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bestglm: Best Subset GLM A. I. McLeod C. Xu University of Western Ontario University of Western Ontario Abstract The function bestglm selects the best subset of inputs for the glm family. The selection methods available include a variety of information criteria as well as cross-validation. Several examples are provided to show that this approach is sometimes more accurate than using the built-in R function step. In the Gaussian case the leaps-and-bounds algorithm in leaps is used provided that there are no factor variables with more than two levels. In the non-Gaussian glm case or when there are factor variables present with three or more levels, a simple exhaustive enumeration approach is used. This vignette also explains how the applications given in our article Xu and McLeod (2010) may easily be reproduced. A separate vignette is available to provide more details about the simulation results reported in Xu and McLeod (2010, Table 2) and to explain how the results may be reproduced. Keywords: best subset GLM, AIC, BIC, extended BIC, cross-validation. 1. Introduction We consider the glm of on inputs, 1 , . . . , . In many cases, can be more parsimoniously modelled and predicted using just a subset of < inputs, 1 , . . . , . The best subset problem is to ﬁnd out of all the 2 subsets, the best subset according to some goodness-of-ﬁt criterion. The built-in R function step may be used to ﬁnd a best subset using a stepwise search. This method is expedient and often works well. When is not too large, step, may be used for a backward search and this typically yields a better result than a forward search. But if is large, then it may be that only a forward search is feasible due to singularity or multicollinearity. In many everyday regression problems we have ≤ 50 and in this case an optimization method known as leaps-and-bounds may be utilized to ﬁnd the best subset. More generally when ≤ 15 a simple direct lexicographic algorithm (Knuth 2005, Algorithm L) may be used to enumerate all possible models. Some authors have criticized the all subsets approach on the grounds that it is too computationally intensive. The term data dredging has been used. This criticism is not without merit since it must be recognized that the signﬁcance level for the -values of the coeﬃcients in the model will be overstated – perhaps even extremely so. Furthermore for prediction purposes, the LASSO or regularization method may outperform the subset model’s prediction. Nevertheless there are several important applications for subset selection methods. In many problems, it is of interest to determine which are the most inﬂuential variables. For many data mining methods such as neural nets or support vector machines, feature selection plays an important role and here too subset selection can help. The idea of data-dredging is somewhat similar to the concern about over-training with artiﬁcal neural nets. In both cases, there does not seem to be any 2 bestglm: Best Subset GLM rigorous justiﬁcation of choosing a suboptimal solution. In the case of glm and linear models our package provides a variety of criterion for choosing a parsimonious subset or collection of possible subsets. In the case of linear regression, Miller (2002) provides a monograph length treatment of this problem while Hastie, Tibshirani, and Friedman (2009, Ch. 3) discuss the subset approach along with other recently developed methods such as lars and lasso. Consider the case of linear regression with observations, (,1 , . . . , , , ), = 1, . . . , we may write the regression, = 0 + 1 ,1 + . . . + , + . (1) When > all possible 2 regressions could be ﬁt and the best ﬁt according to some criterion could be found. When ≤ 25 or thereabouts, an eﬃcient combinatorial algorithm, known as branch-and-bound can be applied to determine the model with the lowest residual sum of squares of size for = 1, . . . , and more generally the lowest subsets for each may also be found. The leaps package (Lumley and Miller 2004) implements the branch-and-bound algorithm as well as other subset selection algorithms. Using the leaps function, regsubsets, the best model of size , = 1, . . . , may be determined in a few seconds when ≤ 25 on a modern personal computer. Even larger models are feasible but since, in the general case, the computer time grows exponentially with , problems with large enough such as > 100, can not be solved by this method. An improved branch-and-bound algorithm is given by Gatu (2006) but the problem with exponential time remains. One well-known and widely used alternative to the best subset approach is the family of stepwise and stagewise algorithms Hastie et al. (2009, Section 3.3). This is often feasible for larger although it may select a sub-optimal model as noted by Miller (2002). For very large Chen and Chen (2008) suggest a tournament algorithm while subselect (Cadima, Cerdeira, Orestes, and Minhoto 2004; Cerdeira, Silva, Cadima, and Minhoto 2009) uses high dimensional optimization algorithms such as genetic search and simulated annealing for such problems. Using subset selection algorithm necessarily involves a high degree of selection bias in the ﬁtted regression. This means that the -values for the regression coeﬃcients are overstated, that is, coeﬃcients may appear to be statistically signﬁcant when they are not. (Wilkinson and Gerard 1981) and the 2 are also inﬂated Rencher and Fu (1980). More generally for the family of glm models similar considerations about selection bias and computational complexity apply. Hosmer, Jovanovic, and Lemeshow (1989) discuss an approximate method for best subsets in logistic regression. No doubt there is scope for the development of more eﬃcient branch-and-bound algorithms for the problem of subset selection in glm models. See Brusco and Stahl (2009) for a recent monograph of the statistical applications of the branch-and-bound algorithm. We use the lexicographical method suggested by Morgan and Tatar (1972) for the all subsets regression problem to enumerate the loglikelihoods for all possible glm model. Assuming there are inputs, there are then 2 possible subsets which may be enumerated by taking = 0, . . . , 2 − 1 and using the base-2 representation of to determine the subset. This method is quite feasible on present PC workstations for not too large. 1.1. Prostate Cancer Example A. I. McLeod, C. Xu 3 As an illustrative example of the subset regression problem we consider the prostate data discussed by Hastie et al. (2009). In this dataset there are 97 observations on men with prostate cancer. The object is to predict and to ﬁnd the inputs most closely related with the outcome variable Prostate-Speciﬁc Antigen (psa). In the general male population, the higher the psa, the greater the chance that prostate cancer is present. To facilitate comparison with the results given in the textbook as well as with other techniques such as LARS, we have standardized all inputs. The standardized prostate data is available in zprostate in our bestglm package and is summarized below, R> library(bestglm) R> data(zprostate) R> str(zprostate) 'data.frame': $ lcavol : num $ lweight: num $ age : num $ lbph : num $ svi : num $ lcp : num $ gleason: num $ pgg45 : num $ lpsa : num $ train : logi 97 obs. of 10 variables: -1.637 -1.989 -1.579 -2.167 -0.508 ... -2.006 -0.722 -2.189 -0.808 -0.459 ... -1.862 -0.788 1.361 -0.788 -0.251 ... -1.02 -1.02 -1.02 -1.02 -1.02 ... -0.523 -0.523 -0.523 -0.523 -0.523 ... -0.863 -0.863 -0.863 -0.863 -0.863 ... -1.042 -1.042 0.343 -1.042 -1.042 ... -0.864 -0.864 -0.155 -0.864 -0.864 ... -0.431 -0.163 -0.163 -0.163 0.372 ... TRUE TRUE TRUE TRUE TRUE TRUE ... The outcome is lpsa which is the logarithm of the psa. In Hastie et al. (2009, Table 3.3) only the training set portion is used. In the training portion there are = 67 observations. Using regsubsets in leaps we ﬁnd subsets of size = 1, . . . , 8 which have the smallest residual sum-of-squares. R> R> R> R> R> R> train <- (zprostate[zprostate[, 10], ])[, -10] X <- train[, 1:8] y <- train[, 9] out <- summary(regsubsets(x = X, y = y, nvmax = ncol(X))) Subsets <- out$which RSS <- out$rss R> cbind(as.data.frame(Subsets), RSS = RSS) 1 2 3 4 5 (Intercept) lcavol lweight age lbph svi lcp gleason pgg45 RSS TRUE TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE 44.52858 TRUE TRUE TRUE FALSE FALSE FALSE FALSE FALSE FALSE 37.09185 TRUE TRUE TRUE FALSE FALSE TRUE FALSE FALSE FALSE 34.90775 TRUE TRUE TRUE FALSE TRUE TRUE FALSE FALSE FALSE 32.81499 TRUE TRUE TRUE FALSE TRUE TRUE FALSE FALSE TRUE 32.06945 4 6 7 8 bestglm: Best Subset GLM TRUE TRUE TRUE TRUE TRUE TRUE TRUE FALSE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE FALSE FALSE TRUE TRUE 30.53978 TRUE 29.43730 TRUE 29.42638 The residual sum-of-squares decreases monotonically as the number of inputs increases. 1.2. Overview of bestglm Package bestglm uses the simple exhaustive search algorithm (Morgan and Tatar 1972) for glm and the regsubsets function in the leaps package to ﬁnd the glm models with smallest sum of squares or deviances for size = 0, 1, . . . , . Size = 0 corresponds to intercept only. The exhaustive search requires more computer time but this is usually not an issue when <= 10. For example, we found that a logistic regression with = 10 requires about 12.47 seconds as compared with only 0.04 seconds for a comparably size linear regression. The timing diﬀerence would not be important in typical data analysis applications but could be a concern in simulation studies. In this case, if a multi-core PC or even better a computer cluster is available, we may use the Rmpi package. Our vignette Xu and McLeod (2009) provides an example of using Rmpi with bestglm. 1.3. Package Options The arguments and their default values are: R> args(bestglm) function (Xy, family = gaussian, IC = "BIC", t = "default", CVArgs = "default", qLevel = 0.99, TopModels = 5, method = "exhaustive", intercept = TRUE, weights = NULL, nvmax = "default", RequireFullEnumerationQ = FALSE, ...) NULL The argument Xy is usually a data-frame containing in the ﬁrst columns the design matrix and in the last column the response. For binomial GLM, the last two columns may represent counts and as in the usual glm function when the family=binomial option is used. When family is set to gaussian, the function regsubsets in leaps is used provided that all inputs are quantitative or that there are no factor inputs with more than two levels. When factor inputs at more than two levels are present, the exhaustive enumeration method is used and in this case the R function lm is used in the gaussian case. For all non-Gaussian models, the R function glm is used with the exhaustive enumeration method. The arguments IC, t, CVArgs, qLevel and TopModels are used with various model selection methods. The model selection methods available are based on either an information criterion or cross-validation. The information criteria and cross-validation methods are are discussed in the Sections 2 and 3. The argument method is simply passed on to the function regsubsets when this function from the leaps package is used. The arguments intercept and nvmax are also passed on to regsubsets or may be used in the exhaustive search with a non-Gaussian GLM model is ﬁt. These two arguments are discussed brieﬂy in Sections 1.4 and 1.5. A. I. McLeod, C. Xu 5 The argument RequireFullEnumerationQ is provided to force the use of the slower exhaustive search algorithm when the faster algorithm in the leaps package would normally be used. This is provided only for checking. The output from bestglm is a list with named components R> Xy <- cbind(as.data.frame(X), lpsa = y) R> out <- bestglm(Xy) R> names(out) [1] "BestModel" [6] "Title" "BestModels" "Bestq" "ModelReport" "qTable" "Subsets" The components BestModel, BestModels, Subsets, qTable and Bestq are of interest and are described in the following table. name description BestModel BestModels Bestq Subsets qTable lm or glm object giving the best model a × logical matrix showing which variables are included in the top models matrix with 2 rows indicating the upper and lower ranges a ( + 1) × logical matrix showing which variables are included for subset sizes = 0, . . . , have the smallest deviance a table showing all possible model choices for diﬀerent intervals of . 1.4. Intercept Term Sometimes it may be desired not to include an intercept term in the model. Usually this occurs when the response to the inputs is thought to be proportional. If the relationship is multiplicative of the form = 1 1 +...+ then a linear regression through the origin of log on 1 , . . . , may be appropriate. Another, but not recommended use, of this option is to set intercept to FALSE and then include a column of 1’s in the design matrix to represent the intercept term. This will enable one to exclude the intercept term if it is not statistically signiﬁcant. Usually the intercept term is always included even if it is not statistically signiﬁcant unless there are prior reasons to suspect that the regression may pass through the origin. Cross-validation methods are not available in the regression through the origin case. 1.5. Limiting the Number of Variables The argument nvmax may be used to limit the number of possible explanatory variables that are allowed to be included. This may be useful when is quite large. Normally the information criterion will eliminate unnecessary variables automatically and so when the default setting is used for nvmax all models up to an including the full model with inputs are considered. Cross-validation methods are not available when nvmax is set to a value less than . 1.6. Forcing Variables to be Included 6 bestglm: Best Subset GLM In some applications, the model builder may wish to require that some variables be included in all models. This could be done by using the residuals from a regression with the required variables as inputs with a design matrix formed from the optional variables. For this reason, the optional argument force.in used in leaps is not implemented in bestglm. 2. Information criteria Information criteria or cross-validation is used to select the best model out of these + 1 model cases, = 0, 1, . . . , . The information criteria include the usual aic and bic as well as two types of extended bic (Chen and Chen 2008; Xu and McLeod 2010). These information criteria are discussed in the Section 2. When the information criterion approach is used, it is possible to select the best models out of all possible models by setting the optional argument TopModels = T. All the information criteria we consider are based on a penalized form of the deviance or minus twice the log-likelihood. In the multiple linear regression the deviance = −2 log ℒ, where ℒ is the maximized log-likelihood, log ℒ = −(/2) log /, where is the residual sum of squares. 2.1. AIC Akaike (1974) showed that aic = + 2, where is the number of parameters, provides an estimate of the entropy. The model with the smallest aic is preferred. Many other criteria which are essentially equivalent to the aic have also been suggested. Several other asymptotically equivalent but more specialized criteria were suggested In the context of autoregressive models, Akaike (1970) suggested the ﬁnal prediction error criterion, fpe = ˆ2 (1 + 2/), where ˆ2 is the estimated residual variance in a model with parameters. and in the subset regression problem, Mallows (1973) suggesed using = /ˆ 2 + 2 − , where is the residual sum-of-squares for a model with inputs and ˆ2 is the residual variance using all inputs. Nishii (1984) showed that minimizing or fpe is equivalent to minimizing the AIC. In practice, with small , these criteria often select the same model. From the results of (Shibata 1981), the aic is asympotically eﬃcient but not consistent. Best AIC Model for Prostate Data R> bestglm(Xy, IC = "AIC") AIC BICq equivalent for q in (0.708764213288624, 0.889919748490004) Best Model: Estimate Std. Error t value Pr(>|t|) (Intercept) 2.4668675 0.08760022 28.160516 6.632457e-36 lcavol 0.6764486 0.12383666 5.462426 9.883880e-07 lweight 0.2652760 0.09363348 2.833132 6.298761e-03 age -0.1450300 0.09756540 -1.486490 1.424742e-01 lbph 0.2095349 0.10128348 2.068796 4.295574e-02 svi 0.3070936 0.12190105 2.519204 1.449125e-02 A. I. McLeod, C. Xu lcp pgg45 7 -0.2872242 0.15300241 -1.877253 6.543004e-02 0.2522850 0.11562030 2.182013 3.310324e-02 The best subset model using aic has 7 variables and two of them are not even signiﬁcant at 5%. 2.2. BIC The bic criterion (Schwarz 1978) can be derived using Bayesian methods as discussed by Chen and Chen (2008). If a uniform prior is assumed of all possible models, the usual bic criterion may be written, bic = + log(). The model with the smallest bic corresponds to the model with maximum posterior probability. The diﬀerence between these criterion is in the penalty. When > 7, the bic penalty is always larger than for the aic and consequently the bic will never select models with more parameters than the aic. In practice, the BIC often selects more parsimonious models than the aic. In time series forecasting experiments, time series models selected using the bic often outperform aic selected models (Noakes, McLeod, and Hipel 1985; Koehler and Murphree 1988; Granger and Jeon 2004). On the other hand, sometimes the bic underﬁts and so in some applications, such as autoregressivespectral density estimation and for generating synthetic riverﬂows and simulations of other types of time series data, it may be preferable to use the aic (Percival and Walden 1993). Best BIC Model for Prostate Data R> bestglm(Xy, IC = "BIC") BIC BICq equivalent for q Best Model: Estimate (Intercept) 2.4773573 lcavol 0.7397137 lweight 0.3163282 in (0.0176493852011195, 0.512566675362627) Std. Error t value Pr(>|t|) 0.09304738 26.624687 2.475214e-36 0.09318316 7.938277 4.141615e-11 0.08830716 3.582135 6.576173e-04 Note that IC="BIC" is the default. 2.3. BICg The notation bicg and bic will be used interchangeably. In mathematical writing bic is preferred but in our R code the parameter is denoted by bicg. Chen and Chen (2008) observed that in large problems, the bic tends to select models with too many parameters and suggested that instead of a prior uniform of all possible models, a prior uniform of models of ﬁxed size. The general form of the bic criterion can be written, ( ) bic = + log() + 2 log (2) where is an adjustable parameter, in the number of possible input variables not counting the bias or intercept term and is the number of parameters in the model. Taking = 0 8 bestglm: Best Subset GLM reduces to the BIC. Notice that mid-sized models have the largest models, while = 0, corresponding to only an intercept term and = corresponding to using all parameters are equally likely a priori. As pointed out in Xu and McLeod (2010) this prior is not reasonable because it is symmetric, giving large models and small models equal prior probability. Best BICg Model for Prostate Data R> bestglm(Xy, IC = "BICg") BICg(g = 1) BICq equivalent for q Best Model: Estimate (Intercept) 2.4773573 lcavol 0.7397137 lweight 0.3163282 in (0.0176493852011195, 0.512566675362627) Std. Error t value Pr(>|t|) 0.09304738 26.624687 2.475214e-36 0.09318316 7.938277 4.141615e-11 0.08830716 3.582135 6.576173e-04 2.4. BICq As with the bic the notation bicq and bic will be used interchangably. The bic criterion (Xu and McLeod 2010) is derived by assuming a Bernouilli prior for the parameters. Each parameter has a priori probability of of being included, where ∈ [0, 1]. With this prior, the resulting information criterion can be written, = + log() − 2 log /(1 − ). (3) When = 1/2, the BICq is equivalent to the BIC while = 0 and = 1 correspond to selecting the models with = and = 0 respectively. Moreover, can be chosen to give results equivalent to the BICg for any or the aic Xu and McLeod (2010). When other information criteria are used with bestglm, the range of the parameter that will produce the same result is shown. For example in 2.3.1, we see that ∈ (0.0176493852011195, 0.512566675362627) produces an equivalent result. For = 0, the penalty is taken to be −∞ and so no parameters are selected and similarly for = 1, the full model with all covariates is selected. Xu and McLeod (2010) derive an interval estimate for that is based on a conﬁdence probability , 0 < < 1. This parameter may be set by the optional argument qLevel = . The default setting is with = 0.99. Numerical Illustration -Interval Computation In Xu and McLeod (2010, Table 1) we provided a brief illustrations of the computation of the intervals for given by our Theorem. R> set.seed(1233211235) R> p <- 5 R> n <- 100 A. I. McLeod, C. Xu R> R> R> R> R> R> R> X <- matrix(rnorm(n * p), ncol = p) err <- rnorm(n) y <- 0.1 * (X[, 1] + X[, 2] + X[, 3]) + err Xy <- as.data.frame(cbind(X, y)) names(Xy) <- c(paste("X", 1:p, sep = ""), "y") ans <- bestglm(Xy) ans$Subsets 0 1* 2 3 4 5 (Intercept) X1 X2 X3 X4 X5 logLikelihood BIC TRUE FALSE FALSE FALSE FALSE FALSE -16.617205 33.23441 TRUE FALSE FALSE FALSE FALSE TRUE -12.933572 30.47231 TRUE FALSE FALSE TRUE FALSE TRUE -11.149821 31.50998 TRUE TRUE FALSE TRUE FALSE TRUE -9.667975 33.15146 TRUE TRUE FALSE TRUE TRUE TRUE -9.608972 37.63862 TRUE TRUE TRUE TRUE TRUE TRUE -9.589967 42.20578 9 R> ans$qTable LogL q1 q2 k [1,] -16.617205 0.0000000 0.2008406 0 [2,] -12.933572 0.2008406 0.6268752 1 [3,] -11.149821 0.6268752 0.6943933 2 [4,] -9.667975 0.6943933 0.9040955 3 [5,] -9.608972 0.9040955 0.9075080 4 [6,] -9.589967 0.9075080 1.0000000 5 In Xu and McLeod (2010, Table 1) we added 20 to the value of the log-likelihood. Best BICq Model for Prostate Data Using the bic with its default choice for the tuning parameter = , R> R> R> R> R> R> data(zprostate) train <- (zprostate[zprostate[, 10], ])[, -10] X <- train[, 1:8] y <- train[, 9] Xy <- cbind(as.data.frame(X), lpsa = y) out <- bestglm(Xy, IC = "BICq") 3. Cross-Validation Cross-validation approaches to model selection are widely used and are also available in the bestglm function The old standard, leave-one-out cross-validation (loocv) is implemented along with the more modern methods: K-fold and delete-d cross-valiation (CV). 10 bestglm: Best Subset GLM All CV methods work by ﬁrst narrowing the ﬁeld to the best models of size for = 0, 1, . . . , and then comparing each of these models + 1 possible models using cross-validation to select the best one. The best model of size is chosen as the one with the smallest deviance. 3.1. Delete-d Cross-Validation The delete-d method was suggested by Shao (1993). In the random sampling version of this algorithm, random samples of size are used as the validation set. Many validation sets are generated in this way and the complementary part of the data is used each time as the training set. Typically 1000 validation sets are used. When = 1, the delete-d is similar to LOOCV (3.4) and should give the same result if enough validation sets are used. Shao (1997) shows that when increases with , this method will be consistent. Note that -fold cross-validation is approximately equivalent taking ≈ /. But Shao (1997) recommends a much larger cross-validation sample than is customarily used in -fold CV. Letting = log as suggested Shao (1997, page 236, 4th line of last paragraph) and using Shao (1997, eqn. 4.5), we obtain ★ = (1 − (log − 1)−1 ), (4) where is the number of observations. Comparison of size of validation samples for various sample sizes using delete- and -fold cross-validation. 50 100 200 500 1000 ★ 33 73 154 405 831 = 10 5 10 20 50 100 =5 10 20 40 100 200 Best Delete-d Model for Prostate Data The default cross-validation method is delete-d with 1000 replications, as with bestglm(Xy, IC="CV". This takes about one minute to run, so in this vignette we set the optional tuning parameter t=10 so only 10 replications are done. The default for IC="CV" is delete-d with as in eqn. (4) but in the example below, we set the optional tuning parameter t=10 R> set.seed(123321123) R> bestglm(Xy, IC = "CV", t = 10) CVd(d = 47, REP = 10) No BICq equivalent Best Model: Estimate Std. Error t value Pr(>|t|) (Intercept) 2.4627121 0.08901202 27.667185 3.167240e-36 A. I. McLeod, C. Xu lcavol lweight lbph svi pgg45 0.5566392 0.2415963 0.1989292 0.2393565 0.1221447 0.11360017 0.09467037 0.10187183 0.11734589 0.10256941 4.899985 2.551974 1.952740 2.039752 1.190849 11 7.408246e-06 1.323253e-02 5.544293e-02 4.571228e-02 2.383261e-01 In practice though at least 1000 replications are usually needed to obtain convergence. 3.2. K-fold Hastie et al. (2009) discuss -fold CV. With this method, the data are divided, randomly, into folds of approximately equal size. For the prostate training data with = 67 and using = 10 folds, R> set.seed(2377723) R> ind <- sample(rep(1:10, length = 67)) R> ind [1] 10 [26] 2 [51] 4 1 7 1 5 10 7 1 8 4 2 8 9 10 4 2 9 4 5 7 10 5 2 3 10 6 9 7 8 9 8 4 5 3 1 7 7 1 8 5 2 9 2 6 4 6 6 3 5 4 1 6 3 5 1 6 8 6 9 7 2 3 3 3 10 We see that the observations in Π1 are, R> (1:67)[1 == ind] [1] 2 13 14 29 43 45 52 and the values of , = 1, . . . , 10 are: R> tabulate(ind) [1] 7 7 7 7 7 7 7 6 6 6 These folds form a partition of the the observations 1, . . . , . We will denote the set of elements in the th partition by Π . One fold is selected as the validation sample and the reamining to into the training sample. The performance is calibrated on the validation sample. This is repeated for each fold. The average performance over the folds is determined. Hastie et al. (2009) suggest using the one-standard-deviation rule with K-fold cross-validation. This makes the model selection more stable than simply selecting the model model with the best overall average performance. This rule was original deﬁned in (Breiman, Freidman, Olshen, and Stone 1984, p. 79, Deﬁnition 3.19)) and used for selecting the best prunned CART. 12 bestglm: Best Subset GLM For subset selection, this approach is implemented as follows. The validation sum-of-squares is computed for each of the validation samples, ∑ = lim (ˆ (−) )2 , (5) ∈Π where (−) denotes the prediction error when the th validation sample is removed, the model ﬁt to the remainder of the data and then used to predict the observations ∈ Π in the validation sample. The ﬁnal cross-validation score is cv = 1∑ (6) =1 where is the number of observations. In each validation sample we may obtain the estimate of the cross-validation mean-square error, cv = / , where is the number of observations in the th validation sample. Let 2 be the sample variance of cv1 , . . . , cv . So an 2 estimate of the sample variance of cv, the mean of cv1 , . . . , cv √ is /. Then an interval estimate for CV, using the one-standard-devation rule, is cv±/ . When applied to model selection, this suggests that instead of selecting the model with the smallest CV, the most parsimonious adequate model will correspond to the model with the best CV score which is still inside this interval. Using this rule greatly improves the stability of k-fold CV. This rule is implemented when the HTF CV method is used in our bestglm function. R> set.seed(2377723) R> out <- bestglm(Xy, IC = "CV", CVArgs = list(Method = "HTF", K = 10, + REP = 1)) R> out CV(K = 10, REP = 1) BICq equivalent for q Best Model: Estimate (Intercept) 2.4773573 lcavol 0.7397137 lweight 0.3163282 in (0.0176493852011195, 0.512566675362627) Std. Error t value Pr(>|t|) 0.09304738 26.624687 2.475214e-36 0.09318316 7.938277 4.141615e-11 0.08830716 3.582135 6.576173e-04 In Figure 1 below we reproduce one of the graphs shown in (Hastie et al. 2009, page 62, Figure 3.3) that illustrates how the one-standard deviation rule works for model selection. R> R> R> R> R> R> R> R> cverrs <- out$Subsets[, "CV"] sdCV <- out$Subsets[, "sdCV"] CVLo <- cverrs - sdCV CVHi <- cverrs + sdCV ymax <- max(CVHi) ymin <- min(CVLo) k <- 0:(length(cverrs) - 1) plot(k, cverrs, xlab = "Subset Size", ylab = "CV Error", ylim = c(ymin, A. I. McLeod, C. Xu + R> R> R> R> R> R> R> R> R> R> R> R> R> R> R> R> R> 13 ymax), type = "n", yaxt = "n") points(k, cverrs, cex = 2, col = "red", pch = 16) lines(k, cverrs, col = "red", lwd = 2) axis(2, yaxp = c(0.6, 1.8, 6)) segments(k, CVLo, k, CVHi, col = "blue", lwd = 2) eps <- 0.15 segments(k - eps, CVLo, k + eps, CVLo, col = "blue", lwd = 2) segments(k - eps, CVHi, k + eps, CVHi, col = "blue", lwd = 2) indBest <- oneSdRule(out$Subsets[, c("CV", "sdCV")]) abline(v = indBest - 1, lty = 2) indMin <- which.min(cverrs) fmin <- sdCV[indMin] cutOff <- fmin + cverrs[indMin] abline(h = cutOff, lty = 2) indMin <- which.min(cverrs) fmin <- sdCV[indMin] cutOff <- fmin + cverrs[indMin] min(which(cverrs < cutOff)) 1.0 ● ● 0.6 CV Error 1.4 [1] 3 0 ● 2 ● ● 4 ● ● 6 ● ● 8 Subset Size Figure 1: Model selection with 10-fold cross-validation and 1-sd rule 3.3. Bias Correction Davison and Hinkley (1997, Algorithm 6.5, p.295) suggested an adjusted CV statistic which corrects for bias but this method has quite variable in small samples. 14 bestglm: Best Subset GLM Running the program 3 times produces 3 diﬀerent results. R> set.seed(2377723) R> bestglm(Xy, IC = "CV", CVArgs = list(Method = "DH", K = 10, REP = 1)) CVAdj(K = 10, REP = 1) No BICq equivalent Best Model: Estimate Std. Error t value Pr(>|t|) (Intercept) 2.4627121 0.08901202 27.667185 3.167240e-36 lcavol 0.5566392 0.11360017 4.899985 7.408246e-06 lweight 0.2415963 0.09467037 2.551974 1.323253e-02 lbph 0.1989292 0.10187183 1.952740 5.544293e-02 svi 0.2393565 0.11734589 2.039752 4.571228e-02 pgg45 0.1221447 0.10256941 1.190849 2.383261e-01 R> bestglm(Xy, IC = "CV", CVArgs = list(Method = "DH", K = 10, REP = 1)) CVAdj(K = 10, REP = 1) No BICq equivalent Best Model: Estimate (Intercept) 2.4511950 lcavol 0.6479821 lweight 0.2412408 lbph 0.1827709 svi 0.3131270 lcp -0.2668206 pgg45 0.2126933 Std. Error t value Pr(>|t|) 0.08783569 27.906596 4.589425e-36 0.12357416 5.243670 2.153436e-06 0.09315188 2.589758 1.203646e-02 0.10067001 1.815545 7.443814e-02 0.12305547 2.544601 1.353129e-02 0.15391392 -1.733570 8.813073e-02 0.11363923 1.871654 6.613158e-02 R> bestglm(Xy, IC = "CV", CVArgs = list(Method = "DH", K = 10, REP = 1)) CVAdj(K = 10, REP = 1) BICq equivalent for q in (0.708764213288624, 0.889919748490004) Best Model: Estimate Std. Error t value Pr(>|t|) (Intercept) 2.4668675 0.08760022 28.160516 6.632457e-36 lcavol 0.6764486 0.12383666 5.462426 9.883880e-07 lweight 0.2652760 0.09363348 2.833132 6.298761e-03 age -0.1450300 0.09756540 -1.486490 1.424742e-01 lbph 0.2095349 0.10128348 2.068796 4.295574e-02 svi 0.3070936 0.12190105 2.519204 1.449125e-02 lcp -0.2872242 0.15300241 -1.877253 6.543004e-02 pgg45 0.2522850 0.11562030 2.182013 3.310324e-02 The results obtained after 1000 simulations are summarized in the table below. A. I. McLeod, C. Xu 15 Number of inputs selected 1 2 3 4 5 6 7 8 Frequency in 1000 simulations 0 0 23 61 64 289 448 115 When REP is increased to 100, the result converges the model with 7 inputs. It takes about 66 seconds. Using REP=100 many times, it was found that models with 7 inputs were selected 95 We conclude that if either this method (Davison and Hinkley 1997, Algorithm 6.5, p.295) or the method of Hastie et al. (2009) is used, many replications are need to obtain a stable result. In view of this, the delete-d of cross-validation is recommended. 3.4. Leave-one-out Cross-Validation For completeness we include leave-one-out CV (loocv) but this method is not recommended because the model selection is not usually as accurate as either of the other CV methods discussed above. This is due to the high variance of this method (Hastie et al. 2009, Section 7.10). In leave-one-out CV (loocv), one observation, say the , is removed, the regression is reﬁt and the prediction error, ˆ() for the missing observation is obtained. This process is repeated for all observations = 1, . . . , and the prediction error sum of squares is obtained, press = ∑ ˆ2() . (7) =1 In the case of linear regression, leave-out-CV can be computed very eﬃciently using the PRESS method (Allen 1971), ˆ() = ˆ where ˆ is the usual regression residual and ℎ, is the -th element on the diagonal of the hat matrix = ′ )−1 ′ . Stone (1977) showed that asymptotically LOOCV is equivalent to the AIC. The computation is very eﬃcient. Best LOOCV Model for Prostate Data R> bestglm(Xy, IC = "LOOCV") LOOCV BICq equivalent for q in (0.708764213288624, 0.889919748490004) Best Model: Estimate Std. Error t value Pr(>|t|) (Intercept) 2.4668675 0.08760022 28.160516 6.632457e-36 lcavol 0.6764486 0.12383666 5.462426 9.883880e-07 lweight 0.2652760 0.09363348 2.833132 6.298761e-03 age -0.1450300 0.09756540 -1.486490 1.424742e-01 lbph 0.2095349 0.10128348 2.068796 4.295574e-02 svi 0.3070936 0.12190105 2.519204 1.449125e-02 lcp -0.2872242 0.15300241 -1.877253 6.543004e-02 pgg45 0.2522850 0.11562030 2.182013 3.310324e-02 4. Examples from our BICq Paper 16 bestglm: Best Subset GLM The following examples were brieﬂy discussed in our paper “Improved Extended Bayesian Information Criterion” (Xu and McLeod 2010). 4.1. Hospital Manpower Data This dataset was used as an example in our paper (Xu and McLeod 2010, Example 1). We commented on the fact that both the AIC and BIC select the same model with 3 variables even though one of the variables is not even signﬁcant at the 5% level and has the incorrect sign. R> data(manpower) R> bestglm(manpower, IC = "AIC") AIC BICq equivalent for q in (0.258049145974038, 0.680450993834175) Best Model: Estimate Std. Error t value Pr(>|t|) (Intercept) 1523.38923568 786.89772473 1.935943 7.492387e-02 Xray 0.05298733 0.02009194 2.637243 2.050318e-02 BedDays 0.97848162 0.10515362 9.305258 4.121293e-07 Stay -320.95082518 153.19222065 -2.095086 5.631250e-02 R> bestglm(manpower, IC = "BIC") BIC BICq equivalent for q in (0.258049145974038, 0.680450993834175) Best Model: Estimate Std. Error t value Pr(>|t|) (Intercept) 1523.38923568 786.89772473 1.935943 7.492387e-02 Xray 0.05298733 0.02009194 2.637243 2.050318e-02 BedDays 0.97848162 0.10515362 9.305258 4.121293e-07 Stay -320.95082518 153.19222065 -2.095086 5.631250e-02 In this case the BIC is completely useless selecting the full model when = 1 or = 0.5. R> bestglm(manpower, IC = "BICg") BICg(g = 1) BICq equivalent for q in (0.801591282573779, 1) Best Model: Estimate Std. Error t value (Intercept) 1962.94815647 1.071362e+03 1.8321993 Load -15.85167473 9.765299e+01 -0.1623266 Xray 0.05593038 2.125828e-02 2.6309923 BedDays 1.58962370 3.092083e+00 0.5140947 AreaPop -4.21866799 7.176557e+00 -0.5878401 Stay -394.31411702 2.096395e+02 -1.8809148 Pr(>|t|) 0.09410839 0.87399215 0.02336582 0.61735574 0.56851117 0.08670281 A. I. McLeod, C. Xu R> bestglm(manpower, IC = "BICg", t = 0.5) BICg(g = 0.5) BICq equivalent for q in (0.258049145974038, 0.680450993834175) Best Model: Estimate Std. Error t value Pr(>|t|) (Intercept) 1523.38923568 786.89772473 1.935943 7.492387e-02 Xray 0.05298733 0.02009194 2.637243 2.050318e-02 BedDays 0.97848162 0.10515362 9.305258 4.121293e-07 Stay -320.95082518 153.19222065 -2.095086 5.631250e-02 Finally, with the BIC with its default choice, = 0.25, R> out <- bestglm(manpower, IC = "BICq") R> out BICq(q = 0.25) BICq equivalent for q in (0.00764992882308291, 0.258049145974038) Best Model: Estimate Std. Error t value Pr(>|t|) (Intercept) -68.31395896 228.44597086 -0.2990377 7.693043e-01 Xray 0.07486591 0.01913019 3.9134953 1.559779e-03 BedDays 0.82287456 0.08295986 9.9189488 1.033117e-07 The optimal range of includes = 0.25, R> out$Bestq q1 q2 selected k BICq1 0.007649929 0.2580491 2 BICq2 0.007649929 0.2580491 2 The calculations for the best may be checked using R> out$Subsets (Intercept) 0 TRUE 1 TRUE 2* TRUE 3 TRUE 4 TRUE 5 TRUE and R> out$qTable Load Xray BedDays AreaPop Stay logLikelihood BICq FALSE FALSE FALSE FALSE FALSE -146.0833 292.1667 FALSE FALSE TRUE FALSE FALSE -115.6360 236.3024 FALSE TRUE TRUE FALSE FALSE -109.3540 228.7688 FALSE TRUE TRUE FALSE TRUE -106.8812 228.8538 FALSE TRUE TRUE TRUE TRUE -106.2205 232.5627 TRUE TRUE TRUE TRUE TRUE -106.2001 237.5525 17 18 [1,] [2,] [3,] [4,] [5,] [6,] bestglm: Best Subset GLM LogL -146.0833 -115.6360 -109.3540 -106.8812 -106.2205 -106.2001 q1 0.000000e+00 2.466916e-13 7.649929e-03 2.580491e-01 6.804510e-01 8.015913e-01 q2 2.466916e-13 7.649929e-03 2.580491e-01 6.804510e-01 8.015913e-01 1.000000e+00 k 0 1 2 3 4 5 4.2. South African Heart Disease The response variable, chd, indicates the presence or absence of coronary heart disease and there are nine inputs. The sample size is 462. Logistic regression is used. The full model is, R> data(SAheart) R> out <- bestglm(SAheart, IC = "BICq", t = 1, family = binomial) Note: in this special case with BICq with t = 1 only fitted model is returned. With t=1, full model is fitted. R> out BICq(q = 1) Best Model: Estimate (Intercept) -6.1507208650 sbp 0.0065040171 tobacco 0.0793764457 ldl 0.1739238981 adiposity 0.0185865682 famhistPresent 0.9253704194 typea 0.0395950250 obesity -0.0629098693 alcohol 0.0001216624 age 0.0452253496 Std. Error z value Pr(>|z|) 1.308260018 -4.70145138 2.583188e-06 0.005730398 1.13500273 2.563742e-01 0.026602843 2.98375801 2.847319e-03 0.059661738 2.91516648 3.554989e-03 0.029289409 0.63458325 5.257003e-01 0.227894010 4.06052980 4.896149e-05 0.012320227 3.21382267 1.309805e-03 0.044247743 -1.42176449 1.550946e-01 0.004483218 0.02713729 9.783502e-01 0.012129752 3.72846442 1.926501e-04 We ﬁnd that the bounding interval for is 0.191 ≤ ≤ 0.901. For values of in this interval a model with 5 inputs: tobacco, ldl, famhist, typea and age and as expected all variables have very low -values. Using in the interval 0.094 < < 0.190 results in a subset of the above model which excludes ldl. Using cross-validation Hastie et al. (2009, §4.4.2) also selected a model for this data with only four inputs but their subset excluded typea instead of ldl. It is interesting that the subset chosen in Hastie et al. (2009, Section 4.4.2) may be found using two other suboptimal procedures. First using the bic with = 0.25 and the R function step, A. I. McLeod, C. Xu R> R> R> R> R> ans <- glm(chd ˜ ., data = SAheart) q <- 0.25 n <- nrow(SAheart) k <- log(n) - 2 * log(q/(1 - q)) step(ans, k = k) Start: AIC=585.74 chd ˜ sbp + tobacco + ldl + adiposity + famhist + typea + obesity + alcohol + age Df Deviance AIC - alcohol 1 79.919 577.49 - adiposity 1 79.945 577.64 - sbp 1 80.187 579.04 - obesity 1 80.350 579.98 <none> 79.904 585.74 - typea 1 81.480 586.43 - ldl 1 81.612 587.18 - tobacco 1 81.962 589.15 - age 1 82.002 589.38 - famhist 1 83.025 595.11 Step: AIC=577.49 chd ˜ sbp + tobacco + ldl + adiposity + famhist + typea + obesity + age - adiposity - sbp - obesity <none> - typea - ldl - tobacco - age - famhist Df Deviance AIC 1 79.957 569.38 1 80.192 570.73 1 80.362 571.71 79.919 577.49 1 81.483 578.11 1 81.677 579.21 1 81.979 580.92 1 82.035 581.23 1 83.025 586.78 Step: AIC=569.38 chd ˜ sbp + tobacco + ldl + famhist + typea + obesity + age - sbp - obesity <none> - typea - ldl - tobacco Df Deviance AIC 1 80.248 562.73 1 80.490 564.12 79.957 569.38 1 81.491 569.83 1 81.921 572.26 1 82.025 572.84 19 20 bestglm: Best Subset GLM - famhist - age 1 1 83.063 578.65 83.232 579.59 Step: AIC=562.73 chd ˜ tobacco + ldl + famhist + typea + obesity + age - obesity <none> - typea - ldl - tobacco - famhist - age Df Deviance AIC 1 80.686 556.91 80.248 562.73 1 81.736 562.88 1 82.223 565.62 1 82.396 566.59 1 83.331 571.81 1 84.416 577.78 Step: AIC=556.91 chd ˜ tobacco + ldl + famhist + typea + age Df Deviance AIC - typea 1 82.043 556.28 <none> 80.686 556.91 - ldl 1 82.322 557.84 - tobacco 1 82.867 560.90 - famhist 1 83.725 565.66 - age 1 84.483 569.82 Step: AIC=556.28 chd ˜ tobacco + ldl + famhist + age <none> - ldl - tobacco - age - famhist Call: Df Deviance 82.043 1 83.914 1 84.351 1 85.309 1 85.368 AIC 556.28 558.36 560.76 565.98 566.30 glm(formula = chd ˜ tobacco + ldl + famhist + age, data = SAheart) Coefficients: (Intercept) -0.237407 tobacco 0.017263 ldl 0.032533 famhistPresent 0.178173 age 0.006836 Degrees of Freedom: 461 Total (i.e. Null); 457 Residual Null Deviance: 104.6 Residual Deviance: 82.04 AIC: 524.6 Even with = 0.1 in the above script only tobacco, famhist and age are selected. And using A. I. McLeod, C. Xu 21 = 0.5 in the above script with step selects the same model the bicselects when exhaustive enumeration is done using bestglm. This example points out that using step for subset selection may produce a suboptimal answer. Yet another way that the four inputs selected by Hastie et al. (2009, Section 4.4.2) could be obtained is to use least squares with bestglm to ﬁnd the model with the best four inputs. R> out <- bestglm(SAheart, IC = "BICq", t = 0.25) Note: binary categorical variables converted to 0-1 so 'leaps' could be used. R> out$Subsets (Intercept) sbp tobacco ldl adiposity famhist typea obesity alcohol 0 TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE 1 TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE 2 TRUE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE 3 TRUE FALSE TRUE FALSE FALSE TRUE FALSE FALSE FALSE 4* TRUE FALSE TRUE TRUE FALSE TRUE FALSE FALSE FALSE 5 TRUE FALSE TRUE TRUE FALSE TRUE TRUE FALSE FALSE 6 TRUE FALSE TRUE TRUE FALSE TRUE TRUE TRUE FALSE 7 TRUE TRUE TRUE TRUE FALSE TRUE TRUE TRUE FALSE 8 TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE FALSE 9 TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE age logLikelihood BICq 0 FALSE 343.1572 -686.3144 1 TRUE 377.7581 -747.1834 2 TRUE 387.4922 -758.3188 3 TRUE 394.0337 -763.0691 4* TRUE 399.2435 -765.1559 5 TRUE 403.0944 -764.5248 6 TRUE 404.3510 -758.7053 7 TRUE 405.1909 -752.0524 8 TRUE 405.3023 -743.9423 9 TRUE 405.3439 -735.6928 5. Other Illustrative Examples 5.1. Nuclear Power Plant Data R> data(znuclear) R> bestglm(znuclear, IC = "AIC") AIC BICq equivalent for q in (0.349204366418954, 0.716418902103358) 22 bestglm: Best Subset GLM Best Model: Estimate Std. Error t value Pr(>|t|) (Intercept) -38.7480703 7.91826983 -4.893502 4.910313e-05 date 0.5620284 0.11445901 4.910303 4.701224e-05 capacity 0.4759804 0.07818015 6.088252 2.310934e-06 NE 0.6588957 0.19616044 3.358963 2.510375e-03 CT 0.3714664 0.15987847 2.323430 2.858187e-02 N -0.2277672 0.10786682 -2.111560 4.489115e-02 PT -0.5982476 0.30044058 -1.991235 5.748951e-02 5.2. Detroit Homicide Data Our analysis will use the six inputs which generate the lowest residual sum of squares. These inputs are 1, 2, 4, 6, 7 and 11 as given in Miller (2002, Table 3.14). We have scaled the inputs, although this is not necessary in this example. Using backward step-wise regression in R, no variables are removed. But note that variables 1, 6 and 7 are all only signiﬁcant at about 5%. Bearing in mind the selection eﬀect, the true signiﬁcance is much less. R> R> R> R> R> R> data(Detroit) X <- as.data.frame(scale(Detroit[, c(1, 2, 4, 6, 7, 11)])) y <- Detroit[, ncol(Detroit)] Xy <- cbind(X, HOM = y) out <- lm(HOM ˜ ., data = Xy) step(out, k = log(nrow(Xy))) Start: AIC=-11.34 HOM ˜ FTP.1 + UEMP.2 + LIC.4 + CLEAR.6 + WM.7 + WE.11 Df Sum of Sq <none> - WM.7 - CLEAR.6 - FTP.1 - WE.11 - UEMP.2 - LIC.4 1 1 1 1 1 1 1.2724 1.3876 1.4376 8.1170 16.3112 20.6368 RSS AIC 1.3659 -11.3357 2.6383 -5.3427 2.7535 -4.7871 2.8035 -4.5533 9.4830 11.2888 17.6771 19.3849 22.0027 22.2305 Call: lm(formula = HOM ˜ FTP.1 + UEMP.2 + LIC.4 + CLEAR.6 + WM.7 + Coefficients: (Intercept) 25.127 WE.11 6.084 FTP.1 1.724 UEMP.2 2.570 Same story with exhaustive search algorithm. LIC.4 5.757 CLEAR.6 -2.329 WE.11, data = Xy) WM.7 -2.452 A. I. McLeod, C. Xu 23 R> out <- bestglm(Xy, IC = "BIC") R> out BIC BICq equivalent for q in (0.115398370069662, 1) Best Model: Estimate Std. Error t value Pr(>|t|) (Intercept) 25.126923 0.1323333 189.875990 1.439772e-12 FTP.1 1.724110 0.6861084 2.512883 4.572467e-02 UEMP.2 2.569527 0.3035648 8.464511 1.485656e-04 LIC.4 5.757015 0.6046682 9.520948 7.657697e-05 CLEAR.6 -2.329338 0.9435019 -2.468822 4.853518e-02 WM.7 -2.452200 1.0372544 -2.364126 5.596776e-02 WE.11 6.083694 1.0188489 5.971144 9.892298e-04 We can use BICq to reduce the number of variables. The qTable let’s choose q for other possible models. R> out$qTable [1,] [2,] [3,] [4,] [5,] [6,] LogL -35.832829 -17.767652 -6.215995 4.237691 8.006726 14.645170 q1 0.000000e+00 5.144759e-08 3.468452e-05 1.039797e-04 7.680569e-02 1.153984e-01 q2 5.144759e-08 3.468452e-05 1.039797e-04 7.680569e-02 1.153984e-01 1.000000e+00 k 0 1 2 3 4 6 This suggest we try q=0.05 R> bestglm(Xy, IC = "BICq", t = 0.05) BICq(q = 0.05) BICq equivalent for q in (0.000103979673982901, 0.0768056921650389) Best Model: Estimate Std. Error t value Pr(>|t|) (Intercept) 25.12692 0.2406075 104.43119 3.435051e-15 UEMP.2 3.38307 0.2601848 13.00257 3.876404e-07 LIC.4 8.20378 0.2802445 29.27365 3.090409e-10 WE.11 10.90084 0.2787164 39.11089 2.321501e-11 Or q=0.0005. R> bestglm(Xy, IC = "BICq", t = 5e-05) 24 bestglm: Best Subset GLM BICq(q = 5e-05) BICq equivalent for q in (3.46845195655643e-05, 0.000103979673982901) Best Model: Estimate Std. Error t value Pr(>|t|) (Intercept) 25.126923 0.5101048 49.258354 2.871539e-13 LIC.4 4.473245 0.6381795 7.009384 3.673796e-05 CLEAR.6 -13.386666 0.6381795 -20.976334 1.346067e-09 The above results agree with Miller (2002, Table 3.14). It is interesting that the subset model of size 2 is not a subset itself of the size 3 model. It is clear that simply adding and/or dropping one variable at a time as in the stepwise and stagewise algorithms will not work in moving either from model 2 to model 3 or vice-versa. Using delete-d CV with d=4 suggests variables 2,4,6,11 R> set.seed(1233211) R> bestglm(Xy, IC = "CV", CVArgs = list(Method = "d", K = 4, REP = 50)) CVd(d = 4, REP = 50) BICq equivalent for q in (0.0768056921650389, 0.115398370069661) Best Model: Estimate Std. Error t value Pr(>|t|) (Intercept) 25.126923 0.1909731 131.573114 1.244969e-14 UEMP.2 2.571151 0.3840754 6.694391 1.535921e-04 LIC.4 7.270181 0.4337409 16.761574 1.624771e-07 CLEAR.6 -3.250371 1.2964006 -2.507227 3.652839e-02 WE.11 8.329213 1.0492726 7.938083 4.617821e-05 5.3. Air Quality Data Here is an example of a dataset with categorical variables at more than 2 levels. First we look at the full model, R> data(AirQuality) R> bestglm(AirQuality, IC = "BICq", t = 1) Note: in this special case with BICq with t = 1 only fitted model is returned. With t=1, full model is fitted. BICq(q = 1) Best Model: Df Sum Sq Mean Sq F value Pr(>F) Solar.R 1 14780 14780 31.9857 1.815e-07 *** Wind 1 39969 39969 86.5007 8.147e-15 *** Temp 1 19050 19050 41.2273 6.239e-09 *** month 11 3713 338 0.7305 0.7066 weekday 6 2703 451 0.9750 0.4469 Residuals 90 41586 462 --Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 A. I. McLeod, C. Xu 25 Next we ﬁnd the best AIC model, R> bestglm(AirQuality, IC = "AIC") Morgan-Tatar search since factors present with more than 2 levels. AIC Best Model: Df Sum Sq Mean Sq F value Pr(>F) Solar.R 1 14780 14780 32.944 8.946e-08 *** Wind 1 39969 39969 89.094 9.509e-16 *** Temp 1 19050 19050 42.463 2.424e-09 *** Residuals 107 48003 449 --Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 5.4. Forest Fires The forest ﬁre data were collected during January 2000 to December 2003 for ﬁres in the Montesinho natural park located in the Northeast region of Portugal. The response variable of interest was area burned in ha. When the area burned as less than one-tenth of a hectare, the response variable as set to zero. In all there were 517 ﬁres and 247 of them recorded as zero. stered, such the time,and date,Morais 'V atiallocation 9 X9 ﬁt gn d (" and yams The datasetwere wasregtprovided byasCortez (2007)within who aalso this data using neural of Figure 2), the type of vegetation involved, the SiX components of the FWI system nets and support machines. total burned ,.-ea. The second database wa; collected by the Brag,."a Polyand the vector technic Institute, containing several weather observations (eg wind speed) that were The region recorded was divided a 10-by-10 coordinates X loc<ted and Ymrunning from 1 to 9 with ainto 30 mmute penod grid by a with meteorologtcal station the center of the Montesmho park. The two databases were stored m tens ofmdividual 'Vreadas shown insheets, the diagram below. Theand categorical variable region in this under distinct a substantial manual xyarea effort wa;indicates perfonnedthe to intemto a smgle datasetWlth a total of 517 entries. This d<tais availille at grid for thegrate ﬁre.them There are 36 diﬀerent regions so xyarea has 35 df. fonn<t~ http.llwww.dsi.uminho.pt/~pcortez/forestfires/ F~2. The map oftheMonte>inho n.tural pork Figure 2: Montesinho Park Table I shows a descnption ofthe selected data features. The first four rows denote the spatial and temporal attributes. Only two geogr~hic features were mc1uded, the X ,.,d Y axlS values where the fire occurred, Slnce the type of vegetation presented a low quality (i. e. more than 80% 0f the values were mlSsmg). After consulting the M ontesmho fire mspector, we selected the month,.,d day of the week temporal vanables Average monthly weather conditions are quite di stinc~ vJ1ile the day 0f the week coul d also mfluence forest fires (e.g. work days vs weekend) smce most fires have a human cause. Next come the four FWI components thit are affected directly by the weather conditions (Figure I, m bold). The BUI and FWI were discarded smce they,.-e dependent of the previous values. From the meteorological station d<tabase, we selected the four weather attributes used by the FWI system. In contrast with the time lags used by FWI, m this case the values denote mst,.,trecords, as given by the station sensors when the fire was detected. The exception lS the rain vanable, which denotes the occumulated preapilati on within the prevIOus 30 mmutes Fitting the best-AIC regression, R> data(Fires) R> bestglm(Fires, IC = "AIC") 26 bestglm: Best Subset GLM Morgan-Tatar search since factors present with more than 2 levels. AIC Best Model: Df Sum Sq Mean Sq F value Pr(>F) month 11 37.37 3.3970 1.7958 0.05195 . DMC 1 6.31 6.3145 3.3381 0.06829 . DC 1 4.85 4.8468 2.5622 0.11008 temp 1 8.92 8.9165 4.7136 0.03039 * wind 1 3.94 3.9384 2.0820 0.14967 Residuals 501 947.72 1.8917 --Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 6. Simulated Data 6.1. Null Regression Example Here we check that our function handles the null regression case where there are no inputs to include in the model. We assume an intercept term only. R> R> R> R> R> set.seed(123312123) X <- as.data.frame(matrix(rnorm(50), ncol = 2, nrow = 25)) y <- rnorm(25) Xy <- cbind(X, y = y) bestglm(Xy) BIC BICq equivalent for q in (0, 0.540989544689166) Best Model: Estimate Std. Error t value Pr(>|t|) (Intercept) -0.3074955 0.2323344 -1.323504 0.1981378 6.2. Logistic Regression As a check we simulate a logistic regression with = 10 inputs. The inputs are all Gaussian white noise with unit variance. So the model equation may be written, is IID Bernouilli distribution with parameter , = ℰ( ) = ℎ(0 + 1 1 + . . . + ) where ℎ() = (1 + − )−1 . Note that ℎ is the inverse of the logit transformation and it may coveniently obtained in R using plogist. In the code below we set 0 = = −1 and 1 = 3, 2 = 2, 3 = 4/3, 4 = 2 32 and = 0, = 5, . . . , 10. Taking = 500 as the sample size we ﬁnd after ﬁt with glm. R> set.seed(231231) R> n <- 500 A. I. McLeod, C. Xu R> R> R> R> R> R> R> R> R> R> 27 K <- 10 a <- -1 b <- c(c(9, 6, 4, 8)/3, rep(0, K - 4)) X <- matrix(rnorm(n * K), ncol = K) L <- a + X %*% b p <- plogis(L) Y <- rbinom(n = n, size = 1, prob = p) X <- as.data.frame(X) out <- glm(Y ˜ ., data = X, family = binomial) summary(out) Call: glm(formula = Y ˜ ., family = binomial, data = X) Deviance Residuals: Min 1Q Median -2.80409 -0.28120 -0.02809 3Q 0.25338 Max 2.53513 Coefficients: Estimate Std. Error z value Pr(>|z|) (Intercept) -0.882814 0.182554 -4.836 1.33e-06 *** V1 3.186252 0.325376 9.793 < 2e-16 *** V2 1.874106 0.242864 7.717 1.19e-14 *** V3 1.500606 0.215321 6.969 3.19e-12 *** V4 2.491092 0.281585 8.847 < 2e-16 *** V5 0.029539 0.165162 0.179 0.858 V6 -0.179920 0.176994 -1.017 0.309 V7 -0.047183 0.172862 -0.273 0.785 V8 -0.121629 0.168903 -0.720 0.471 V9 -0.229848 0.161735 -1.421 0.155 V10 -0.002419 0.177972 -0.014 0.989 --Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 (Dispersion parameter for binomial family taken to be 1) Null deviance: 685.93 Residual deviance: 243.86 AIC: 265.86 on 499 on 489 degrees of freedom degrees of freedom Number of Fisher Scoring iterations: 7 6.3. Binomial Regression As a further check we ﬁt a binomial regression taking = 500 with = 10 inputs and with Bernouilli number of trials = 100. So in this case the model equation may be 28 bestglm: Best Subset GLM written, is IID binomially distributed with number of trials = 10 and parameter , = ℰ( ) = ℎ(0 + 1 1 + . . . + ) where ℎ() = (1 + − )−1 . We used the same ’s as in Section 6.2. R> R> R> R> R> R> R> R> R> R> R> R> R> R> R> set.seed(231231) n <- 500 K <- 8 m <- 100 a <- 2 b <- c(c(9, 6, 4, 8)/10, rep(0, K - 4)) X <- matrix(rnorm(n * K), ncol = K) L <- a + X %*% b p <- plogis(L) Y <- rbinom(n = n, size = m, prob = p) Y <- cbind(Y, m - Y) dimnames(Y)[[2]] <- c("S", "F") X <- as.data.frame(X) out <- glm(Y ˜ ., data = X, family = binomial) summary(out) Call: glm(formula = Y ˜ ., family = binomial, data = X) Deviance Residuals: Min 1Q -2.77988 -0.70691 Median 0.07858 3Q 0.75158 Max 2.70323 Coefficients: Estimate Std. Error z value Pr(>|z|) (Intercept) 2.025629 0.016967 119.383 <2e-16 *** V1 0.898334 0.015175 59.197 <2e-16 *** V2 0.607897 0.012987 46.809 <2e-16 *** V3 0.429355 0.013609 31.551 <2e-16 *** V4 0.835002 0.014962 55.807 <2e-16 *** V5 -0.006607 0.013867 -0.476 0.634 V6 -0.011497 0.013596 -0.846 0.398 V7 0.022112 0.013660 1.619 0.105 V8 0.000238 0.013480 0.018 0.986 --Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 (Dispersion parameter for binomial family taken to be 1) Null deviance: 10752.23 Residual deviance: 507.19 AIC: 2451.8 on 499 on 491 degrees of freedom degrees of freedom A. I. McLeod, C. Xu 29 Number of Fisher Scoring iterations: 4 In this example, one input V6 is signﬁcant at level 0.03 even though its correct coeﬃcient is zero. R> Xy <- cbind(X, Y) R> bestglm(Xy, family = binomial) Morgan-Tatar search since family is non-gaussian. BIC BICq equivalent for q in (0, 0.870630550022155) Best Model: Estimate Std. Error z value Pr(>|z|) (Intercept) 2.0247237 0.01689493 119.84211 0.000000e+00 V1 0.8995804 0.01513694 59.42948 0.000000e+00 V2 0.6063199 0.01287612 47.08872 0.000000e+00 V3 0.4290062 0.01360140 31.54132 2.358219e-218 V4 0.8349437 0.01485556 56.20412 0.000000e+00 Using the default selection method, BIC, the correct model is selected. 6.4. Binomial Regression With Factor Variable An additional check was done to incorporate a factor variable. We include a factor input representing the day-of-week eﬀect. The usual corner-point method was used to parameterize this variable and large coeﬃcients chosen, so that this factor would have a strong eﬀect. Using the corner-point method, means that the model matrix will have six additional columns of indicator variables. We used four more columns of numeric variables and then added the six columns for the indicators to simulate the model. R> R> R> R> R> R> + R> R> R> R> R> R> R> R> R> set.seed(33344111) n <- 500 K <- 4 m <- 100 a <- 2 dayNames <- c("Sunday", "Monday", "Tuesday", "Wednesday", "Friday", "Saturday") Days <- data.frame(d = factor(rep(dayNames, n))[1:n]) Xdays <- model.matrix(˜d, data = Days) bdays <- c(7, 2, -7, 0, 2, 7)/10 Ldays <- Xdays %*% bdays b <- c(c(9, 6)/10, rep(0, K - 2)) X <- matrix(rnorm(n * K), ncol = K) L <- a + X %*% b L <- L + Ldays p <- plogis(L) 30 R> R> R> R> R> R> R> bestglm: Best Subset GLM Y <- rbinom(n = n, size = m, prob = p) Y <- cbind(Y, m - Y) dimnames(Y)[[2]] <- c("S", "F") X <- as.data.frame(X) X <- data.frame(X, days = Days) out <- glm(Y ˜ ., data = X, family = binomial) anova(out, test = "Chisq") Analysis of Deviance Table Model: binomial, link: logit Response: Y Terms added sequentially (first to last) Df Deviance Resid. Df Resid. Dev P(>|Chi|) NULL 499 5609.2 V1 1 2992.36 498 2616.9 <2e-16 *** V2 1 1253.89 497 1363.0 <2e-16 *** V3 1 2.49 496 1360.5 0.1145 V4 1 1.91 495 1358.6 0.1668 d 5 797.44 490 561.1 <2e-16 *** --Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 After ﬁtting with glm we ﬁnd the results as expected. The factor variable is highly signﬁcant as well as the ﬁrst two quantiative variables. Using bestglm, we ﬁnd it selects the correct model. R> Xy <- cbind(X, Y) R> out <- bestglm(Xy, IC = "BICq", family = binomial) Morgan-Tatar search since family is non-gaussian. Note: factors present with more than 2 levels. R> out BICq(q = 0.25) Best Model: Response S : Df Sum Sq Mean Sq F value Pr(>F) V1 1 24841.5 24841.5 792.61 < 2.2e-16 *** V2 1 9110.0 9110.0 290.67 < 2.2e-16 *** d 5 7473.4 1494.7 47.69 < 2.2e-16 *** A. I. McLeod, C. Xu Residuals 492 15419.9 31.3 --Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Response F : Df Sum Sq Mean Sq F value V1 1 24841.5 24841.5 792.61 V2 1 9110.0 9110.0 290.67 d 5 7473.4 1494.7 47.69 Residuals 492 15419.9 31.3 --Signif. codes: 0 '***' 0.001 '**' 0.01 Pr(>F) < 2.2e-16 *** < 2.2e-16 *** < 2.2e-16 *** '*' 0.05 '.' 0.1 ' ' 1 6.5. Poisson Regression R> R> R> R> R> R> R> R> R> R> R> R> set.seed(231231) n <- 500 K <- 4 a <- -1 b <- c(c(1, 0.5), rep(0, K - 2)) X <- matrix(rnorm(n * K), ncol = K) L <- a + X %*% b lambda <- exp(L) Y <- rpois(n = n, lambda = lambda) X <- as.data.frame(X) out <- glm(Y ˜ ., data = X, family = poisson) summary(out) Call: glm(formula = Y ˜ ., family = poisson, data = X) Deviance Residuals: Min 1Q Median -2.1330 -0.8084 -0.4853 3Q 0.4236 Max 2.6320 Coefficients: Estimate Std. Error z value Pr(>|z|) (Intercept) -0.92423 0.07933 -11.650 <2e-16 *** V1 0.97841 0.05400 18.119 <2e-16 *** V2 0.51967 0.05707 9.107 <2e-16 *** V3 0.03773 0.05525 0.683 0.495 V4 0.03085 0.04646 0.664 0.507 --Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 31 32 bestglm: Best Subset GLM (Dispersion parameter for poisson family taken to be 1) Null deviance: 880.73 Residual deviance: 427.49 AIC: 913.41 on 499 on 495 degrees of freedom degrees of freedom Number of Fisher Scoring iterations: 5 As expected the ﬁrst two variables are highly signﬁcant and the next two are not. R> Xy <- data.frame(X, y = Y) R> bestglm(Xy, family = poisson) Morgan-Tatar search since family is non-gaussian. BIC BICq equivalent for q in (0, 0.947443940310683) Best Model: Estimate Std. Error z value Pr(>|z|) (Intercept) -0.9292265 0.07955966 -11.679619 1.620199e-31 V1 0.9897770 0.05284046 18.731421 2.744893e-78 V2 0.5302822 0.05588287 9.489171 2.328782e-21 Our function bestglm selects the correct model. 6.6. Gamma Regression To simulate a Gamma regression we ﬁrst write a function GetGammaParameters that translates mean and standard deviation into the shape and scale parameters for the function rgamma. R> GetGammaParameters <- function(muz, sdz) { + phi <- (sdz/muz)ˆ2 + nu <- 1/phi + lambda <- muz/nu + list(shape = nu, scale = lambda) + } R> set.seed(321123) R> test <- rnorm(20) R> n <- 500 R> b <- c(0.25, 0.5, 0, 0) R> b0 <- 0.3 R> K <- length(b) R> sdz <- 1 R> X <- matrix(rnorm(n * K), ncol = K) R> L <- b0 + X %*% b R> muHat <- exp(L) R> gp <- GetGammaParameters(muHat, sdz) R> zsim <- rgamma(n, shape = gp$shape, scale = gp$scale) A. I. McLeod, C. Xu 33 R> Xy <- data.frame(as.data.frame.matrix(X), y = zsim) R> out <- glm(y ˜ ., data = Xy, family = Gamma(link = log)) R> summary(out) Call: glm(formula = y ˜ ., family = Gamma(link = log), data = Xy) Deviance Residuals: Min 1Q Median -5.6371 -0.7417 -0.1968 3Q 0.2237 Max 3.2105 Coefficients: Estimate Std. Error t value (Intercept) 0.313964 0.044124 7.115 V1 0.191957 0.040983 4.684 V2 0.558321 0.042485 13.142 V3 0.018709 0.044939 0.416 V4 0.004252 0.043367 0.098 --Signif. codes: 0 '***' 0.001 '**' 0.01 Pr(>|t|) 3.93e-12 *** 3.64e-06 *** < 2e-16 *** 0.677 0.922 '*' 0.05 '.' 0.1 ' ' 1 (Dispersion parameter for Gamma family taken to be 0.944972) Null deviance: 792.56 Residual deviance: 621.67 AIC: 1315.9 on 499 on 495 degrees of freedom degrees of freedom Number of Fisher Scoring iterations: 9 R> bestglm(Xy, family = Gamma(link = log)) Morgan-Tatar search since family is non-gaussian. BIC BICq equivalent for q in (0.000599916119599198, 0.953871171759292) Best Model: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.3110431 0.04353445 7.144757 3.229210e-12 V1 0.1931868 0.04098936 4.713096 3.172312e-06 V2 0.5560244 0.04226866 13.154531 4.011556e-34 As expected, bestglm selects the correct model. 7. Simulation Experiment Please see the separate vignette Xu and McLeod (2009) for a discussion of how the simulation experiment reported in Xu and McLeod (2010, Table 2) was carried out as well for more 34 bestglm: Best Subset GLM detailed results of the simulation results themselves. The purpose of the simulation experiment reported on in Xu and McLeod (2010, Table 2) and described in more detail in the accompanying vignette Xu and McLeod (2009) was to compare diﬀerent information criteria used in model selection. Similar simulation experiments were used by Shao (1993) to compare cross-valiation criteria for linear model selection. In the simulation experiment reported by Shao (1993), the performance of various CV methods for linear model selection were investigated for the linear regression, = 2 + 2 2 + 3 3 + 4 4 + 5 5 + , (8) where nid(0, 1). A ﬁxed sample size of = 40 was used and the design matrix used is given in (Shao 1993, Table 1) and the four diﬀerent values of the ’s are shown in the table below, Experiment 2 3 4 5 1 0 0 4 0 2 0 0 4 8 3 9 0 4 8 4 9 6 4 8 The table below summarizes the probability of correct model selection in the experiment reported by Shao (1993, Table 2). Three model selection methods are compared: LOOCV (leave-one-out CV), CV(d=25) or the delete-d method with d=25 and APCV which is a very eﬃcient computation CV method but specialized to the case of linear regression. Experiment LOOCV CV(d=25) APCV 1 0.484 0.934 0.501 2 0.641 0.947 0.651 3 0.801 0.965 0.818 4 0.985 0.948 0.999 The CV(d=25) outperforms LOOCV in all cases and it also outforms APCV by a large margin in Experiments 1, 2 and 3 but in case 4 APCV is slightly better. In the code below we show how to do our own experiments to compare model selection using the bic, bic and bic criteria. R> testCorrect <- function(ans, NB) { + NBfit <- names(coef(ans))[-1] + ans <- ifelse(length(NBfit) == length(NB) & (!any(is.na(match(NBfit, + NB)))), 1, 0) + ans + } R> NSIM <- 5 R> data(Shao) R> set.seed(123321123) R> X <- as.matrix.data.frame(Shao) R> BETA <- list(b1 = c(0, 0, 4, 0), b2 = c(0, 0, 4, 8), b3 = c(9, + 0, 4, 8), b4 = c(9, 6, 4, 8)) R> NamesBeta <- list(b1 = c("x4"), b2 = c("x4", "x5"), b3 = c("x2", + "x4", "x5"), b4 = c("x2", "x3", "x4", "x5")) R> hitsBIC <- hitsEBIC <- hitsQBIC <- numeric(4) A. I. McLeod, C. Xu 35 R> startTime <- proc.time()[1] R> for (iB in 1:4) { + b <- BETA[[iB]] + NB <- NamesBeta[[iB]] + for (iSIM in 1:NSIM) { + y <- 2 + X %*% b + rnorm(40) + Xy <- cbind(Shao, y) + hitsBIC[iB] <- hitsBIC[iB] + testCorrect(bestglm(Xy, + IC = "BIC")$BestModel, NB) + hitsEBIC[iB] <- hitsEBIC[iB] + testCorrect(bestglm(Xy, + IC = "BICg")$BestModel, NB) + hitsQBIC[iB] <- hitsQBIC[iB] + testCorrect(bestglm(Xy, + IC = "BICq")$BestModel, NB) + } + } R> endTime <- proc.time()[1] R> totalTime <- endTime - startTime R> ans <- matrix(c(hitsBIC, hitsEBIC, hitsQBIC), byrow = TRUE, ncol = 4) R> dimnames(ans) <- list(c("BIC", "BICg", "BICq"), 1:4) R> ans <- t(ans)/NSIM R> ans 1 2 3 4 BIC BICg BICq 0.6 0.8 0.8 1.0 0.8 1.0 1.0 0.8 1.0 1.0 1.0 1.0 R> totalTime user.self 1.01 Increasing the number of simulations so NSIM=10000, the following result was obtained, 1 2 3 4 BIC 0.8168 0.8699 0.9314 0.9995 BICg 0.8666 0.7741 0.6312 0.9998 BICq 0.9384 0.9566 0.9761 0.9974 8. Controlling Type 1 Error Rate Consider the case where there are input variables and it we wish to test the null hypothesis ℋ0 : the output is not related to any inputs. By adjusting in the bic criterion, we can control the Type 1 error rate. Using simulation, we can determine for any particular and 36 bestglm: Best Subset GLM , what value of is needed to achieve a Type 1 error rate for a particular level, such as = 0.05. We compare the performance of information selection criteria in the case of a null model with = 25 inputs and = 30 observations. Using 50 simulations takes about 30 seconds. Since there is no relation between the inputs and the output, the correct choice is the null model with no parameters. Using the BICq criterion with = 0.05 works better than AIC, BIC or BICg. We may consider the number of parameters selected as the frequency of Type 1 errors in an hypothesis testing framework. By adjusting we may adjust the Type 1 error rate to any desired level. This suggests a possible bootstrapping approach to the problem of variable selection. R> set.seed(123321123) R> startTime <- proc.time()[1] R> NSIM <- 5 R> p <- 25 R> n <- 30 R> ans <- numeric(4) R> names(ans) <- c("AIC", "BIC", "BICg", "BICq") R> for (iSim in 1:NSIM) { + X <- matrix(rnorm(n * p), ncol = p) + y <- rnorm(n) + Xy <- as.data.frame(cbind(X, y)) + names(Xy) <- c(paste("X", 1:p, sep = ""), "y") + bestAIC <- bestglm(Xy, IC = "AIC") + bestBIC <- bestglm(Xy, IC = "BIC") + bestEBIC <- bestglm(Xy, IC = "BICg") + bestQBIC <- bestglm(Xy, IC = "BICq", t = 0.05) + ans[1] <- ans[1] + length(coef(bestAIC$BestModel)) - 1 + ans[2] <- ans[2] + length(coef(bestBIC$BestModel)) - 1 + ans[3] <- ans[3] + length(coef(bestEBIC$BestModel)) - 1 + ans[4] <- ans[4] + length(coef(bestQBIC$BestModel)) - 1 + } R> endTime <- proc.time()[1] R> totalTime <- endTime - startTime R> totalTime user.self 4.52 R> ans AIC 58 BIC BICg BICq 13 0 0 9. Concluding Remarks A. I. McLeod, C. 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