Statistical learning and selective inference

Forward Stepwise Regression

Supposed that we have a dataset with N observations (xi,yi),i=1,2,…N, yi being the outcome measurement and each xi a vector of p predictors (or features). We wish to build a model for predicting yi from xi. This is known as the supervised learning problem, because the outcome y supervises (or guides) the prediction process.^† The standard linear regression model has the formyi=β0+∑jxijβj+ϵi,[1]with the errors ϵi being independent and distributed as N(0,σ2), where β is a vector of p unknown coefficients (weights) to be estimated.

With a moderate or large number of predictors, it is common to use forward stepwise regression to select a good subset of predictors. This procedure enters predictors one at a time, choosing the predictor that most decreases the residual sum of squares RSS=∑(yi−y^)2 at each stage (y^i being the predicted value for observation i). Classic statistical theory for assessing the strength each predictor would go as follows. Defining RSSk to be the residual sum of squares for the model containing k predictors, we use the change in residual sum of squares to form a test statisticRk=1σ2(RSSk−1−RSSk)[2](with σ assumed known here), and compare it to a χ² distribution with one degree of freedom. There is a big problem with this approach, however. This classic theory assumes that the models being compared were prespecified before seeing the data. However, this is clearly not the case here: We have “cherry-picked” the best predictor at each stage to maximize Rk. Hence, we would expect that even if the new predictor had no effect in the population, the distribution of values of Rk will be larger than a χ12 distribution.

Fig. 1 quantifies this. With a sample size of N=100, we simulated Gaussian datasets from two scenarios: one with p=10 predictors and the other with p=50. There is no signal in the underlying population. We are examining just the first step of forward stepwise regression. The horizontal axis shows the nominal (desired) P value (set by the statistician) and the vertical axis is the actual P value from this process, which properly accounts for the selection of the best predictor among the p. For p=10, we see that, for example, a test at nominal type I error of 5% would actually have a type I error of 20%. For p=50, things become worse: The actual type I error is about 70%.

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Fig. 1.

A simulation example with N=100 observations and p=10 or 50 predictors and null effects (β=0). Shown are the nominal and actual P values for entering the strongest predictor at the first step; the 45° dashed line is included for reference.

The correct P values in Fig. 1 were estimated by simulating from the Gaussian distribution and recording the maximal value for Rk achieved each time. However, we would like to obtain a correct P value not just for the first step of forward stepwise regression, but for all steps.

As an example, Rhee et al. (2) studied six nucleoside reverse transcriptase inhibitors that are used to treat HIV-1. The target of these drugs can become resistant through mutation, and Rhee et al. (2) compared a collection of models for predicting the log susceptibility, a measure of drug resistance based on the location of mutations. We chose one of the inhibitors, with a total of n=1,073 samples and p=240 mutation sites. To make the problem more manageable computationally, we randomly selected 100 of the samples and chose the 30 sites with highest variation in mutation status across all samples. Forward stepwise regression entered the predictors in the order (5,9,25,8,16,21,⋯). Fig. 2 gives a summary of the P values.

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Fig. 2.

P values for HIV data, as predictors are entered successively into the model by forward stepwise regression .The black points are the naive P values that do not correct of the selection effects, and the green points are selection-adjusted.

The naive P values suggest that there are six strong predictors (P value less than 0.05), but the more realistic selection-adjusted P values yield only two or three significant predictors.

How were these adjusted P values obtained? Trying to obtain them by naive Monte Carlo sampling would be tedious, and perhaps infeasible for a large number of predictors, because we would have to generate many samples and keep only those for which the sequence of selected variables is the same as that obtained in our original data. Fortunately, there is a simple mathematical construction that gives us these selection-adjusted P values in closed form.

Recall that our data vector y is normally distributed with mean μ=β0+∑jxijβj and variance σ2. Consider again the HIV data and suppose that we have taken two steps of forward stepwise regression, entering the variables x5 and x9. Assume that we are interested, for example, in the true partial regression coefficient of x9 in this model. Then, our estimate β^ would be the least squares coefficient of x9 in the fitted model. Standard statistical theory would tell us that β^ has a normal distribution with mean β and some variance τ2:β^∼N(β,τ2).[3]However, this theory assumes that we had only these two predictors available to us at the onset. This is not the case here, because we have actually selected the strongest two predictors from among the 30 predictors available. This selection should change how we view the strength of the apparent association between x9 and y. In particular, the number of predictors from which we chose these two should affect how we assess the apparent association. If we had started with 300 rather than 30 predictors, we should have set a higher bar for calling x9 significant in our model.

How do we adjust for the effects of selection? It turns out that for forward stepwise regression and many other procedures the selection events can be written in the “polyhedral” form Ay≤b for some matrix A and vector b (3, 4). We can understand this statement as follows. Suppose that we consider any new vector of outcomes, say y*, in place of our actual data y. We run our forward stepwise procedure and keep track of the predictors entered at each stage. These will not likely be the same as our original list because the data have changed. However, the polyhedral form says that the set of new data vectors y* that would yield the same list of predictors (up to some step, such as step 2) can be described by the set Ay*≤b. The quantities A and b depend on the data and the selected variables. Roughly speaking, each stage of forward stepwise regression represents a competition among all p variables, and A and b simply reconstruct this competition and check whether y* yields the same result. What does this polyhedral selection give us? It turns out that under the selection Ay≤b the naive expression (Eq. 3) is replaced by a truncated normal distribution:β^∼TNc,d(β,τ2).[4]This is just a normal distribution truncated to lie in the interval (c,d), The limits (c,d) depend on the data and the selection events that led to the model in question. The formulae for these limits are somewhat complicated but are easily computable. Details can be found in refs. 3 and 4. All of this has given us the key result (Eq. 4), which exactly accounts for the selection process through the truncation limits (c,d). We used this result to get the selection-adjusted P values in Fig. 1.

Let’s dig more deeply to understand how this works. The limits (c,d) express the fact that the selection events in forward stepwise regression restrict the possible range of values of β^ that we might obtain after the selection has been made. For example, consider again the second step of forward stepwise regression and let β^ be the regression coefficient for x9. To simplify matters, suppose that all of the predictors were uncorrelated and standardized to have mean 0 and variance 1. Then, having selected x5 at the first step, we know that the regression coefficient for x9 must be larger than the coefficients of all of the other 28 predictors not yet chosen (because it won the competition and was entered at step 2) and can be no larger than the coefficient of x5 (because the latter won the competition at the first stage). These two facts define the limits c and d, respectively. With correlated predictors—as in our data—the definitions of c and d are more complicated, but the idea is the same.

Fig. 3 depicts this situation, showing a truncated normal distribution, with truncation limits (c,d).

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Fig. 3.

HIV data: truncated normal distribution for the coefficient of x9, the ninth feature measuring mutation status at a given site. The standardized coefficient for x9 equals 5.1 and is indicated by the vertical blue line in each plot. The truncation limits implied by the selection are c=4.3,d=6.3. The lower panel zooms in on the area of interest in the tail of the distribution.

Ignoring the selection effects, under the null hypothesis that the true coefficient for x9 was actually 0, and the error variance was 1, then β^ would have a standard normal distribution. Hence we would expect β^ to fall between, say, −2 and +2 and would consider it to be evidence against the null hypothesis if it fell outside this range. Indeed, the observed value of 5.1 would be considered extremely unlikely under the null hypothesis. However, accounting properly for the fact that x9 has been selected at the second stage of forward stepwise regression changes the way in which we judge the observed value of β^. The selection events imply that the estimate must lie between c and d (4.3 and 6.3 in the Fig. 3, respectively.) Hence, the value of 5.1 represents moderate but not overwhelming evidence that the coefficient is nonzero.

False Discovery Rates and a Sequential Stopping Rule

Looking at the sequence of adjusted P values (green points) in Fig. 1, a natural question arises: When should we stop adding variables? Should we stop at, say, two or three predictors, beyond which the P values are above 0.05? If we do stop there, what can we say about the resulting model? The notion of false discovery rate (FDR) is useful for this purpose. This concept has been an important one for large-scale testing, especially in the biomedical sciences (see ref. 5).

Let’s review this concept and other related, more traditional ones. Table 1 summarizes the theoretical outcomes of m hypothesis tests.

The quantity V counts false-positive calls, and much of statistical testing focuses on Prob(V≥1), the probability of at least one false positive. This is called the familywise error rate (FWER), and many statistical methods try to ensure that FWER is less than some level (such as 0.05). This is a useful approach when the number of tests m is not too large (say, <50). However, with a larger number of tests, common in today’s big data problems, we expect that V will be much great than 1 and this approach is too stringent. Instead we shift our focus to the FDR:FDR=E(V/R).[5]This is the average proportion of false-positive calls that we make, among the R tests that we reject, that is, effects that we call significant. (V/R is defined to be zero when R = 0.)

Now, it turns out that there is a simple stopping rule for sequences of P values as in Fig. 1 that gives us automatic control of the FDR of the resulting model (Eq. 6). Choosing a target FDR level of α, and denoting the successive P values by pv1,pv2,…, the ForwardStop rule (ref. 6) is defined byk^=max{k:−1k∑i=1klog(1−pvi)≤α}.[6]Essentially, we stop at the last time that the average P value up to that point is below some target FDR level α. This average, however, is taken on the (complementary) log scale. Using the ForwardStop rule, the final model contains the predictors entered at the first k^ steps and has FDR at most α.

In the HIV example, for α=0.05 we get k^=3, meaning that we should stop after three steps. If we do so, the expected number of false positives is 0.05⋅3=0.15. Alternatively, if we set α=0.10, we obtain k^=5; there will be 0.10⋅5=0.5 false positives on average.

The Lasso

We now broaden our discussion to a more modern approach to supervised learning, one that can be applied to large datasets. We give a brief description of the method and then show how selective inference can be applied in this more general framework.

The lasso, or ℓ1 penalization, recasts the regression problem as a convex optimization.^‡ The lasso solves the problemminimizeβ0,β[12∑i=1N(yi−β0−∑jxijβj)2+λ∑j=1p|βj|].[7]The tuning parameter λ effectively balances the tradeoff between two objectives: the goodness of fit to the data (sum squares in the first term) with the complexity of model (second term). It turns out that this is equivalent to minimizing the sum of squares with a “budget” constraint ∑|βj|≤s. The value of s depends on λ: Larger values of λ imply a smaller budget s. The best value for λ or s depends on the data and is usually chosen by cross-validation (described below).

Because of the absolute value function appearing in the penalty, over a range of λ values the solution to this problem is sparse, that is, many of the weights βj are set to 0. Hence, like forward stepwise regression, the lasso selects the most informative predictors among the p available predictors. However, because the problem has been cast as a convex optimization, the search throughout the possible models can be carried out much more effectively. Lasso and ℓ1 penalization methods are now widely used in statistics, engineering, and other sciences, for example in signal processing and compressed sensing (7–9).

For a fixed choice of the tuning parameter λ, the solution to Eq. 7 has nonzero values β^j on a subset of the predictors, what we call the active set. These are the most informative predictors, as judged by the lasso. Fig. 4 shows the profiles of the solutions to the lasso problem for the HIV data.

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Fig. 4.

Paths of estimated coefficients from the lasso applied to the HIV data. Each coefficient path corresponds to the estimated β for a mutation number, with some of mutation site numbers with larger coefficients on the right. The estimated optimal budget is indicated by the vertical green line.

Each profile corresponds to one predictor (mutation site). The different points on each profile correspond to different values of the penalty parameter λ, with this value decreasing as we move from left to right. For interpretability, we have not plotted against λ on the horizontal axis, but the “parameter budget” ∑|β^j| implied by each value of λ. On the left, λ is very large and the effective budget is zero, so that all parameter estimates are zero. As we move to the right, λ decreases and the effective budget increases, so that more parameters become nonzero. The vertical dotted lines mark the places where a coefficient becomes nonzero, as we move from left to right. On the right, the value of λ is zero so that there is no budget constraint on the parameters. Hence, the values at the right end of the plot are the usual least squares estimates.

Also shown on the plot is the optimal value of λ (green line) estimated using cross-validation. Cross-validation works as follows. (i) We divide the samples into roughly 10 equal-sized parts at random. (ii) For each part K, leave out this part, fit the lasso to the other nine parts over a range of λ values, and then record the prediction error over the left-out part. (iii) Repeat step ii for K=1,2,…10, and for each value of λ, compute the average prediction error over the 10 parts. Finally our estimate λ^ is the value yielding the smallest average prediction error. The cross-validated choice of λ yielded a model with nine predictors (Fig. 4).

It turns out that this selection of predictors can again be described as a polyhedral region of the form Ay≤b. That is, for fixed predictors and value λ, the vector of response values y* that would yield the same active set after applying the lasso can be written in the form Ay*≤b. Here A and b depend on the predictors, the active set and λ, but not y. This fact gives us what we need to construct selection-adjusted intervals for the parameters of the fitted model, using the same arguments as above for forward stepwise regression.

Fig. 5 shows naive least squares confidence intervals and the selection-adjusted intervals for the nine underlying parameters. We see that some of the selection-adjusted intervals are much wider than their naive counterparts.

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Fig. 5.

Selection intervals for the HIV data, from the fitted lasso model.

Principal Components and Beyond

Principal components analysis (PCA) is a classic statistical method developed in the early 1900s but now used more widely used than ever. For example, PCA was a core method used by most leading competitors in the Netflix movie prediction competition (10). PCA is a method for unsupervised learning that seeks to discover the important correlation patterns among a set of features. It works by computing the sets of linear combinations of features that are maximally correlated. It can be thought as a method for determining the K leading eigenvectors and eigenvalues of the sample correlation matrix of the features.

In PCA analysis one computes the leading eigenvectors and eigenvalues (one for each component) and then must decide on the number of components K that are “significant.” Traditionally, this is done through the so-called scree plot, which is simply a plot of the eigenvalues in order from smallest to largest. Fig. 6, Left shows an example from some simulated data with nine features. The underlying correlation matrix, from which the data were simulated, has two strong components.

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Fig. 6.

Eigenvalue (scree) plots for two simulated data matrices, each with nine features. (Left) There are two strong leading components in the underlying population. (Right) The two leading components are only moderate in strength.

The scree plot shows a nice “elbow” and would lead us to choose the correct number of components (two). In Fig. 6, Right, we have reduced the signal-to-noise ratio, so that the population eigenvalues of the first two eigenvectors are only moderately larger than the others. Now the scree plot shows no elbow, and we would be hard pressed to guess the number of components.

However, selective inference can come to our rescue. Choosing the leading eigenvectors of a covariance matrix is similar in spirit to forward stepwise regression, and with somewhat more complicated mathematics one can derive selection-adjusted P values for each successive increase in the rank (11). For the example in Fig. 6, Right, the adjusted P values are (0.030, 0.064, 0.222, 0.286, 0.197, 0.831, 0.510, 0.185, and 0.1260), and hence this gives a moderately strong indication of the correct rank. Although this may seem too good to true based on the scree plot, one must remember that the P value estimation procedure uses more information in the correlation matrix than just the eigenvalues shown here.