Package 'MXM'

Description

'MXM' stands for Mens eX Machina, meaning 'Mind from the Machine' in Latin. The package provides source code for the SES algorithm and for some appropriate statistical conditional independence tests. (Fisher and Spearman correlation, G-square test are some examples. Currently the response variable can be univariate or multivariate Euclidean, proportions within 0 and 1, compositional data without zeros and ones, binary, nominal or ordinal multinomial, count data (handling also overdispersed and with more zeros than expected), longitudinal, clustered data, survival and case-control. Robust versions are also available in some cases and a K-fold cross validation is offered. Bayesian network related algorithms and ridge reression are also included. Read the package's help pages for more details.

MMPC and SES can handle even thousands of variables and for some tests, even many sample sizes of tens of thousands. The user is best advised to check his variables in the beginning. For some regressions, logistic and Poisson for example, we have used C++ codes for speed reasons.

For more information the reader is addressed to

Lagani V., Athineou G., Farcomeni A., Tsagris M. and Tsamardinos I. (2017). Feature Selection with the R Package MXM: Discovering Statistically Equivalent Feature Subsets. Journal of Statistical Software, 80(7), doi:10.18637/jss.v080.i07 and

Tsagris, M. and Tsamardinos, I. (2019). Feature selection with the R package MXM. F1000Research 7: 1505.

Details

Maintainer

Konstantina Biza [email protected].

Note

Acknowledgments: The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n. 617393.

Michail Tsagris would like to express his acknowledgments to Marios Dimitriadis and Manos Papadakis, undergraduate students in the department of computer science, university of Crete, for their programming tips and advice. Their help has been very valuable. Dr Uwe Ligges and Prof Kurt Hornik from the CRAN team are greatly acknowledged for their assistance. Prof Achim Zeileis is greatly acknowledged for this help with the quasi Poisson and quasi binomial regression models. Christina Chatzipantsiou and Kleio Maria Verrou are acknowledged for their suggestions. Nikolaos Pandis from the University of Bern is acknowledged for his suggestion of the AFT (regression) models and for his suggestions. Michail is grateful to James Dalgleish from Columbia University who suggested that we mention, in various places, that most algorithms return the logarithm of the p-values and not the p-values. Stavros Lymperiadis provided a very useful example where weights are used in a regression model; in surveys when stratified random sampling has been applied. Dr David Gomez Cabrero Lopez is also acknowledged. Margarita Rebolledo is acknowledged for spotting a bug. Zurab Khasidashvili from Intel Israel is acknowledged for spotting a bug in the function mmmb(). Teny Handhayani (PhD student at the University of York) spotted a bug in the conditional independence tests with mixed data and she is acknowledged for that. Dr. Kjell Jorner (Postdoctoral Fellow at the Department of Chemistry, University of Toronto) spotted a bug in two performance metrics of the bbc() function and he is acknowledged for that.

Disclaimer: Professor Tsamardinos is the creator of this package and Dr Lagani supervised Mr Athineou build it. Dr Tsagris is the current maintainer.

Author(s)

Ioannis Tsamardinos <[email protected]>, Vincenzo Lagani <[email protected]>, Giorgos Athineou <[email protected]>, Michail Tsagris <[email protected]>, Giorgos Borboudakis <[email protected]>, Anna Roumpelaki <[email protected]>, Konstantina Biza <[email protected]>.

References

Tsagris, M., Papadovasilakis, Z., Lakiotaki, K., & Tsamardinos, I. (2022). The $\gamma$-OMP algorithm for feature selection with application to gene expression data. IEEE/ACM Transactions on Computational Biology and Bioinformatics, 19(2): 1214-1224.

Tsagris, M. and Tsamardinos, I. (2019). Feature selection with the R package MXM. F1000Research 7: 1505

Borboudakis G. and Tsamardinos I. (2019). Forward-backward selection with early dropping. Journal of Machine Learning Research, 20(8): 1-39.

Tsagris, M. (2019). Bayesian Network Learning with the PC Algorithm: An Improved and Correct Variation. Applied Artificial Intelligence, 33(2):101-123.

Tsagris, M., Lagani, V. and Tsamardinos, I. (2018). Feature selection for high-dimensional temporal data. BMC Bioinformatics, 19:17.

Tsagris, M., Borboudakis, G., Lagani, V. and Tsamardinos, I. (2018). Constraint-based causal discovery with mixed data. International Journal of Data Science and Analytics, 6(1): 19-30.

Tsagris, M., Papadovasilakis, Z., Lakiotaki, K. and Tsamardinos, I. (2018). Efficient feature selection on gene expression data: Which algorithm to use? BioRxiv preprint.

Lagani V., Athineou G., Farcomeni A., Tsagris M. and Tsamardinos I. (2017). Feature Selection with the R Package MXM: Discovering Statistically Equivalent Feature Subsets. Journal of Statistical Software, 80(7), doi:10.18637/jss.v080.i07.

Chen S., Billings S. A., and Luo W. (1989). Orthogonal least squares methods and their application to non-linear system identification. International Journal of control, 50(5), 1873-1896. http://eprints.whiterose.ac.uk/78100/1/acse

Davis G. (1994). Adaptive Nonlinear Approximations. PhD thesis. http://www.geoffdavis.net/papers/dissertation.pdf

Demidenko E. (2013). Mixed Models: Theory and Applications with R, 2nd Edition. New Jersey: Wiley & Sons.

Gharavi-Alkhansari M., anz Huang T. S. (1998, May). A fast orthogonal matching pursuit algorithm. In Acoustics, Speech and Signal Processing, 1998. Proceedings of the 1998 IEEE International Conference on (Vol. 3, pp. 1389-1392).

Lagani V., Kortas G. and Tsamardinos I. (2013), Biomarker signature identification in "omics" with multiclass outcome. Computational and Structural Biotechnology Journal, 6(7):1-7.

Liang K.Y. and Zeger S.L. (1986). Longitudinal data analysis using generalized linear models. Biometrika, 73(1): 13-22.

Mallat S. G. & Zhang Z. (1993). Matching pursuits with time-frequency dictionaries. IEEE Transactions on signal processing, 41(12), 3397-3415. https://www.di.ens.fr/~mallat/papiers/MallatPursuit93.pdf

Paik M.C. (1988). Repeated measurement analysis for nonnormal data in small samples. Communications in Statistics-Simulation and Computation, 17(4): 1155-1171.

Pati Y. C., Rezaiifar R. and Krishnaprasad P. S. (1993). Orthogonal matching pursuit: Recursive function approximation with applications to wavelet decomposition. In Signals, Systems and Computers. 1993 Conference Record of The Twenty-Seventh Asilomar Conference on. IEEE.

Prentice R.L. and Zhao L.P. (1991). Estimating equations for parameters in means and covariances of multivariate discrete and continuous responses. Biometrics, 47(3): 825-839.

Spirtes P., Glymour C. and Scheines R. (2001). Causation, Prediction, and Search. The MIT Press, Cambridge, MA, USA, 3nd edition.

Tsamardinos I., Greasidou E. and Borboudakis G. (2018). Bootstrapping the out-of-sample predictions for efficient and accurate cross-validation. Machine Learning 107(12): 1895-1922.

Tsamardinos I., Lagani V. and Pappas D. (2012) Discovering multiple, equivalent biomarker signatures. In proceedings of the 7th conference of the Hellenic Society for Computational Biology & Bioinformatics - HSCBB12.

Tsamardinos, Brown and Aliferis (2006). The max-min hill-climbing Bayesian network structure learning algorithm. Machine learning, 65(1), 31-78.

Tsamardinos I., Aliferis C. F. and Statnikov, A. (2003). Time and sample efficient discovery of Markov blankets and direct causal relations. In Proceedings of the 9th ACM SIGKDD international conference on Knowledge discovery and data mining p. 673-678.

Yan J. and Fine J. (2004). Estimating equations for association structures. Statistics in medicine, 23(6): 859-874

Zhang, Jiji. (2008). On the completeness of orientation rules for causal discovery in the presence of latent confounders and selection bias. Artificial Intelligence 172(16): 1873–1896.

Ziegler A., Kastner C., Brunner D. and Blettner M. (2000). Familial associations of lipid profiles: A generalised estimating equations approach. Statistics in medicine, 19(24): 3345-3357

See Also

SES, MMPC, fbed.reg, gomp, pc.sel, censIndCR,testIndFisher, testIndLogistic, gSquare, testIndRQ

Description

Returns and plots, if asked, the descendants or ancestors of one or all node(s) (or variable(s))

Usage

findDescendants(G, node = NULL, graph = FALSE)
findAncestors(G, node = NULL, graph = FALSE)

Arguments

Details

The functions searches for the descendants of some node. This is an S3 class output.

Value

Author(s)

Anna Roumpelaki

R implementation and documentation: Anna Roumpelaki <[email protected]>

See Also

plotnetwork, nei, mb, pc.or

Examples

# simulate a dataset with continuous data
y = rdag(1000, 10, 0.3)
tru = y$G 
x = y$x
mod = pc.con(x)
G = pc.or(mod)$G
plotnetwork(G)
findDescendants(G, 4, graph = FALSE)
findAncestors(G, 4, graph = FALSE)
findAncestors(G)

Description

Backward phase of MMPC.

Usage

mmpcbackphase(target, dataset, max_k = 3, threshold = 0.05, test = NULL,
wei = NULL, R = 1)

Arguments

Details

For each of the selected variables (dataset) the function performs conditional independence tests where the conditioning sets are formed from the other variables. All possible combinations are tried until the variable becomes non significant. The maximum size of the conditioning set is equal to max_k. This is called in the MMPC when the backward phase is requested.

Value

A list including:

Author(s)

Ioannis Tsamardinos, Michail Tsagris

R implementation and documentation: Michail Tsagris [email protected]

References

Tsamardinos, Brown and Aliferis (2006). The max-min hill-climbing Bayesian network structure learning algorithm. Machine learning, 65(1), 31-78.

See Also

MMPC, mmhc.skel, CondIndTests, cv.mmpc

Examples

set.seed(123)

#simulate a dataset with continuous data
dataset <- matrix(runif(500 * 100, 1, 100), ncol = 100)

#define a simulated class variable 
target <- 3 * dataset[, 10] + 2 * dataset[, 100] + 3 * dataset[, 20] + rnorm(500, 0, 5)

# MMPC algorithm 
m1 <- MMPC(target, dataset, max_k = 3, threshold = 0.05, test="testIndFisher");
m2 <- MMPC(target, dataset, max_k = 3, threshold = 0.05, test="testIndFisher", backward = TRUE);
x <- dataset[, m1@selectedVars]
mmpcbackphase(target, x, test = testIndFisher)

Description

Variable selection in regression models with backward selection

Usage

bs.reg(target, dataset, threshold = 0.05, wei = NULL, test = NULL, user_test = NULL)

Arguments

Details

If the sample size is less than the number of variables a meesage will appear and no backward regression is performed.

Value

The output of the algorithm is S3 object including:

Author(s)

Michail Tsagris

R implementation and documentation: Michail Tsagris [email protected]

See Also

glm.fsreg, lm.fsreg, bic.fsreg, bic.glm.fsreg, CondIndTests, MMPC, SES

Examples

set.seed(123)
dataset <- matrix( runif(200 * 10, 1, 100), ncol = 10 )
target <- rnorm(200)
a <- bs.reg(target, dataset, threshold = 0.05, test = "testIndRQ") 
b <- bs.reg(target, dataset, threshold = 0.05, test = "testIndReg") 
b2 <- bs.reg(target, dataset, threshold = 0.05, test = "testIndFisher")

Description

Backward selection regression for GLMM

Usage

glmm.bsreg(target, dataset, id, threshold = 0.05, wei = NULL, test = "testIndGLMMReg")

Arguments

Details

If the sample size is less than the number of variables a meesage will appear and no backward regression is performed.

Value

The output of the algorithm is S3 object including:

Author(s)

Michail Tsagris

R implementation and documentation: Michail Tsagris [email protected]

References

Eugene Demidenko (2013). Mixed Models: Theory and Applications with R, 2nd Edition. New Jersey: Wiley & Sons.

See Also

fbed.glmm.reg, ebic.glmm.bsreg, MMPC.glmm

Examples

## Not run: 
require(lme4)
data(sleepstudy)
reaction <- sleepstudy$Reaction
days <- sleepstudy$Days
subject <- sleepstudy$Subject
x <- matrix(rnorm(180 * 200),ncol = 200) ## unrelated predictor variables
m1 <- glmm.bsreg(Reaction, x, subject) 
m2 <- MMPC.glmm(target = reaction, group = subject, dataset = x)

## End(Not run)

Description

Backward selection regression for GLMM using the eBIC

Usage

ebic.glmm.bsreg(target, dataset, id, wei = NULL, gam = NULL, test = "testIndGLMMReg")

Arguments

Details

The algorithm is a variation of the usual forward selection. At every step, the most significant variable enters the selected variables set. In addition, only the significant variables stay and are further examined. The non signifcant ones are dropped. This goes until no variable can enter the set. The user has the option to redo this step 1 or more times (the argument K). In the end, a backward selection is performed to remove falsely selected variables.

Value

A list including:

Author(s)

Michail Tsagris

R implementation and documentation: Michail Tsagris [email protected]

References

Borboudakis G. and Tsamardinos I. (2019). Forward-backward selection with early dropping. Journal of Machine Learning Research, 20(8): 1-39.

Eugene Demidenko (2013). Mixed Models: Theory and Applications with R, 2nd Edition. New Jersey: Wiley & Sons.

See Also

fbed.glmm.reg, glmm.bsreg, MMPC.glmm

Examples

## Not run: 
require(lme4)
data(sleepstudy)
reaction <- sleepstudy$Reaction
days <- sleepstudy$Days
subject <- sleepstudy$Subject
x <- matrix(rnorm(180 * 20),ncol = 20) ## unrelated preidctor variables
m1 <- ebic.glmm.bsreg(reaction, x, id = subject) 
m2 <- MMPC.glmm(reaction, group = subject, dataset = x)

## End(Not run)

Description

Backward selection regression using the eBIC

Usage

ebic.bsreg(target, dataset, test = NULL, wei = NULL, gam = NULL)

Arguments

Details

The algorithm is a variation of the usual forward selection. At every step, the most significant variable enters the selected variables set. In addition, only the significant variables stay and are further examined. The non signifcant ones are dropped. This goes until no variable can enter the set. The user has the option to redo this step 1 or more times (the argument K). In the end, a backward selection is performed to remove falsely selected variables.

Value

A list including:

Author(s)

Michail Tsagris

R implementation and documentation: Michail Tsagris [email protected]

References

Borboudakis G. and Tsamardinos I. (2019). Forward-backward selection with early dropping. Journal of Machine Learning Research, 20(8): 1-39.

See Also

fs.reg, bic.fsreg, MMPC

Examples

dataset <- matrix( runif(100 * 15, 1, 100), ncol = 15 )
target <- rt(100, 10)
a1 <- ebic.bsreg(target, dataset, test = "testIndReg") 
target <- rpois(100, 10)
a2 <- ebic.bsreg(target, dataset, test = "testIndPois")

Description

Variable selection in generalised linear regression models with backward selection

Usage

glm.bsreg(target, dataset, threshold = 0.05, wei = NULL, test = NULL)
glm.bsreg2(target, dataset, threshold = 0.05, wei = NULL, test = NULL)

Arguments

Details

This functions currently implements only linear, binomial, binary logistic and Poisson regression. If the sample size is less than the number of variables a meesage will appear and no backward regression is performed.

Value

The output of the algorithm is S3 object including:

Author(s)

Michail Tsagris

R implementation and documentation: Michail Tsagris [email protected]

See Also

fs.reg, lm.fsreg, bic.fsreg, bic.glm.fsreg, CondIndTests, MMPC, SES

Examples

set.seed(123)

#simulate a dataset with continuous data
dataset <- matrix( runif(200 * 10, 1, 100), ncol = 10 )

#define a simulated class variable 
target <- rpois(200, 10)
a <- glm.bsreg(target, dataset, threshold = 0.05) 

target <- rbinom(200, 1, 0.6)
b <- glm.bsreg(target, dataset, threshold = 0.05)

target <- rgamma(200, 1, 2)
b1 <- glm.bsreg2(target, dataset, threshold = 0.05, test = "testIndGamma")
b2 <- glm.bsreg2(target, dataset, threshold = 0.05, test = "testIndNormLog")

Description

Bayesian Network construction using a hybrid of MMPC and PC.

Usage

mmpc.or(x, max_k = 5, threshold = 0.01, test = "testIndFisher", backward = TRUE, 
symmetry = TRUE, ini.pvalue = NULL)

Arguments

Details

The MMPC is run on every variable. The backward phase (see Tsamardinos et al., 2006) can then take place. After all variables have been used, the matrix is checked for inconsistencies and they are corrected if you want. The "symmetry" argument. Do you want the edge to stay if it was discovered from both variables when they were considered as responses?

Value

A list including:

Bear in mind that the values can be extracted with the $ symbol, i.e. this is an S3 class output.

Author(s)

Michail Tsagris

R implementation and documentation: Giorgos Athineou <[email protected]> and Michail Tsagris [email protected]

References

Tsamardinos, Brown and Aliferis (2006). The max-min hill-climbing Bayesian network structure learning algorithm. Machine learning, 65(1), 31-78.

Spirtes P., Glymour C. and Scheines R. (2001). Causation, Prediction, and Search. The MIT Press, Cambridge, MA, USA, 3nd edition.

See Also

pc.skel, pc.or, corfs.network

Examples

y <- rdag2(500, p = 20, nei = 3)
ind <- sample(1:20, 20)
x <- y$x[, ind]
a1 <- mmpc.or(x, max_k = 3, threshold = 0.01, test = "testIndFisher" ) 
b <- pc.skel( x, alpha = 0.01 ) 
a2 <- pc.or(b)

Description

Beta regression.

Usage

beta.mod(target, dataset, wei = NULL, xnew= NULL)

Arguments

Details

The beta regression is fitted. The "beta.reg" is an internal wrapper function and is used for speed up purposes. It is not to be called directly by the user unless they know what they are doing.

Value

A list including:

Author(s)

Michail Tsagris

R implementation and documentation: Michail Tsagris [email protected]

References

Ferrari S.L.P. and Cribari-Neto F. (2004). Beta Regression for Modelling Rates and Proportions. Journal of Applied Statistics, 31(7): 799-815.

See Also

beta.regs, testIndBeta, reg.fit, ridge.reg

Examples

y <- rbeta(300, 3, 5)
x <- matrix( rnorm(300 * 2), ncol = 2)
a1 <- beta.mod(y, x)
w <- runif(300)
a2 <- beta.mod(y, x, w)

Description

Variable selection in regression models with forward selection using BIC

Usage

bic.fsreg(target, dataset, test = NULL, wei = NULL, tol = 2, ncores = 1)

Arguments

Details

If the current 'test' argument is defined as NULL or "auto" and the user_test argument is NULL then the algorithm automatically selects the best test based on the type of the data. Particularly:

• if target is a factor, the multinomial or the binary logistic regression is used. If the target has two values only, binary logistic regression will be used.

• if target is a ordered factor, the ordinal regression is used.

• if target is a numerical vector or a matrix with at least two columns (multivariate) linear regression is used.

• if target is discrete numerical (counts), the poisson regression conditional independence test is used. If there are only two values, the binary logistic regression is to be used.

• if target is a Surv object, the Survival conditional independence test (Cox regression) is used.

Value

The output of the algorithm is S3 object including:

Author(s)

Michail Tsagris

R implementation and documentation: Giorgos Athineou <[email protected]> Michail Tsagris [email protected]

References

Tsamardinos I., Aliferis C. F. and Statnikov, A. (2003). Time and sample efficient discovery of Markov blankets and direct causal relations. In Proceedings of the 9th ACM SIGKDD international conference on Knowledge discovery and data mining (pp. 673-678).

See Also

glm.fsreg, lm.fsreg, bic.glm.fsreg, CondIndTests, MMPC, SES

Examples

set.seed(123)
dataset <- matrix( runif(200 * 20, 1, 100), ncol = 20 )
target <- 3 * dataset[, 10] + 2 * dataset[, 15] + 3 * dataset[, 20] + rnorm(200, 0, 5)

a1 <- bic.fsreg(target, dataset, tol = 4, ncores = 1, test = "testIndReg" ) 
a3 <- MMPC(target, dataset, ncores = 1)
target <- round(target)
b1 <- bic.fsreg(target, dataset, tol = 2, ncores = 1, test = "testIndReg" )

Description

Variable selection in generalised linear models with forward selection based on BIC

Usage

bic.glm.fsreg( target, dataset, wei = NULL, tol = 0, ncores = 1) 

bic.mm.fsreg( target, dataset, wei = NULL, tol = 0, ncores = 1) 

bic.gammafsreg(target, dataset, wei = NULL, tol = 0, ncores = 1) 

bic.normlog.fsreg(target, dataset, wei = NULL, tol = 0, ncores = 1)

Arguments

Details

Forward selection via the BIC is implemented. A variable which results in a reduction of BIC will be included, until the reduction is below a threshold set by the user (argument "tol").

Value

The output of the algorithm is S3 object including:

Author(s)

Michail Tsagris

R implementation and documentation: Giorgos Aathineou <[email protected]> Michail Tsagris [email protected]

See Also

fs.reg, lm.fsreg, bic.fsreg, CondIndTests, MMPC, SES

Examples

set.seed(123)
dataset <- matrix( runif(200 * 20, 1, 100), ncol = 20 )
target <- 3 * dataset[, 10] + 2 * dataset[, 15] + 3 * dataset[, 20] + rnorm(200, 0, 5)
a1 <- bic.glm.fsreg(target, dataset, tol = 2, ncores = 1 ) 
a2 <- bic.glm.fsreg( round(target), dataset, tol = 2, ncores = 1 ) 
y <- target   ;   me <- median(target)  ;   y[ y < me ] <- 0   ;   y[ y >= me ] <- 1
a3 <- bic.glm.fsreg( y, dataset, tol = 2, ncores = 1 )

Description

Bootstrap bias correction for the performance of the cross-validation procedure.

Usage

bbc(predictions, target, metric = "auc.mxm", conf = 0.95, B = 1000)

Arguments

Details

Upon completion of the cross-validation, the predicted values produced by all predictive models across all folds is collected in a matrix $P$ of dimensions $n \times M$, where $n$ is the number of samples and $M$ the number of trained models or configurations. Sampled with replacement a fraction of rows (predictions) from $P$ are denoted as the in-sample values. On average, the newly created set will be comprised by 63.2% of the original individuals (The probability of sampling, with replacement, a sample of $n$ numbers from a set of $n$ numbers is $1-\left(1-\frac{1}{n} \right)^n \simeq 1-\frac{1}{e}=0.632$), whereas the rest 36.8% will be random copies of them. The non re-sampled rows are denoted as out-of-sample values. The performance of each model in the in-sample rows is calculated and the model (or configuration) with the optimal performance is selected, followed by the calculation of performance in the out-of-sample values. This process is repeated B times and the average performance is returned.

Note, that the only computational overhead is with the repetitive re-sampling and calculation of the predictive performance, i.e. no model is fitted nor trained. The final estimated performance usually underestimates the true performance, but this negative bias is smaller than the optimistic uncorrected performance.

Note, that all metrics are for maximization. For this reason "mse.mxm", "mae.mxm", "ord_mae.mxm", "poisdev.mxm", "binomdev.mxm" are multiplied by -1.

Value

A list including:

Author(s)

R implementation and documentation: Michail Tsagris [email protected].

References

Ioannis Tsamardinos, Elissavet Greasidou and Giorgos Borboudakis (2018). Bootstrapping the out-of-sample predictions for efficient and accurate cross-validation. Machine Learning (To appear).

https://link.springer.com/article/10.1007/s10994-018-5714-4

See Also

cv.ses, cv.gomp

Examples

predictions <- matrix(rbinom(200 * 100, 1, 0.7), ncol = 100) 
target <- rbinom(200, 1, 0.5)
bbc(predictions, target, metric = "auc.mxm")

Description

Calculation of the constant and slope for each subject over time.

Usage

group.mvbetas(x, id, reps)

Arguments

Details

This function is used internally in SES and MMPC and does calculations required bys the first step of the Static-Longitudinal scenario of Tsagris, Lagani and Tsamardinos (2018). The measurements of each subject are regressed againt time. So, for each subject, we get the constant and interecept over time and this is repated for very feature.

Value

A matrix. The first r ( = length( unique(id) ), the nubmer of subjects ) rows contain the constants and the other r rows contain the slopes.

Author(s)

Michail Tsagris

R implementation and documentation: Michail Tsagris [email protected]

References

Tsagris M., Lagani V., & Tsamardinos I. (2018). Feature selection for high-dimensional glmm data. BMC bioinformatics, 19(1), 17.

McCullagh, Peter, and John A. Nelder. Generalized linear models. CRC press, USA, 2nd edition, 1989.

See Also

fbed.gee.reg, glmm.bsreg, MMPC.glmm

Examples

## assume these are longitudinal data, each column is a variable (or feature)
x <- matrix( rnorm(100 * 30), ncol = 30 ) 
id <- rep(1:20, each = 5)  ## 20 subjects
reps <- rep( seq(4, 12, by = 2), 20)  ## 5 time points for each subject
a <- group.mvbetas(x, id, reps)
dim(a)  ## 5  100
## these are the regression coefficients of the first subject's values on the 
## reps (which is assumed to be time in this example)
a[c(1, 21), 1] 
coef( lm( x[id == 1, 1] ~ reps[1:5] ) )

Description

Information on why one ore more variables were not selected.

Usage

certificate.of.exclusion(xIndex, sesObject = NULL, mmpcObject = NULL) 
certificate.of.exclusion2(xIndex, mmpc2object)

Arguments

Value

A list with the conditioning variables (if any), the test statistic and the logarithm of the p-value. In case a variable has been selected a message appears.

Author(s)

Michail Tsagris

R implementation and documentation: Michail Tsagris [email protected]

See Also

MMPC

Examples

set.seed(123)
#simulate a dataset with continuous data
dataset <- matrix(runif(100 * 100, 1, 100), ncol = 100)
#define a simulated class variable 
target <- 3 * dataset[, 10] + 2 * dataset[, 100] + 3 * dataset[, 20] + rnorm(100, 0, 5)
# define some simulated equivalences
dataset[, 15] <- dataset[, 10] + rnorm(100, 0, 2)
dataset[, 100] <- dataset[, 100] + rnorm(100, 0, 2) 
dataset[, 20] <- dataset[, 100] + rnorm(100, 0, 2)
# run the SES algorithm
mod1 <- SES(target, dataset, max_k = 5, threshold = 0.05, test = "testIndFisher", 
hash = TRUE, hashObject = NULL);
mod2 <- MMPC(target, dataset, max_k = 5, threshold = 0.05, test = "testIndFisher", 
hash = TRUE, hashObject = NULL);
certificate.of.exclusion(c(10, 15, 30, 45, 20), mod1)
certificate.of.exclusion(c(10, 15, 30, 45, 20), NULL, mod2)

Description

Check Markov equivalence of two DAGs.

Usage

equivdags(g1, g2)

Arguments

Details

Two DAGs are Markov equivalent if a) they have the same adjancencies (regardlsee of the mark, arrowhead, tail or nothing) and b) they have the same unshielded colliders.

Value

A list including:

Author(s)

Michail Tsagris

R implementation and documentation: Giorgos Athineou <[email protected]> and Michail Tsagris [email protected]

References

Tsamardinos, Brown and Aliferis (2006). The max-min hill-climbing Bayesian network structure learning algorithm. Machine learning, 65(1), 31-78.

See Also

pc.or, mmhc.skel, pc.con

Examples

y <- rdag(1000, 10, 0.3)
tru <- y$G 
x <- y$x
mod <- pc.con(x)

eg <- dag2eg(y$G) ## make it essential graph first 
est <- pc.or(mod)$G

equivdags(est, tru)

Description

Check whether a directed graph is acyclic.

Usage

is.dag(dag)

Arguments

Details

The topological sort is performed. If it cannot be performed, NAs are returned. Hence, the functions checks for NAs.

Value

A logical value, TRUE if the matrix represents a DAG and FALSE otherwise.

Author(s)

Michail Tsagris

R implementation and documentation: Michail Tsagris [email protected]

References

Chickering, D.M. (1995). A transformational characterization of equivalent Bayesian network structures. Proceedings of the 11th Conference on Uncertainty in Artificial Intelligence, Montreal, Canada, 87-98.

See Also

topological_sort, dag2eg, pc.or

Examples

# simulate a dataset with continuous data
# simulate a dataset with continuous data
G <- rdag(100, 20, 0.3)$G
is.dag(G)  ## TRUE

Description

Currently the MXM package supports numerous tests for different types of target (dependent) and predictor (independent) variables. The target variable can be of continuous, discrete, categorical and of survival type. As for the predictor variables, they can be continuous, categorical or mixed.

The testIndFisher and the gSquare tests have two things in common. They do not use a model implicitly (i.e. estimate some beta coefficients), even though there is an underlying assumed one. Secondly they are pure tests of independence (again, with assumptions required).

As for the other tests, they share one thing in common. For all of them, two parametric models must be fit. The null model containing the conditioning set of variables alone and the alternative model containing the conditioning set and the candidate variable. The significance of the new variable is assessed via a log-likelihood ratio test with the appropriate degrees of freedom. All of these tests which are available for SES and MMPC are summarized in the below table.

Details

These tests can be called by SES, MMPC, wald.mmpc or individually by the user. In all regression cases, there is an option for weights.

Log-likelihood ratio tests

1. testIndFisher. This is a standard test of independence when both the target and the set of predictor variables are continuous (continuous-continuous). When the joint multivariate normality of all the variables is assumed, we know that if a correlation is zero this means that the two variables are independent. Moving in this spirit, when the partial correlation between the target variable and the new predictor variable conditioning on a set of (predictor) variables is zero, then we have evidence to say they are independent as well. An easy way to calculate the partial correlation between the target and a predictor variable conditioning on some other variables is to regress the both the target and the new variable on the conditioning set. The correlation coefficient of the residuals produced by the two regressions equals the partial correlation coefficient. If the robust option is selected, the two aforementioned regression models are fitted using M estimators (Marona et al., 2006). If the target variable consists of proportions or percentages (within the (0, 1) interval), the logit transformation is applied beforehand.

2. testIndSpearman. This is a non-parametric alternative to testIndFisher test. It is a bit slower than its competitor, yet very fast and suggested when normality assumption breaks down or outliers are present. In fact, within SES, what happens is that the ranks of the target and of the dataset (predictor variables) are computed and the testIndSpearman is aplied. This is faster than applying Fisher with M estimators as described above. If the target variable consists of proportions or percentages (within the (0, 1) interval), the logit transformation is applied beforehand.

3. testIndReg. In the case of target-predictors being continuous-mixed or continuous-categorical, the suggested test is via the standard linear regression. In this case, two linear regression models are fitted. One with the conditioning set only and one with the conditioning set plus the new variable. The significance of the new variable is assessed via the F test, which calculates the residual sum of squares of the two models. The reason for the F test is because the new variable may be categorical and in this case the t test cannot be used. It makes sense to say, that this test can be used instead of the testIndFisher, but it will be slower. If the robust option is selected, the two models are fitted using M estimators (Marona et al. 2006). If the target variable consists of proportions or percentages (within the (0, 1) interval), the logit transformation is applied beforehand.

4. testIndRQ. An alternative to testIndReg for the case of continuous-mixed (or continuous-continuous) variables is the testIndRQ. Instead of fitting two linear regression models, which model the expected value, one can choose to model the median of the distribution (Koenker, 2005). The significance of the new variable is assessed via a rank based test calibrated with an F distribution (Gutenbrunner et al., 1993). The reason for this is that we performed simulation studies and saw that this type of test attains the type I error in contrast to the log-likelihood ratio test. The benefit of this regression is that it is robust, in contrast to the classical linear regression. If the target variable consists of proportions or percentages (within the (0, 1) interval), the logit transformation is applied beforehand.

5. testIndBeta. When the target is proportion (or percentage, i.e., between 0 and 1, not inclusive) the user can fit a regression model assuming a beta distribution. The predictor variables can be either continuous, categorical or mixed. The procedure is the same as in the testIndReg case.

6. Alternatives to testIndBeta. Instead of testIndBeta the user has the option to choose all the previous to that mentioned tests by transforming the target variable with the logit transformation. In this way, the support of the target becomes the whole of R^d and then depending on the type of the predictors and whether a robust approach is required or not, there is a variety of alternative to beta regression tests.

7. testIndIGreg. When you have non negative data, i.e. the target variable takes positive values (including 0), a suggested regression is based on the the inverse gaussian distribution. The link function is not the inverse of the square root as expected, but the logarithm. This is to ensure that the fitted values will be always be non negative. The predictor variables can be either continuous, categorical or mixed. The significance between the two models is assessed via the log-likelihood ratio test. Alternatively, the user can use the Weibull regression (censIndWR), gamma regression (testIndGamma) or Gaussian regression with log link (testIndNormLog).

8. testIndGamma. This is an alternative to testIndIGreg.

9. testIndNormLog. This is a second alternative to testIndIGreg.

10. testIndPois. When the target is discrete, and in specific count data, the default test is via the Poisson regression. The predictor variables can be either continuous, categorical or mixed. The procedure is the same as in all the previously regression model based tests, i.e. the log-likelihood ratio test is used to assess the conditional independence of the variable of interest.

11. testIndNB. As an alternative to the Poisson regression, we have included the Negative binomial regression to capture cases of overdispersion. The predictor variables can be either continuous, categorical or mixed.

12. testIndQPois. This is a better alternative for discrete target, better than the testIndPois and than the testIndNB, because it can capture both cases of overdispersion and undersidpesrion.

13. testIndZIP. When the number of zeros is more than expected under a Poisson model, the zero inflated poisson regression is to be employed. The predictor variables can be either continuous, categorical or mixed.

14. testIndLogistic. When the target is categorical with only two outcomes, success or failure for example, then a binary logistic regression is to be used. Whether regression or classification is the task of interest, this method is applicable. The advantage of this over a linear or quadratic discriminant analysis is that it allows for categorical predictor variables as well and for mixed types of predictors.

15. testIndQBinom. This is an alternative to either the testIndLogistic or especially the testIndBeta.

16. testIndMultinom. If the target has more than two outcomes, but it is of nominal type, there is no ordering of the outcomes, multinomial logistic regression will be employed. Again, this regression is suitable for classification purposes as well and it to allows for categorical predictor variables.

17. testIndOrdinal. This is a special case of multinomial regression, in which case the outcomes have an ordering, such as not satisfied, neutral, satisfied. The appropriate method is ordinal logistic regression.

18. testIndBinom. When the target variable is a matrix of two columns, where the first one is the number of successes and the second one is the number of trials, binomial regression is to be used.

19. gSquare. If all variables, both the target and predictors are categorical the default test is the G-square test of independence. It is similar to the chi-squared test of independence, but instead of using the chi-squared metric between the observed and estimated frequencies in contingency tables, the Kullback-Leibler divergence of the observed from the estimated frequencies is used. The asymptotic distribution of the test statistic is a chi-squared distribution on some appropriate degrees of freedom. The target variable can be either ordered or unordered with two or more outcomes.

20. Alternatives to gSquare. An alternative to the gSquare test is the testIndLogistic. Depending on the nature of the target, binary, un-ordered multinomial or ordered multinomial the appropriate regression model is fitted.

21. censIndCR. For the case of time-to-event data, a Cox regression model is employed. The predictor variables can be either continuous, categorical or mixed. Again, the log-likelihood ratio test is used to assess the significance of the new variable.

22. censIndWR. A second model for the case of time-to-event data, a Weibull regression model is employed. The predictor variables can be either continuous, categorical or mixed. Again, the log-likelihood ratio test is used to assess the significance of the new variable. Unlike the semi-parametric Cox model, the Weibull model is fully parametric.

23. censIndER. A third model for the case of time-to-event data, an exponential regression model is employed. The predictor variables can be either continuous, categorical or mixed. Again, the log-likelihood ratio test is used to assess the significance of the new variable. This is a special case of the Weibull model.

24. testIndClogit. When the data come from a case-control study, the suitable test is via conditional logistic regression.

25. testIndMVreg. In the case of multivariate continuous targets, the suggested test is via a multivariate linear regression. The target variable can be compositional data as well. These are positive data, whose vectors sum to 1. They can sum to any constant, as long as it the same, but for convenience reasons we assume that they are normalised to sum to 1. In this case the additive log-ratio transformation (multivariate logit transformation) is applied beforehand.

26. testIndSPML. With a circular target, the projected bivariate normal distribution (Presnell et al., 1998) is used to perform regression.

Tests for clustered/longitudinal data

1. testIndGLMMReg, testIndGLMM, testIndGLMMPois & testIndGLMMLogistic. In the case of a longitudinal or clustered targets (continuous, proportions, binary or counts), the suggested test is via a (generalised) linear mixed model. testIndGLMMCR stands for mixed effects Cox regression.

2. testIndGEEReg, testIndGEELogistic, testIndGEEPois, testIndGEENormLog and testIndGEEGamma. In the case of a longitudinal or clustered targets (continuous, proportions, binary, counts, positive, strictly positive), the suggested test is via GEE (Generalised Estimating Equations).

Wald based tests

The available tests for wald.ses and wald.mmpc are listed below. Note, that only continuous predictors are allowed.

Permutation based tests

The available tests for perm.ses and perm.mmpc are listed below. Note, that only continuous predictors are allowed.

Author(s)

Michail Tsagris <[email protected]>

References

Aitchison J. (1986). The Statistical Analysis of Compositional Data, Chapman & Hall; reprinted in 2003, with additional material, by The Blackburn Press.

Brown P.J. (1994). Measurement, Regression and Calibration. Oxford Science Publications.

Cox D.R. (1972). Regression models and life-tables. J. R. Stat. Soc., 34, 187-220.

Demidenko E. (2013). Mixed Models: Theory and Applications with R, 2nd Edition. New Jersey: Wiley & Sons.

Draper, N.R. and Smith H. (1988). Applied regression analysis. New York, Wiley, 3rd edition.

Fieller E.C. and Pearson E.S. (1961). Tests for rank correlation coefficients: II. Biometrika, 48(1 & 2): 29-40.

Ferrari S.L.P. and Cribari-Neto F. (2004). Beta Regression for Modelling Rates and Proportions. Journal of Applied Statistics, 31(7): 799-815.

Gail, M.H., Jay H.L., and Lawrence V.R. (1981). Likelihood calculations for matched case-control studies and survival studies with tied death times. Biometrika 68(3): 703-707.

Gutenbrunner C., Jureckova J., Koenker R. and Portnoy S. (1993). Tests of Linear Hypothesis based on Regression Rank Scores, Journal of NonParametric Statistics 2, 307-331.

Hoerl A.E. and Kennard R.W. (1970). Ridge regression: Biased estimation for nonorthogonal problems. Technometrics, 12(1): 55-67.

Joseph M.H. (2011). Negative Binomial Regression. Cambridge University Press, 2nd edition.

Koenker R.W. (2005). Quantile Regression. Cambridge University Press.

Lagani V., Kortas G. and Tsamardinos I. (2013). Biomarker signature identification in "omics" with multiclass outcome. Computational and Structural Biotechnology Journal, 6(7): 1-7.

Lagani V. and Tsamardinos I. (2010). Structure-based variable selection for survival data. Bioinformatics Journal 16(15): 1887-1894.

Lambert D. (1992). Zero-inflated Poisson regression, with an application to defects in manufacturing. Technometrics 34(1)1: 1-14.

Liang K.Y. and Zeger S.L. (1986). Longitudinal data analysis using generalized linear models. Biometrika, 73(1): 13-22.

Mardia K.V., Kent J.T. and Bibby J.M. (1979). Multivariate Analysis. Academic Press, New York, USA.

Maronna R.D. Yohai M.V. (2006). Robust Statistics, Theory and Methods. Wiley.

McCullagh P. and Nelder J.A. (1989). Generalized linear models. CRC press, USA, 2nd edition.

Paik M.C. (1988). Repeated measurement analysis for nonnormal data in small samples. Communications in Statistics-Simulation and Computation, 17(4): 1155-1171.

Pinheiro J., and D. Bates. Mixed-effects models in S and S-PLUS. Springer Science & Business Media, 2006.

Prentice R.L. and Zhao L.P. (1991). Estimating equations for parameters in means and covariances of multivariate discrete and continuous responses. Biometrics, 47(3): 825-839.

Presnell Brett, Morrison Scott P. and Littell Ramon C. (1998). Projected multivariate linear models for directional data. Journal of the American Statistical Association, 93(443): 1068-1077.

Scholz, F. W. (2001). Maximum likelihood estimation for type I censored Weibull data including covariates. ISSTECH-96-022, Boeing Information & Support Services.

Smith, R. L. (1991). Weibull regression models for reliability data. Reliability Engineering & System Safety, 34(1), 55-76.

Spirtes P., Glymour C. and Scheines R. (2001). Causation, Prediction, and Search. The MIT Press, Cambridge, MA, USA, 3nd edition.

Szekely G.J. and Rizzo, M.L. (2014). Partial distance correlation with methods for dissimilarities. The Annals of Statistics, 42(6): 2382–2412.

Szekely G.J. and Rizzo M.L. (2013). Energy statistics: A class of statistics based on distances. Journal of Statistical Planning and Inference 143(8): 1249–1272.

Therneau T.M., Grambsch P.M. and Pankratz V.S. (2003). Penalized Survival Models and Frailty, Journal of Computational and Graphical Statistics, 12(1):156-175.

Yan J. and Fine J. (2004). Estimating equations for association structures. Statistics in medicine, 23(6): 859-874

Ziegler A., Kastner C., Brunner D. and Blettner M. (2000). Familial associations of lipid proles: A generalised estimating equations approach. Statistics in medicine, 19(24): 3345-3357

Description

Conditional independence regression based tests.

Usage

cond.regs(target, dataset, xIndex, csIndex, test = NULL, wei = NULL, ncores = 1)

glmm.condregs(target, reps = NULL, id, dataset, xIndex, csIndex, test, wei = NULL, 
slopes = FALSE, ncores = 1)

gee.condregs(target, reps = NULL, id, dataset, xIndex, csIndex, test, wei = NULL, 
correl = "echangeable", se = "jack", ncores = 1)

Arguments

Details

This function is more as a help function for MMPC, but it can also be called directly by the user. In some, one should specify the regression model to use and the function will perform all simple regressions, i.e. all regression models between the target and each of the variables in the dataset. The function does not check for zero variance columns, only the "univregs" and related functions do.

If you want to use the GEE methodology, make sure you load the library geepack first.

Value

A list including:

Author(s)

Michail Tsagris

R implementation and documentation: Michail Tsagris [email protected]

References

Chen J. and Chen Z. (2008). Extended Bayesian information criteria for model selection with large model spaces. Biometrika, 95(3): 759-771.

Eugene Demidenko (2013). Mixed Models: Theory and Applications with R, 2nd Edition. New Jersey: Wiley & Sons.

McCullagh, Peter, and John A. Nelder. Generalized linear models. CRC press, USA, 2nd edition, 1989.

Presnell Brett, Morrison Scott P. and Littell Ramon C. (1998). Projected multivariate linear models for directional data. Journal of the American Statistical Association, 93(443): 1068-1077.

See Also

univregs, SES, MMPC, CondIndTests, reg.fit, ridge.reg

Examples

y <- rpois(100, 15)
x <- matrix( rnorm(100 * 10), ncol = 10)
a1 <- univregs(y, x, test = testIndPois)
a4 <- cond.regs(y, as.data.frame(x), xIndex = 1:9, csIndex = 10, test = testIndPois)

Description

The main task of this test is to provide a p-value PVALUE for the null hypothesis: feature 'X' is independent from 'TARGET' given a conditioning set CS. The pvalue is calculated by comparing a logistic model based on the conditioning set CS against a model whose regressor are both X and CS. The comparison is performed through a chi-square test with the aproprirate degrees of freedom on the difference between the deviances of the two models.

Usage

testIndLogistic(target, dataset, xIndex, csIndex, wei = NULL,
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL)

testIndMultinom(target, dataset, xIndex, csIndex, wei = NULL,
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL)

testIndOrdinal(target, dataset, xIndex, csIndex, wei = NULL,
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL)

testIndQBinom(target, dataset, xIndex, csIndex, wei = NULL,  
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL)

permLogistic(target, dataset, xIndex, csIndex, wei = NULL, 
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL, 
threshold = 0.05, R = 999)

permMultinom(target, dataset, xIndex, csIndex, wei = NULL, 
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL, 
threshold = 0.05, R = 999)

permOrdinal(target, dataset, xIndex, csIndex, wei = NULL, 
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL, 
threshold = 0.05, R = 999)

waldLogistic(target, dataset, xIndex, csIndex, wei = NULL, 
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL)

waldQBinom(target, dataset, xIndex, csIndex, wei = NULL, 
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL)

waldOrdinal(target, dataset, xIndex, csIndex, wei = NULL, 
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL)

Arguments

Details

The testIndQBinom does quasi binomial regression which is suitable for binary targets and proportions including 0 and 1.

If hash = TRUE, testIndLogistic requires the arguments 'stat_hash' and 'pvalue_hash' for the hash-based implementation of the statistic test. These hash Objects are produced or updated by each run of SES (if hash == TRUE) and they can be reused in order to speed up next runs of the current statistic test. If "SESoutput" is the output of a SES run, then these objects can be retrieved by SESoutput@hashObject$stat_hash and the SESoutput@hashObject$pvalue_hash.

Important: Use these arguments only with the same dataset that was used at initialization. For all the available conditional independence tests that are currently included on the package, please see "?CondIndTests".

The log-likelihood ratio test used in "testIndLogistic" requires the fitting of two models. The Wald test used in waldBinary" and "waldOrdinal" requires fitting of only one model, the full one. The significance of the variable is examined only. Only continuous (or binary) predictor variables are currently accepted in this test.

The testIndQBinom can also be used with percentages, see testIndBeta for example.

Value

A list including:

Note

This test uses the function multinom (package nnet) for multinomial logistic regression, the function clm (package ordinal) for ordinal logit regression and the function glm (package stats) for binomial regression.

Author(s)

Vincenzo Lagani and Ioannis Tsamardinos

R implementation and documentation: Vincenzo Lagani <[email protected]>, Giorgos Athineou <[email protected]> and Michail Tsagris [email protected]

References

Vincenzo Lagani, George Kortas and Ioannis Tsamardinos (2013), Biomarker signature identification in "omics" with multiclass outcome. Computational and Structural Biotechnology Journal, 6(7):1-7.

McCullagh, Peter, and John A. Nelder. Generalized linear models. CRC press, USA, 2nd edition, 1989.

See Also

SES, gSquare, CondIndTests

Examples

#simulate a dataset with categorical data 
dataset_m <- matrix( sample(c(0, 1, 2), 20 * 100, replace = TRUE), ncol = 20)
#initialize categorical target
target_m <- dataset_m[, 20]
#remove target from the dataset
dataset_m <- dataset_m[, -20]

#run the conditional independence test for the nominal class variable
testIndMultinom(target_m, dataset_m, xIndex = 14, csIndex = c(1, 2) )

########################################################################
#run the conditional independence test for the ordinal class variable
testIndOrdinal( factor(target_m, ordered = TRUE), dataset_m, 
xIndex = 14, csIndex = c(1, 2) )

#run the SES algorithm using the testIndLogistic conditional independence test 
#for the ordinal class variable
sesObject <- SES(factor(target_m, ordered=TRUE), dataset_m, max_k = 3, 
threshold = 0.05, test = "testIndOrdinal")

########################################################################
#simulate a dataset with binary data
dataset_b <- matrix(sample(c(0,1), 50 * 20, replace = TRUE), ncol = 20)
#initialize binary target
target_b <- dataset_b[, 20]
#remove target from the dataset
dataset_b <- dataset_b[, -20]

#run the conditional independence test for the binary class variable
testIndLogistic( target_b, dataset_b, xIndex = 14, csIndex = c(1, 2) )

#run the MMPC algorithm using the testIndLogistic conditional independence test
#for the binary class variable
mmpcObject <- MMPC(target_b, dataset_b, max_k = 3, threshold = 0.05, 
test = "testIndLogistic")

Description

The main task of this test is to provide a p-value PVALUE for the null hypothesis: feature 'X' is independent from 'TARGET' given a conditioning set CS. The pvalue is calculated by comparing a conditional logistic regression model based on the conditioning set CS against a model whose regressors are both X and CS. The comparison is performed through a chi-square test with the appropriate degrees of freedom on the difference between the deviances of the two models. This is suitable for a case control design

Usage

testIndClogit(target, dataset, xIndex, csIndex, wei = NULL, 
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL)

permClogit(target, dataset, xIndex, csIndex, wei = NULL,  
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL, 
threshold = 0.05, R = 999)

Arguments

Details

If hash = TRUE, testIndClogit requires the arguments 'stat_hash' and 'pvalue_hash' for the hash-based implementation of the statistic test. These hash Objects are produced or updated by each run of SES (if hash == TRUE) and they can be reused in order to speed up next runs of the current statistic test. If "SESoutput" is the output of a SES run, then these objects can be retrieved by SESoutput@hashObject$stat_hash and the SESoutput@hashObject$pvalue_hash.

Important: Use these arguments only with the same dataset that was used at initialization. For all the available conditional independence tests that are currently included on the package, please see "?CondIndTests".

This is for case control studies. The log-likelihood for a conditional logistic regression model equals the log-likelihood from a Cox model with a particular data structure. When a well tested Cox model routine is available many packages use this "trick" rather than writing a new software routine from scratch, and this is what the "clogit" function in the "survival" package does. In detail, a stratified Cox model with each case/control group assigned to its own stratum, time set to a constant, status of 1=case 0=control, and using the exact partial likelihood has the same likelihood formula as a conditional logistic regression. The "clogit" routine creates the necessary dummy variable of times (all 1) and the strata, then calls the function "coxph".

Note that this function is a bit sensitive and prone to breaking down for some reason which I have not yet figured out. In case of singular design matrix it will work, it identifies the problem and handles it internally like most regression functions do. However, problems can still occur.

Value

A list including:

Author(s)

Vincenzo Lagani, Ioannis Tsamardinos, Giorgos Athineou and Michail Tsagris

R implementation and documentation: Giorgos Athineou <[email protected]>, Vincenzo Lagani <[email protected]> and Michail Tsagris [email protected]

References

Michell H. Gail, Jay H. Lubin and Lawrence V. Rubinstein (1980). Likelihood calculations for matched case-control studies and survival studies with tied death times. Biometrika 68:703-707.

See Also

SES, testIndLogistic, censIndCR, censIndWR

Examples

#simulate a dataset with continuous data
dataset <- matrix( rnorm(100 * 7), nrow = 100 ) 
#the target feature is the last column of the dataset as a vector
case <- rbinom(100, 1, 0.6)
ina <- which(case == 1)
ina <- sample(ina, 50)
case[-ina] = 0 
id <- rep(1:50, 2)
target <- cbind(case, id)

results <- testIndClogit(target, dataset, xIndex = 4, csIndex = 1)
results

#run the SES algorithm using the testIndClogit conditional independence test
a <- MMPC(target, dataset, max_k = 3, threshold = 0.01, test = "testIndClogit")

Description

The main task of this test is to provide a p-value PVALUE for the null hypothesis: feature 'X' is independent from 'TARGET' given a conditioning set CS. The pvalue is calculated by comparing a Beta regression model based on the conditioning set CS against a model whose regressor are both X and CS. The comparison is performed through a log-likelihood ratio test using circular regression with 2 degrees of freedom.

Usage

testIndSPML(target, dataset, xIndex, csIndex, wei = NULL, univariateModels = NULL, 
hash = FALSE, stat_hash = NULL, pvalue_hash = NULL)

Arguments

Details

If hash = TRUE, testIndSPML requires the arguments 'stat_hash' and 'pvalue_hash' for the hash-based implementation of the statistic test. These hash Objects are produced or updated by each run of SES (if hash == TRUE) and they can be reused in order to speed up next runs of the current statistic test. If "SESoutput" is the output of a SES run, then these objects can be retrieved by SESoutput@hashObject$stat_hash and the SESoutput@hashObject$pvalue_hash.

Important: Use these arguments only with the same dataset that was used at initialization. For all the available conditional independence tests that are currently included on the package, please see "?CondIndTests".

The log-likelihood ratio test used in "testIndSPML" requires the fitting of two models. The significance of the variable is examined and Only continuous (or binary) predictor variables are currently accepted in this test.

Value

A list including:

Author(s)

Michail Tsagris

R implementation and documentation: Michail Tsagris [email protected]

References

Presnell Brett, Morrison Scott P. and Littell Ramon C. (1998). Projected multivariate linear models for directional data. Journal of the American Statistical Association, 93(443): 1068-1077.

See Also

univregs, SES, testIndReg, CondIndTests

Examples

y <- runif(100, - pi, pi)  ## suppose these are radians
x <- matrix( rnorm(100 * 3), ncol = 3)
testIndSPML(y1, x, csIndex = 1, xIndex = 2)
## alternatively
y1 <- cbind(cos(y), sin(y)) ## a matrix with two columns
testIndSPML(y1, x, csIndex = 1, xIndex = 2)

Description

The main task of this test is to provide a p-value PVALUE for the null hypothesis: feature 'X' is independent from 'TARGET' given a conditioning set CS. The pvalue is calculated by comparing a linear model based on the conditioning set CS against a model with both X and CS. The comparison is performed through an F test the appropriate degrees of freedom on the difference between the deviances of the two models. This test accepts a longitudinal target and longitudinal, categorical, continuous or mixed data as predictor variables.

Usage

testIndGEEReg(target, reps = NULL, group, dataset, xIndex, csIndex,  wei =  NULL, 
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL, 
correl = "exchangeable", se = "jack")

testIndGEELogistic(target, reps = NULL, group, dataset, xIndex, csIndex,  wei =  NULL, 
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL, 
correl = "exchangeable", se = "jack")

testIndGEEPois(target, reps = NULL, group, dataset, xIndex, csIndex,  wei =  NULL, 
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL, 
correl = "exchangeable", se = "jack")

testIndGEEGamma(target, reps = NULL, group, dataset, xIndex, csIndex,  wei =  NULL, 
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL, 
correl = "exchangeable", se = "jack")

testIndGEENormLog(target, reps = NULL, group, dataset, xIndex, csIndex,  wei =  NULL, 
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL, 
correl = "exchangeable", se = "jack")

Arguments

Details

If hash = TRUE, testIndGEE requires the arguments 'stat_hash' and 'pvalue_hash' for the hash-based implementation of the statistic test. These hash Objects are produced or updated by each run of SES (if hash == TRUE) and they can be reused in order to speed up next runs of the current statistic test. If "SESoutput" is the output of a SES.temp run, then these objects can be retrieved by SESoutput@hashObject$stat_hash and the SESoutput@hashObject$pvalue_hash.

Important: Use these arguments only with the same dataset that was used at initialization. For all the available conditional independence tests that are currently included on the package, please see "?CondIndTests".

This test is for longitudinal and clustered data. Bear in mind that the time effect, for the longitudinal data case, is linear. It could be of higer order as well, but this would be a hyper-parameter, increasing the complexity of the models to be tested.

Make sure you load the library geepack first.

Value

A list including:

Author(s)

Michail Tsagris

R implementation and documentation: Michail Tsagris [email protected]

References

Liang K.Y. and Zeger S.L. (1986). Longitudinal data analysis using generalized linear models. Biometrika, 73(1): 13-22.

Prentice R.L. and Zhao L.P. (1991). Estimating equations for parameters in means and covariances of multivariate discrete and continuous responses. Biometrics, 47(3): 825-839.

Heagerty P.J. and Zeger S.L. (1996) Marginal regression models for clustered ordinal measurements. Journal of the American Statistical Association, 91(435): 1024-1036.

Paik M.C. (1988). Repeated measurement analysis for nonnormal data in small samples. Communications in Statistics-Simulation and Computation, 17(4): 1155-1171.

Ziegler A., Kastner C., Brunner D. and Blettner M. (2000). Familial associations of lipid profiles: A generalised estimating equations approach. Statistics in medicine, 19(24): 3345-3357

Yan J. and Fine J. (2004). Estimating equations for association structures. Statistics in medicine, 23(6): 859-874.

Eugene Demidenko (2013). Mixed Models: Theory and Applications with R, 2nd Edition. New Jersey: Wiley & Sons.

Tsagris, M., Lagani, V., & Tsamardinos, I. (2018). Feature selection for high-dimensional glmm data. BMC bioinformatics, 19(1), 17.

See Also

SES.glmm, MMPC.glmm, CondIndTests

Examples

library("geepack", quietly = TRUE)
y <- rnorm(150)
x <- matrix(rnorm(150 * 5), ncol = 5)
id <- sample(1:20, 150, replace = TRUE)
testIndGEEReg(y, group = id, dataset = x, xIndex = 1, csIndex = 3)

Description

The main task of this test is to provide a p-value PVALUE for the null hypothesis: feature 'X' is independent from 'TARGET' given a conditioning set CS. The pvalue is calculated by comparing a linear model based on the conditioning set CS against a model with both X and CS. The comparison is performed through an F test the appropriate degrees of freedom on the difference between the deviances of the two models. This test accepts a longitudinal target and longitudinal, categorical, continuous or mixed data as predictor variables.

Usage

testIndGLMMReg(target, reps = NULL, group, dataset, xIndex, csIndex, wei = NULL,
univariateModels = NULL, hash = FALSE, stat_hash = NULL, 
pvalue_hash = NULL, slopes = FALSE)

testIndGLMMLogistic(target, reps = NULL, group, dataset, xIndex, csIndex, wei = NULL,
univariateModels = NULL, hash = FALSE, stat_hash = NULL, 
pvalue_hash = NULL, slopes = FALSE)

testIndGLMMPois(target, reps = NULL, group, dataset, xIndex, csIndex, wei = NULL,
univariateModels = NULL, hash = FALSE, stat_hash = NULL, 
pvalue_hash = NULL, slopes = FALSE)

testIndGLMMNB(target, reps = NULL, group, dataset, xIndex, csIndex, wei = NULL,
univariateModels = NULL, hash = FALSE, stat_hash = NULL, 
pvalue_hash = NULL, slopes = FALSE)

testIndGLMMGamma(target, reps = NULL, group, dataset, xIndex, csIndex, wei = NULL,
univariateModels = NULL, hash = FALSE, stat_hash = NULL, 
pvalue_hash = NULL, slopes = FALSE)

testIndGLMMNormLog(target, reps = NULL, group, dataset, xIndex, csIndex, wei = NULL,
univariateModels = NULL, hash = FALSE, stat_hash = NULL, 
pvalue_hash = NULL, slopes = FALSE)

testIndGLMMOrdinal(target, reps = NULL, group, dataset, xIndex, csIndex, wei = NULL,
univariateModels = NULL, hash = FALSE, stat_hash = NULL, 
pvalue_hash = NULL, slopes = FALSE)

testIndGLMMCR(target, reps = NULL, group, dataset, xIndex, csIndex, wei = NULL,
univariateModels = NULL, hash = FALSE, stat_hash = NULL, 
pvalue_hash = NULL, slopes = FALSE)

testIndLMM(target, reps = NULL, group, dataset, xIndex, csIndex, wei = NULL,
univariateModels = NULL, hash = FALSE, stat_hash = NULL, 
pvalue_hash = NULL, slopes = FALSE)

Arguments

Details

If hash = TRUE, testIndGLMM requires the arguments 'stat_hash' and 'pvalue_hash' for the hash-based implementation of the statistic test. These hash Objects are produced or updated by each run of SES (if hash == TRUE) and they can be reused in order to speed up next runs of the current statistic test. If "SESoutput" is the output of a SES.temp run, then these objects can be retrieved by SESoutput@hashObject$stat_hash and the SESoutput@hashObject$pvalue_hash.

Important: Use these arguments only with the same dataset that was used at initialization. For all the available conditional independence tests that are currently included on the package, please see "?CondIndTests".

This test is for longitudinal and clustered data. Bear in mind that the time effect, for the longitudinal data case, is linear. It could be of higer order as well, but this would be a hyper-parameter, increasing the complexity of the models to be tested.

The testIndLMM is used for linear mixed models with no weights, no slopes and no reps. This is a random intercepts model only. The advantage of this function is that it is tens of times faster than testIndGLMM.

Value

A list including:

Author(s)

Vincenzo Lagani, Ioannis Tsamardinos, Michail Tsagris and Giorgos Athineou

R implementation and documentation: Giorgos Athineou <[email protected]>, Vincenzo Lagani <[email protected]> and Michail Tsagris [email protected]

References

Eugene Demidenko (2013). Mixed Models: Theory and Applications with R, 2nd Edition. New Jersey: Wiley & Sons.

Jose Pinheiro Jose and Douglas Bates. Mixed-effects models in S and S-PLUS. Springer Science & Business Media, 2006.

See Also

SES.glmm, MMPC.glmm, CondIndTests

Examples

y <- rnorm(150)
x <- matrix(rnorm(150 * 5), ncol = 5)
id <- sample(1:20, 150, replace = TRUE)
testIndGLMMReg(y, group = id, dataset = x, xIndex = 1, csIndex = 3)
testIndLMM(y, group = id, dataset = x, xIndex = 1, csIndex = 3)

Description

The main task of this test is to provide a p-value PVALUE for the null hypothesis: feature 'X' is independent from 'TARGET' given a conditioning set CS. The pvalue is calculated by comparing a Beta regression model based on the conditioning set CS against a model whose regressor are both X and CS. The comparison is performed through a chi-square test with the appropriate degrees of freedom on the difference between the deviances of the two models.

Usage

testIndBeta(target, dataset, xIndex, csIndex, wei = NULL,  
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL)

permBeta(target, dataset, xIndex, csIndex, wei = NULL,  
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL, 
threshold = 0.05, R = 999)

waldBeta(target, dataset, xIndex, csIndex, wei = NULL, 
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL)

Arguments

Details

If hash = TRUE, testIndBeta requires the arguments 'stat_hash' and 'pvalue_hash' for the hash-based implementation of the statistic test. These hash Objects are produced or updated by each run of SES (if hash == TRUE) and they can be reused in order to speed up next runs of the current statistic test. If "SESoutput" is the output of a SES run, then these objects can be retrieved by SESoutput@hashObject$stat_hash and the SESoutput@hashObject$pvalue_hash.

Important: Use these arguments only with the same dataset that was used at initialization. For all the available conditional independence tests that are currently included on the package, please see "?CondIndTests". An alternative regression to this is "testIndReg" and "testIndRQ" .In these two latter cases, the logit transformation is first applied to the target variable.

The log-likelihood ratio test used in "testIndBeta" requires the fitting of two models. The Wald test used in "waldBeta" requires fitting of only one model, the full one. The significance of the variable is examined only. Only continuous (or binary) predictor variables are currently accepted in this test.

The testIndQBinom is another alternative to use.

Value

A list including:

Author(s)

Vincenzo Lagani, Ioannis Tsamardinos, Michail Tsagris and Giorgos Athineou

R implementation and documentation: Giorgos Athineou <[email protected]>, Vincenzo Lagani <[email protected]> and Michail Tsagris [email protected]

References

Ferrari S.L.P. and Cribari-Neto F. (2004). Beta Regression for Modelling Rates and Proportions. Journal of Applied Statistics, 31(7): 799-815.

See Also

testIndQBinom, SES, testIndReg, testIndRQ, testIndFisher, CondIndTests

Examples

#simulate a dataset with continuous data
dataset <- matrix(runif(100 * 10, 1, 1000), ncol = 10 ) 
#the target feature is the last column of the dataset as a vector
target <- dataset[, 10]
dataset <- dataset[, -10]
target <- target / (max(target) + 2 )

testIndBeta(target, dataset, xIndex = 4, csIndex = 9)

#run the MMPC algorithm using the testIndBeta conditional independence test
mmpcObject <- MMPC(target, dataset, max_k = 3, threshold = 0.05, test = "testIndBeta")

Description

The main task of this test is to provide a p-value PVALUE for the null hypothesis: feature 'X' is independent from 'TARGET' given a conditioning set CS. The pvalue is calculated by comparing a logistic model based on the conditioning set CS against a model whose regressor are both X and CS. The comparison is performed through a chi-square test with the aproprirate degrees of freedom on the difference between the deviances of the two models.

Usage

testIndTimeLogistic(target, dataset, xIndex, csIndex, wei = NULL,
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL)

testIndTimeMultinom(target, dataset, xIndex, csIndex, wei = NULL,
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL)

Arguments

Details

This conditional independence test is devised for the static-longitudinal scenario of Tsagris, Lagani and Tsamardinos (2018). The idea is that you have many features of longitudinal data for many subjects. For each subject you have calculated the coefficients of a simple linear regression over time and this is repeated for each feature. In the end, assuming p features, you have p constants and p slopes for each subject, each constant and slope refers to a feature for a subject.

Value

A list including:

Note

This test uses the function multinom (package nnet) for multinomial logistic regression, the function clm (package ordinal) for ordinal logit regression and the function glm (package stats) for binomial regression.

Author(s)

Michail Tsagris

R implementation and documentation: Michail Tsagris [email protected]

References

Tsagris M., Lagani V., & Tsamardinos I. (2018). Feature selection for high-dimensional temporal data. BMC bioinformatics, 19(1), 17.

Vincenzo Lagani, George Kortas and Ioannis Tsamardinos (2013), Biomarker signature identification in "omics" with multiclass outcome. Computational and Structural Biotechnology Journal, 6(7):1-7.

McCullagh, Peter, and John A. Nelder. Generalized linear models. CRC press, USA, 2nd edition, 1989.

See Also

SES, gSquare, CondIndTests

Examples

## assume these are longitudinal data, each column is a variable (or feature)
x <- matrix( rnorm(400 * 50), ncol = 50 ) 
id <- rep(1:80, each = 5)  ## 80 subjects
reps <- rep( seq(4, 12, by = 2), 80)  ## 5 time points for each subject
dataset <- group.mvbetas(x, id, reps)
## these are the regression coefficients of the first subject's values on the 
## reps (which is assumed to be time in this example)
target <- rbinom(80, 1, 0.5)
testIndTimeLogistic(target, dataset, xIndex = 1, csIndex = 0)
testIndTimeLogistic(target, dataset, xIndex = 1, csIndex = 2)

Description

Many conditional independence tests counting the number of times a possible collider d-separates two nodes .

Usage

condis(ind1, ind2, cs1, cs2, Var, dat, type = "pearson", rob = FALSE, max_k = 2, R = 1 )

Arguments

Details

This is to be used in the conservative version of Rule 0 of the Pc algorithm. When one wants to know whether a variable is a possible collider, Ramsey, Spirtes and Zhang (2005) propose to perform the following action. For every unshilded triple (X, Y, Z) check all subsets of X's possible parents and of Z's poissible parents. a) If Y is NOT in any such set conditional on which, X and Z are independent orient X - Y - Z as X -> Y <- Z. b) If Y is in ALL sets conditional on which, X and Z are independent, leave X - Y - Z as it is, i.e. a non-collider. c) Mark the triple X - Y - Z as "unfaitfull" otherwise. This modification leads to the so called conservative PC (CPC) algorithm.

A few years later, Colombo and Maathuis (2014) suggested a modification of the previous action, called the majority rule. If Y is less than 50% of thet sets that render X and Z independent orient X - Y - Z as X -> Y <- Z. If Y is found in exactly 50% of sets that render X and Z independent, this triple is marked as "ambiuous". This modification leads the so called majority rule PC (MPC) algorithm.

This function we have implemented here, does exactly this. It applies tests to many subsets and returns a matrix with two columns. The first one contains 0 or 1 and the second is the p-value. A value of 0 indicates absenece of the possible collider from the set that produced that p-value, whereas a value of 1 indicates its presence in the set.

This way, we can measure the proportion of times the possible collider Y was in a subset that rendered X and Z independent.

Value

A matrix with two columns. The second one is the logarithm of the p-value. The first one contains 0s and 1s. The value of 0 means that the candidate collider was not in that set which produced the relevant p-value, whereas a value of 1 indicates that it was a member of that conditioning set.

Author(s)

Michail Tsagris

R implementation and documentation: Michail Tsagris [email protected]

References

Ramsey, J., Zhang, J., Spirtes, P., 2006. Adjacency-faithfulness and conservative causal inference. Proceedings of the 22nd Annual Conference on Uncertainty in Artificial Intelligence (UAI 2006). https://arxiv.org/ftp/arxiv/papers/1206/1206.6843.pdf

Colombo, Diego, and Marloes H. Maathuis (2014). Order-independent constraint-based causal structure learning. The Journal of Machine Learning Research 15(1): 3741–3782.

See Also

SES, MMPC, testIndLogistic

Examples

x <- rdag2(1000, p = 10, nei = 5)
G <- x$G
dat <- x$x
cs1 <- which(G[6, ] > 0  |  G[, 6] > 0)
cs2 <- which(G[7, ] > 0  |  G[, 7] > 0)
cs1 <- setdiff( cs1, c(7, 3) )
cs2 <- setdiff( cs2, c(6, 3) ) 
condis(6, 7, cs1, cs2, 3, dat, type = "pearson", rob = FALSE, max_k = 3, R = 1 )

Description

The main task of this test is to provide a p-value PVALUE for the null hypothesis: feature 'X' is independent from 'TARGET' given a conditioning set CS. The pvalue is calculated by comparing a linear regression model based on the conditioning set CS against a model whose regressor are both X and CS. The comparison is performed through an F test the appropriate degrees of freedom on the difference between the deviances of the two models.

Usage

testIndReg(target, dataset, xIndex, csIndex, wei = NULL, 
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL)

testIndRQ(target, dataset, xIndex, csIndex, wei = NULL,
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL)

testIndMVreg(target, dataset, xIndex, csIndex, wei = NULL,
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL)

testIndMMReg(target, dataset, xIndex, csIndex, wei = NULL,  
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL)

waldMMReg(target, dataset, xIndex, csIndex, wei = NULL, 
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL) 

permReg(target, dataset, xIndex, csIndex, wei = NULL, 
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL, 
threshold = 0.05, R = 999)

permMMReg(target, dataset, xIndex, csIndex, wei = NULL, 
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL, 
threshold = 0.05, R = 999)

permRQ(target, dataset, xIndex, csIndex, wei = NULL, 
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL, 
threshold = 0.05, R = 999)

permMVreg(target, dataset, xIndex, csIndex, wei = NULL,
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL, 
threshold = 0.05, R = 999)

Arguments

Details

If hash = TRUE, all three tests require the arguments 'stat_hash' and 'pvalue_hash' for the hash-based implementation of the statistic test. These hash Objects are produced or updated by each run of SES (if hash == TRUE) and they can be reused in order to speed up next runs of the current statistic test. If "SESoutput" is the output of a SES run, then these objects can be retrieved by SESoutput@hashObject$stat_hash and the SESoutput@hashObject$pvalue_hash.

Important: Use these arguments only with the same dataset that was used at initialization.

TestIndReg offers linear regression.

testIndMMReg offers robust linear MM estimation regression.

TestIndRQ offers quantile (median) regression as a robust alternative to linear regression.

In both cases, if the dependent variable consists of proportions (values between 0 and 1) the logit transformation is applied and the tests are applied then.

testIndMVreg is for multivariate continuous response variables. Compositional data are positive multivariate data and each vector (observation) sums to the same constant, usually taken 1 for convenience. A check is performed and if such data are found, the additive log-ratio (multivariate logit) transformation (Aitchison, 1986) is applied beforehand. Zeros are not allowed.

For all the available conditional independence tests that are currently included on the package, please see "?CondIndTests".

The Wald test used in "waldMMReg" requires fitting of only one model, the full one. The significance of the variable is examined only. Only continuous (or binary) predictor variables are currently accepted in this test.

Value

A list including:

Author(s)

Vincenzo Lagani, Ioannis Tsamardinos, Michail Tsagris and Giorgos Athineou

R implementation and documentation: Giorgos Athineou <[email protected]>, Vincenzo Lagani <[email protected]> and Michail Tsagris [email protected]

References

Draper, N.R. and Smith H. (1988). Applied regression analysis. New York, Wiley, 3rd edition.

Hampel F. R., Ronchetti E. M., Rousseeuw P. J., and Stahel W. A. (1986). Robust statistics: the approach based on influence functions. John Wiley & Sons.

Koenker R.W. (2005). Quantile regression. New York, Cambridge University Press.

Sadovski A. (1974). L1-norm fit of a straight line. Applied Statistics, 23(2):244-248.

Yohai, V. J. (1987). High breakdown-point and high efficiency robust estimates for regression. The Annals of Statistics, 15(2): 642-656.

Mardia, Kanti, John T. Kent and John M. Bibby. Multivariate analysis. Academic press, 1979.

John Aitchison. The Statistical Analysis of Compositional Data, Chapman & Hall; reprinted in 2003, with additional material, by The Blackburn Press.

See Also

testIndRQ, testIndFisher, testIndSpearman, CondIndTests

Examples

#simulate a dataset with continuous data
dataset <- matrix(runif(100 * 50, 1, 100), ncol = 50 )
#the target feature is the last column of the dataset as a vector
target <- dataset[, 50]
dataset <- dataset[, -50]

testIndReg(target, dataset, xIndex = 44, csIndex = 10)
testIndMMReg(target, dataset, xIndex = 44, csIndex = 10)
testIndRQ(target, dataset, xIndex = 44, csIndex = 10)
testIndIGreg(target, dataset, xIndex = 44, csIndex = 10)

#run the MMPC algorithm using the testIndReg conditional independence test
m1 <- MMPC(target, dataset, max_k = 3, threshold = 0.05, test = "testIndReg")
m2 <- MMPC(target, dataset, max_k = 3, threshold = 0.05, test = "testIndRQ")

Description

The main task of this test is to provide a p-value PVALUE for the null hypothesis: feature 'X' is independent from 'TARGET' given a conditioning set CS. The pvalue is calculated by comparing a Poisson regression model based on the conditioning set CS against a model whose regressor are both X and CS. The comparison is performed through a chi-square test with the appropriate degrees of freedom on the difference between the deviances of the two models. The models supported here are poisson, zero inlftaed poisson and negative binomial.

Usage

testIndPois(target, dataset, xIndex, csIndex, wei = NULL,  
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL)

testIndNB(target, dataset, xIndex, csIndex, wei = NULL, 
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL)

testIndZIP(target, dataset, xIndex, csIndex, wei = NULL, 
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL)

testIndQPois(target, dataset, xIndex, csIndex, wei = NULL,
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL)

permPois(target, dataset, xIndex, csIndex, wei = NULL,
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL, 
threshold = 0.05, R = 999)

permNB(target, dataset, xIndex, csIndex, wei = NULL,  
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL, 
threshold = 0.05, R = 999)

permZIP(target, dataset, xIndex, csIndex, wei = NULL,  
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL, 
threshold = 0.05, R = 999)

waldPois(target, dataset, xIndex, csIndex, wei = NULL,  
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL) 

waldNB(target, dataset, xIndex, csIndex, wei = NULL, 
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL) 

waldZIP(target, dataset, xIndex, csIndex, wei = NULL,  
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL)

Arguments

Details

If hash = TRUE, all three tests require the arguments 'stat_hash' and 'pvalue_hash' for the hash-based implementation of the statistic test. These hash Objects are produced or updated by each run of SES (if hash == TRUE) and they can be reused in order to speed up next runs of the current statistic test. If "SESoutput" is the output of a SES run, then these objects can be retrieved by SESoutput@hashObject$stat_hash and the SESoutput@hashObject$pvalue_hash.

Important: Use these arguments only with the same dataset that was used at initialization. For all the available conditional independence tests that are currently included on the package, please see "?CondIndTests".

If you have overdispersion, the variance is higher than the mean, a negative binomial is to be used. If you have more zeros than expected under a Poisson model, not overdispersion, then zero inlfated Poisson is to be used. Bear in mind that if you have a small number of zeros, there is no reason to use this model. If for example you have count data but no, or 1 zeros, this will not work.

The log-likelihood ratio test used in "testIndPois", "testIndNB" and "testIndZIP" requires the fitting of two models. The Wald test used in "waldPois", "waldNB" and "waldZIP" requires fitting of only one model, the full one. The significance of the variable is examined only. Only continuous (or binary) predictor variables are currently accepted in this test.

The testIndQPois does quasi Poisson regression. The benefit of this regression is that it works for over and under dispersed data. Negative Binomial works for over dispersed data, but not for under dispersed data. In addition it is fast.

Value

A list including:

Author(s)

Vincenzo Lagani, Ioannis Tsamardinos, Michail Tsagris and Giorgos Athineou

R implementation and documentation: Giorgos Athineou <[email protected]>, Vincenzo Lagani <[email protected]> and Michail Tsagris [email protected]

References

McCullagh P., and Nelder J.A. (1989). Generalized linear models. CRC press, USA, 2nd edition.

Lambert D. (1992). Zero-inflated Poisson regression, with an application to defects in manufacturing. Technometrics, 34(1):1-14.

Joseph M.H. (2011). Negative Binomial Regression. Cambridge University Press, 2nd edition.

See Also

testIndReg, testIndNB, testIndZIP, gSquare, CondIndTests

Examples

#simulate a dataset with continuous data
dataset <- matrix(runif(100 * 20, 1, 50), ncol = 20 ) 
#the target feature is the last column of the dataset as a vector
target <- rpois(100, 10)
results <- testIndPois(target, dataset, xIndex = 14, csIndex = 10)
results

#run the SES algorithm using the testIndPois conditional independence test
m1 <- MMPC(target, dataset, max_k = 3, threshold = 0.05, test = "testIndPois");
m2 <- MMPC(target, dataset, max_k = 3, threshold = 0.05, test = "testIndNB");

Description

The main task of this test is to provide a p-value PVALUE for the null hypothesis: feature 'X' is independent from 'TARGET' given a conditioning set CS. This test can based on the Cox (semi-parametric) regression or on the Weibull (parametric) regression.

Usage

testIndTobit(target, dataset, xIndex, csIndex, wei = NULL, 
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL)

waldTobit(target, dataset, xIndex, csIndex, wei = NULL,  
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL)

permTobit(target, dataset, xIndex, csIndex, wei = NULL, 
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL, 
threshold = 0.05, R = 999)

Arguments

Details

Tobit regression is performed. The implied model is Gaussian with left censored data.

If hash = TRUE, censIndCR, censIndWR and censIndER require the arguments 'stat_hash' and 'pvalue_hash' for the hash-based implementation of the statistic test. These hash Objects are produced or updated by each run of SES (if hash == TRUE) and they can be reused in order to speed up next runs of the current statistic test. If "SESoutput" is the output of a SES run, then these objects can be retrieved by SESoutput@hashObject$stat_hash and the SESoutput@hashObject$pvalue_hash.

Important: Use these arguments only with the same dataset that was used at initialization.

For all the available conditional independence tests that are currently included on the package, please see "?CondIndTests".

The log-likelihood ratio test used in "testIndTobit" requires the fitting of two models. The Wald test used in "waldTobit", requires fitting of only one model, the full one. The significance of the variable is examined only. Only continuous (or binary) predictor variables are currently accepted in this test.

Value

A list including:

Note

This test uses the functions "survreg" and Surv of the package survival and the function anova (analysis of variance) of the package stats.

Author(s)

R implementation and documentation: Michail Tsagris [email protected]

References

Tobin James (1958). Estimation of relationships for limited dependent variables. Econometrica. 26(1): 24-36.

See Also

SES, censIndWR, testIndFisher, gSquare, testIndLogistic, Surv, anova, CondIndTests

Examples

require(survival, quietly = TRUE)

x <- matrix( rnorm(100 * 30), ncol = 30)
y <- x[, 1] - x[, 2] + rnorm(100, 5)
y[y < 0 ] <- 0
y <- survival::Surv(y, y>0, type = 'left') 
  
#run the censIndCR   conditional independence test
testIndTobit(y, x, xIndex = 12, csIndex = c(5, 7, 4) )
waldTobit(y, x, xIndex = 12, csIndex = c(5, 7, 4) )
permTobit(y, x, xIndex = 12, csIndex = c(5, 7, 4), R = 499 )
  
#run the SES algorithm using the censIndCR conditional independence
#test for the survival class variable
a <- MMPC(y, x, max_k = 2, threshold = 0.05, test = "testIndTobit")

Description

The main task of this test is to provide a p-value PVALUE for the null hypothesis: feature 'X' is independent from 'TARGET' given a conditioning set CS. The pvalue is calculated by comparing a Poisson regression model based on the conditioning set CS against a model whose regressor are both X and CS. The comparison is performed through a chi-square test with the appropriate degrees of freedom on the difference between the deviances of the two models. The models supported here are poisson, zero inlftaed poisson and negative binomial.

Usage

testIndGamma(target, dataset, xIndex, csIndex, wei = NULL, 
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL)
  
testIndNormLog(target, dataset, xIndex, csIndex, wei = NULL,  
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL)

testIndIGreg(target, dataset, xIndex, csIndex, wei = NULL,
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL)

permGamma(target, dataset, xIndex, csIndex, wei = NULL, 
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL, 
threshold = 0.05, R = 999)

permNormLog(target, dataset, xIndex, csIndex, wei = NULL, 
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL, 
threshold = 0.05, R = 999)

permIGreg(target, dataset, xIndex, csIndex, wei = NULL,  
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL, 
threshold = 0.05, R = 999)

waldGamma(target, dataset, xIndex, csIndex, wei = NULL, 
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL) 

waldNormLog(target, dataset, xIndex, csIndex, wei = NULL, 
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL) 

waldIGreg(target, dataset, xIndex, csIndex, wei = NULL,  
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL)

Arguments

Details

If hash = TRUE, all three tests require the arguments 'stat_hash' and 'pvalue_hash' for the hash-based implementation of the statistic test. These hash Objects are produced or updated by each run of SES (if hash = TRUE) and they can be reused in order to speed up next runs of the current statistic test. If "SESoutput" is the output of a SES run, then these objects can be retrieved by SESoutput@hashObject$stat_hash and the SESoutput@hashObject$pvalue_hash.

Important: Use these arguments only with the same dataset that was used at initialization. For all the available conditional independence tests that are currently included on the package, please see "?CondIndTests".

For the testIndGamma and testIndNormLog the F test is used and not the log-likelihood ratio test because both of these regression models have a nuisance parameter. The testIndNormLog can be seen as a non linear Gaussian model where the conditional mean is related with the covariate(s) via an exponential function.

TestIndIGreg fits an inverse gaussian distribution with a log link. The testIndIGreg has some problems due to problems in R's implementation of the inverse gaussian regression with a log link.

Value

A list including:

Author(s)

Michail Tsagris

R implementation and documentation: Michail Tsagris [email protected]

References

McCullagh P., and Nelder J.A. (1989). Generalized linear models. CRC press, USA, 2nd edition.

See Also

testIndReg, testIndNB, testIndZIP, gSquare, CondIndTests

Examples

#simulate a dataset with continuous data
dataset <- matrix( rnorm(200 * 20, 1, 5), ncol = 20 ) 
#the target feature is the last column of the dataset as a vector
target <- rgamma(200, 1, 3)
testIndGamma(target, dataset, xIndex = 14, csIndex = 10)
testIndNormLog(target, dataset, xIndex = 14, csIndex = 10)
#run the MMPC algorithm using the testIndPois conditional independence test
m1 <- MMPC(target, dataset, max_k = 3, threshold = 0.05, test = "testIndGamma");
m2 <- MMPC(target, dataset, max_k = 3, threshold = 0.05, test = "testIndNormLog");

Description

The main task of this test is to provide a p-value PVALUE for the null hypothesis: feature 'X' is independent from 'TARGET' given a conditioning set CS. The pvalue is calculated by comparing a binomial logistic regression model based on the conditioning set CS against a model whose regressor are both X and CS. The comparison is performed through a chi-square test with the appropriate degrees of freedom on the difference between the deviances of the two models.

Usage

testIndBinom(target, dataset, xIndex, csIndex, wei = NULL, 
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL)

permBinom(target, dataset, xIndex, csIndex, wei = NULL, 
univariateModels = NULL, hash = FALSE, stat_hash = NULL, pvalue_hash = NULL, 
threshold = 0.05, R = 999)

waldBinom(target, dataset, xIndex, csIndex, wei = NULL,  
univariateModels = NULL , hash = FALSE, stat_hash = NULL, pvalue_hash = NULL)

Arguments

Details

If hash = TRUE, all three tests require the arguments 'stat_hash' and 'pvalue_hash' for the hash-based implementation of the statistic test. These hash Objects are produced or updated by each run of SES (if hash == TRUE) and they can be reused in order to speed up next runs of the current statistic test. If "SESoutput" is the output of a SES run, then these objects can be retrieved by SESoutput@hashObject$stat_hash and the SESoutput@hashObject$pvalue_hash.

Important: Use these arguments only with the same dataset that was used at initialization. For all the available conditional independence tests that are currently included on the package, please see "?CondIndTests".

If you have overdispersion, the variance is higher than the mean, a negative binomial is to be used. If you have more zeros than expected under a Poisson model, not overdispersion, then zero inlfated Poisson is to be used. This is in fact a logistic reression where the target is the ratio of successes divided by the totals and the weights are the totals.

The log-likelihood ratio test used in "testIndBinom" requires the fitting of two models. The Wald test used in "waldBinom" requires fitting of only one model, the full one. The significance of the variable is examined only. Only continuous (or binary) predictor variables are currently accepted in this test.

Value

A list including:

Author(s)

Vincenzo Lagani, Ioannis Tsamardinos, Giorgos Athineou and Michail Tsagris.

R implementation and documentation: Giorgos Athineou <[email protected]>, Vincenzo Lagani <[email protected]> and Michail Tsagris [email protected]

References

McCullagh P., and Nelder J.A. (1989). Generalized linear models. CRC press, USA, 2nd edition.

See Also

testIndLogistic, testIndBeta, testIndReg, CondIndTests

Examples

#simulate a dataset with continuous data
dataset <- matrix(runif(200 * 20, 1, 50), ncol = 20 ) 
#the target feature is the last column of the dataset as a vector
y <- rbinom(200, 10, 0.6)
N <- sample(11:20, 200, replace = TRUE)
target <- cbind(y, N)
testIndBinom(target, dataset, xIndex = 14, csIndex = 10)

#run the MMPC algorithm using the testIndPois conditional independence test
a <- MMPC(target, dataset, max_k = 3, threshold = 0.05, test = "testIndBinom")