Bootstrap Test for the Ergodicity Information Index

Tests the Ergodicity Information Index obtained in the empirical sample with a distribution of EII obtained by a variant of bootstrap sampling (see Details for the procedure)

Usage

boot.ergoInfo(
  dynEGA.object,
  EII,
  use = c("edge.list", "unweighted", "weighted"),
  shuffles = 5000,
  iter = 100,
  ncores,
  verbose = TRUE
)

Arguments

dynEGA.object

A dynEGA or a dynEGA.ind.pop object. If a dynEGA object, then level = c("individual", "population") is required

EII

A ergoInfo object used to estimate the Empirical Ergodicity Information Index or the estimated value of EII estimated using the ergoInfo function. Inherits use from ergoInfo. If no ergoInfo object is provided, then it is estimated

use

Character (length = 1). A string indicating what network element will be used to compute the algorithm complexity, the list of edges or the weights of the network. Defaults to use = "unweighted". Current options are:

"edge.list" — Calculates the algorithm complexity using the list of edges
"unweighted" — Calculates the algorithm complexity using the binary weights of the encoded prime transformed network. 0 = edge absent and 1 = edge present
"weighted" — Calculates the algorithm complexity using the weights of encoded prime-weight transformed network

shuffles

Numeric. Number of shuffles used to compute the Kolmogorov complexity. Defaults to 5000

iter

Numeric (length = 1). Number of replica samples to generate from the bootstrap analysis. Defaults to 100 (1000 for robustness)

ncores

Numeric (length = 1). Number of cores to use in computing results. Defaults to ceiling(parallel::detectCores() / 2) or half of your computer's processing power. Set to 1 to not use parallel computing

If you're unsure how many cores your computer has, then type: parallel::detectCores()

verbose

Boolean (length = 1). Should progress be displayed? Defaults to TRUE. Set to FALSE to not display progress

Value

Returns a list containing:

empirical.ergoInfo: Empirical Ergodicity Information Index
boot.ergoInfo: The values of the Ergodicity Information Index obtained in the bootstrap
p.value: The two-sided p-value of the bootstrap test for the Ergodicity Information Index. The null hypothesis is that the empirical Ergodicity Information index is equal to or greater than the expected value of the EII with small variation in the population structure
effect: Indicates wheter the empirical EII is greater or less then the bootstrap distribution of EII.
interpretation: How you can interpret the result of the test in plain English

Details

In traditional bootstrap sampling, individual participants are resampled with replacement from the empirical sample. This process is time consuming when carried out across v number of variables, n number of participants, t number of time points, and i number of iterations. Instead, boot.ergoInfo uses the premise of an ergodic process to establish more efficient test that works directly on the sample's networks.

With an ergodic process, the expectation is that all individuals will have a systematic relationship with the population. Destroying this relationship should result in a significant loss of information. Following this conjecture, boot.ergoInfo shuffles a random subset of edges that exist in the population that is equal to the number of shared edges it has with an individual. An individual's unique edges remain the same, controlling for their unique information. The result is a replicate individual that contains the same total number of edges as the actual individual but its shared information with the population has been scrambled.

This process is repeated over each individual to create a replicate sample and is repeated for X iterations (e.g., 100). This approach creates a sampling distribution that represents the expected information between the population and individuals when a random process generates the shared information between them. If the shared information between the population and individuals in the empirical sample is sufficiently meaningful, then this process should result in significant information loss.

How to interpret the results: the result of boot.ergoInfo is a sampling distribution of EII values that would be expected if the process was random (null distribution). If the empirical EII value is greater than or not significantly different from the null distribution, then the empirical data can be expected to be generated from an nonergodic process and the population structure is not sufficient to describe all individuals. If the empirical EII value is significantly lower than the null distribution, then the empirical data can be described by the population structure – the population structure is sufficient to describe all individuals.

References

Original Implementation
Golino, H., Nesselroade, J. R., & Christensen, A. P. (2022). Toward a psychology of individuals: The ergodicity information index and a bottom-up approach for finding generalizations. PsyArXiv.

Author

Hudson Golino <hfg9s at virginia.edu> & Alexander P. Christensen <alexander.christensen at Vanderbilt.Edu>

Examples

# Obtain simulated data
sim.data <- sim.dynEGA

if (FALSE) { # \dontrun{
# Dynamic EGA individual and population structures
dyn1 <- dynEGA.ind.pop(
  data = sim.dynEGA[,-26], n.embed = 5, tau = 1,
  delta = 1, id = 25, use.derivatives = 1,
  model = "glasso", ncores = 2, corr = "pearson"
)

# Empirical Ergodicity Information Index
eii1 <- ergoInfo(dynEGA.object = dyn1, use = "unweighted")

# Bootstrap Test for Ergodicity Information Index
testing.ergoinfo <- boot.ergoInfo(
  dynEGA.object = dyn1, EII = eii1,
  ncores = 2, use = "unweighted"
)

# Plot result
plot(testing.ergoinfo)

# Example using `dynEGA`
dyn2 <- dynEGA(
  data = sim.dynEGA, n.embed = 5, tau = 1,
  delta = 1, use.derivatives = 1, ncores = 2,
  level = c("individual", "population")
)

# Empirical Ergodicity Information Index
eii2 <- ergoInfo(dynEGA.object = dyn2, use = "unweighted")

# Bootstrap Test for Ergodicity Information Index
testing.ergoinfo2 <- boot.ergoInfo(
  dynEGA.object = dyn2, EII = eii2,
  ncores = 2
)

# Plot result
plot(testing.ergoinfo2)} # }