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A natural technique to select variables in the context of generalized linear models is to use a stepŵise procedure. It is natural, but contreversial, as discussed by Frank Harrell  in a great post, clearly worth reading. Frank mentioned about 10 points against a stepwise procedure.

• It yields R-squared values that are badly biased to be high.
• The F and chi-squared tests quoted next to each variable on the printout do not have the claimed distribution.
• The method yields confidence intervals for effects and predicted values that are falsely narrow (see Altman and Andersen (1989)).
• It yields p-values that do not have the proper meaning, and the proper correction for them is a difficult problem.
• It gives biased regression coefficients that need shrinkage (the coefficients for remaining variables are too large (see Tibshirani (1996)).
• It has severe problems in the presence of collinearity.
• It is based on methods (e.g., F tests for nested models) that were intended to be used to test prespecified hypotheses.
• Increasing the sample size does not help very much (see Derksen and Keselman (1992)).
• It allows us to not think about the problem.
• It uses a lot of paper.

In order to illustrate that issue of variable selection, consider a dataset I’ve been using many times on the blog,

MYOCARDE=read.table(
"http://freakonometrics.free.fr/saporta.csv",
head=TRUE,sep=";")

where we have observations from people entering E.R., because of a (potential) infarctus, and we want to understand who did survive, and to build a predictive model.

What if we use a forward stepwise logistic regression here? I want to use a forward construction since it usually yields to models with less explanatory variables. We can use Akaike Information Criterion

> reg0=glm(PRONO~1,data=MYOCARDE,family=binomial)
> reg1=glm(PRONO~.,data=MYOCARDE,family=binomial)
> step(reg0,scope=formula(reg1),
+ direction="forward",k=2)                  # AIC
Start:  AIC=98.03
PRONO ~ 1

Df Deviance    AIC
+ REPUL  1   46.884 50.884
+ INSYS  1   51.865 55.865
+ INCAR  1   53.313 57.313
+ PRDIA  1   78.503 82.503
+ PAPUL  1   82.862 86.862
+ PVENT  1   87.093 91.093
96.033 98.033
+ FRCAR  1   94.861 98.861

Step:  AIC=50.88
PRONO ~ REPUL

Df Deviance    AIC
+ INCAR  1   44.530 50.530
+ PVENT  1   44.703 50.703
+ INSYS  1   44.857 50.857
46.884 50.884
+ PAPUL  1   45.274 51.274
+ PRDIA  1   46.322 52.322
+ FRCAR  1   46.540 52.540

Step:  AIC=50.53
PRONO ~ REPUL + INCAR

Df Deviance    AIC
44.530 50.530
+ PVENT  1   43.134 51.134
+ PRDIA  1   43.772 51.772
+ INSYS  1   44.305 52.305
+ PAPUL  1   44.341 52.341
+ FRCAR  1   44.521 52.521

Call:  glm(formula = PRONO ~ REPUL + INCAR, family = binomial, data = MYOCARDE)

Coefficients:
(Intercept)        REPUL
1.633668    -0.003564
INCAR
1.618479

Degrees of Freedom: 70 Total (i.e. Null);  68 Residual
Null Deviance:	    96.03
Residual Deviance: 44.53 	AIC: 50.53
> step(reg0,scope=formula(reg1),
+ direction="forward",k=log(n))           # BIC
Start:  AIC=98.11
PRONO ~ 1

Df Deviance    AIC
+ REPUL  1   46.884 51.043
+ INSYS  1   51.865 56.024
+ INCAR  1   53.313 57.472
+ PRDIA  1   78.503 82.662
+ PAPUL  1   82.862 87.021
+ PVENT  1   87.093 91.252
96.033 98.113
+ FRCAR  1   94.861 99.020

Step:  AIC=51.04
PRONO ~ REPUL

Df Deviance    AIC
+ INCAR  1   44.530 50.768
+ PVENT  1   44.703 50.942
46.884 51.043
+ INSYS  1   44.857 51.095
+ PAPUL  1   45.274 51.512
+ PRDIA  1   46.322 52.561
+ FRCAR  1   46.540 52.778

Step:  AIC=50.77
PRONO ~ REPUL + INCAR

Df Deviance    AIC
44.530 50.768
+ PVENT  1   43.134 51.452
+ PRDIA  1   43.772 52.089
+ INSYS  1   44.305 52.623
+ PAPUL  1   44.341 52.659
+ FRCAR  1   44.521 52.838

Call:  glm(formula = PRONO ~ REPUL + INCAR, family = binomial, data = MYOCARDE)

Coefficients:
(Intercept)        REPUL
1.633668    -0.003564
INCAR
1.618479

Degrees of Freedom: 70 Total (i.e. Null);  68 Residual
Null Deviance:	    96.03
Residual Deviance: 44.53 	AIC: 50.53

With those two approaches, we have the same story: the most important variable (or say with the highest predictive value) is REPUL. And we can improve the model by adding INCAR. And that’s it. We can get a good model with those two covariates.

Now, what about using cross-validation here? We should keep in ming that AIC is asymptotically equivalent to One-Leave-Out Cross Validation (see Stone (1977)),  while BIC is equivalent to -fold Cross Validation (see Shao (1997)), where

• Using Leave-One-Out Cross Validation

In order to select the first variable, consider 7 logistic regression, each on a single different variable. Each time, we estimate the model on  observations and get a prediction on the remaining one,

on. Set . The function to get those values is

> name_var=names(MYOCARDE)
> pred_i=function(i,k){
+ fml = paste(name_var,"~",name_var[k],sep="")
+ reg=glm(fml,data=MYOCARDE[-i,],family=binomial)
+ predict(reg,newdata=MYOCARDE[i,],
+ type="response")
+ }

then for each variable , we get the ROC curve using and,

> library(AUC)
> ROC=function(k){
+  Y=MYOCARDE[,8]=="Survival"
+  S=Vectorize(function(i) pred_i(i,k))
+ (1:length(Y))
+  R=roc(S,as.factor(Y))
+  return(list(roc=cbind(R$fpr,R$tpr),
+              auc=AUC::auc(R)))
+ }

Here, for each variable, we compute the area under the curve (AUC criterion)

> AUC=rep(NA,7)
> for(k in 1:7){
+   AUC[k]=ROC(k)$auc + cat("Variable ",k,"(",name_var[k],") :", + AUC[k],"n") } Variable 1 ( FRCAR ) : 0.4934319 Variable 2 ( INCAR ) : 0.8965517 Variable 3 ( INSYS ) : 0.909688 Variable 4 ( PRDIA ) : 0.7487685 Variable 5 ( PAPUL ) : 0.7134647 Variable 6 ( PVENT ) : 0.6584565 Variable 7 ( REPUL ) : 0.9154351 But we can also visualize those curves, > plot(0:1,0:1,col="white",xlab="",ylab="") > for(k in 1:7) + lines(ROC(k)$roc,type="s",col=CL[k])
> legend(.8,.45,name_var,col=CL,lty=1,cex=.8) (there is no PRONO here, there is a typo  in the Legend)

where here colors were obtained using

> library(RColorBrewer)
> CL=brewer.pal(8, "Set1")[-7]

Here ROC curves were obtained using a Leave-one-Out strategy. And the best variable (if we should keep one, and one only) is

> k0=which.max(AUC)
> name_var[k0]
 "REPUL"

Now, consider a stepwise procedure: we keep that ‘best’ variable in our model, and we try to add another one.

> pred_i=function(i,k){
+ vk=c(k0,k)
+ fml = paste(name_var,"~",paste(name_var[vk],
+ collapse="+"),sep="")
+ reg=glm(fml,data=MYOCARDE[-i,],family=binomial)
+ predict(reg,newdata=MYOCARDE[i,],
+ type="response")
+ }
> library(AUC)
> ROC=function(k){
+ Y=MYOCARDE[,8]=="Survival"
+ S=Vectorize(function(i) pred_i(i,k))
+ (1:length(Y))
+ R=roc(S,as.factor(Y))
+ return(list(roc=cbind(R$fpr,R$tpr),
+ auc=AUC::auc(R)))
+ }
> plot(0:1,0:1,col="white",xlab="",ylab="")
> for(k in (1:7)[-k0]) lines(ROC(k)$roc,type="s",col=CL[k]) > segments(0,0,1,1,lty=2,col="grey") > legend(.8,.45, + name_var[-k0], + col=CL[-k0],lty=1,cex=.8) We were already quite good. And we might expect to find another variable that will increase the predictive power of our classifier. > AUC=rep(NA,7) > for(k in (1:7)[-k0]){ + AUC[k]=ROC(k)$auc
+  cat("Variable ",k,"(",name_var[k],") :",
+  AUC[k],"n")
+ }
Variable  1 ( FRCAR ) : 0.9064039
Variable  2 ( INCAR ) : 0.9195402
Variable  3 ( INSYS ) : 0.9187192
Variable  4 ( PRDIA ) : 0.9137931
Variable  5 ( PAPUL ) : 0.9187192
Variable  6 ( PVENT ) : 0.9137931

And, of course, we can move foreward, add another variable, etc,

> k0=c(k0,which.max(AUC))

> pred_i=function(i,k){
+   vk=c(k0,k)
+   fml = paste(name_var,"~",paste(
+ name_var[vk],collapse="+"),sep="")
+ reg=glm(fml,data=MYOCARDE[-i,],family=binomial)
+ predict(reg,newdata=MYOCARDE[i,],
+ type="response")
+ }
> library(AUC)
> ROC=function(k){
+  Y=MYOCARDE[,8]=="Survival"
+  S=Vectorize(function(i) pred_i(i,k))
+  (1:length(Y))
+  R=roc(S,as.factor(Y))
+  return(list(roc=cbind(R$fpr,R$tpr),
+  auc=AUC::auc(R)))
+ }
>
> plot(0:1,0:1,col="white",xlab="",ylab="")
> for(k in (1:7)[-k0]) lines(ROC(k)$roc,type="s",col=CL[k]) > segments(0,0,1,1,lty=2,col="grey") > legend(.8,.45,name_var[-k0], + col=CL[-k0],lty=1,cex=.8) But here, the gain is rather small (if any) > AUC=rep(NA,7) > for(k in (1:7)[-k0]){ + AUC[k]=ROC(k)$auc
+ cat("Variable ",k,"(",name_var[k],") :",
+ AUC[k],"n")
+ }
Variable  1 ( FRCAR ) : 0.9121511
Variable  3 ( INSYS ) : 0.9170772
Variable  4 ( PRDIA ) : 0.910509
Variable  5 ( PAPUL ) : 0.907225
Variable  6 ( PVENT ) : 0.909688

With that stepwise algorithm, the best strategy is to keep, first, REPUL, and then to add INCAR. Which is consistent with the stepwise procedure using Akaike Information Criterion.

An alternative could be to select the best pair among all possible pairs. It will be time consuming, but it can be used to avoid the stepwise drawback.

> pred_i=function(i,k){
+ fml = paste(name_var,"~",paste(name_var[
+ as.integer(k)],collapse="+"),sep="")
+ reg=glm(fml,data=MYOCARDE[-i,],family=binomial)
+ predict(reg,newdata=MYOCARDE[i,],
+ type="response")
+ }
> library(AUC)
> ROC=function(k){
+   Y=MYOCARDE[,8]=="Survival"
+   L=list()
+   n=length(Y)
+   nk=trunc(n/trunc(n/10))
+   for(i in 1:(nk-1)) L[[i]]=((i-1)*
+     trunc(n/10)+1:(n/10))
+   L[[nk]]=((nk-1)*trunc(n/10)+1):n
+   S=unlist(Vectorize(function(i)
+     pred_i(L[[i]],k))(1:nk))
+   R=roc(S,as.factor(Y))
+   return(AUC::auc(R))
+ }

> v=data.frame(k1=rep(1:7,each=7),k2=rep(1:7,7))
> v=v[v$k1 v$auc=NA
> for(i in 1:nrow(v)) v$auc[i]=ROC(v[i,1:2]) > v k1 k2 auc 2 1 2 0.9047619 3 1 3 0.9047619 4 1 4 0.6990969 5 1 5 0.6395731 6 1 6 0.6334154 7 1 7 0.8817734 10 2 3 0.9072250 11 2 4 0.9088670 12 2 5 0.8940887 13 2 6 0.8801314 14 2 7 0.8899836 18 3 4 0.8916256 19 3 5 0.8817734 20 3 6 0.9014778 21 3 7 0.8768473 26 4 5 0.6925287 27 4 6 0.7138752 28 4 7 0.8825944 34 5 6 0.6912972 35 5 7 0.8834154 42 6 7 0.8834154 Here the best pair is > v[which.max(v$auc),]
k1 k2      auc
11  2  4 0.908867
> name_var[as.integer(v[which.max(v$auc),1:2])]  "INCAR" "PRDIA" which is different, compared with the one we got above. What is odd here is that we get a smaller AUC than the ones we got at step 2 in the stepwise procedure. Nevertheless, even with a few observations (our dataset is rather small here), it is time consuming to look at all ROC curves, for all pairs. An alternative might be to use Fold Cross Validation. • Using -Fold Cross Validation Here we consider a partition of indices, , and we define based on observations. For all, set . Then, we can use the stepwise method described above > pred_i=function(i,k){ + fml = paste(name_var,"~",name_var[k],sep="") + reg=glm(fml,data=MYOCARDE[-i,],family=binomial) + predict(reg,newdata=MYOCARDE[i,], + type="response") + } > library(AUC) > ROC=function(k){ + Y=MYOCARDE[,8]=="Survival" + L=list() + n=length(Y) + nk=trunc(n/trunc(n/10)) + for(i in 1:(nk-1)) L[[i]]=((i-1)* + trunc(n/10)+1:(n/10)) + L[[nk]]=((nk-1)*trunc(n/10)+1):n + S=unlist(Vectorize(function(i) + pred_i(L[[i]],k))(1:nk)) + R=roc(S,as.factor(Y)) + return(list(roc=cbind(R$fpr,R$tpr), + auc=AUC::auc(R))) + } > plot(0:1,0:1,col="white",xlab="",ylab="") > for(k in (1:7)) lines(ROC(k)$roc,col=CL[k])
> segments(0,0,1,1,lty=2,col="grey")
> legend(.8,.45,name_var,col=CL,lty=1,cex=.8) with

> AUC=rep(NA,7)
> for(k in 1:7){
+ AUC[k]=ROC(k)$auc + cat("Variable ",k,"(",name_var[k],") :", + AUC[k],"n") + } Variable 1 ( FRCAR ) : 0.3932677 Variable 2 ( INCAR ) : 0.8940887 Variable 3 ( INSYS ) : 0.908046 Variable 4 ( PRDIA ) : 0.7278325 Variable 5 ( PAPUL ) : 0.6756979 Variable 6 ( PVENT ) : 0.63711 Variable 7 ( REPUL ) : 0.8834154 So, this time, INSYS is probably the best covariate to use. Now, if we keep that variable, and move forward, > k0=which.max(AUC) > pred_i=function(i,k){ + vk=c(k0,k) + fml = paste(name_var,"~",paste(name_var[vk], + collapse="+"),sep="") + reg=glm(fml,data=MYOCARDE[-i,],family=binomial) + predict(reg,newdata=MYOCARDE[i,], + type="response") + } > library(AUC) > ROC=function(k){ + Y=MYOCARDE[,8]=="Survival" + L=list() + n=length(Y) + nk=trunc(n/trunc(n/10)) + for(i in 1:(nk-1)) L[[i]]=((i-1)* + trunc(n/10)+1:(n/10)) + L[[nk]]=((nk-1)*trunc(n/10)+1):n + S=unlist(Vectorize(function(i) + pred_i(L[[i]],k))(1:nk)) + R=roc(S,as.factor(Y)) + return(list(roc=cbind(R$fpr,R$tpr), + auc=AUC::auc(R))) + } > plot(0:1,0:1,col="white",xlab="",ylab="") > for(k in (1:7)[-k0]) + lines(ROC(k)$roc,col=CL[k])
> segments(0,0,1,1,lty=2,col="grey")
> legend(.8,.45,name_var[-k0],
+        col=CL[-k0],lty=1,cex=.8) and our best choice for the second variable would be INCAR

> AUC=rep(NA,7)
> for(k in (1:7)[-k0]){
+   AUC[k]=ROC(k)$auc + cat("Variable ",k,"(",name_var[k],") :", + AUC[k],"n") + } Variable 1 ( FRCAR ) : 0.9047619 Variable 2 ( INCAR ) : 0.907225 Variable 4 ( PRDIA ) : 0.8916256 Variable 5 ( PAPUL ) : 0.8817734 Variable 6 ( PVENT ) : 0.9014778 Variable 7 ( REPUL ) : 0.8768473 > which.max(AUC)  2 Here again, it is possible to look at all pairs > pred_i=function(i,k){ + fml = paste(name_var,"~",paste(name_var[ + as.integer(k)],collapse="+"),sep="") + reg=glm(fml,data=MYOCARDE[-i,],family=binomial) + predict(reg,newdata=MYOCARDE[i,], + type="response") + } > library(AUC) > ROC=function(k){ + Y=MYOCARDE[,8]=="Survival" + L=list() + n=length(Y) + nk=trunc(n/trunc(n/10)) + for(i in 1:(nk-1)) L[[i]]=((i-1)* + trunc(n/10)+1:(n/10)) + L[[nk]]=((nk-1)*trunc(n/10)+1):n + S=unlist(Vectorize(function(i) + pred_i(L[[i]],k))(1:nk)) + R=roc(S,as.factor(Y)) + return(AUC::auc(R)) + } > v=data.frame(k1=rep(1:7,each=7),k2=rep(1:7,7)) > v=v[v$k1 v$auc=NA > for(i in 1:nrow(v)) v$auc[i]=ROC(v[i,1:2])
> v[which.max(v$auc),] k1 k2 auc 11 2 4 0.908867 > name_var[as.integer(v[which.max(v$auc),1:2])]
 "INCAR" "PRDIA"

which is the same as what we got using an One-Leave-Out strategy: we have again the same two covariates. And again, the AUC is lower than the one we got using a stepwise procedure (I still don’t understand how this is possible).   An alternative for the code would be to store all the regression models in a list,

> L=list()
> n=nrow(MYOCARDE)
> nk=trunc(n/trunc(n/10))
> for(i in 1:(nk-1)) L[[i]]=((i-1)*trunc(n/10)+
+  1:(n/10))
> L[[nk]]=((nk-1)*trunc(n/10)+1):n
> REG=list()
> for(k in 1:7){
+   REG[[k]]=list()
+   fml = paste(name_var,"~",
+ paste(name_var[as.integer(k)],collapse="+"),
+ sep="")
+ for(i in 1:nk) REG[[k]][[i]]=reg=glm(fml,
+  data=MYOCARDE[-L[[i]],],family=binomial)
+ }

and then to call them, properly

> pred_i=function(i,k){
+  I=which(sapply(1:10,function(j) i%in%L[[j]]))
+  predict(REG[[k]][[I]],newdata=MYOCARDE[i,],
+  type="response")
+ }

One has to check about the efficiency, especially with a large dataset.

• Using Trees and Random Forests

Another quick, but popular (from what I’ve seen), technique is to use trees. Important variables should appear in the output,

> library(rpart)
> tree=rpart(PRONO~.,data=MYOCARDE)
> library(rpart.plot)
> rpart.plot(tree) Here, the first variable that appears in the tree construction is INSYS, and the second on is REPUL. Which is rather different with what we got above. But using one tree is maybe not sufficient. One can use the variable importance function (described in a previous post) obtained using random forests.

> library(randomForest)
> rf=randomForest(PRONO~.,data=MYOCARDE,
+                 importance=TRUE)
> rf$importance[,4] FRCAR INCAR INSYS 1.042006 7.363255 8.954898 PRDIA PAPUL PVENT 3.149235 2.571267 3.152619 REPUL 7.510110 Here, we have the same story as the one we got with a simple tree: the ‘most important’ variable is INSYS while the second one is REPUL. But here, we consider tree based predictors. And not a logistic regression. • Using Dedicated R functions It is possible to use some dedicated R functions for variable selection. For instance, since we consider a logistic regression, use > library(bestglm) > Xy=as.data.frame(MYOCARDE) > Xy[,8]=(Xy[,8]=="Death")*1 > names(Xy)=names(MYOCARDE) > B=bestglm(Xy) > B$Subsets[,2:8]
FRCAR INCAR INSYS PRDIA PAPUL PVENT REPUL
0  FALSE FALSE FALSE FALSE FALSE FALSE FALSE
1  FALSE FALSE FALSE FALSE FALSE FALSE  TRUE
2* FALSE  TRUE FALSE FALSE FALSE FALSE  TRUE
3  FALSE  TRUE FALSE FALSE FALSE  TRUE  TRUE
4   TRUE  TRUE FALSE FALSE FALSE  TRUE  TRUE
5   TRUE  TRUE  TRUE FALSE FALSE  TRUE  TRUE
6   TRUE  TRUE FALSE  TRUE  TRUE  TRUE  TRUE
7   TRUE  TRUE  TRUE  TRUE  TRUE  TRUE  TRUE

With only one variable, we should consider REPUL (row 1 of the matrix), while with two variables, we should consider REPUL and INCAR (and that is the best model, based on some Bayesian Information Criterion). Here, cross validation techniques can be used also,

> B=bestglm(Xy, IC = "CV", CVArgs =
+ list(Method = "HTF", K = 10, REP = 1))
>  cverrs = B$Subsets[, "CV"] > sdCV = B$Subsets[, "sdCV"]
>  CVLo = cverrs - sdCV
>  CVHi = cverrs + sdCV
>  ymax = max(CVHi)
>  ymin = min(CVLo)
>  k = 0:(length(cverrs) - 1)
>  plot(k, cverrs, ylim = c(ymin,
+  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)
>  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))
 2 If we summarize, here,

• stepwise, AIC : REPUL + INCAR
• stepwise, BIC : REPUL + INCAR
• One-leave-Out CV stepwise : REPUL + INCAR
• One-leave-Out CV pairs : INCARPRDIA
• -fold CV stepwise : INSYS + INCAR
• -fold CV pairs : INCARPRDIA
• Tree : INSYS + REPUL
• Variable Importance (RF) : INSYS + INCAR
• Best GLM : REPUL + INCAR

That is the lovely part with statistical tools: there are usually multiple (valid) answers. And this is why machine learning is difficult: if there was a single answer, any machine could built up a model that works well. But obviously, it has to be more complicated…