mcgoftest  R Documentation 
To accomplish the nonlinear fit of a probability distribution function (PDF), different optimization algorithms can be used. Each algorithm will return a different set of estimated parameter values. AIC and BIC are not useful (in this case) to decide which parameter set of values is the best. The goodnessoffit tests (GOF) can help in this case. Please, see below the examples on how to use this function.
mcgoftest(varobj, model, ...) ## S4 method for signature 'numeric_OR_matrix,missing' mcgoftest( varobj, model = NULL, distr = NULL, pars = NULL, num.sampl = 999, sample.size = NULL, stat = c("ks", "ad", "sw", "rmse", "chisq", "hd"), min.val = NULL, breaks = NULL, par.names = NULL, seed = 1, num.cores = 1L, tasks = 0L, verbose = TRUE ) ## S4 method for signature 'numeric,CDFmodel' mcgoftest( varobj, model, num.sampl = 999, sample.size = NULL, stat = c("ks", "ad", "sw", "rmse", "chisq", "hd"), min.val = NULL, breaks = NULL, par.names = NULL, seed = 1, num.cores = 1L, tasks = 0L, verbose = TRUE ) ## S4 method for signature 'numeric_OR_matrix,CDFmodelList' mcgoftest( varobj, model, num.sampl = 999, sample.size = NULL, stat = c("ks", "ad", "sw", "rmse", "chisq", "hd"), min.val = NULL, breaks = NULL, par.names = NULL, seed = 1, num.cores = 1L, tasks = 0L, verbose = TRUE ) ## S4 method for signature 'ANY,NLM' mcgoftest( varobj, model, num.sampl = 999, sample.size = NULL, stat = c("ks", "ad", "sw", "rmse", "chisq", "hd"), min.val = NULL, breaks = NULL, par.names = NULL, seed = 1, num.cores = 1L, tasks = 0L, verbose = TRUE ) ## S4 method for signature 'ANY,nls.lm' mcgoftest( varobj, model, distr, num.sampl = 999, sample.size = NULL, stat = c("ks", "ad", "sw", "rmse", "chisq", "hd"), min.val = NULL, breaks = NULL, par.names = NULL, seed = 1, num.cores = 1L, tasks = 0L, verbose = TRUE )
varobj 
A a vector containing observations, the variable for which the CDF parameters was estimated or the discrete absolute frequencies of each observation category. 
model 
A nonlinear regression model from one of the following classes: "CDFmodel", "CDFmodelList", "NLM", and "nls.lm". 
distr 
The possible options are:

pars 
CDF model parameters. A list of parameters to evaluate the CDF. 
num.sampl 
Number of resamplings. 
sample.size 
Size of the samples used for each sampling. 
stat 
One string denoting the statistic to used in the testing:

min.val 
A number denoting the lower bound of the domain where CDF is defined. 
breaks 
Default is NULL. Basically, the it is same as in function

par.names 
(Optional) The names of the parameters from distr function. Some distribution functions would require to provide the names of their parameters. 
seed 
An integer used to set a 'seed' for random number generation. 
num.cores, tasks 
Parameters for parallel computation using package

verbose 
if verbose, comments and progress bar will be printed. 
The test is intended mostly for continuous distributions. Basically, given the set of parameter values pars from distribution distr, num.sampl sets of random samples will be generated, each one of them with sample.size element. The selected statistic pars will be computed for each randomly generated set (b_stats) and for sample varobj (s_stat). Next, the bootstrap pvalue will be computed as: mean(c(s_stat, b_stats) >= s_stat).
If both variables, varobj and distr, are
numerical vectors, then tableBoots
function is applied. That
is, the problem is confronted as a n x m contingency independence test,
since there is no way to prove that two arbitrary sequences of integer
numbers would follow the same probability distribution without provide
additional information/knowledge.
If sampling size is lesser the size of the sample, then the test becomes a Monte Carlo test. The test is based on the use of measures of goodness of fit, statistics. The following statistics are available (and some limitations for their application to continuous variables are given):
Kolmogorov Smirnov statistic ('ks'). Limitations: sensitive to ties [1]. Only the parametric Monte Carlo resampling (provided that there is not ties in the data) can be used.
Anderson–Darling statistic ('ad') [2]. Limitation: by construction, it depends on the sample size. So, the size of the sampling must be close to the sample size if Monte Carlo resampling is used, which could be a limitation if the sample size is too large [2]. In particular, could be an issue in some genomic applications. It is worth highlighting that, for the current application, the Anderson–Darling statistic is not standardized as typically done in testing GoF for normal distribution with Anderson–Darling test. It is not required since, the statistic is not compared with a corresponding theoretical value. In addition, since the computation of this statistic requires for the data to be put in order [2], it does not make sense to perform a permutation test. That is, the maximum sampling size is the sample size less 1.
ShapiroWilk statistic ('sw') [5]. A Jackknife resampling is applied, instead of a Monte Carlo resampling. Simulation studies suggest that ShapiroWilk test is the most powerful normality test, followed by AndersonDarling test, Lillie/ors test and KolmogorovSmirnov test [6]. In this case, a jackknife resampling is applied (leaveoneout) and the pvalue for the mean of ShapiroWilk statistic is returned.
Pearson's Chisquared statistic ('chisq'). Limitation: the
sample must be discretized (partitioned into bins), which is
could be a source of bias that leads to the rejection of the
null hypothesis. Here, the discretization is done using
function the resources from function
hist
.
Root Mean Square statistic ('rmse'). Limitation: the same as 'chisq'.
Hellinger Divergence statistic ('hd'). Limitation: the same as 'chisq'.
If the argument distr must be defined in environmentnamespace from any package or the environment defined by the user. if missing( sample.size ), then sample.size < length(varobj)  1.
Notice that 'chisq', 'rmse', and 'hd' tests can be applied to testing two discrete probability distributions as well. However, here, mcgoftest function is limited to continuous probability distributions.
Additionally, the only supported ndimensional probability distribution is
Dirichlet Distribution (Dir). The GOF for Dir is based on the fact
that if a variable x = (x_1, x_2, ...x_n) follows Dirichlet
Distribution with parameters α = α_1, ... , α_n (all
positive reals), in short, x ~ Dir(α), then x_i ~
Beta(α_i, α_0  α_i), where Beta(.) stands for the Beta
distribution and α_0 = ∑ α_i (see Detail section,
dirichlet
function, and example 5).
A numeric vector with the following data:
Statistic value.
mc_p.value: the probability of finding the observed, or more extreme, results when the null hypothesis H_0 of a study question is true obtained Monte Carlo resampling approach.
Robersy Sanchez (https://genomaths.com).
Feller, W. On the KolmogorovSmirnov Limit Theorems for Empirical Distributions. Ann. Math. Stat. 19, 177–189 (1948).
Anderson, T. . & Darling, D. A. A Test Of Goodness Of Fit. J. Am. Stat. Assoc. 49, 765–769 (1954).
Watson, G. S. On ChiSquare GoodnessOfFit Tests for Continuous Distributions. J. R. Stat. Soc. Ser. B Stat. Methodol. 20, 44–72 (1958).
A. Basu, A. Mandal, L. Pardo, Hypothesis testing for two discrete populations based on the Hellinger distance. Stat. Probab. Lett. 80, 206–214 (2010).
Patrick Royston (1982). An extension of Shapiro and Wilk's W test for normality to large samples. Applied Statistics, 31, 115–124. doi: 10.2307/2347973.
Y. Bee Wah, N. Mohd Razali, Power comparisons of ShapiroWilk, KolmogorovSmirnov, Lilliefors and AndersonDarling tests. J. Stat. Model. Anal. 2, 21–33 (2011).
Distribution fitting: fitMixDist
,
fitdistr
, fitCDF
, and
bicopulaGOF
.
## ======== Example 1 ======= # Let us generate a random sample a from a specified Weibull distribution: # Set a seed set.seed(1) # Random sample from Weibull( x  shape = 0.5, scale = 1.2 ) x < rweibull(10000, shape = 0.5, scale = 1.2) # MC KS test accept the null hypothesis that variable x comes # from Weibull(x  shape = 0.5, scale = 1.2), while the standard # KolmogorovSmirnov test reject the Null Hypothesis. mcgoftest(x, distr = "weibull", pars = c(0.5, 1.2), num.sampl = 500, sample.size = 1000, num.cores = 4 ) ## ========= Example 2 ====== # Let us generate a random sample a random sample from a specified Normal # distribution: # Set a seed set.seed(1) x < rnorm(10000, mean = 1.5, sd = 2) # MC KS test accept the null hypothesis that variable x comes # from N(x  mean = 0.5, sd = 1.2), while the standard # KolmogorovSmirnov test reject the Null Hypothesis. This an old KS issue, # well known by statisticians since the 1970s. mcgoftest(x, distr = "norm", pars = c(1.5, 2), num.sampl = 500, sample.size = 1000, num.cores = 1 ) ## ========= Example 3 ====== ## Define a Weibull 3parameter distribution function pwdist < function(x, pars) { pweibull(x  pars[1], shape = pars[2], scale = pars[3] ) } rwdist < function(n, pars) { rweibull(n, shape = pars[2], scale = pars[3] ) + pars[1] } ## A random generation from Weibull3P set.seed(123) pars < c(mu = 0.9, shape = 1.4, scale = 3.7) w < rwdist(200, pars = pars) ## Testing GoF mcgoftest( varobj = w, distr = "wdist", pars = list(pars), num.sampl = 100, sample.size = 199, stat = "chisq", num.cores = 4, breaks = 100, seed = 123 ) ## ========= Example 4 ====== ##  Testing GoF of a mixture distribution.  ## Define a mixture distribution to be evaluated with functions 'mixtdistr' ## (see ?mixtdistr). In the current case, it will be mixture of a LogNormal ## and a Weibull distributions: phi < c(0.37, 0.63) # Mixture proportions args < list( lnorm = c(meanlog = 0.837, sdlog = 0.385), weibull = c(shape = 2.7, scale = 5.8) ) ## Sampling from the specified mixture distribution set.seed(123) x < rmixtdistr(n = 1e5, phi = phi, arg = args) hist(x, 100, freq = FALSE) x1 < seq(0, 10, by = 0.001) lines(x1, dmixtdistr(x1, phi = phi, arg = args), col = "red") ## The GoF for the simulated sample pars < c(list(phi = phi), arg = list(args)) mcgoftest( varobj = x, distr = "mixtdistr", pars = pars, num.sampl = 999, sample.size = 999, stat = "chisq", num.cores = 4, breaks = 200, seed = 123 ) #' ## ========= Example 5 ====== ## ShapiroWilk test of normality. set.seed(151) r < rnorm(21, mean = 5, sd = 1) shapiro.test(r) # Classical test mcgoftest(r, stat = "sw") ## ========= Example 6 ====== ## GoF for Dirichlet Distribution (these examples can be run, ## the 'notrun' is only to prevent time consuming in R checking) ## Not run: set.seed(1) alpha < c(2.1, 3.2, 3.3) x < rdirichlet(n = 100, alpha = alpha) mcgoftest( varobj = x, distr = "dirichlet", pars = alpha, num.sampl = 999, sample.size = 100, stat = "chisq", par.names = "alpha", num.cores = 4, breaks = 50, seed = 1 ) ## Now, adding some noise to the sample set.seed(1) x < x + replicate(3, runif(100, max = 0.1)) x < x / rowSums(x) mcgoftest( varobj = x, distr = "dirichlet", pars = alpha, num.sampl = 999, sample.size = 100, stat = "chisq", par.names = "alpha", num.cores = 4, breaks = 50, seed = 1 ) ## End(Not run) ## ========= Example 7 ====== ## Testing multinomially distributed vectors ## A vector of probability parameters set.seed(123) prob < round(runif(12, min = 0.1, max = 0.8), 2) prob < prob / sum(prob) # To normalize the probability vector ## Generate multinomially distributed random numeric vectors r < rmultinom(12, size = 120, prob = prob) ## A list the parameter values must be provided pars < list(size = 120, prob = prob) mcgoftest(r, distr = "multinom", stat = "chisq", pars = pars, sample.size = 20, num.sampl = 50 ) ## ========= Example 8 ====== ## Testing whether two discrete probabilities vectors ## would come from different probability distributions. set.seed(1) ## Vector of absolute frequencies n < 12 alpha < round(runif(n, min = 0.1, max = 0.8) * 10, 1) x1 < round(runif(n = n) * 100) ## Add some little noise x2 < x1 + round(runif(n, min = 0.1, max = 0.2) * 10) mcgoftest( varobj = x1, distr = x2, num.sampl = 999, stat = "chisq", seed = 1 ) ## Compare it with Chisquare test from R package 'stat' chisq.test(rbind(x1, x2), simulate.p.value = TRUE, B = 2e3)$p.value ## Add bigger noise to 'x1' x3 < x1 + round(rnorm(n, mean = 5, sd = 1) * 10) mcgoftest( varobj = x1, distr = x3, num.sampl = 999, stat = "chisq", seed = 1 ) ## Chisquare test from R package 'stat' is consistent with 'mcgoftest' chisq.test(rbind(x1, x3), simulate.p.value = TRUE, B = 2e3)$p.value ## We will always fail in to detect differences between 'crude' probability ## vectors without additional information. chisq.test(x = x1, y = x3, simulate.p.value = TRUE, B = 2e3)$p.value
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