boot_ci {snowboot}

Confidence Intervals from Bootstrapped Network Degree Distribution

Description

The function calculates bootstrap confidence intervals for the parameters of network degree distribution: probabilities of node degrees f(k) and mean degree \mu, where k = 0, 1, \ldots are the degrees.

Usage

boot_ci(x, prob = 0.95, method = c("percentile", "basic"))

Arguments

x

a list with bootstrapped results – output of boot_dd.

prob

confidence level for the intervals. Default is 0.95 (i.e., 95% confidence).

method

method for calculating the bootstrap intervals. Default is "percentile" (see Details).

Details

Currently, the bootstrap intervals can be calculated with two alternative methods: "percentile" or "basic". The "percentile" intervals correspond to Efron's 100\cdotprob% intervals (see Efron 1979, also Equation 5.18 by Davison and Hinkley 1997 and Equation 3 by Gel et al. 2017, Chen et al. 2018):

(\theta^*_{[B\alpha/2]}, \theta^*_{[B(1-\alpha/2)]}),

where \theta^*_{[B\alpha/2]} and \theta^*_{[B(1-\alpha/2)]} are empirical quantiles of the bootstrap distribution with B bootstrap replications for parameter \theta (\theta can be the f(k) or \mu), and \alpha = 1 - prob.

The "basic" method produces intervals (see Equation 5.6 by Davison and Hinkley 1997):

(2\hat{\theta} - \theta^*_{[B(1-\alpha/2)]}, 2\hat{\theta} - \theta^*_{[B\alpha/2]}),

where \hat{\theta} is the sample estimate of the parameter. Note that this method can lead to negative confidence bounds, especially when \hat{\theta} is close to 0.

Value

A list object of class "snowboot" with the following elements:

fk_ci

A matrix of dimensions 2 \timeslength(x$fk), where the number of columns corresponds to the number of probabilities f(k) estimated from an LSMI sample. Each column of the matrix is a confidence interval for a corresponding f(k). I.e., the first row of the matrix gives the lower bounds, while the second row contains all upper bounds.

mu_ci

A numeric vector of length 2 with lower and upper confidence bounds for the network mean degree \mu.

prob

Confidence level for the intervals.

method

Method that was used for calculating the bootstrap intervals.

fk

A vector with an estimate of the degree distribution, copied from the input x$fk.

mu

An estimate of the mean degree, copied from the input x$mu.

References

Chen Y, Gel YR, Lyubchich V, Nezafati K (2018). “Snowboot: bootstrap methods for network inference.” The R Journal, 10(2), 95–113. doi: 10.32614/RJ-2018-056.

Davison AC, Hinkley DV (1997). Bootstrap Methods and Their Application. Cambridge University Press, Cambridge.

Efron B (1979). “Bootstrap methods: Another look at the jackknife.” The Annals of Statistics, 7(1), 1–26. doi: 10.1214/aos/1176344552.

Gel YR, Lyubchich V, Ramirez Ramirez LL (2017). “Bootstrap quantification of estimation uncertainties in network degree distributions.” Scientific Reports, 7, 5807. doi: 10.1038/s41598-017-05885-x.

See Also

boot_dd

Examples

net <- artificial_networks[[1]]
lsmiEstimate <- lsmi_dd(net = net, n.seed = 5, n.wave = 3)
bootEstimates <- boot_dd(lsmiEstimate, B = 10)
bootIntervals1 <- boot_ci(bootEstimates)

#Another version of the intervals:
bootIntervals2 <- boot_ci(bootEstimates, method = "basic")