Test the necessity of fusion

Function for testing the null hypothesis that all population precision matrices are equal and thus the necessity for the fusion penalty. Note, the test performed is conditional on the supplied penalties and targets.

fused.test(Ylist, Tlist, lambda, n.permutations = 100, verbose = FALSE, ...)

Arguments

Ylist: A list of length \(G\) of observations matrices for each class. Variables are assumed to correspond to the columns.
Tlist: A list of target matrices for each class. Should be same length as Ylist-
lambda: A non-negative, symmetric \(G\) by \(G\) matrix giving the ridge and fusion penalties.
n.permutations: The number of permutations to approximate the null distribution. Default is 100. Should be increased if sufficient computing power is available.
verbose: Print out extra progress information
...: Arguments passed to ridgeP.fused.

Value

Returns a list values containing the observed test statistic and the test statistic under the null distribution.

Details

The function computes the observed score statistic \(U_obs\) using the fused ridge estimator on the given data. Next, the score statistic is computed a number of times (given by n.permutations) under the null-hypothesis by effectively permuting the class labels of the data.

References

Bilgrau, A.E., Peeters, C.F.W., Eriksen, P.S., Boegsted, M., and van Wieringen, W.N. (2020). Targeted Fused Ridge Estimation of Inverse Covariance Matrices from Multiple High-Dimensional Data Classes. Journal of Machine Learning Research, 21(26): 1-52.

Author

Anders Ellern Bilgrau, Carel F.W. Peeters <carel.peeters@wur.nl>, Wessel, N. van Wieringen

Examples

ns <- c(10, 5, 23)
Ylist <- createS(ns, p = 15, topology = "banded", dataset = TRUE)

# Use the identity target matrix for each class
Tlist <- replicate(length(ns), diag(15), simplify = FALSE)

# Do the test
lm <- matrix(10, 3, 3)
diag(lm) <- 1
ft <- fused.test(Ylist, Tlist, lambda = lm,
                 n.permutations = 500)
print(ft)
#> 
#> Score-based permutation test
#> 
#> Null hypothesis: Population precision matrices are equal
#> Alternative:     Population precision matrices are not equal
#> 
#> Observed statistic: U = 206098.963, p-value = 0.01
#> Summary of null distribution obtained by permutation:
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#>  122300  146200  155400  156900  164300  220400 

# Summary spits out a bit more information
summary(ft)
#> 
#> Score-based permutation test
#> 
#> Null hypothesis: Population precision matrices are equal
#> Alternative:     Population precision matrices are not equal
#> 
#> Observed statistic: U = 206098.963, p-value = 0.01
#> Summary of null distribution obtained by permutation:
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#>  122300  146200  155400  156900  164300  220400 
#> 
#> The number of extreme observations under the null hypothesis
#> was 5 out of 500 permutations.

# The returned object can alo be plotted via
hist(ft)
# or via the alias
plot(ft)


# Customization and parameters work a usual:
hist(ft, col = "steelblue", main = "Null distribution", add.extra = FALSE,
     xlab = "Score statistic", freq = FALSE)