Structure dependence test

Test whether or not a network estimates can be considered structurally dependent, i.e. a function of the network structure. By rewiring the graph and calculating a particular statistic $t$, the test compares the observed mean of $t$ against the empirical distribution of it obtained from rewiring the network.

n_rewires(graph, p = c(20L, rep(0.1, nslices(graph) - 1)))

struct_test(graph, statistic, R, rewire.args = list(), ...)

# S3 method for class 'diffnet_struct_test'
c(..., recursive = FALSE)

# S3 method for class 'diffnet_struct_test'
print(x, ...)

# S3 method for class 'diffnet_struct_test'
hist(
  x,
  main = "Empirical Distribution of Statistic",
  xlab = expression(Values ~ of ~ t),
  breaks = 20,
  annotated = TRUE,
  b0 = expression(atop(plain("") %up% plain("")), t[0]),
  b = expression(atop(plain("") %up% plain("")), t[]),
  ask = TRUE,
  ...
)

struct_test_asymp(graph, Y, statistic_name = "distance", p = 2, ...)

Arguments

graph: A diffnet graph.
p: Either a Numeric scalar or vector of length nslices(graph)-1 with the number of rewires per links.
statistic: A function that returns either a scalar or a vector.
R: Integer scalar. Number of repetitions.
rewire.args: List. Arguments to be passed to rewire_graph
...: Further arguments passed to the method (see details).
recursive: Ignored
x: A diffnet_struct_test class object.
main: Character scalar. Title of the histogram.
xlab: Character scalar. x-axis label.
breaks: Passed to hist.
annotated: Logical scalar. When TRUE marks the observed data average and the simulated data average.
b0: Character scalar. When annotated=TRUE, label for the value of b0.
b: Character scalar. When annotated=TRUE, label for the value of b.
ask: Logical scalar. When TRUE, asks the user to type <Enter> to see each plot (as many as statistics where computed).
Y: Numeric vector of length $n$.
statistic_name: Character scalar. Name of the metric to compute. Currently this can be either "distance",">","<","==", ">=", or "<=".

Value

A list of class diffnet_struct_test containing the following:

graph: The graph passed to struct_test.
p.value: The resulting p-value of the test (see details).
t0: The observed value of the statistic.
mean_t: The average value of the statistic applied to the simulated networks.
R: Number of simulations.
statistic: The function statistic passed to struct_test.
boot: A boot class object as return from the call to boot.
rewire.args: The list rewire.args passed to struct_test.

Details

struct_test computes the test by generating the null distribution using Monte Carlo simulations (rewiring). struct_test_asymp computes the test using an asymptotic approximation. While available, we do not recommend using the asymptotic approximation since it has not shown good results when compared to the MC approximation. Furthermore, the asymptotic version has only been implemented for graph as static graph.

The output from the hist method is the same as hist.default.

struct_test is a wrapper for the function boot from the boot package. Instead of resampling data–vertices or edges–in each iteration the function rewires the original graph using rewire_graph and applies the function defined by the user in statistic.

The default values to rewire_graph via rewire.args are:

`p`	Number or Integer with default `n_rewires(graph)`.
`undirected`	Logical scalar with default `getOption("diffnet.undirected", FALSE)`.
`copy.first`	Logical scalar with `TRUE`.
`algorithm`	Character scalar with default `"swap"`.

In struct_test ... are passed to boot, otherwise are passed to the corresponding method (hist for instance).

From the print method, p-value for the null of the statistic been equal between graph and its rewired versions is computed as follows

$$% p(\tau)=2\times\min\left(\mbox{Pr}(t\leq\tau), \mbox{Pr}(t\geq\tau)\right) % $$

Where $\mbox{Pr}\{\cdot\}$ is approximated using the Empirical Distribution Function retrieved from the simulations.

For the case of the asymptotic approximation, under the null we have

$$% \sqrt{n}\left(\hat\beta(Y,G)-\mu_\beta\right)\sim^d\mbox{N}\left(0,\sigma_\beta^2\right) $$

The test is actually on development by Vega Yon and Valente. A copy of the working paper can be distributed upon request to g.vegayon@gmail.com.

The function n_rewires proposes a vector of number of rewirings that are performed in each iteration.

References

Vega Yon, George G. and Valente, Thomas W. (On development).

Davidson, R., & MacKinnon, J. G. (2004). Econometric Theory and Methods. New York: Oxford University Press.

Author

George G. Vega Yon

Examples

# Creating a random graph
set.seed(881)
diffnet <- rdiffnet(100, 5, seed.graph="small-world")
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.

# Testing structure-dependency of threshold
res <- struct_test(
  diffnet,
  function(g) mean(threshold(g), na.rm=TRUE),
  R=100
)
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.

res
#> 
#> Structure dependence test
#> # Simulations     : 100
#> # nodes           : 100
#> # of time periods : 5
#> --------------------------------------------------------------------------------
#>  H0: E[beta(Y,G)|G] - E[beta(Y,G)] = 0 (no structure dependency)
#>     observed    expected       p.val
#>       0.3947      0.1261      0.0000
hist(res)

# Adding a legend
legend("topright", bty="n",
 legend=c(
   expression(t[0]:~Baseline),
   expression(t:~Rewired~average)
 )
 )


# Concatenating results
c(res, res)
#> 
#> Structure dependence test
#> # Simulations     : 200
#> # nodes           : 100
#> # of time periods : 5
#> --------------------------------------------------------------------------------
#>  H0: E[beta(Y,G)|G] - E[beta(Y,G)] = 0 (no structure dependency)
#>     observed    expected       p.val
#>       0.3947      0.1261      0.0000

# Running in parallel fashion
res <- struct_test(
  diffnet, function(g) mean(threshold(g), na.rm=TRUE),
  R=100, ncpus=2, parallel="multicore"
)
#> Warning: The option -copy.first- is set to TRUE. In this case, the first graph will be treated as a baseline, and thus, networks after T=1 will be replaced with T-1.

res
#> 
#> Structure dependence test
#> # Simulations     : 100
#> # nodes           : 100
#> # of time periods : 5
#> --------------------------------------------------------------------------------
#>  H0: E[beta(Y,G)|G] - E[beta(Y,G)] = 0 (no structure dependency)
#>     observed    expected       p.val
#>       0.3947      0.1158      0.0000

hist(res)