compare_distributions {poweRlaw} | R Documentation |
Vuong's test for non-nested models
Description
Since it is possible to fit power law models to any data set,
it is recommended that alternative distributions are considered.
A standard technique is to use Vuong's test.
This is a likelihood ratio test for model selection using the
Kullback-Leibler criteria.
The test statistic, R
, is the ratio of the
log-likelihoods of the data between the two competing models.
The sign of R
indicates which model is better.
Since the value of R
is esimated,
we use the method proposed by Vuong, 1989 to select the model.
Usage
compare_distributions(d1, d2)
Arguments
d1 |
A distribution object |
d2 |
A distribution object |
Details
This function compares two models. The null hypothesis is that both classes of distributions are equally far from the true distribution. If this is true, the log-likelihood ratio should (asymptotically) have a Normal distribution with mean zero. The test statistic is the sample average of the log-likelihood ratio, standardized by a consistent estimate of its standard deviation. If the null hypothesis is false, and one class of distributions is closer to the "truth", the test statistic goes to +/-infinity with probability 1, indicating the better-fitting class of distributions.
Value
This function returns
test_statistic
The test statistic.
p_one_sided
A one-sided p-value, which is an upper limit on getting that small a log-likelihood ratio if the first distribution,
d1
, is actually true.p_two_sided
A two-sided p-value, which is the probability of getting a log-likelihood ratio which deviates that much from zero in either direction, if the two distributions are actually equally good.
ratio
A data frame with two columns. The first column is the
x
value and second column is the difference in log-likelihoods.
Note
Code initially based on R code developed by Cosma Rohilla Shalizi (http://bactra.org/). Also see Appendix C in Clauset et al, 2009.
References
Vuong, Quang H. (1989): "Likelihood Ratio Tests for Model Selection and Non-Nested Hypotheses", Econometrica 57: 307–333.
Examples
########################################################
# Example data #
########################################################
x = rpldis(100, xmin=2, alpha=3)
########################################################
##Continuous power law #
########################################################
m1 = conpl$new(x)
m1$setXmin(estimate_xmin(m1))
########################################################
##Exponential
########################################################
m2 = conexp$new(x)
m2$setXmin(m1$getXmin())
est2 = estimate_pars(m2)
m2$setPars(est2$pars)
########################################################
##Vuong's test #
########################################################
comp = compare_distributions(m1, m2)
plot(comp)