Simulate asynchronous longitudinal data

Draw a sparse longitudinal response and a sparse longitudinal covariate whose observation times are independent, so the two are never seen together. This is the design of Cao, Zeng and Fine (2015), Section 4: response times and covariate times are independent Poisson streams on \((0, 1)\), the covariate is a mean-zero Gaussian process, and the error is a second, independent Gaussian process.

Usage

sim_async_data(
  n,
  beta = c(0.5, 1.5),
  lambda_y = 5,
  lambda_x = 5,
  link = c("identity", "log", "logistic"),
  x_cov = function(s, t) exp(-abs(s - t)),
  e_cov = function(s, t) 2^(-abs(s - t))
)

Arguments

n

Number of subjects.

beta

Either a numeric vector c(intercept, slopes) or a function of a numeric time vector returning a matrix with length(beta(t)) columns and one row per time.

lambda_y, lambda_x

Poisson rates for the response and covariate observation streams on (0, 1). Each subject is forced to have at least one observation of each.

link

One of "identity", "log" or "logistic", matching kee_async(). Under "logistic" the response is Bernoulli and no error process is added; under "log" the response is Poisson.

x_cov, e_cov

Covariance functions of two times, vectorised, for the covariate and error processes.

Value

A list of two tibbles on different time grids:

y

id, time, y – one row per response occasion.

x

id, time, and one column per covariate coordinate – named x when there is a single covariate, x1, x2, ... otherwise. One row per covariate occasion, on a time grid independent of y's.

Details

The default covariate covariance \(\exp(-|s - t|)\) is the Ornstein- Uhlenbeck covariance. Its diagonal has a kink, so it is only one-sided differentiable, which is exactly the case Cao, Zeng and Fine's Section 6 leaves open. Supply a smooth x_cov, for instance function(s, t) exp(-(s - t)^2), to generate data inside their stated assumptions instead.

beta may be a numeric vector, giving time-invariant coefficients suitable for kee_async(), or a function of time returning one row per supplied time, giving a coefficient curve suitable for kee_async_td(). In both cases the first element is the intercept and the remaining p elements multiply the p covariate coordinates.

References

Cao, H., Zeng, D. and Fine, J. P. (2015). Regression analysis of sparse asynchronous longitudinal data. Journal of the Royal Statistical Society, Series B 77, 755-776.

Examples

set.seed(1)
d <- sim_async_data(n = 50)
str(d)
#> List of 2
#>  $ y: tibble [257 × 3] (S3: tbl_df/tbl/data.frame)
#>   ..$ id  : int [1:257] 1 1 1 1 2 2 2 2 2 2 ...
#>   ..$ time: num [1:257] 0.202 0.573 0.898 0.908 0.108 ...
#>   ..$ y   : num [1:257] -2.945 -0.819 0.211 0.33 2.494 ...
#>  $ x: tibble [258 × 3] (S3: tbl_df/tbl/data.frame)
#>   ..$ id  : int [1:258] 1 1 1 1 2 2 2 2 2 2 ...
#>   ..$ time: num [1:258] 0.0618 0.6291 0.6608 0.9447 0.0233 ...
#>   ..$ x   : num [1:258] -0.864 0.345 0.437 0.253 0.292 ...

# The two tables share `id` and nothing else; their `time` grids are
# independent draws, which is what makes the data asynchronous.
head(d$y, 3)
#> # A tibble: 3 × 3
#>      id  time      y
#>   <int> <dbl>  <dbl>
#> 1     1 0.202 -2.95 
#> 2     1 0.573 -0.819
#> 3     1 0.898  0.211
head(d$x, 3)
#> # A tibble: 3 × 3
#>      id   time      x
#>   <int>  <dbl>  <dbl>
#> 1     1 0.0618 -0.864
#> 2     1 0.629   0.345
#> 3     1 0.661   0.437

# A coefficient curve, for kee_async_td().
d2 <- sim_async_data(n = 50, beta = function(tt) cbind(0.5, 1 + tt))