Août 6 2020
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This is called a K-fold cross-validation. Conventional tests of statistical significance are based on the probability that an observation arose by chance, and necessarily accept some risk of mistaken test results, called the significance. The lasso parameter estimates are given by Trevor Hastie & Friedman (2001) as:Least Angle Regression was developed by Efron, Hastie, Johnstone & Tibshirani (2004). This index can, for example, be based on Akaike Information Criterion weights given byWhere the delta_i are differences in ordered AIC and K is the number of models. It begins by centering all the variables and scaling the covariates. Initially, all parameters are set to 0, and then parameters are added based on correlations with current residuals.Cross-validation is a resampling method, like the bootstrap or the jackknife, which takes yet another approach to model evaluation. We can test the strategy combining forward and backward at the same time:The results of 10 000 simulations generating 10 000 different data bases using 3 different stepwise strategies can be sumarized in the following graphs: In none of the simulation, stepwise regression is able to find the true model. When enough hypotheses are tested, it is virtually certain that some falsely appear statistically significant, since almost every data set with any degree of randomness is likely to contain some spurious correlations. The functionfwd.model = stepAIC(min.model, direction=’forward’, scope=biggest,trace=0)As you can see on the results of the forward selection strategy using AIC as indicator, 11 variables are considered as significant with the p-value that are too small, the confidence intervals that are too narrow to be true, while none of these variable has a relationship with the DP.One can argue that the forward is not the best stragegy because it doesn’t take into account the impact of the presence of one variable on the significance of the other variables. This bias-variance tradeoff is central to the selection of a good method and a good model.Another excellent alternative that is often overlooked is using substantive knowledge to guide variable selection. In my first lesson at the university, feature selection was about the p-value. Next, I show these methods violate statistical theory; then I show that the theoretical violations have important practical consequences in commonly encountered situations. And see how it behaves.And lets test first the forward strategy using AIC as the selection criteria. This is what is done in exploratory research after all. It gives biased regression coefficients that need shrinkage (the coefficients for remaining variables are too large; see Tibshirani [1996]). One way of looking at this is to note that principal component regression is based on the spectral decomposition of X’X, partial least squares is based on the decomposition of X’Y’.The lasso is one of a class of shrinkage methods (perhaps the best-known shrinkage method is ridge regression). The Fand chi-squared tests quoted next to each... Intuitive explanation. So this is thePROC GLMSELECT was introduced early in version 9, and is now standard in SAS. In stepwise regression, this assumption is grossly violated in ways that are difficult to determine. Here are some of the problems with stepwise variable selection. Put in another way, for a data analyst to use stepwise methods is equivalent to telling his or her boss that his or her salary should be cut. This value of probability is proper to help concluding if this specific man is criminal or not.But if, the criminal DNA is used to compare with the sample of 20 000 men, then, the probability of finding the matching DNA is:So in this case, if we find a person with his DNA matching the criminal DNA, his probability of crime is not This resumes exactly (but not all) the problem of stepwise regression. There are no solutions to the problems that stepwise regression methods have. If you have 10 people each toss a coin ten times, and one of them gets 10 heads, you are less suspicious, but you can still quantify the likelihood. Miller (2002)) — this is the price paid for the decreased bias in the predicted values. Cross-validation typically takes K replicate samples of the data, each one using (K-1)/K of the data to build the model and the remaining 1/K of the data to test the model in some way. First, I discuss methods that are not automatic,but that rely on judgement. Results were not encouraging: Stepwise led to 10 IVs with 5 significant at 0.05; forward to 28 IVs, with 5 significant at 0.05, and backward to 10 IVs, with 8 significant at 0.05. The F and chi-squared tests quoted next to each variable on the printout do not have the claimed distribution. Therefor it is suggested to use it only in exploratory research. “Stepwise regression is one of these things, like outlier detection and pie charts, which appear to be popular among non-statisticans but are considered by statisticians to be a bit of a joke.” Tibshirani and Hastie in their recent Statistical Learning MOOC were quite positive about stepwise regression, in particular forward stepwise selection for variable selection. It results in the p-value that is too small, the confidence interval that is too narrow, and AIC, BIC that are too good to be true.Another explanation of the fallacy of stepwise can be resumed by the use of data mining to uncover patterns in data that can be presented as statistically significant, without first devising a specific hypothesis as to the underlying causality.
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