LogicReg (version 1.6.5)

# logreg: Logic Regression

## Description

Fit one or a series of Logic Regression models, carry out cross-validation or permutation tests for such models, or fit Monte Carlo Logic Regression models.

Logic regression is a (generalized) regression methodology that is primarily applied when most of the covariates in the data to be analyzed are binary. The goal of logic regression is to find predictors that are Boolean (logical) combinations of the original predictors. Currently the Logic Regression methodology has scoring functions for linear regression (residual sum of squares), logistic regression (deviance), classification (misclassification), proportional hazards models (partial likelihood), and exponential survival models (log-likelihood). A feature of the Logic Regression methodology is that it is easily possible to extend the method to write ones own scoring function if you have a different scoring function. `logreg.myown` contains information on how to do so.

## Usage

```logreg(resp, bin, sep, wgt, cens, type, select, ntrees, nleaves,
penalty, seed, kfold, nrep, oldfit, anneal.control, tree.control,
mc.control)```

## Value

An object of the class `logreg`. This contains virtually all input parameters, and in addition

If `select = 1:`

an object of class `logregmodel`: the Logic Regression model. This model contains a list of `ntrees` objects of class `logregtree`.

If `select = 2` or `select = 6`:

`nmodels:`

the number of models fitted.

`allscores:`

a matrix with 3 columns, containing the scores of all models. Column 1 contains the score, column 2 the number of leaves and column 3 the number of trees.

`alltrees:`

a list with `nmodels` objects of class `logregmodel`.

If `select = 3:`

`cvscores:`

a matrix with the results of cross validation. The `train.ave`

and `test.ave` columns for train and test contain running averages of the scores for individual validation sets. As such these scores are of most interest for the rows where `k=kfold`.

If `select = 4:`

`nullscore:`

score of the null-model.

`bestscore:`

score of the best model.

`randscores:`

scores of the permutations; vector of length `nrep`.

If `select = 5:`

`bestscore:`

score of the best model.

`randscores:`

scores of the permutations; each column corresponds to one model size.

If `select = 7:`

`size:` a matrix with two columns, indicating which size models were fit how often.

`single:` a vector with as many elements as there are binary predictors. `single[i]` shows how often predictor `i` is in any of the MCMC models. Note that when a predictor is twice in the same model, it is only counted once, thus, in particular, `sum(size[,1]*size[,2]` will typically be slightly larger than `sum(single)`.

`double:` square matrix with as size the number of binary predictors. `double[i,j]` shows how often predictors `i` and `j` are in the same tree of the same MCMC model if `i>j`, if `i<=j`

`double[i,j]` equals zero. Note that for models with several logic trees two predictors can both be in the model but not be in the same tree.

`triple:` square 3D array with as size the number of binary predictors. See `double`, but here `triple[i,j,k]` shows how often three predictors are jointly in one logic tree.

In addition, the file `slogiclisting.tmp`

in the current working directory can be created. This file contains a compact listing of all models visited. Column 1: proportional to the log posterior probability of the model; column 2: score (log-likelihood); column 3: how often was this model visited, column 4 through 3 + maximum number of leaves: summary of the first tree, if there are two trees, column 4 + maximum number of leaves through 3 + twice the maximum number of leaves contains the second tree, and so on.

In this compact notation, leaves are in the same sequence as the rows in a `logregtree` object; a zero means that the leave is empty, a 1000 means an ``and'' and a 2000 an ``or'', any other positive number means a predictor and a negative number means ``not'' that predictor.

The `mc.control` element `output` can be used to surppress the creation of `double`, `triple`, and/or `slogiclisting.tmp`. There doesn't seem to be too much use in surppressing `double`. Surpressing `triple` speeds up computations a bit (in particular on machines with limited memory when there are many binary predictors), and reduces the size of both the code and the object, surppressing `slogiclisting.tmp`

saves the creation of a possibly very large file, which can slow down the code considerably. See `logreg.mc.control` for details.

## Arguments

resp

vector with the response variables. Let `n1` be the length of this column.

bin

matrix or data frame with binary data. Let `n2` be the number of columns of this object. `bin` should have `n1` rows.

sep

(optional) matrix or data frame that is fitted additively in the logic regression model. `sep` should have `n1` rows. When exponential survival models (`type = 5`) are used, the additive predictors have to be binary. When logistic regression models (`type = 3`) are used `logreg` is much faster when all additive predictors are binary.

wgt

(optional) vector of length `n1` with case weights; default is `rep(1,n1)`.

cens

(optional) an indicator variable with censoring indicators if `type` equals 4 (proportional hazards model) or 5 (exponential survival model); default is `rep(1,n1)`.

type

type of model to be fit: (1) classification, (2) regression, (3) logistic regression, (4) proportional hazards model (Cox regression), (5) exponential survival model, or (0) your own scoring function. If `type = 0`, the code needs to be recompiled, uncompiled `type = 0` results in a constant score of 0, which may be useful to generate a sample from the prior when `select = 7` (Monte Carlo Logic Regression).

select

type of model selection to be carried out: (1) fit a single model, (2) fit multiple models, (3) cross-validation, (4) null-model permutation test, (5) conditional permutation test, (6) a greedy stepwise algorithm, or (7) Monte Carlo Logic Regression (using MCMC). See details below.

ntrees

number of logic trees to be fit. A single number if you select to fit a single model (`select = 1`), carry out the null-model permutation test (`select = 4`), carry out greedy stepwise selection (`select = 6`) or, select using MCMC (`select = 7`), or a range (e.g. `c(ntreeslow,ntreeshigh)`) for any of the other selection options. In our applications, we usually ended up with models having between one and four trees. In general, fitting one and two trees in the initial exploratory analysis is a good idea.

nleaves

maximum number of leaves to be fit in all trees combined. A single number if you select to fit a single model (`select = 1`) carry out the null-model permutation test (`select = 4`), carry out greedy stepwise selection (`select = 6`) or, select using MCMC (`select = 7`), or a range (e.g. `c(nleaveslow,nleaveshigh)`) for any of the other selection options. If `select` is 1, 4, 6, or 7, the default is `-1`, which is expanded to become ```ntrees * tree.control\\$treesize ```.

penalty

specifying the penalty parameter allows you to penalize the score of larger models. The penalty takes the form `penalty` times the number of leaves in the model. (For some score functions, we compute average scores: the penalty is naturally adjusted.) Thus `penalty = 2` is somewhat comparable to AIC. Note, however, that there is no relation between the size of the model and the number of parameters (dimension) of the model, as is usual the case for AIC like penalties. `penalty` is only relevant when `select = 1`.

seed

a seed for the random number generator. The random seed is taken to be `abs(seed)`. \ For the cross-validation version, if `seed < 0` the sequence of the cases is not permuted for division in training-test sets. This is useful if you already permuted the sequence, and wish to compare results with other approaches, or if there is a relation between the sequence of the cases, for example for a matched case-control study.

kfold

the number of groups the cases are randomly assigned to. In turn, the model is trained on `(kfold - 1)` of those groups, and scored on the group left out. Common choices are `kfold = 5` and `kfold = 10`. Only relevant for cross-validation (`select = 3`).

nrep

the number of runs on permuted data for each model size. We recommend first running this program with a small number of repetitions (e.g. 10 or 25) before sending off a big job. Only relevant for the null-model test (`select = 4`) or the permutation test (`select = 5`).

oldfit

object of class `logreg`, typically the result of a previous call to `logreg`. All options that are not specified default to the value used in `oldfit`. For the permutation test (`select = 5`) an `oldfit` object obtained with `select = 2` (fit multiple models) is mandatory, as the best models of each size need to be in place.

anneal.control

simulated annealing parameters - best set using the function `logreg.anneal.control`.

tree.control

several secondary parameters - best set using the function `logreg.tree.control`.

mc.control

Markov chain Monte Carlo parameters - best set using the function `logreg.mc.control`.

## Author

Ingo Ruczinski ingo@jhu.edu and Charles Kooperberg clk@fredhutch.org.

## Details

Logic Regression is an adaptive regression methodology that attempts to construct predictors as Boolean combinations of binary covariates.

In most regression problems a model is developed that only relates the main effects (the predictors or transformations thereof) to the response. Although interactions between predictors are considered sometimes as well, those interactions are usually kept simple (two- to three-way interactions at most). But often, especially when all predictors are binary, the interaction between many predictors is what causes the differences in response. This issue often arises in the analysis of SNP microarray data or in data mining problems. Given a set of binary predictors X, we try to create new, better predictors for the response by considering combinations of those binary predictors. For example, if the response is binary as well (which is not required in general), we attempt to find decision rules such as ``if X1, X2, X3 and X4 are true'', or ``X5 or X6 but not X7 are true'', then the response is more likely to be in class 0. In other words, we try to find Boolean statements involving the binary predictors that enhance the prediction for the response. In more specific terms: Let X1,...,Xk be binary predictors, and let Y be a response variable. We try to fit regression models of the form g(E[Y]) = b0 + b1 L1+ ...+ bn Ln, where Lj is a Boolean expression of the predictors X, such as Lj=[(X2 or X4c) and X7]. The above framework includes many forms of regression, such as linear regression (g(E[Y])=E[Y]) and logistic regression (g(E[Y])=log(E[Y]/(1-E[Y]))). For every model type, we define a score function that reflects the ``quality'' of the model under consideration. For example, for linear regression the score could be the residual sum of squares and for logistic regression the score could be the deviance. We try to find the Boolean expressions in the regression model that minimize the scoring function associated with this model type, estimating the parameters bj simultaneously with the Boolean expressions Lj. In general, any type of model can be considered, as long as a scoring function can be defined. For example, we also implemented the Cox proportional hazards model, using the partial likelihood as the score.

Since the number of possible Logic Models we can construct for a given set of predictors is huge, we have to rely on some search algorithms to help us find the best scoring models. We define the move set by a set of standard operations such as splitting and pruning the tree (similar to the terminology introduced by Breiman et al (1984) for CART). We investigated two types of algorithms: a greedy and a simulated annealing algorithm. While the greedy algorithm is very fast, it does not always find a good scoring model. The simulated annealing algorithm usually does, but computationally it is more expensive. Since we have to be certain to find good scoring models, we usually carry out simulated annealing for our case studies. However, as usual, the best scoring model generally over-fits the data, and methods to separate signal and noise are needed. To assess the over-fitting of large models, we offer the option to fit a model of a specific size. For the model selection itself we developed and implemented permutation tests and tests using cross-validation. If sufficient data is available, an analysis using a training and a test set can also be carried out. These tests are rather complicated, so we will not go into detail here and refer you to Ruczinski I, Kooperberg C, LeBlanc ML (2003), cited below.

There are two alternatives to the simulated annealing algorithm. One is a stepwise greedy selection of models. This is when setting `select = 6`, and yields a sequence of models from size 1 through a maximum size. At each time among all the models that are one larger than the current model the best model is selected, yielding a sequence of models of different sizes. Usually these models are not the best possible, and, if the simulated annealing chain is long enough, you should expect that the models selected using `select = 2` are better.

The second alternative is to run a Markov Chain Monte Carlo (MCMC) algorithm. This is what is done in Monte Carlo Logic Regression. The algorithm used is a reversible jump MCMC algorithm, due to Green (1995). Other than the length of the Markov chain, the only parameter that needs to be set is a parameter for the geometric prior on model size. Other than in many MCMC problems, the goal in Monte Carlo Logic Regression is not to yield one single best predicting model, but rather to provide summaries of all models. These are exactly the elements that are shown above as the output when `select = 7`.

MONITORING

The help file for `logreg.anneal.control`, contains more information on how to monitor the simulated annealing optimization for logreg. Here is some general information.

Find the best scoring model of any size `(select = 1)`

During the iterations the following information is printed out:

 log-temp current score best score acc / rej / sing current parameters -1.000 1.494 1.494 0( 0) 0 0 2.88 -1.99 0.00 -1.120 1.150 1.043 655( 54) 220 71 3.63 0.15 -1.82 -1.240 1.226 1.043 555( 49) 316 80 3.83 0.05 -1.71 ... -2.320 0.988 0.980 147( 36) 759 58 3.00 -2.11 1.11 -2.440 0.982 0.980 25( 31) 884 60 2.89 -2.12 1.24 -2.560 0.988 0.979 35( 61) 850 51 3.00 -2.11 1.11 ... -3.760 0.964 0.964 2( 22) 961 15 2.57 -2.15 1.55 -3.880 0.964 0.964 0( 17) 961 22 2.57 -2.15 1.55 -4.000 0.964 0.964 0( 13) 970 17 2.57 -2.15 1.55

log-temp:
logarithm (base 10) of the temperature at the last iteration before the print out.
current score:
the score after the last iterations.
best score:
the single lowest score seen at any iteration.
acc:
the number of proposed moves that were accepted since the last print out for which the model changed, within parenthesis, the number of those that were identical in score to the move before acceptance.
rej:
the number of proposed moves that gave numerically acceptable results, but were rejected by the simulated annealing algorithm since the last print out.
sing:
the number of proposed moves that were rejected because they gave numerically unacceptable results, for example because they yielded a singular system.
current parameters:
the values of the coefficients (first for the intercept, then for the linear (separate) components, then for the logic trees).

This information can be used to judge the convergence of the simulated annealing algorithm, as described in the help file of `logreg.anneal.control`. Typically we want (i) the number of acceptances to be high in the beginning, (ii) the number of acceptances with different scores to be low at the end, and (iii) the number of iterations when the fraction of acceptances is moderate to be as large as possible.

Find the best scoring models for various sizes `(select = 2)`

During the iterations the same information as for find the best scoring model of any size, for each size model considered.

Carry out cross-validation for model selection `(select = 3)`

Information about the simulated annealing as described above can be printed out. Otherwise, during the cross-validation process information like

``` Step 5 of 10 [ 2 trees; 4 leaves] The CV score is 1.127 1.120 1.052 1.122 ```

The first of the four scores is the training-set score on the current validation sample, the second score is the average of all the training-set scores that have been processed for this model size, the third is the test-set score on the current validation sample, and the fourth score is the average of all the test-set scores that have been processed for this model size. Typically we would prefer the model with the lowest test-set score average over all cross-validation steps.

Carry out a permutation test to check for signal in the data `(select = 4)`

Information about the simulated annealing as described above can be printed out. Otherwise, first the score of the model of size 0 (typically only fitting an intercept) and the score of the best model are printed out. Then during the permutation lines like

``` Permutation number 21 out of 50 has score 1.47777 ```

are printed. Each score is the result of fitting a logic tree model, on data where the response has been permuted. Typically we would believe that there is signal in the data if most permutations have worse (higher) scores than the best model, but we may believe that there is substantial over-fitting going on if these permutation scores are much better (lower) than the score of the model of size 0.

Carry out a permutation test for model selection `(select = 5)`

To be able to run this option, an object of class `logreg` that was run with `(select = 2)` needs to be in place. Information about the simulated annealing as described above can be printed out. Otherwise, lines like

``` Permutation number 8 out of 25 has score 1.00767 model size 3 with 2 tree(s)```

are printed. We can compare these scores to the tree of the same size and the best tree. If the scores are about the same as the one for the best tree, we think that the ``true'' model size may be the one that is being tested, while if the scores are much worse, the true model size may be larger. The comparison with the model of the same size suggests us again how much over-fitting may be going on. `plot.logreg` generates informative histograms.

Greedy stepwise selection of Logic Regression models `(select = 6)`

The scores of the best greedy models of each size are printed.

Monte Carlo Logic Regression `(select = 7)`

A status line is printed every so many iterations. This information is probably not very useful, other than that it helps you figure out how far the code is.

PARAMETERS

As Logic Regression is written in Fortran 77 some parameters had to be hard coded in. The default values of these parameters are

maximum number of columns in the input file: 1000
maximum number of leaves in a logic tree: 128
maximum number of logic trees: 5
maximum number of separate parameters: 50
maximum number of total parameters(separate + trees): 55

If these parameters are not large enough (an error message will let you know this), you need to reset them in slogic.f and recompile. In particular, the statements defining these parameters are

`PARAMETER (LGCn2MAX = 1000)`
`PARAMETER (LGCnknMAX = 128)`
`PARAMETER (LGCntrMAX = 5)`
`PARAMETER (LGCnsepMAX = 50)`
`PARAMETER (LGCbetaMAX = 55)`

The unfortunate thing is that you will have to change these parameter definitions several times in the code. So search until you have found them all.

## References

Ruczinski I, Kooperberg C, LeBlanc ML (2003). Logic Regression, Journal of Computational and Graphical Statistics, 12, 475-511.

Ruczinski I, Kooperberg C, LeBlanc ML (2002). Logic Regression - methods and software. Proceedings of the MSRI workshop on Nonlinear Estimation and Classification (Eds: D. Denison, M. Hansen, C. Holmes, B. Mallick, B. Yu), Springer: New York, 333-344.

Kooperberg C, Ruczinski I, LeBlanc ML, Hsu L (2001). Sequence Analysis using Logic Regression, Genetic Epidemiology, 21, S626-S631.

Kooperberg C, Ruczinki I (2005). Identifying interacting SNPs using Monte Carlo Logic Regression, Genetic Epidemiology, 28, 157-170.

Selected chapters from the dissertation of Ingo Ruczinski, available from https://research.fredhutch.org/content/dam/stripe/kooperberg/ingophd-logic.pdf

`eval.logreg`, `frame.logreg`, `plot.logreg`, `print.logreg`, `predict.logreg`, `logregtree`, `plot.logregtree`, `print.logregtree`, `logregmodel`, `plot.logregtree`, `print.logregtree`, `logreg.myown`, `logreg.anneal.control`, `logreg.tree.control`, `logreg.mc.control`, `logreg.testdat`

## Examples

Run this code
``````data(logreg.savefit1,logreg.savefit2,logreg.savefit3,logreg.savefit4,
logreg.savefit5,logreg.savefit6,logreg.savefit7,logreg.testdat)

myanneal <- logreg.anneal.control(start = -1, end = -4, iter = 500, update = 100)
# in practie we would use 25000 iterations or far more - the use of 500 is only
# to have the examples run fast
if (FALSE) myanneal <- logreg.anneal.control(start = -1, end = -4, iter = 25000, update = 500)
fit1 <- logreg(resp = logreg.testdat[,1], bin=logreg.testdat[, 2:21], type = 2,
select = 1, ntrees = 2, anneal.control = myanneal)
# the best score should be in the 0.95-1.10 range
plot(fit1)
# you'll probably see X1-X4 as well as a few noise predictors
# use logreg.savefit1 for the results with 25000 iterations
plot(logreg.savefit1)
print(logreg.savefit1)
z <- predict(logreg.savefit1)
plot(z, logreg.testdat[,1]-z, xlab="fitted values", ylab="residuals")
# there are some streaks, thanks to the very discrete predictions
#
# a bit less output
myanneal2 <- logreg.anneal.control(start = -1, end = -4, iter = 500, update = 0)
# in practie we would use 25000 iterations or more - the use of 500 is only
# to have the examples run fast
if (FALSE) myanneal2 <- logreg.anneal.control(start = -1, end = -4, iter = 25000, update = 0)
#
# fit multiple models
fit2 <- logreg(resp = logreg.testdat[,1], bin=logreg.testdat[, 2:21], type = 2,
select = 2, ntrees = c(1,2), nleaves =c(1,7), anneal.control = myanneal2)
# equivalent
if (FALSE) fit2 <- logreg(select = 2, ntrees = c(1,2), nleaves =c(1,7), oldfit = fit1,
anneal.control = myanneal2)
plot(fit2)
# use logreg.savefit2 for the results with 25000 iterations
plot(logreg.savefit2)
print(logreg.savefit2)
# After an initial steep decline, the scores only get slightly better
# for models with more than four leaves and two trees.
#
# cross validation
fit3 <- logreg(resp = logreg.testdat[,1], bin=logreg.testdat[, 2:21], type = 2,
select = 3, ntrees = c(1,2), nleaves=c(1,7), anneal.control = myanneal2)
# equivalent
if (FALSE) fit3 <- logreg(select = 3, oldfit = fit2)
plot(fit3)
# use logreg.savefit3 for the results with 25000 iterations
plot(logreg.savefit3)
# 4 leaves, 2 trees should top
# null model test
fit4 <- logreg(resp = logreg.testdat[,1], bin=logreg.testdat[, 2:21], type = 2,
select = 4, ntrees = 2, anneal.control = myanneal2)
# equivalent
if (FALSE) fit4 <- logreg(select = 4, anneal.control = myanneal2, oldfit = fit1)
plot(fit4)
# use logreg.savefit4 for the results with 25000 iterations
plot(logreg.savefit4)
# A histogram of the 25 scores obtained from the permutation test. Also shown
# are the scores for the best scoring model with one logic tree, and the null
# model (no tree). Since the permutation scores are not even close to the score
# of the best model with one tree (fit on the original data), there is overwhelming
# evidence against the null hypothesis that there was no signal in the data.
fit5 <- logreg(resp = logreg.testdat[,1], bin=logreg.testdat[, 2:21], type = 2,
select = 5, ntrees = c(1,2), nleaves=c(1,7), anneal.control = myanneal2,
nrep = 10, oldfit = fit2)
# equivalent
if (FALSE) fit5 <- logreg(select = 5, nrep = 10, oldfit = fit2)
plot(fit5)
# use logreg.savefit5 for the results with 25000 iterations and 25 permutations
plot(logreg.savefit5)
# The permutation scores improve until we condition on a model with two trees and
# four leaves, and then do not change very much anymore. This indicates that the
# best model has indeed four leaves.
#
# greedy selection
fit6 <- logreg(select = 6, ntrees = 2, nleaves =c(1,12), oldfit = fit1)
plot(fit6)
# use logreg.savefit6 for the results with 25000 iterations
plot(logreg.savefit6)
#
# Monte Carlo Logic Regression
fit7 <- logreg(select = 7, oldfit = fit1, mc.control=
logreg.mc.control(nburn=500, niter=2500, hyperpars=log(2), output=-2))
# we need many more iterations for reasonable results
if (FALSE) logreg.savefit7 <- logreg(select = 7, oldfit = fit1, mc.control=
logreg.mc.control(nburn=1000, niter=100000, hyperpars=log(2)))
#
plot(fit7)
# use logreg.savefit7 for the results with 100,000 iterations
plot(logreg.savefit7)

``````

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