algstat is a collection of tools to help you use algebraic statistical methods in R. It is intended to be an end user package for that purpose, and consequently depends on other packages that make connections to math software to do key computations. Currently, the latter package is used to connect R to LattE with 4ti2 for lattice problems and the computation of Markov bases, and the m2r package connects R to Macaulay2 for algebraic computations. In the future, bertini will connect R to Bertini, which is used to numerically solve systems of polynomial equations. m2r can be used for that purpose to some extent currently by using Macaulay2's connection to PHCPack, so be sure to look there for now if that's what you're trying to do.
If you have something you're wanting implemented here, please file an issue!
Exact inference with log-linear models
Note: this section assumes you have latter installed and working on your machine.
One of the most well-developed parts of the package allows users to perform (conditional) exact tests for log-linear models. There are several great references on the math behind this, such as Diaconis and Sturmfels' original paper, the Lectures on Algebraic Statistics, and Markov Bases in Algebraic Statistics, so we'll keep the technical discussion to a minimum.
Fisher's exact test
We'll begin by doing Fisher's exact test on a built-in dataset called politics.
library(algstat) # Loading required package: mpoly # Loading required package: stringr # Loading required package: latter # LattE found in /Applications/latte/dest/bin # 4ti2 found in /Applications/latte/dest/bin # Loading required package: m2r # M2 found in /Applications/Macaulay2-1.9.2/bin data(politics) politics # Party # Personality Democrat Republican # Introvert 3 7 # Extrovert 6 4
Here's how you typically do Fisher's exact test in R:
fisher.test(politics) # # Fisher's Exact Test for Count Data # # data: politics # p-value = 0.3698 # alternative hypothesis: true odds ratio is not equal to 1 # 95 percent confidence interval: # 0.03005364 2.46429183 # sample estimates: # odds ratio # 0.305415
Since the independence model is log-linear, this exact same procedure can be done with algstat. The go-to function here is
loglinear(~ Personality + Party, data = politics) # Computing Markov moves (4ti2)... done. # Running chain (C++)... done. # Call: # loglinear(model = ~Personality + Party, data = politics) # # Fitting method: # Iterative proportional fitting (with stats::loglin) # # MCMC details: # N = 10000 samples (after thinning), burn in = 1000, thinning = 10 # # Distance Stat SE p.value SE mid.p.value # P(samp) 0.3715 0.0048 0.2204 # Pearson X^2 1.8182 0.0145 0.3715 0.0048 0.2204 # Likelihood G^2 1.848 0.0154 0.3715 0.0048 0.2204 # Freeman-Tukey 1.8749 0.0165 0.3715 0.0048 0.2204 # Cressie-Read 1.8247 0.0147 0.3715 0.0048 0.2204 # Neyman X^2 2.0089 0.0227 0.3715 0.0048 0.2972
Exact inference in algebraic statistics is done using MCMC to sample from the conditional distribution of the data given its sufficient statistics under the model. Consequently, the p-values estimated are only determined up to Monte Carlo error. The standard p-value is given under the column
p.value in the row labeled
The asymptotic test of independence analogous to Fisher's exact test (which does not condition on the marginals being known) can be done in either of two ways.
The first way uses the
loglin() function from the stats package. It outputs the likelihood ratio statistic (
Likelihood G^2 in the output above) and Pearson's chi-squared statistic (
Pearson X^2 above), but you have to calculate the p-value yourself.
(loglinMod <- stats::loglin(politics, list(1, 2))) # 2 iterations: deviation 0 # $lrt #  1.848033 # # $pearson #  1.818182 # # $df #  1 # # $margin # $margin[] #  "Personality" # # $margin[] #  "Party" pchisq(loglinMod$pearson, df = 1, lower.tail = FALSE) #  0.1775299
The second way is the
loglm() function in the MASS package, which is a nice wrapper of
loglin() (in fact, algstat's
loglinear() function uses the IPF implementation from
loglin(), although it doesn't need to). It's syntax looks identical to
MASS::loglm(~ Personality + Party, data = politics) # Call: # MASS::loglm(formula = ~Personality + Party, data = politics) # # Statistics: # X^2 df P(> X^2) # Likelihood Ratio 1.848033 1 0.1740123 # Pearson 1.818182 1 0.1775299
Fisher's exact test on RxC tables
Doing Fisher's exact test on larger problems is a significantly more complicated problem. The documentation for
fisher.test() illustrates how it can be used on RxC tables in general, not just on 2x2 tables. Here's an example from its documentation drawn from Agresti (2002, p.57):
Job <- matrix( c(1,2,1,0, 3,3,6,1, 10,10,14,9, 6,7,12,11), nrow = 4, ncol = 4, dimnames = list( income = c("< 15k", "15-25k", "25-40k", "> 40k"), satisfaction = c("VeryD", "LittleD", "ModerateS", "VeryS") ) ) Job # satisfaction # income VeryD LittleD ModerateS VeryS # < 15k 1 3 10 6 # 15-25k 2 3 10 7 # 25-40k 1 6 14 12 # > 40k 0 1 9 11 fisher.test(Job) # # Fisher's Exact Test for Count Data # # data: Job # p-value = 0.7827 # alternative hypothesis: two.sided
Here's the algstat counterpart:
loglinear(~ income + satisfaction, data = Job) # Care ought be taken with tables with sampling zeros to ensure the MLE exists. # Computing Markov moves (4ti2)... done. # Running chain (C++)... done. # Call: # loglinear(model = ~income + satisfaction, data = Job) # # Fitting method: # Iterative proportional fitting (with stats::loglin) # # MCMC details: # N = 10000 samples (after thinning), burn in = 1000, thinning = 10 # # Distance Stat SE p.value SE mid.p.value # P(samp) 0.7706 0.0042 0.7701 # Pearson X^2 5.9655 0.0407 0.7602 0.0043 0.7602 # Likelihood G^2 6.7641 0.0437 0.7639 0.0042 0.7639 # Freeman-Tukey 8.6189 0.0597 0.7645 0.0042 0.7645 # Cressie-Read 6.0752 0.0401 0.7616 0.0043 0.7616 # Neyman X^2 6.2442 0.0472 0.5821 0.0049 0.5821
The asymptotic test can be performed as well. The chi-square approximation is actually very good here:
MASS::loglm(~ income + satisfaction, data = Job) # Call: # MASS::loglm(formula = ~income + satisfaction, data = Job) # # Statistics: # X^2 df P(> X^2) # Likelihood Ratio 6.764053 9 0.6616696 # Pearson 5.965515 9 0.7433647
Fisher's exact test on multi-way tables
fisher.test() does not work with multi-way tables and is prone to crashing even in large-celled two-way tables (see
?loglinear for an example). Thus, the only way to do exact inference in multi-way tables is to use
loglinear(). We'll illustrate this using the drugs dataset from
loglinear()'s documentation, taken from Agresti (2002, p.322), on which we'll test the no-three-way interaction model:
data(drugs) ftable(drugs) # Alcohol Yes No # Cigarette Marijuana # Yes Yes 991 3 # No 538 43 # No Yes 44 2 # No 456 279 loglinear(subsets(1:3, 2), data = drugs) # Computing Markov moves (4ti2)... done. # Running chain (C++)... done. # Call: # loglinear(model = subsets(1:3, 2), data = drugs) # # Fitting method: # Iterative proportional fitting (with stats::loglin) # # MCMC details: # N = 10000 samples (after thinning), burn in = 1000, thinning = 10 # # Distance Stat SE p.value SE mid.p.value # P(samp) 0.5969 0.0049 0.4606 # Pearson X^2 0.5279 0.0142 0.5969 0.0049 0.4606 # Likelihood G^2 0.4845 0.0153 0.5969 0.0049 0.4606 # Freeman-Tukey 0.4672 0.0242 0.5969 0.0049 0.4606 # Cressie-Read 0.512 0.0137 0.5969 0.0049 0.4606 # Neyman X^2 0.4294 0.0132 0.5969 0.0049 0.4606
Note that here we've used the more concise syntax of facet specification; if you want to understand the model specification better, read the documentation in
?loglinear. You can perform the asymptotic test with
loglm() like this:
MASS::loglm(~ 1*2 + 2*3 + 1*3, data = drugs) # Call: # MASS::loglm(formula = ~1 * 2 + 2 * 3 + 1 * 3, data = drugs) # # Statistics: # X^2 df P(> X^2) # Likelihood Ratio 0.4845145 1 0.4863845 # Pearson 0.5279994 1 0.4674492
Statistical applications of LattE
Note: this section assumes you have latter installed and working on your machine.
There are many statistical applications and potential applications of LattE in R. One example is found in the
count program, implemented in
latter::count() counts the number of integer points in a convex polytope. This can be useful for counting the number of contingency tables with fixed marginals. algstat uses
latter::count() in the
countTables() function, which determines the number of contingency tables in the fiber (isostatistical region) of a table given an exponential family model.
countTables(politics) # the independence model is the default #  10
For example, we can determine the number of tables with the same row sums of
politics as follows:
(A <- hmat(varlvls = c(2, 2), facets = 1:2)[1:2,]) # 11 12 21 22 # 1+ 1 1 0 0 # 2+ 0 0 1 1 countTables(politics, A) #  121
Numerically solving systems of polynomial equations
Note: this section assumes you have Bertini installed and algstat has registered it.
algstat also provides back-end connections to Bertini to solve systems of polynomial equations. While this work is still being implemented, here's a peak at what it can currently do.
First, algstat can run raw Bertini programs using
bertini(). It also has a nice print method to display the results. For example, here's how you would find the intersection of the line f(x) = x and the unit circle using Bertini:
code <- " INPUT variable_group x, y; function f, g; f = x^2 + y^2 - 1; g = y - x; END; " bertini(code) # 2 solutions (x,y) found. (2 real, 0 complex; 2 nonsingular, 0 singular.) # (-0.707,-0.707) (R) # ( 0.707, 0.707) (R)
Even better, algstat can team up with mpoly (working under the hood) to solve systems of polynomial equations using
curve(x^2, from = -2, to = 2) curve(2 - x^2, from = -2, to = 2, add = TRUE)
polySolve(c("y = x^2", "y = 2 - x^2"), varOrder = c("x", "y")) # 2 solutions (x,y) found. (2 real, 0 complex; 2 nonsingular, 0 singular.) # (-1,1) (R) # ( 1,1) (R)
There are currently two ways to get algstat. Here's the preferred way until we push a new release to CRAN:
- From Github:
if(!requireNamespace("devtools")) install.packages("devtools") install.packages(c("mpoly", "m2r")) devtools::install_github("dkahle/latter") devtools::install_github("dkahle/algstat")
- From CRAN:
install.packages("algstat")(don't get this now; it's currently out of date)
Installing supporting software
Coming soon! See the links above for direct information.