 36 1 0 0 0 0 0
#> 37 1 0 0 0 0 0
#> 38 1 0 0 0 0 0
#> 39 1 0 0 0 0 0
#> 40 1 0 0 0 0 0
#> week5:month0 week0:month1 week1:month1 week2:month1 week3:month1
#> 1 0 0 0 0 0
#> 2 0 0 0 0 0
#> 3 0 0 0 0 0
#> 4 0 0 0 0 0
#> 5 0 0 0 0 0
#> 6 0 0 0 0 0
#> 7 0 0 0 0 0
#> 8 0 0 0 0 0
#> 9 0 0 0 0 0
#> 10 0 0 0 0 0
#> 11 0 0 0 0 0
#> 12 0 0 0 0 0
#> 13 0 0 0 0 0
#> 14 0 0 0 0 0
#> 15 0 0 0 0 0
#> 16 0 0 0 0 0
#> 17 0 0 0 0 0
#> 18 0 0 0 0 0
#> 19 0 0 0 0 0
#> 20 0 0 0 0 0
#> 21 0 0 0 0 0
#> 22 0 0 0 0 0
#> 23 0 0 0 0 0
#> 24 0 0 0 0 0
#> 25 0 0 0 0 0
#> 26 0 0 0 0 0
#> 27 0 0 0 0 0
#> 28 0 0 0 0 0
#> 29 0 0 0 0 0
#> 30 0 0 0 0 0
#> 31 0 0 0 0 0
#> 32 0 0 0 0 0
#> 33 0 0 0 0 0
#> 34 0 0 0 0 0
#> 35 0 0 0 0 0
#> 36 0 0 0 0 0
#> 37 0 0 0 0 0
#> 38 0 0 0 0 0
#> 39 0 0 0 0 0
#> 40 0 0 0 0 0
#> week4:month1 week5:month1 week0:month0:day week1:month0:day week2:month0:day
#> 1 0 0 0 0 0
#> 2 0 0 1 0 0
#> 3 0 0 2 0 0
#> 4 0 0 3 0 0
#> 5 0 0 4 0 0
#> 6 0 0 5 0 0
#> 7 0 0 6 0 0
#> 8 0 0 0 7 0
#> 9 0 0 0 8 0
#> 10 0 0 0 9 0
#> 11 0 0 0 10 0
#> 12 0 0 0 11 0
#> 13 0 0 0 12 0
#> 14 0 0 0 13 0
#> 15 0 0 0 0 14
#> 16 0 0 0 0 15
#> 17 0 0 0 0 16
#> 18 0 0 0 0 17
#> 19 0 0 0 0 18
#> 20 0 0 0 0 19
#> 21 0 0 0 0 20
#> 22 0 0 0 0 0
#> 23 0 0 0 0 0
#> 24 0 0 0 0 0
#> 25 0 0 0 0 0
#> 26 0 0 0 0 0
#> 27 0 0 0 0 0
#> 28 0 0 0 0 0
#> 29 0 0 0 0 0
#> 30 0 0 0 0 0
#> 31 1 0 0 0 0
#> 32 1 0 0 0 0
#> 33 1 0 0 0 0
#> 34 1 0 0 0 0
#> 35 1 0 0 0 0
#> 36 0 1 0 0 0
#> 37 0 1 0 0 0
#> 38 0 1 0 0 0
#> 39 0 1 0 0 0
#> 40 0 1 0 0 0
#> week3:month0:day week4:month0:day week5:month0:day week0:month1:day
#> 1 0 0 0 0
#> 2 0 0 0 0
#> 3 0 0 0 0
#> 4 0 0 0 0
#> 5 0 0 0 0
#> 6 0 0 0 0
#> 7 0 0 0 0
#> 8 0 0 0 0
#> 9 0 0 0 0
#> 10 0 0 0 0
#> 11 0 0 0 0
#> 12 0 0 0 0
#> 13 0 0 0 0
#> 14 0 0 0 0
#> 15 0 0 0 0
#> 16 0 0 0 0
#> 17 0 0 0 0
#> 18 0 0 0 0
#> 19 0 0 0 0
#> 20 0 0 0 0
#> 21 0 0 0 0
#> 22 21 0 0 0
#> 23 22 0 0 0
#> 24 23 0 0 0
#> 25 24 0 0 0
#> 26 25 0 0 0
#> 27 26 0 0 0
#> 28 27 0 0 0
#> 29 0 28 0 0
#> 30 0 29 0 0
#> 31 0 0 0 0
#> 32 0 0 0 0
#> 33 0 0 0 0
#> 34 0 0 0 0
#> 35 0 0 0 0
#> 36 0 0 0 0
#> 37 0 0 0 0
#> 38 0 0 0 0
#> 39 0 0 0 0
#> 40 0 0 0 0
#> week1:month1:day week2:month1:day week3:month1:day week4:month1:day
#> 1 0 0 0 0
#> 2 0 0 0 0
#> 3 0 0 0 0
#> 4 0 0 0 0
#> 5 0 0 0 0
#> 6 0 0 0 0
#> 7 0 0 0 0
#> 8 0 0 0 0
#> 9 0 0 0 0
#> 10 0 0 0 0
#> 11 0 0 0 0
#> 12 0 0 0 0
#> 13 0 0 0 0
#> 14 0 0 0 0
#> 15 0 0 0 0
#> 16 0 0 0 0
#> 17 0 0 0 0
#> 18 0 0 0 0
#> 19 0 0 0 0
#> 20 0 0 0 0
#> 21 0 0 0 0
#> 22 0 0 0 0
#> 23 0 0 0 0
#> 24 0 0 0 0
#> 25 0 0 0 0
#> 26 0 0 0 0
#> 27 0 0 0 0
#> 28 0 0 0 0
#> 29 0 0 0 0
#> 30 0 0 0 0
#> 31 0 0 0 30
#> 32 0 0 0 31
#> 33 0 0 0 32
#> 34 0 0 0 33
#> 35 0 0 0 34
#> 36 0 0 0 0
#> 37 0 0 0 0
#> 38 0 0 0 0
#> 39 0 0 0 0
#> 40 0 0 0 0
#> week5:month1:day
#> 1 0
#> 2 0
#> 3 0
#> 4 0
#> 5 0
#> 6 0
#> 7 0
#> 8 0
#> 9 0
#> 10 0
#> 11 0
#> 12 0
#> 13 0
#> 14 0
#> 15 0
#> 16 0
#> 17 0
#> 18 0
#> 19 0
#> 20 0
#> 21 0
#> 22 0
#> 23 0
#> 24 0
#> 25 0
#> 26 0
#> 27 0
#> 28 0
#> 29 0
#> 30 0
#> 31 0
#> 32 0
#> 33 0
#> 34 0
#> 35 0
#> 36 35
#> 37 36
#> 38 37
#> 39 38
#> 40 39
#>
#> $fixed$index
#> [1] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
#> [26] 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 1 2 3 4 5 6 7 8 9 10
#> [51] 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
#> [76] 36 37 38 39 40 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
#> [101] 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 1 2 3 4 5
#> [126] 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
#> [151] 31 32 33 34 35 36 37 38 39 40 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
#> [176] 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
#> [201] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
#> [226] 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 1 2 3 4 5 6 7 8 9 10
#> [251] 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
#> [276] 36 37 38 39 40
#>
#>
#> $random
#> $random$formula
#> [1] "~0 + fixed + week__month0 + week__month1 + day__week__month0 + day__week__month1"
#>
#> $random$design
#> fixed week__month0 week__month1 day__week__month0 day__week__month1
#> 1 0 1 0 0 0
#> 2 0 1 0 0 0
#> 3 0 1 0 0 0
#> 4 0 1 0 0 0
#> 5 0 1 0 0 0
#> 6 0 1 0 0 0
#> 7 0 0 1 0 0
#> 8 0 0 1 0 0
#> 9 0 0 1 0 0
#> 10 0 0 1 0 0
#> 11 0 0 1 0 0
#> 12 0 0 1 0 0
#> 13 0 0 0 1 0
#> 14 0 0 0 1 0
#> 15 0 0 0 1 0
#> 16 0 0 0 1 0
#> 17 0 0 0 1 0
#> 18 0 0 0 1 0
#> 19 0 0 0 0 1
#> 20 0 0 0 0 1
#> 21 0 0 0 0 1
#> 22 0 0 0 0 1
#> 23 0 0 0 0 1
#> 24 0 0 0 0 1
#> attr(,"assign")
#> [1] 1 2 3 4 5
#>
#> $random$index
#> [1] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
#>
#>
#> attr(,"class")
#> [1] "enw_formula" "list"