Generate synthetic data (group difference)
Usage
generate_synthetic_data(
n_subjects = 100,
n_timepoints = 11,
prop_observed = 0.7,
observation_variance = 1,
random_effect_variance_ratio = 1,
random_effect_ar1_correlation = 0.9,
effect_size = 1,
grpdiff_function = "sigmoid",
missingness = "uniform",
seed = 1
)Arguments
- n_subjects
Number of subjects (default 100). Half of them are in group 0 and the other half in group 1 (on average).
- n_timepoints
Number of time points at which the data is sampled spread regularly on the \([0,1]\) interval (default 11).
- prop_observed
Proportion of observed time points (default 0.7).
- observation_variance
Variance of the observation error (default 1).
- random_effect_variance_ratio
Ratio of the variance of the random effect to the variance of the observation error (default 1).
- random_effect_ar1_correlation
Correlation between two random effects at two consecutive timepoints (default 0.9).
- effect_size
Effect size of the group (default 1).
- grpdiff_function
Group difference function (implemented: "sigmoid" (default), "sine".) Can also be a callable function, provided evaluation is vectorized.
- missingness
Missing observation mechanism (implemented: "uniform" (default), "contiguous", "sqrt", "fixed_uniform")
- seed
Seed for the random number generator (default 1).
Value
A list containing the following elements:
- data
A data frame containing the longitudinal data in long format.
- data_wide
A data frame containing the longitudinal data in wide format.
- data_wide_imputed
A data frame containing the longitudinal data in wide format with imputed missing values.
- true_values
A data frame containing the true values of the fixed effects.
- times
A vector of timepoints at which the data is sampled.
- colnames
A list containing the column names of the data frames.
Details
The data is generated according to the following model:
$$y_{ij} = \beta_0(t_{ij}) + \beta_1(t_{ij}) g_i + \theta_{ij} + \epsilon_{ij}$$
where \(y_{ij}\) is the response of subject \(i\) at time \(j\) at time \((t_{ij})\),
\(g_i\) is the group of subject \(i\),
\(\theta_{ij}\) is the random effect of subject \(i\) at time \(j\),
\(\epsilon_{ij}\) is the observation error of subject \(i\) at time \(j\),
\(\beta_0(\cdot)\) is the intercept function, and \(\beta_1(\cdot)\) is the effect size.
The random effects are generated according to a multivariate normal distribution
with mean 0 and covariance matrix \(\Sigma\) such that \(\Sigma_{ij} = r\sigma^2\rho^{|i-j|}\)
(\(\rho\) defined by random_effect_ar1_correlation and \(r\) by random_effect_variance_ratio).
The observation errors are generated according to a multivariate normal distribution
with mean 0 and covariance matrix \(\sigma^2 I\)
(\(\sigma^2\) defined by observation_variance).
The random effects and observation errors are independent.
The response is generated according to a logistic function of time and group:
$$f(t) = \frac{1}{1 + \exp((0.6-t)20)}$$
The observations are then sampled uniformly at a proportion prop_observed of the timepoints.
An imputed version of the data is also provided where the missing values are imputed using
the function [refund::fpca.sc].
The data is returned in three formats:
- data
A data frame containing the longitudinal data in long format.
- data_wide
A data frame containing the longitudinal data in wide format.
- data_wide_imputed
A data frame containing the longitudinal data in wide format with imputed missing values.