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Data Science and Predictive Analytics (UMich HS650)
Regularized Linear Modeling and Controlled Variable Selection
SOCR/MIDAS (Ivo Dinov)
August 2021
Many biomedical and biosocial studies involve large amounts of complex data, including cases where the number of features (\(k\)) is large and may exceed the number of cases (\(n\)). In such situations, parameter estimates are difficult to compute or may be unreliable as the system is underdetermined. Regularization provides one approach to improve model reliability, prediction accuracy, and result interpretability. It is based on augmenting the fidelity term of the objective function used in the model fitting process with a regularization term that provides restrictions on the parameter space.
Classical techniques for choosing important covariates to include in a model of complex multivariate data rely on various types of stepwise variable selection processes, see Chapter 16. These tend to improve prediction accuracy in certain situations, e.g., when a small number of features are strongly predictive, or heavily associated, with the clinical outcome or the specific biosocial trait. However, the prediction error may be large when the model relies purely on a fidelity term. Including an additional regularization term in the optimization of the cost function improves the prediction accuracy. For example, below we show that by shrinking large regression coefficients, ridge regularization reduces overfitting and improves prediction error. Similarly, the least absolute shrinkage and selection operator (LASSO) employs regularization to perform simultaneous parameter estimation and variable selection. LASSO enhances the prediction accuracy and provides a natural interpretation of the resulting model. Regularization refers to forcing certain characteristics on the model, or the corresponding scientific inference. Examples include discouraging complex models or extreme explanations, even if they fit the data, enforcing model generalizability to prospective data, or restricting model overfitting of accidental samples.
In this chapter, we extend the mathematical foundation we presented in Chapter 4 and (1) discuss computational protocols for handling complex high-dimensional data, (2) illustrate model estimation by controlling the false-positive rate of selection of salient features, and (3) derive effective forecasting models.
1 Questions
Applications of regularized linear modeling techniques will help us address problems like:
- How to deal with extremely high-dimensional data (hundreds or thousands of features)?
- Why mix fidelity (model fit) and regularization (model interpretability) terms in objective function optimization?
- How to reduce the false-positive rate, increase scientific validation, and improve result reproducibility (e.g., Knockoff filtering)?
2 Matrix notation
We should review the basics of matrix notation, linear algebra, and matrix computing we covered in Chapter 4. At the core of matrix manipulations are scalars, vectors and matrices.
- \({y}_i\): output or response variable, \(i = 1, ..., n\) (cases, subjects, units, etc.)
- \(x_{ij}\): input, predictor, or feature variable, \(1\leq j \leq k,\ 1\leq i \leq n.\)
\[{y} = \begin{pmatrix} y_1 \\ y_2 \\ \vdots \\ y_n \end{pmatrix} \quad ,\] and \[\quad {X} = \begin{pmatrix} x_{1,1} & x_{1,2} & \cdots & x_{1,k} \\ x_{2,1} & x_{2,2} & \cdots & x_{2,k} \\ \vdots & \vdots & \cdots & \vdots \\ x_{n,1} & x_{n,2} & \cdots & x_{n,k} \end{pmatrix}_{cases\times features}.\]
3 Regularized Linear Modeling
If we assume that the covariates are orthonormal, i.e., we have a special kind of a design matrix \(X^T X = I\), then:
- The ordinary least squares (
OLS) estimates minimize the following objective function: \[\min_{ \beta \in \mathbb{R}^k } \left\{ \frac{1}{N} \left\| y - X \beta \right\|_2^2\right\}, \]
and are defined in general by \[\hat{\beta}^{OLS} = (X^T X)^{-1} X^T y.\]
LASSOestimates minimize a modified cost function \[\min_{ \beta \in \mathbb{R}^k } \left\{ \frac{1}{N} \left\| y - X \beta \right\|_2^2 + \lambda \| \beta \|_1 \right\}, \]
and may be expressed as a soft-thresholding function of the OLS estimates: \[\hat{\beta}_j = S_{N \lambda}( \hat{\beta}^\text{OLS}_j ) = \hat{\beta}^\text{OLS}_j \max \left( 0, 1 - \frac{ N \lambda }{ |\hat{\beta}^{OLS}_j| } \right), \]
where \(S_{N \lambda}\) is a soft thresholding operator translating values towards zero. This is different from the hard thresholding operator, which sets smaller values to zero and leaves larger ones unchanged.
Ridgeregression minimizes a similar objective function (using a different norm): \[\min_{ \beta \in \mathbb{R}^k } \left\{ \frac{1}{N} \| y - X \beta \|_2^2 + \lambda \| \beta \|_2^2 \right\}, \]
which yields estimates \(\hat{\beta}_j = ( 1 + N \lambda )^{-1} \hat{\beta}^{OLS}_j\). Thus, ridge regression shrinks all coefficients by a uniform factor, \((1 + N \lambda)^{-1}\), and does not set any coefficients to zero.
Best subsetselection regression, also known as orthogonal matching pursuit (OMP), minimizes the same cost function with respect to the zero-norm: \[\min_{ \beta \in \mathbb{R}^k } \left\{ \frac{1}{N} \left\| y - X \beta \right\|_2^2 + \lambda \| \beta \|_0 \right\}, \]
where \(\|.\|_0\) is the “\(\ell^0\) norm”, defined for \(z\in R^d\) as \(\| z \|_o = m\), where exactly \(m\) components of \(z\) are nonzero. In this case, a closed form of the parameter estimates is: \[\hat{\beta}_j = H_{ \sqrt{ N \lambda } } \left( \hat{\beta}^{OLS}_j \right) = \hat{\beta}^{OLS}_j I \left( \left| \hat{\beta}^{OLS}_j \right| \geq \sqrt{ N \lambda } \right), \]
where \(H_\alpha\) is a hard-thresholding function and \(I\) is an indicator function (it is 1 if its argument is true, and 0 otherwise).
The LASSO estimates may share similar features selection/estimates with both Ridge and Best (OMP). This is because they both shrink the magnitude of all the coefficients, like ridge regression, but also set some of them to zero, as in the best subset selection case. Ridge regression scales all of the coefficients by a constant factor, whereas LASSO translates the coefficients towards zero by a constant value and then sets the small values to zero.
3.1 Ridge Regression
Ridge regression relies on \(L^2\) regularization to improve the model prediction accuracy. It improves prediction error by shrinking large regression coefficients and reduce overfitting. By itself, ridge regularization does not perform variable selection and does not really help with model interpretation.
Let’s show an example using the MLB dataset 01a_data.txt, which includes, player’s Name, Team, Position, Height, Weight, and Age. We may fit in any regularized linear mode, e.g., \(Weight \sim Age + Height\).
# install.packages("doParallel")
library("doParallel")
library(plotly)
library(tidyr)
# Data: https://umich.instructure.com/courses/38100/files/folder/data (01a_data.txt)
data <- read.table('https://umich.instructure.com/files/330381/download?download_frd=1', as.is=T, header=T)
attach(data); str(data)## 'data.frame': 1034 obs. of 6 variables:
## $ Name : chr "Adam_Donachie" "Paul_Bako" "Ramon_Hernandez" "Kevin_Millar" ...
## $ Team : chr "BAL" "BAL" "BAL" "BAL" ...
## $ Position: chr "Catcher" "Catcher" "Catcher" "First_Baseman" ...
## $ Height : int 74 74 72 72 73 69 69 71 76 71 ...
## $ Weight : int 180 215 210 210 188 176 209 200 231 180 ...
## $ Age : num 23 34.7 30.8 35.4 35.7 ...# Training Data
# Full Model: x <- model.matrix(Weight ~ ., data = data[1:900, ])
# Reduced Model
x <- model.matrix(Weight ~ Age + Height, data = data[1:900, ])
# creates a design (or model) matrix, and adds 1 column for outcome according to the formula.
y <- data[1:900, ]$Weight
# Testing Data
x.test <- model.matrix(Weight ~ Age + Height, data = data[901:1034, ])
y.test <- data[901:1034, ]$Weight
# install.packages("glmnet")
library("glmnet")
library(doParallel)
cl <- makePSOCKcluster(6)
registerDoParallel(cl); getDoParWorkers()## [1] 6# getDoParName(); getDoParVersion()
cv.ridge <- cv.glmnet(x, y, type.measure="mse", alpha=0, parallel=T)
## alpha =1 for lasso only, alpha = 0 for ridge only, and 0<alpha<1 to blend ridge & lasso penalty !!!!
# plot(cv.ridge)
plotCV.glmnet <- function(cv.glmnet.object, name="") {
df <- as.data.frame(cbind(x=log(cv.glmnet.object$lambda), y=cv.glmnet.object$cvm,
errorBar=cv.glmnet.object$cvsd), nzero=cv.glmnet.object$nzero)
featureNum <- cv.glmnet.object$nzero
xFeature <- log(cv.glmnet.object$lambda)
yFeature <- max(cv.glmnet.object$cvm)+max(cv.glmnet.object$cvsd)
dataFeature <- data.frame(featureNum, xFeature, yFeature)
plot_ly(data = df) %>%
# add error bars for each CV-mean at log(lambda)
add_trace(x = ~x, y = ~y, type = 'scatter', mode = 'markers',
name = 'CV MSE', error_y = ~list(array = errorBar)) %>%
# add the lambda-min and lambda 1SD vertical dash lines
add_lines(data=df, x=c(log(cv.glmnet.object$lambda.min), log(cv.glmnet.object$lambda.min)),
y=c(min(cv.glmnet.object$cvm)-max(df$errorBar), max(cv.glmnet.object$cvm)+max(df$errorBar)),
showlegend=F, line=list(dash="dash"), name="lambda.min", mode = 'lines+markers') %>%
add_lines(data=df, x=c(log(cv.glmnet.object$lambda.1se), log(cv.glmnet.object$lambda.1se)),
y=c(min(cv.glmnet.object$cvm)-max(df$errorBar), max(cv.glmnet.object$cvm)+max(df$errorBar)),
showlegend=F, line=list(dash="dash"), name="lambda.1se") %>%
# Add Number of Features Annotations on Top
add_trace(dataFeature, x = ~xFeature, y = ~yFeature, type = 'scatter', name="Number of Features",
mode = 'text', text = ~featureNum, textposition = 'middle right',
textfont = list(color = '#000000', size = 9)) %>%
# Add top x-axis (non-zero features)
# add_trace(data=df, x=~c(min(cv.glmnet.object$nzero),max(cv.glmnet.object$nzero)),
# y=~c(max(y)+max(errorBar),max(y)+max(errorBar)), showlegend=F,
# name = "Non-Zero Features", yaxis = "ax", mode = "lines+markers", type = "scatter") %>%
layout(title = paste0("Cross-Validation MSE (", name, ")"),
xaxis = list(title=paste0("log(",TeX("\\lambda"),")"), side="bottom", showgrid=TRUE), # type="log"
hovermode = "x unified", legend = list(orientation='h'), # xaxis2 = ax,
yaxis = list(title = cv.glmnet.object$name, side="left", showgrid = TRUE))
}
plotCV.glmnet(cv.ridge, "Ridge")coef(cv.ridge)## 4 x 1 sparse Matrix of class "dgCMatrix"
## s1
## (Intercept) -39.3383693
## (Intercept) .
## Age 0.5831697
## Height 3.0427340sqrt(cv.ridge$cvm[cv.ridge$lambda == cv.ridge$lambda.1se])## [1] 18.07047#plot variable feature coefficients against the shrinkage parameter lambda.
glmmod <-glmnet(x, y, alpha = 0)
plot(glmmod, xvar="lambda")
grid()
# for plot_glmnet with ridge/lasso coefficient path labels
# install.packages("plotmo")
library(plotmo)
plot_glmnet(glmmod, lwd=4) #default colors
# More elaborate plots can be generated using:
# plot_glmnet(glmmod,label=2,lwd=4) #label the 2 biggest final coefs
# specify color of each line
# g <- "blue"
# plot_glmnet(glmmod, lwd=4, col=c(2,g))
# report the model coefficient estimates
coef(glmmod)[, 1]## (Intercept) (Intercept) Age Height
## 2.016556e+02 0.000000e+00 8.327372e-37 4.789383e-36cv.glmmod <- cv.glmnet(x, y, alpha=0)
mod.ridge <- cv.glmnet(x, y, alpha = 0, thresh = 1e-12, parallel = T)
lambda.best <- mod.ridge$lambda.min
lambda.best## [1] 1.086267ridge.pred <- predict(mod.ridge, newx = x.test, s = lambda.best)
ridge.RMS <- mean((y.test - ridge.pred)^2); ridge.RMS## [1] 263.8461ridge.test.r2 <- 1 - mean((y.test - ridge.pred)^2)/mean((y.test - mean(y.test))^2)
#plot(cv.glmmod)
plotCV.glmnet(cv.glmmod, "Ridge")best_lambda <- cv.glmmod$lambda.min
best_lambda## [1] 1.086267In the plots above, different colors represents the vector of features, and the corresponding coefficients, displayed as a function of the regularization parameter, \(\lambda\). The top horizontal axis indicates the number of nonzero coefficients at the current value of \(\lambda\). For LASSO regularization, this top-axis corresponds to the effective degrees of freedom (df) for the model.
Notice the usefulness of Ridge regularization for model estimation in highly ill-conditioned problems (\(n<<k\)) where slight feature perturbations may cause disproportionate alterations of the corresponding weight calculations. When \(\lambda\) is very large, the regularization effect dominates the optimization of the objective function and the coefficients tend to zero. At the other extreme, as \(\lambda\longrightarrow 0\), the resulting model solution tends towards the ordinary least squares (OLS) and the coefficients exhibit large oscillations. In practice, we often may need to tune \(\lambda\) to balance this tradeoff.
Also note that in the cv.glmnet call, the extreme values of the parameter \(\alpha = 0\) (ridge) and \(\alpha = 1\) (LASSO) correspond to different types of regularization, and intermediate values of \(0<\alpha<1\) corresponds to elastic net blended regularization.
3.2 Least Absolute Shrinkage and Selection Operator (LASSO) Regression
Estimating the linear regression coefficients in a linear regression model using LASSO involves minimizing an objective function that includes an \(L^1\) regularization term which tends to shrink the number of features. A descriptive representation of the fidelity (left) and regularization (right) terms of the objective function are shown below:
\[\underbrace{\sum_{i=1}^n \left [ y_i - \beta_0 - \sum_{j=1}^k \beta_j x_{ij} \right ]^2}_{\text{fidelity term}} + \underbrace{\lambda\sum_{j=1}^{k}|\beta_j|}_{\text{regilarization term}}.\] LASSO jointly achieves model quality, reliability and variable selection by penalizing the sum of the absolute values of the regression coefficients. This forces the shrinkage of certain coefficients effectively acting as a variable selection process. This is similar to ridge regression’s penalty on the sum of the squares of the regression coefficients, although ridge regression only shrinks the magnitude of the coefficients without truncating them to \(0\).
Let’s show how to select the regularization weight parameter \(\lambda\) using training data and report the error using testing data.
mod.lasso <- cv.glmnet(x, y, alpha = 1, thresh = 1e-12, parallel = T)
## alpha =1 for lasso only, alpha = 0 for ridge only, and 0<alpha<1 for elastic net, a blend ridge & lasso penalty !!!!
lambda.best <- mod.lasso$lambda.min
lambda.best## [1] 0.05406379lasso.pred <- predict(mod.lasso, newx = x.test, s = lambda.best)
LASSO.RMS <- mean((y.test - lasso.pred)^2); LASSO.RMS## [1] 261.8045Let’s retrieve the estimates of the model coefficients.
mod.lasso <- glmnet(x, y, alpha = 1)
predict(mod.lasso, s = lambda.best, type = "coefficients")## 4 x 1 sparse Matrix of class "dgCMatrix"
## s1
## (Intercept) -182.1429000
## (Intercept) .
## Age 0.9667182
## Height 4.8309312lasso.test.r2 <- 1 - mean((y.test - lasso.pred)^2)/mean((y.test - mean(y.test))^2)Perhaps obtain a classical OLS linear model, as well.
lm.fit <- lm(Weight ~ Age + Height, data = data[1:900, ])
summary(lm.fit)##
## Call:
## lm(formula = Weight ~ Age + Height, data = data[1:900, ])
##
## Residuals:
## Min 1Q Median 3Q Max
## -50.602 -12.399 -0.718 10.913 74.446
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -184.3736 19.4232 -9.492 < 2e-16 ***
## Age 0.9799 0.1335 7.341 4.74e-13 ***
## Height 4.8561 0.2551 19.037 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 17.5 on 897 degrees of freedom
## Multiple R-squared: 0.3088, Adjusted R-squared: 0.3072
## F-statistic: 200.3 on 2 and 897 DF, p-value: < 2.2e-16The OLS linear (unregularized) model has slightly larger coefficients and greater MSE than LASSO, which attests to the shrinkage of LASSO.
lm.pred <- predict(lm.fit, newx = x.test)
LM.RMS <- mean((y - lm.pred)^2); LM.RMS## [1] 305.1995lm.test.r2 <- 1 - mean((y - lm.pred)^2) / mean((y.test - mean(y.test))^2)
# barplot(c(lm.test.r2, lasso.test.r2, ridge.test.r2), col = "red", names.arg = c("OLS", "LASSO", "Ridge"), main = "Testing Data Derived R-squared")
plot_ly(x = c("OLS", "LASSO", "Ridge"), y = c(lm.test.r2, lasso.test.r2, ridge.test.r2),
name = paste0("Model ", TeX("R^2") ," Performance"), type = "bar") %>%
layout(title=paste0("Model ", TeX("R^2") ," Performance"))Compare the results of the three alternative models (LM, LASSO and Ridge) for these data and contrast the derived RSS results.
library(knitr) # kable function to convert tabular R-results into Rmd tables
# create table as data frame
RMS_Table = data.frame(LM=LM.RMS, LASSO=LASSO.RMS, Ridge=ridge.RMS)
# convert to markdown
kable(RMS_Table, format="pandoc", caption="Test Dataset RSS Results", align=c("c", "c", "c"))| LM | LASSO | Ridge |
|---|---|---|
| 305.1995 | 261.8045 | 263.8461 |
stopCluster(cl)As both the inputs (features or predictors) and the output (response) are observed for the testing data, we can build a learner examining the relationship between the two types of features (controlled covariates and observable responses). Most often, we are interested in forecasting or predicting responses based on prospective (new, testing, or validation) data.
4 Predictor Standardization
Prior to fitting regularized linear modeling and estimating the effects, covariates may be standardized. Scaling the features ensures the measuring units of the features do not bias the distance measures or norm estimates. Standardization can be accomplished by using the classic “z-score” formula. This puts each predictor on the same scale (unitless quantities) - the mean is 0 and the variance is 1. We use \(\hat{\beta_0} = \bar{y}\), for the mean intercept parameter, and estimate the coefficients of the remaining predictors. To facilitate interpretation of the results, after the model is estimated, in the context of the specific case-study, we can transform the results back to the original scale/units.
5 Estimation Goals
The basic problem is this: given a set of predictors \({X}\), find a function, \(f({X})\), to model or predict the outcome \(Y\).
Let’s denote the objective (loss or cost) function by \(L(y, f({X}))\). It determines adequacy of the fit and allows us to estimate the squared error loss: \[L(y, f({X})) = (y - f({X}))^2 . \]
We are looking to find \(f\) that minimizes the expected loss: \[ E[(Y - f({X}))^2] \Rightarrow f = E[Y | {X} = {x}].\]
6 Linear Regression
For a linear model: \[Y_i = \beta_0 + x_{i,1}\beta_1 + x_{i,2}\beta_2 + \dots + x_{i,k}\beta_k + \epsilon, \] Let’s assume that:
- The model shorthand matrix notation is: \(Y = X\beta + \epsilon.\)
- And the expectation of the observed outcome given the data, \(E[Y | {X} = {x}]\), is a linear function, which in certain situations can be expressed as: \[\arg\min_{{\beta}} \sum_{i=1}^n \left (y_i - \sum_{j=1}^{k} x_{ij} \beta_j \right )^2 = \arg\min_{{\beta}} \sum_{i=1}^{n} (y_i - x_{i}^T \beta)^2.\]
Multiplying both hand-sides on the left by \(X^T=X'\), which is the transpose of the design matrix \(X\) (recall that matrix multiplication is not always commutative), yields: \[X^T Y = X^T (X\beta) = (X^TX)\beta.\]
To solve for the effect-size coefficients, \(\beta\), we can multiply both sides of the equation by the inverse of its (right hand side) multiplier: \[(X^TX)^{-1} (X^T Y) = (X^TX)^{-1} (X^TX)\beta = \beta.\]
The ordinary least squares (OLS) estimate of \({\beta}\) is given by: \[\hat{{\beta}} = \arg\min_{{\beta}} \sum_{i=1}^n (y_i - \sum_{j=1}^{k} x_{ij} \beta_j)^2 = \arg\min_{{\beta}} \| {y} - {X}{\beta} \|^2_2 \Rightarrow\] \[\hat{{\beta}}^{OLS} = ({X}'{X})^{-1} {X}'{y} \Rightarrow \hat{f}({x}_i) = {x}_i'\hat{{\beta}}.\]
6.1 Drawbacks of Linear Regression
Despite its wide use and elegant theory, linear regression has some shortcomings.
Prediction accuracy - Often can be improved upon;
Model interpretability - Linear model does not automatically do variable selection.
6.2 Assessing Prediction Accuracy
Given a new input, \({x}_0\), how do we assess our prediction \(\hat{f}({x}_0)\)?
The Expected Prediction Error (EPE) is: \[ \begin{aligned} EPE({x}_0) &= E[(Y_0 - \hat{f}({x}_0))^2] \\ &= \text{Var}(\epsilon) + \text{Var}(\hat{f}({x}_0)) + \text{Bias}(\hat{f}({x}_0))^2 \\ &= \text{Var}(\epsilon) + MSE(\hat{f}({x}_0)) \end{aligned} .\]
where
- \(\text{Var}(\epsilon)\): irreducible error variance
- \(\text{Var}(\hat{f}({x}_0))\): sample-to-sample variability of \(\hat{f}({x}_0)\) , and
- \(\text{Bias}(\hat{f}({x}_0))\): average difference of \(\hat{f}({x}_0)\) & \(f({x}_0)\).
6.3 Estimating the Prediction Error
One common approach to estimating prediction error include:
- Randomly splitting the data into “training” and “testing” sets, where the testing data has \(m\) observations that will be used to independently validate the model quality. We estimate/calculate \(\hat{f}\) using training data;
- Estimating prediction error using the testing set MSE \[ \hat{MSE}(\hat{f}) = \frac{1}{m} \sum_{i=1}^{m} (y_i - \hat{f}(x_i))^2.\]
Ideally, we want our model/predictions to perform well with new or prospective data.
6.4 Improving the Prediction Accuracy
If \(f(x) \approx \text{linear}\), \(\hat{f}\) will have low bias but possibly high variance, e.g., in high-dimensional setting due to correlated predictors, when \(k\ \text{features} \ll n\ \text{cases}\), or under-determination, when \(k > n\). The goal is to minimize total error by trading off bias and precision: \[MSE(\hat{f}(x)) = \text{Var}(\hat{f}(x)) +\text{Bias}(\hat{f}(x))^2.\] We can sacrifice bias to reduce variance, which may lead to decrease in \(MSE\). So, regularization allows us to tune this tradeoff.
We aim to predict the outcome variable, \(Y_{n\times1}\), in terms of other features \(X_{n,k}\). Assume a first-order relationship relating \(Y\) and \(X\) is of the form \(Y=f(X)+\epsilon\), where the error term \(\epsilon \sim N(0,\sigma)\). An estimate model \(\hat{f}(X)\) can be computed in many different ways (e.g., using least squares calculations for linear regressions, Newton-Raphson, steepest descent, stochastic gradient descent, or other methods). Then, we can decompose the expected squared prediction error at \(x\) as:
\[E(x)=E[(Y-\hat{f}(x))^2] = \underbrace{\left ( E[\hat{f}(x)]-f(x) \right )^2}_{Bias^2} + \underbrace{E\left [\left (\hat{f}(x)-E[\hat{f}(x)] \right )^2 \right]}_{\text{precision (variance)}} + \underbrace{\sigma^2}_{\text{irreducible error (noise)}}.\]
When the true \(Y\) vs. \(X\) relation is not known, infinite data may be necessary to calibrate the model \(\hat{f}\) and it may be impractical to jointly reduce both the model bias and variance. In general, minimizing the bias at the same time as minimizing the variance may not be possible.
The figure below illustrates diagrammatically the dichotomy between bias and precision (variance), additional information is available in the SOCR SMHS EBook.
6.5 Variable Selection
Oftentimes, we are only interested in using a subset of the original features as model predictors. Thus, we need to identify the most relevant predictors, which usually capture the big picture of the process. This helps us avoid overly complex models that may be difficult to interpret. Typically, when considering several models that achieve similar results, it’s natural to select the simplest of them.
Linear regression does not directly determine the importance of features to predict a specific outcome. The problem of selecting critical predictors is therefore very important.
Automatic feature subset selection methods should directly determine an optimal subset of variables. Forward or backward stepwise variable selection and forward stagewise are examples of classical methods for choosing the best subset by assessing various metrics like \(MSE\), \(C_p\), AIC, or BIC, see Chapter 16.
7 Regularization Framework
As before, we start with a given \({X}\) and look for a (linear) function, \(f({X})=\sum_{j=1}^{p} {x_{j} \beta_j}\), to model or predict \(y\) subject to certain objective cost function, e.g., squared error loss. Adding a second term to the cost function minimization process yields (model parameter) estimates expressed as:
\[\hat{{\beta}}(\lambda) = \arg\min_{\beta} \left\{\sum_{i=1}^n (y_i - \sum_{j=1}^{k} {x_{ij} \beta_j})^2 + \lambda J({\beta})\right\}\]
In the above expression, \(\lambda \ge 0\) is the regularization (tuning or penalty) parameter, \(J({\beta})\) is a user-defined penalty function - typically, the intercept is not penalized.
7.1 Role of the Penalty Term
Consider \(\arg\min J({\beta}) = \sum_{j=1}^k \beta_j^2 =\| {\beta} \|^2_2\) (Ridge Regression, RR).
Then, the formulation of the regularization framework is: \[\hat{{\beta}}(\lambda)^{RR} = \arg\min_{{\beta}} \left\{\sum_{i=1}^n \left (y_i - \sum_{j=1}^{k} x_{ij} \beta_j\right )^2 + \lambda \sum_{j=1}^k \beta_j^2 \right\}.\]
Or, alternatively:
\[\hat{{\beta}}(t)^{RR} = \arg\min_{{\beta}} \sum_{i=1}^n \left (y_i - \sum_{j=1}^{k} x_{ij} \beta_j\right )^2, \] subject to \[\sum_{j=1}^k \beta_j^2 \le t .\]
7.2 Role of the Regularization Parameter
The regularization parameter \(\lambda\geq 0\) directly controls the bias-variance trade-off:
- \(\lambda = 0\) corresponds to OLS, and
- \(\lambda \rightarrow \infty\) puts more weight on the penalty function and results in more shrinkage of the coefficients, i.e., we introduce bias at the sake of reducing the variance.
The choice of \(\lambda\) is crucial and will be discussed below as each \(\lambda\) results in a different solution \(\hat{{\beta}}(\lambda)\).
7.3 LASSO
The LASSO (Least Absolute Shrinkage and Selection Operator) regularization relies on: \[\arg\min J({\beta}) = \sum_{j=1}^k |\beta_j| = \| {\beta} \|_1,\] which leads to the following objective function: \[\hat{{\beta}}(\lambda)^{L} = \arg\min_{\beta} \left\{\sum_{i=1}^n \left (y_i - \sum_{j=1}^{k} x_{ij} \beta_j\right )^2 + \lambda \sum_{j=1}^k |\beta_j| \right\}.\]
In practice, subtle changes in the penalty terms frequently lead to big differences of the results. Not only does the regularization term shrink coefficients towards zero, but it sets some of them to be exactly zero. Thus, it performs continuous variable selection, hence the name, Least Absolute Shrinkage and Selection Operator (LASSO).
For further details, see the Tibshirani’s LASSO website.
7.4 General Regularization Framework
The general regularization framework involves optimization of a more general objective function:
\[\min_{f \in {H}} \sum_{i=1}^n \left\{L(y_i, f(x_i)) + \lambda J(f)\right\}, \]
where \(\mathcal{H}\) is a space of possible functions, \(L\) is the fidelity term, e.g., squared error, absolute error, zero-one, negative log-likelihood (GLM), hinge loss (support vector machines), and \(J\) is the regularizer, e.g., ridge regression, LASSO, adaptive LASSO, group LASSO, fused LASSO, thresholded LASSO, generalized LASSO, constrained LASSO, elastic-net, Dantzig selector, SCAD, MCP, smoothing splines, etc.
This represents a very general and flexible framework that allows us to incorporate prior knowledge (sparsity, structure, etc.) into the model estimation.
8 Likelihood Ratio Test (LRT), False Discovery Rate (FDR), and Logistic Transform
These two concepts will be important in the theoretical model development as well as applications we show below.
8.1 Likelihood Ratio Test (LRT)
The Likelihood Ratio Test (LRT) compares the data fit of two models. For instance, removing predictor variables from a model may reduce the model quality (i.e., a model will have a lower log likelihood). To statistically assess whether the observed difference in model fit is significant, the LRT compares the difference of the log likelihoods of the two models. When this difference is statistically significant, the full model (the one with more variables) represents a better fit to the data, compared to the reduced model. LRT is computed using the log likelihoods (\(ll\)) of the two models:
\[LRT = -2 \ln\left (\frac{L(m_1)}{L(m_2)}\right ) = 2(ll(m_2)-ll(m_1)), \] where:
- \(m_1\) and \(m_2\) are the reduced and the full models, respectively,
- \(L(m_1)\) and \(L(m_2)\) denote the likelihoods of the 2 models, and
- \(ll(m_1)\) and \(ll(m_2)\) represent the log likelihood (natural log of the model likelihood function).
As \(n\longrightarrow \infty\), the distribution of the LRT is asymptotically chi-squared with degrees of freedom equal to the number of parameters that are reduced (i.e., the number of variables removed from the model). In our case, \(LRT \sim \chi_{df=2}^2\), as we have an intercept and one predictor (SE), and the null model is empty (no parameters).
8.2 False Discovery Rate (FDR)
The FDR rate measures the performance of a test:
\[\underbrace{FDR}_{\text{False Discovery Rate}} =\underbrace{E}_{\text{expectation}} \underbrace{\left( \frac{\# False Positives}{\text{total number of selected features}}\right )}_{\text{False Discovery Proportion}}.\]
The Benjamini-Hochberg (BH) FDR procedure involves ordering the p-values, specifying a target FDR, calculating and applying the threshold. Below we show how this is accomplished in R.
# List the p-values (these are typicaly computed by some statistical
# analysis, later these will be ordered from smallest to largest)
pvals <- c(0.9, 0.35, 0.01, 0.013, 0.014, 0.19, 0.35, 0.5, 0.63, 0.67, 0.75, 0.81, 0.01, 0.051)
length(pvals)## [1] 14#enter the target FDR
alpha.star <- 0.05
# order the p-values small to large
pvals <- sort(pvals); pvals## [1] 0.010 0.010 0.013 0.014 0.051 0.190 0.350 0.350 0.500 0.630 0.670 0.750
## [13] 0.810 0.900#calculate the threshold for each p-value
# threshold[i] = alpha*(i/n), where i is the index of the ordered p-value
threshold<-alpha.star*(1:length(pvals))/length(pvals)
# for each index, compare the p-value against its threshold and display the results
cbind(pvals, threshold, pvals<=threshold)## pvals threshold
## [1,] 0.010 0.003571429 0
## [2,] 0.010 0.007142857 0
## [3,] 0.013 0.010714286 0
## [4,] 0.014 0.014285714 1
## [5,] 0.051 0.017857143 0
## [6,] 0.190 0.021428571 0
## [7,] 0.350 0.025000000 0
## [8,] 0.350 0.028571429 0
## [9,] 0.500 0.032142857 0
## [10,] 0.630 0.035714286 0
## [11,] 0.670 0.039285714 0
## [12,] 0.750 0.042857143 0
## [13,] 0.810 0.046428571 0
## [14,] 0.900 0.050000000 0Starting with the smallest p-value and moving up, we find that the largest \(k\) for which the corresponding p-value is less than its threshold, \(\alpha^*\), which yields an index \(\hat{k}=4\).
Next, the algorithm rejects the null hypotheses for the tests that correspond to p-values with indices \(k\leq \hat{k}=4\), i.e., se determine that \(p_{(1)}, p_{(2)}, p_{(3)}, p_{(4)}\) survive FDR correction for multiple testing.
Note: Since we controlled FDR at \(\alpha^*=0.05\), we expect that on average only 5% of the tests that we rejected are spurious. In other words, of the FDR-corrected p-values, only about \(\alpha^*=0.05\) are expected to represent false-positives, e.g., features chosen to be salient, that are in fact not really important.
As a comparison, the Bonferroni corrected \(\alpha\)-value for these data is \(\frac{0.05}{14} = 0.0036\). Note that Bonferroni coincides with the 1-st threshold value corresponding to the smallest p-value. If we had used this correction for multipe testing, then we would have concluded that none of our \(14\) results were significant!
8.2.1 Graphical Interpretation of the Benjamini-Hochberg (BH) Method
There’s an intuitive graphical interpretation of the BH calculations.
- Sort the p-values from largest to smallest.
- Plot the ordered p-values \(p_{(k)}\) on the y-axis versus their indices on the x-axis.
- Superimpose on this plot a line that passes through the origin and has slope \(\alpha^*\).
Any p-value that falls on or below this line corresponds to a significant result.
#generate the "relative-indices" (i/n) that will be plotted on the x-axis
x.values<-(1:length(pvals))/length(pvals)
#select observations that are less than threshold
for.test <- cbind(1:length(pvals), pvals)
pass.test <- for.test[pvals <= 0.05*x.values, ]
pass.test## pvals
## 4.000 0.014#use largest k to color points that meet Benjamini-Hochberg FDR test
last<-ifelse(is.vector(pass.test), pass.test[1], pass.test[nrow(pass.test), 1])
#widen right margin to make room for labels
par(mar=c(4.1, 4.1, 1.1, 4.1))
#plot the points (relative-index vs. probability-values)
# we can also plot the y-axis on a log-scale to spread out the values
# plot(x.values, pvals, xlab=expression(i/n), ylab="log(p-value)", log = 'y')
# plot(x.values, pvals, xlab=expression(i/n), ylab="p-value")
# #add FDR line
# abline(a=0, b=0.05, col=2, lwd=2)
# #add naive threshold line
# abline(h=.05, col=4, lty=2)
# #add Bonferroni-corrected threshold line
# abline(h=.05/length(pvals), col=4, lty=2)
# #label lines
# mtext(c('naive', 'Bonferroni'), side=4, at=c(.05, .05/length(pvals)), las=1, line=0.2)
# #use largest k to color points that meet Benjamini-Hochberg FDR test
# points(x.values[1:last], pvals[1:last], pch=19, cex=1.5)
plot_ly(x=~x.values, y=~pvals, type="scatter", mode="markers", marker=list(size=15),
name="observed p-values", symbols='o') %>%
# add bounding horizontal lines
# add naive threshold line
add_lines(x=~c(0,1), y=~c(0.05, 0.05), mode="lines", line=list(dash='dash'), name="p=0.05") %>%
# add conservative Bonferonni line
add_lines(x=~c(0,1), y=~c(0.05/length(pvals), 0.05/length(pvals)), mode="lines",
line=list(dash='dash'), name="Bonferonni (p=0.05/n)") %>%
# add FDR line
add_lines(x=~c(0,1), y=~c(0, 0.05), mode="lines", line=list(dash='dash'), name="FDR Line") %>%
# highlight the largest k to color points meeting the Benjamini-Hochberg FDR test
add_trace(x=~x.values[1:last], y=~pvals[1:last], mode="markers",symbols='0', name="FDR Test Points") %>%
layout (title="Benjamini-Hochberg FDR Test", legend = list(orientation='h'),
xaxis=list(title=expression(i/n)), yaxis=list(title="p-value"))8.2.2 FDR adjusting the p-values
R can automatically performs the Benjamini-Hochberg procedure. The adjusted p-values are obtained by
pvals.adjusted <- p.adjust(pvals, "BH")
pvals.adjusted## [1] 0.0490000 0.0490000 0.0490000 0.0490000 0.1428000 0.4433333 0.6125000
## [8] 0.6125000 0.7777778 0.8527273 0.8527273 0.8723077 0.8723077 0.9000000The adjusted p-values indicate the corresponding null hypothesis we need to reject to preserve the initial \(\alpha^*\) false-positive rate. We can also compute the adjusted p-values as follows:
# manually calculate the thresholds for hte ordered p-values list
test.p <- length(pvals)/(1:length(pvals))*pvals # test.p
# loop through each p-value and carry out the manual FDR adjustment for multipel testing
adj.p <- numeric(14)
for(i in 1:14) {
adj.p[i]<-min(test.p[i:length(test.p)])
ifelse(adj.p[i]>1, 1, adj.p[i])
}
adj.p## [1] 0.0490000 0.0490000 0.0490000 0.0490000 0.1428000 0.4433333 0.6125000
## [8] 0.6125000 0.7777778 0.8527273 0.8527273 0.8723077 0.8723077 0.9000000Note that the manually computed (adj.p) and the automatically computed (pvals.adjusted) adjusted-p-values are the same.
8.3 Logistic Transformation
For binary outcome variables, or ordinal categorical variables, we may need to employ the logistic curve to transform the polytomous outcomes into real values.
The Logistic curve is \(y=f(x)= \frac{1}{1+e^{-x}}\), where y and x represent probability and quantitative-predictor values, respectively. A slightly more general form is: \(y=f(x)= \frac{K}{1+e^{-x}}\), where the covariate \(x \in (-\infty, \infty)\) and the response \(y \in [0, K]\). For example,
library("ggplot2")
k=7
x <- seq(-10, 10, 1)
# plot(x, k/(1+exp(-x)), xlab="X-axis (Covariate)", ylab="Outcome k/(1+exp(-x)), k=7", type="l")
plot_ly(x=~x, y=~k/(1+exp(-x)), type="scatter", mode="line", name="Logistic model") %>%
layout (title="Logistic Model Y=k/(1+exp(-x)), k=7",
xaxis=list(title="x"), yaxis=list(title="Y=k/(1+exp(-x))"))The point of this logistic transformation is that: \[y= \frac{1}{1+e^{-x}} \Longleftrightarrow x=\ln\frac{y}{1-y},\] which represents the log-odds (when \(y\) is the probability of an event of interest)!!!
We use the logistic regression equation model to estimate the probability of specific outcomes:
(Estimate of)\(P(Y=1| x_1, x_2, ., x_l)= \frac{1}{1+e^{-(a_o+\sum_{k=1}^l{a_k x_k })}}\), where the coefficients \(a_o\) (intercept) and effects \(a_k, k = 1, 2, ..., l\), are estimated using GLM according to a maximum likelihood approach. Using this model allows us to estimate the probability of the dependent (clinical outcome) variable \(Y=1\) (CO), i.e., surviving surgery, given the observed values of the predictors \(X_k, k = 1, 2, ..., l\).
8.3.1 Example: Heart Transplant Surgery
Let’s look at an example of estimating the probability of surviving a heart transplant based on surgeon’s experience. Suppose a group of 20 patients undergo heart transplantation with different surgeons having experience in the range {0(least), 2, …, 10(most)}, representing 100’s of operating/surgery hours. How does the surgeon’s experience affect the probability of the patient survival?
The data below shows the outcome of the surgery (1=survival) or (0=death) according to the surgeons’ experience in 100’s of hours of practice.
| Surgeon’s Experience (SE) | 1 | 1.5 | 2 | 2.5 | 3 | 3.5 | 3.5 | 4 | 4.5 | 5 | 5.5 | 6 | 6.5 | 7 | 8 | 8.5 | 9 | 9.5 | 10 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Clinical Outcome (CO) | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 |
mydata <- read.csv("https://umich.instructure.com/files/405273/download?download_frd=1") # 01_HeartSurgerySurvivalData.csv
# estimates a logistic regression model for the clinical outcome (CO), survival, using the glm
# (generalized linear model) function.
# convert Surgeon's Experience (SE) to a factor to indicate it should be treated as a categorical variable.
# mydata$rank <- factor(mydata$SE)
# mylogit <- glm(CO ~ SE, data = mydata, family = "binomial")
# library(ggplot2)
# ggplot(mydata, aes(x=SE, y=CO)) + geom_point() +
# stat_smooth(method="glm", method.args=list(family = "binomial"), se=FALSE)
mylogit <- glm(CO ~ SE, data=mydata, family = "binomial")
plot_ly(data=mydata, x=~SE, y=~CO, type="scatter", mode="markers", name="Data", marker=list(size=15)) %>%
add_trace(x=~SE, y=~mylogit$fitted.values, type="scatter", mode="lines", name="Logit Model") %>%
layout (title="Logistic Model Clinical Outcome ~ Surgeon's Experience",
xaxis=list(title="SE"), yaxis=list(title="CO"), hovermode = "x unified")## A marker object has been specified, but markers is not in the mode
## Adding markers to the mode...Graph of a logistic regression curve showing probability of surviving the surgery versus surgeon’s experience.
The graph shows the probability of the clinical outcome, survival, (Y-axis) versus the surgeon’s experience (X-axis), with the logistic regression curve fitted to the data.
mylogit <- glm(CO ~ SE, data = mydata, family = "binomial")
summary(mylogit)##
## Call:
## glm(formula = CO ~ SE, family = "binomial", data = mydata)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -1.7131 -0.5719 -0.0085 0.4493 1.8220
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -4.1030 1.7629 -2.327 0.0199 *
## SE 0.7583 0.3139 2.416 0.0157 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 27.726 on 19 degrees of freedom
## Residual deviance: 16.092 on 18 degrees of freedom
## AIC: 20.092
##
## Number of Fisher Scoring iterations: 5The output indicates that surgeon’s experience (SE) is significantly associated with the probability of surviving the surgery (0.0157, Wald test). The output also provides the coefficients for:
- Intercept = -4.1030 and SE = 0.7583.
These coefficients can then be used in the logistic regression equation model to estimate the probability of surviving the heart surgery:
Probability of surviving heart surgery \(CO =1/(1+exp(-(-4.1030+0.7583\times SE)))\)
For example, for a patient who is operated by a surgeon with 200 hours of operating experience (SE=2), we plug in the value 2 in the equation to get an estimated probability of survival, \(p=0.07\):
SE=2
CO =1/(1+exp(-(-4.1030+0.7583*SE)))
CO## [1] 0.07001884[1] 0.07001884
Similarly, a patient undergoing heart surgery with a doctor that has 400 operating hours experience (SE=4), the estimated probability of survival is p=0.26:
SE=4; CO =1/(1+exp(-(-4.1030+0.7583*SE))); CO## [1] 0.2554411CO## [1] 0.2554411for (SE in c(1:5)) {
CO <- 1/(1+exp(-(-4.1030+0.7583*SE)));
print(c(SE, CO))
}## [1] 1.00000000 0.03406915
## [1] 2.00000000 0.07001884
## [1] 3.0000000 0.1384648
## [1] 4.0000000 0.2554411
## [1] 5.0000000 0.4227486[1] 0.2554411
The table below shows the probability of surviving surgery for several values of surgeons’ experience.
| Surgeon’s Experience (SE) | Probability of patient survival (Clinical Outcome) |
|---|---|
| 1 | 0.034 |
| 2 | 0.07 |
| 3 | 0.14 |
| 4 | 0.26 |
| 5 | 0.423 |
The output from the logistic regression analysis gives a p-value of \(p=0.0157\), which is based on the Wald z-score. In addition to the Wald method, we can calculate the p-value for logistic regression using the Likelihood Ratio Test (LRT), which for these data yields \(0.0006476922\).
mylogit <- glm(CO ~ SE, data = mydata, family = "binomial")
summary(mylogit)##
## Call:
## glm(formula = CO ~ SE, family = "binomial", data = mydata)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -1.7131 -0.5719 -0.0085 0.4493 1.8220
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -4.1030 1.7629 -2.327 0.0199 *
## SE 0.7583 0.3139 2.416 0.0157 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 27.726 on 19 degrees of freedom
## Residual deviance: 16.092 on 18 degrees of freedom
## AIC: 20.092
##
## Number of Fisher Scoring iterations: 5| . | Estimate | Std. Error | z value | \(Pr(\gt|z|)\) Wald |
|---|---|---|---|---|
| SE | 0.7583 | 0.3139 | 2.416 | 0.0157 * |
The logit of a number \(0\leq p\leq 1\) is given by the formula: \(logit(p)=log\frac{p}{1-p}\), and represents the log-odds ratio (of survival in this case).
confint(mylogit) ## Waiting for profiling to be done...## 2.5 % 97.5 %
## (Intercept) -8.6083535 -1.282692
## SE 0.2687893 1.576912So, why exponentiating the coefficients? Because,
\[logit(p)=log\frac{p}{1-p} \longrightarrow e^{logit(p)} =e^{log\frac{p}{1-p}}\longrightarrow RHS=\frac{p}{1-p}, \ \text{(odds-ratio, OR)}.\]
exp(coef(mylogit)) # exponentiated logit model coefficients## (Intercept) SE
## 0.01652254 2.13474149- exp(coef(mylogit))
| (Intercept) | SE |
|---|---|
| 0.01652254 | 2.13474149 == exp(0.7583456) |
- coef(mylogit) # raw logit model coefficients
| (Intercept) | SE |
|---|---|
| -4.1030298 | 0.7583456 |
exp(cbind(OR = coef(mylogit), confint(mylogit)))## Waiting for profiling to be done...## OR 2.5 % 97.5 %
## (Intercept) 0.01652254 0.0001825743 0.277290
## SE 2.13474149 1.3083794719 4.839986| . | OR | 2.5% | 97.5% |
|---|---|---|---|
| (Intercept) | 0.01652254 | 0.0001825743 | 0.277290 |
| SE | 2.13474149 | 1.3083794719 | 4.839986 |
We can compute the LRT and report its p-value by using the with() function:
with(mylogit, df.null - df.residual)
with(mylogit, pchisq(null.deviance - deviance, df.null - df.residual, lower.tail = FALSE))## [1] 0.0006476922# mylogit$null.deviance - mylogit$deviance # 11.63365
# mylogit$df.null - mylogit$df.residual # 1
# [1] 0.0006476922
# CONFIRM THE RESULT
# qchisq(1-with(mylogit, pchisq(null.deviance - deviance, df.null - df.residual, lower.tail = FALSE)), 1)
# qchisq(1-0.0006476922, 1)LRT p-value < 0.001 tells us that our model as a whole fits significantly better than an empty model. The deviance residual mylogit$deviance is -2*log likelihood, and we can report the model’s log likelihood by:
mylogit$deviance # model residual diviance## [1] 16.09223-2*logLik(mylogit) # -2 * model_ll## 'log Lik.' 16.09223 (df=2)9 Implementation of Regularization
Before we dive into the theoretical formulation of model regularization, let’s start with a specific application that will ground the subsequent analytics.
9.1 Example: Neuroimaging-genetics study of Parkinson’s Disease Dataset
More information about this specific study and the included derived neuroimaging biomarkers is available here. A link to the data and a brief summary of the features are included below:
- 05_PPMI_top_UPDRS_Integrated_LongFormat1.csv
- Data elements include: FID_IID, L_insular_cortex_ComputeArea, L_insular_cortex_Volume, R_insular_cortex_ComputeArea, R_insular_cortex_Volume, L_cingulate_gyrus_ComputeArea, L_cingulate_gyrus_Volume, R_cingulate_gyrus_ComputeArea, R_cingulate_gyrus_Volume, L_caudate_ComputeArea, L_caudate_Volume, R_caudate_ComputeArea, R_caudate_Volume, L_putamen_ComputeArea, L_putamen_Volume, R_putamen_ComputeArea, R_putamen_Volume, Sex, Weight, ResearchGroup, Age, chr12_rs34637584_GT, chr17_rs11868035_GT, chr17_rs11012_GT, chr17_rs393152_GT, chr17_rs12185268_GT, chr17_rs199533_GT, UPDRS_part_I, UPDRS_part_II, UPDRS_part_III, time_visit
Note that the dataset includes missing values and repeated measures.
The goal of this demonstration is to use OLS, ridge regression, and LASSO to find the best predictive model for the clinical outcomes – UPRDR score (vector) and Research Group (factor variable), in terms of demographic, genetics, and neuroimaging biomarkers.
We can utilize the glmnet package in R for most calculations.
#### Initial Stuff ####
# clean up
rm(list=ls())
# load required packages
# install.packages("arm")
library(glmnet)
library(arm)
library(knitr) # kable function to convert tabular R-results into Rmd tables
# pick a random seed, but set.seed(seed) only effects next block of code!
seed = 1234
#### Organize Data ####
# load dataset
# Data: https://umich.instructure.com/courses/38100/files/folder/data
# (05_PPMI_top_UPDRS_Integrated_LongFormat1.csv)
data1 <- read.table('https://umich.instructure.com/files/330397/download?download_frd=1', sep=",", header=T)
# we will deal with missing values using multiple imputation later. For now, let's just ignore incomplete cases
data1.completeRowIndexes <- complete.cases(data1); table(data1.completeRowIndexes)## data1.completeRowIndexes
## FALSE TRUE
## 609 1155prop.table(table(data1.completeRowIndexes))## data1.completeRowIndexes
## FALSE TRUE
## 0.3452381 0.6547619attach(data1)
# View(data1[data1.completeRowIndexes, ])
# define response and predictors
y <- data1$UPDRS_part_I + data1$UPDRS_part_II + data1$UPDRS_part_III
table(y) # Show Clinically relevant classification## y
## 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
## 54 20 25 12 8 7 11 16 16 9 21 16 13 13 22 25 21 31 25 29 29 28 20 25 28 26
## 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51
## 35 41 23 34 32 31 37 34 28 36 29 27 22 19 17 18 18 19 16 9 10 12 9 11 7 10
## 52 53 54 55 56 57 58 59 60 61 62 63 64 66 68 69 71 75 80 81 82
## 11 5 7 4 1 5 9 4 3 2 1 6 1 2 1 2 1 1 2 3 1y <- y[data1.completeRowIndexes]
# X = scale(data1[,]) # Explicit Scaling is not needed, as glmnet auto standardizes predictors
# X = as.matrix(data1[, c("R_caudate_Volume", "R_putamen_Volume", "Weight", "Age", "chr17_rs12185268_GT")]) # X needs to be a matrix, not a data frame
drop_features <- c("FID_IID", "ResearchGroup", "UPDRS_part_I", "UPDRS_part_II",
"UPDRS_part_III", "time_visit")
X <- data1[ , !(names(data1) %in% drop_features)]
X = as.matrix(X) # remove columns: index, ResearchGroup, and y=(PDRS_part_I + UPDRS_part_II + UPDRS_part_III)
X <- X[data1.completeRowIndexes, ]
summary(X)## L_insular_cortex_ComputeArea L_insular_cortex_Volume
## Min. : 50.03 Min. : 22.63
## 1st Qu.:2174.57 1st Qu.: 5867.23
## Median :2522.52 Median : 7362.90
## Mean :2306.89 Mean : 6710.18
## 3rd Qu.:2752.17 3rd Qu.: 8483.80
## Max. :3650.81 Max. :13499.92
## R_insular_cortex_ComputeArea R_insular_cortex_Volume
## Min. : 40.92 Min. : 11.84
## 1st Qu.:1647.69 1st Qu.:3559.74
## Median :1931.21 Median :4465.12
## Mean :1758.64 Mean :4127.87
## 3rd Qu.:2135.57 3rd Qu.:5319.13
## Max. :2791.92 Max. :8179.40
## L_cingulate_gyrus_ComputeArea L_cingulate_gyrus_Volume
## Min. : 127.8 Min. : 57.33
## 1st Qu.:2847.4 1st Qu.: 6587.07
## Median :3737.7 Median : 8965.03
## Mean :3411.3 Mean : 8265.03
## 3rd Qu.:4253.7 3rd Qu.:10815.06
## Max. :5944.2 Max. :17153.19
## R_cingulate_gyrus_ComputeArea R_cingulate_gyrus_Volume L_caudate_ComputeArea
## Min. : 104.1 Min. : 47.67 Min. : 1.782
## 1st Qu.:2829.4 1st Qu.: 6346.31 1st Qu.: 318.806
## Median :3719.4 Median : 9094.15 Median : 710.779
## Mean :3368.4 Mean : 8194.07 Mean : 657.442
## 3rd Qu.:4261.8 3rd Qu.:10832.53 3rd Qu.: 951.868
## Max. :6593.7 Max. :19761.77 Max. :1453.506
## L_caudate_Volume R_caudate_ComputeArea R_caudate_Volume
## Min. : 0.1928 Min. : 1.782 Min. : 0.193
## 1st Qu.: 264.0013 1st Qu.: 660.696 1st Qu.: 893.637
## Median : 998.2269 Median :1063.046 Median :1803.281
## Mean : 992.2892 Mean : 894.806 Mean :1548.739
## 3rd Qu.:1568.3643 3rd Qu.:1183.659 3rd Qu.:2152.509
## Max. :2746.6208 Max. :1684.563 Max. :3579.373
## L_putamen_ComputeArea L_putamen_Volume R_putamen_ComputeArea
## Min. : 6.76 Min. : 1.228 Min. : 13.93
## 1st Qu.: 775.73 1st Qu.:1234.601 1st Qu.:1255.62
## Median :1029.17 Median :1911.089 Median :1490.05
## Mean : 959.15 Mean :1864.390 Mean :1332.01
## 3rd Qu.:1260.56 3rd Qu.:2623.722 3rd Qu.:1642.41
## Max. :2129.67 Max. :4712.661 Max. :2251.41
## R_putamen_Volume Sex Weight Age
## Min. : 3.207 Min. :1.000 Min. : 43.20 Min. :31.18
## 1st Qu.:2474.041 1st Qu.:1.000 1st Qu.: 69.90 1st Qu.:53.87
## Median :3510.249 Median :1.000 Median : 80.90 Median :62.16
## Mean :3083.007 Mean :1.347 Mean : 82.06 Mean :61.25
## 3rd Qu.:3994.733 3rd Qu.:2.000 3rd Qu.: 90.70 3rd Qu.:68.83
## Max. :7096.580 Max. :2.000 Max. :135.00 Max. :83.03
## chr12_rs34637584_GT chr17_rs11868035_GT chr17_rs11012_GT chr17_rs393152_GT
## Min. :0.00000 Min. :0.0000 Min. :0.0000 Min. :0.0000
## 1st Qu.:0.00000 1st Qu.:0.0000 1st Qu.:0.0000 1st Qu.:0.0000
## Median :0.00000 Median :1.0000 Median :0.0000 Median :0.0000
## Mean :0.01212 Mean :0.6364 Mean :0.3654 Mean :0.4468
## 3rd Qu.:0.00000 3rd Qu.:1.0000 3rd Qu.:1.0000 3rd Qu.:1.0000
## Max. :1.00000 Max. :2.0000 Max. :2.0000 Max. :2.0000
## chr17_rs12185268_GT chr17_rs199533_GT
## Min. :0.0000 Min. :0.0000
## 1st Qu.:0.0000 1st Qu.:0.0000
## Median :0.0000 Median :0.0000
## Mean :0.4268 Mean :0.4052
## 3rd Qu.:1.0000 3rd Qu.:1.0000
## Max. :2.0000 Max. :2.0000# randomly split data into training (80%) and test (20%) sets
set.seed(seed)
train = sample(1 : nrow(X), round((4/5) * nrow(X)))
test = -train
# subset training data
yTrain = y[train]
XTrain = X[train, ]
XTrainOLS = cbind(rep(1, nrow(XTrain)), XTrain)
# subset test data
yTest = y[test]
XTest = X[test, ]
#### Model Estimation & Selection ####
# Estimate models
fitOLS = lm(yTrain ~ XTrain) # Ordinary Least Squares
# glmnet automatically standardizes the predictors
fitRidge = glmnet(XTrain, yTrain, alpha = 0) # Ridge Regression
fitLASSO = glmnet(XTrain, yTrain, alpha = 1) # The LASSOReaders are encouraged to compare and contract the resulting ridge and LASSO models.
9.2 Computational Complexity
Recall that the regularized regression estimates depend on the regularization parameter \(\lambda\). Fortunately, efficient algorithms for choosing optimal \(\lambda\) parameters do exist. Examples of solution path algorithms include:
- LARS Algorithm for the LASSO (Efron et al., 2004)
- Piecewise linearity (Rosset & Zhu, 2007)
- Generic path algorithm (Zhou & Wu, 2013)
- Pathwise coordinate descent (Friedman et al., 2007)
- Alternating Direction Method of Multipliers (ADMM) (Boyd et al. 2011)
We will show how to visualize the relations between the regularization parameter (\(\ln(\lambda)\)) and the number and magnitude of the corresponding coefficients for each specific regularized regression method.
9.3 LASSO and Ridge Solution Paths
The plot for the LASSO results can be obtained via:
library(RColorBrewer)
### Plot Solution Path ###
# LASSO
# plot(fitLASSO, xvar="lambda", label="TRUE")
# # add label to upper x-axis
# mtext("LASSO regularizer: Number of Nonzero (Active) Coefficients", side=3, line=2.5)
plot.glmnet <- function(glmnet.object, name="") {
df <- as.data.frame(t(as.matrix(glmnet.object$beta)))
df$loglambda <- log(glmnet.object$lambda)
df <- as.data.frame(df)
data_long <- gather(df, Variable, coefficient, 1:(dim(df)[2]-1), factor_key=TRUE)
plot_ly(data = data_long) %>%
# add error bars for each CV-mean at log(lambda)
add_trace(x = ~loglambda, y = ~coefficient, color=~Variable,
colors=colorRampPalette(brewer.pal(10,"Spectral"))(dim(df)[2]), # "Dark2",
type = 'scatter', mode = 'lines',
name = ~Variable) %>%
layout(title = paste0(name, " Model Coefficient Values"),
xaxis = list(title = paste0("log(",TeX("\\lambda"),")"), side="bottom", showgrid = TRUE),
hovermode = "x unified", legend = list(orientation='h'),
yaxis = list(title = ' Model Coefficient Values', side="left", showgrid = TRUE))
}
plot.glmnet(fitLASSO, name="LASSO")Similarly, the plot for the Ridge regularization can be obtained by:
### Plot Solution Path ###
# Ridge
# plot(fitRidge, xvar="lambda", label="TRUE")
# # add label to upper x-axis
# mtext("Ridge regularizer: Number of Nonzero (Active) Coefficients", side=3, line=2.5)
plot.glmnet(fitRidge, name="Ridge")9.4 Regression Solution Paths - Ridge vs. LASSO
Let’s try to compare the paths of the LASSO and Ridge regression solutions. Below, you will see that the curves of LASSO are steeper and non-differentiable at some points, which is the result of using the \(L_1\) norm. On the other hand, the Ridge path is smoother and asymptotically tends to \(0\) as \(\lambda\) increases.
Let’s start by examining the joint objective function (including LASSO and Ridge terms):
\[\min_\beta \left (\sum_i (y_i-x_i\beta)^2+\frac{1-\alpha}{2}||\beta||_2^2+\alpha||\beta||_1 \right ),\]
where \(||\beta||_1 = \sum_{j=1}^{p}|\beta_j|\) and \(||\beta||_2 = \sqrt{\sum_{j=1}^{p}||\beta_j||^2}\) are the norms of \(\boldsymbol\beta\) corresponding to the \(L_1\) and \(L_2\) distance measures, respectively. When \(\alpha=0\) and \(\alpha=1\) correspond to Ridge and LASSO regularization. The following two natural questions raise:
- What if \(0 <\alpha<1\)?
- How does the regularization penalty term affect the optimal solution?
In Chapter 9, we explored the minimal SSE (Sum of Square Error) for the OLS (without penalty) where the feasible parameter (\(\beta\)) spans the entire real solution space. In penalized optimization problems, the best solution may actually be unachievable. Therefore, we look for solutions that are “closest”, within the feasible region, to the enigmatic best solution.
The effect of the penalty term on the objective function is separate from the fidelity term (OLS solution). Thus, the effect of \(0\leq \alpha \leq 1\) is limited to the size and shape of the penalty region. Let’s try to visualize the feasible region as:
- centrosymmetric topology, when \(\alpha=0\), and
- super diamond topology, then \(\alpha=1\).
Below is a hands-on demonstration of that process using the following simple quadratic equation solver.
library(needs)
# Constructing Quadratic Formula
quadraticEquSolver <- function(a,b,c){
if(delta(a,b,c) > 0){ # first case D>0
x_1 = (-b+sqrt(delta(a,b,c)))/(2*a)
x_2 = (-b-sqrt(delta(a,b,c)))/(2*a)
result = c(x_1,x_2)
# print(result)
}
else if(delta(a,b,c) == 0){ # second case D=0
result = -b/(2*a)
# print(result)
}
else {"There are no real roots."} # third case D<0
}
# Constructing delta
delta<-function(a,b,c){
b^2-4*a*c
}To make this realistic, we will use the MLB dataset to first fit an OLS model. The dataset contains \(1,034\) records of heights and weights for some current and recent Major League Baseball (MLB) Players.
- Height: Player height in inch,
- Weight: Player weight in pounds,
- Age: Player age at time of record.
Then, we can obtain the SSE for any \(||\boldsymbol\beta||\):
\[SSE = ||Y-\hat Y||^2 = (Y-\hat Y)^{T}(Y-\hat Y)=Y^TY - 2\beta^TX^TY + \beta^TX^TX\beta.\]
Next, we will compute the contours for SSE in several situations.
library("ggplot2")
# load data
mlb<- read.table('https://umich.instructure.com/files/330381/download?download_frd=1', as.is=T, header=T)
str(mlb)## 'data.frame': 1034 obs. of 6 variables:
## $ Name : chr "Adam_Donachie" "Paul_Bako" "Ramon_Hernandez" "Kevin_Millar" ...
## $ Team : chr "BAL" "BAL" "BAL" "BAL" ...
## $ Position: chr "Catcher" "Catcher" "Catcher" "First_Baseman" ...
## $ Height : int 74 74 72 72 73 69 69 71 76 71 ...
## $ Weight : int 180 215 210 210 188 176 209 200 231 180 ...
## $ Age : num 23 34.7 30.8 35.4 35.7 ...fit<-lm(Height~Weight+Age-1, data = as.data.frame(scale(mlb[,4:6])))
points = data.frame(x=c(0,fit$coefficients[1]),y=c(0,fit$coefficients[2]),z=c("(0,0)","OLS Coef"))
Y=scale(mlb$Height)
X = scale(mlb[,c(5,6)])
beta1=seq(-0.556, 1.556, length.out = 100)
beta2=seq(-0.661, 0.3386, length.out = 100)
df <- expand.grid(beta1 = beta1, beta2 = beta2)
b = as.matrix(df)
df$sse <- rep(t(Y)%*%Y,100*100) - 2*b%*%t(X)%*%Y + diag(b%*%t(X)%*%X%*%t(b))base <- ggplot(df) +
stat_contour(aes(beta1, beta2, z = sse),breaks = round(quantile(df$sse, seq(0, 0.2, 0.03)), 0),
size = 0.5,color="darkorchid2",alpha=0.8)+
scale_x_continuous(limits = c(-0.4,1))+
scale_y_continuous(limits = c(-0.55,0.4))+
coord_fixed(ratio=1)+
geom_point(data = points,aes(x,y))+
geom_text(data = points,aes(x,y,label=z),vjust = 2,size=3.5)+
geom_segment(aes(x = -0.4, y = 0, xend = 1, yend = 0),colour = "grey46",
arrow = arrow(length=unit(0.30,"cm")),size=0.5,alpha=0.8)+
geom_segment(aes(x = 0, y = -0.55, xend = 0, yend = 0.4),colour = "grey46",
arrow = arrow(length=unit(0.30,"cm")),size=0.5,alpha=0.8)plot_alpha = function(alpha=0,restrict=0.2,beta1_range=0.2,annot=c(0.15,-0.25,0.205,-0.05)){
a=alpha; t=restrict; k=beta1_range; pos=data.frame(V1=annot[1:4])
text=paste("(",as.character(annot[3]),",",as.character(annot[4]),")",sep = "")
K = seq(0,k,length.out = 50)
y = unlist(lapply((1-a)*K^2/2+a*K-t, quadraticEquSolver,
a=(1-a)/2,b=a))[seq(1,99,by=2)]
fills = data.frame(x=c(rev(-K),K), y1=c(rev(y),y), y2=c(-rev(y),-y))
p<-base+geom_line(data=fills,aes(x = x,y = y1),colour = "salmon1",alpha=0.6,size=0.7)+
geom_line(data=fills,aes(x = x,y = y2),colour = "salmon1",alpha=0.6,size=0.7)+
geom_polygon(data = fills, aes(x, y1),fill = "red", alpha = 0.2)+
geom_polygon(data = fills, aes(x, y2), fill = "red", alpha = 0.2)+
geom_segment(data=pos,aes(x = V1[1] , y = V1[2], xend = V1[3], yend = V1[4]),
arrow = arrow(length=unit(0.30,"cm")),alpha=0.8,colour = "magenta")+
ggplot2::annotate("text", x = pos$V1[1]-0.01, y = pos$V1[2]-0.11,
label = paste(text,"\n","Point of Contact \n i.e., Coef of", "alpha=",fractions(a)),size=3)+
xlab(expression(beta[1]))+
ylab(expression(beta[2]))+
ggtitle(paste("alpha =",as.character(fractions(a))))+
theme(legend.position="none")
}# $\alpha=0$ - Ridge
p1 <- plot_alpha(alpha=0,restrict=(0.21^2)/2,beta1_range=0.21,annot=c(0.15,-0.25,0.205,-0.05))
p1 <- p1 + ggtitle(expression(paste(alpha, "=0 (Ridge)")))
# $\alpha=1/9$
p2 <- plot_alpha(alpha=1/9,restrict=0.046,beta1_range=0.22,annot =c(0.15,-0.25,0.212,-0.02))
p2 <- p2 + ggtitle(expression(paste(alpha, "=1/9")))
# $\alpha=1/5$
p3 <- plot_alpha(alpha=1/5,restrict=0.063,beta1_range=0.22,annot=c(0.13,-0.25,0.22,0))
p3 <- p3 + ggtitle(expression(paste(alpha, "=1/5")))
# $\alpha=1/2$
p4 <- plot_alpha(alpha=1/2,restrict=0.123,beta1_range=0.22,annot=c(0.12,-0.25,0.22,0))
p4 <- p4 + ggtitle(expression(paste(alpha, "=1/2")))
# $\alpha=3/4$
p5 <- plot_alpha(alpha=3/4,restrict=0.17,beta1_range=0.22,annot=c(0.12,-0.25,0.22,0))
p5 <- p5 + ggtitle(expression(paste(alpha, "=3/4")))# $\alpha=1$ - LASSO
t=0.22
K = seq(0,t,length.out = 50)
fills = data.frame(x=c(-rev(K),K),y1=c(rev(t-K),c(t-K)),y2=c(-rev(t-K),-c(t-K)))
p6 <- base +
geom_segment(aes(x = 0, y = t, xend = t, yend = 0),colour = "salmon1",alpha=0.1,size=0.2)+
geom_segment(aes(x = 0, y = t, xend = -t, yend = 0),colour = "salmon1",alpha=0.1,size=0.2)+
geom_segment(aes(x = 0, y = -t, xend = t, yend = 0),colour = "salmon1",alpha=0.1,size=0.2)+
geom_segment(aes(x = 0, y = -t, xend = -t, yend = 0),colour = "salmon1",alpha=0.1,size=0.2)+
geom_polygon(data = fills, aes(x, y1),fill = "red", alpha = 0.2)+
geom_polygon(data = fills, aes(x, y2), fill = "red", alpha = 0.2)+
geom_segment(aes(x = 0.12 , y = -0.25, xend = 0.22, yend = 0),colour = "magenta",
arrow = arrow(length=unit(0.30,"cm")),alpha=0.8)+
ggplot2::annotate("text", x = 0.11, y = -0.36,
label = "(0.22,0)\n Point of Contact \n i.e Coef of LASSO",size=3)+
xlab( expression(beta[1]))+
ylab( expression(beta[2]))+
theme(legend.position="none")+
ggtitle(expression(paste(alpha, "=1 (LASSO)")))Then, let’s add the six feasible regions corresponding to \(\alpha=0\) (Ridge), \(\alpha=\frac{1}{9}\), \(\alpha=\frac{1}{5}\), \(\alpha=\frac{1}{2}\), \(\alpha=\frac{3}{4}\) and \(\alpha=1\) (LASSO).
This figure provides some intuition into the continuum from Ridge to LASSO regularization. The feasible regions are drawn as ellipse contours of the SSE in red. Curves around the corresponding feasible regions represent the boundary of the constraint function \(\frac{1-\alpha}{2}||\beta||_2^2+\alpha||\beta||_1\leq t\).
In this example, \(\beta_2\) shrinks to \(0\) for \(\alpha=\frac{1}{5}\), \(\alpha=\frac{1}{2}\), \(\alpha=\frac{3}{4}\) and \(\alpha=1\).
We observe that it is almost impossible for the contours of Ridge regression to touch the circle at any of the coordinate axes. This is also true in higher dimensions (\(nD\)), where the \(L_1\) and \(L_2\) metrics are unchanged and the 2D ellipse representations of the feasibility regions become hyper-ellipsoidal shapes.
Generally, as \(\alpha\) goes from \(0\) to \(1\). The coefficients of more features tend to shrink towards \(0\). This specific property makes LASSO useful for variable selection.
By Lagrangian duality, any solution of \(\min_\beta {||Y-X\beta||^2_2} +\lambda ||\beta||_2\) and \(\min_\beta {||Y-X\beta||^2_1} +\lambda ||\beta||_1\) must also represent a solution to the corresponding Ridge (\(\hat{\beta}^{RR}\)) or LASSO (\(\hat{\beta}^{L}\)) optimization problems:
\[\min_\beta {||Y-X\beta||^2_2},\ \ \text{subject to}\ \ ||\beta||_2 \leq||\hat{\beta}^{RR}||_2, \] \[\min_\beta {||Y-X\beta||^2_2},\ \ \text{subject to}\ \ ||\beta||_1 \leq ||\hat{\beta}^{L}||_1, \]
Suppose we actually know hte value of \(||\hat{\beta}^{RR}||_2\) and \(||\hat{\beta}^{L}||_1\), then we can pictorialy represent the optimizaiton problem and illstrate the complementary model-fitting, variable selection and shrinkage of the Ridge and LASSO regularization.
The topologies of the solution (domain) regions are dfferent for Ridge and LASSO. Ridge regularization corresponds with ball topology and LASSO with diamond topology. This is because the solution regions are defined by \(||\hat{\beta}||_2\leq ||\hat{\beta}^{RR}||_2\) and \(||\hat{\beta}||_1\leq ||\hat{\beta}^{L}||_1\), respectively.
On the other hand, the topology of the fidelity term \(||Y-X\beta||^2_2\) is elipsoidal, centered at the OLS estimate, \(\hat{\beta}^{OLS}\). To solve the optimizaiton problem, we look for the tightest contour around \(\hat{\beta}^{OLS}\) that hits the solution domain (ball for Ridge or diamond for LASSO). This intersection point would represent the solution estimate \(\hat{\beta}\). As the LASSO domain space (\(l_1\) unit ball) has these corners, the solution estimate \(\hat{\beta}\) is likely to be at the corners. Hence LASSO solutions tend to inlcude many zeroes, whereas Ridge regression solutions (constraint set is a round ball) may not.
Let’s compare the feasibility regions corresponding to Ridge (top, \(p1\)) and LASSO (bottom, \(p6\)) regularization.
plot(p1)
SSE Contour and Penalty Region (Ridge).
plot(p6)
SSE Contour and Penalty Region (LASSO).
Then, we can plot the progression from Ridge to LASSO. (This composite plot is intense and may take several minutes to render!)
library("gridExtra")
grid.arrange(p1,p2,p3,p4,p5,p6,nrow=3)
SSE Contour and Penalty Region for 6 values of Alpha.
9.5 Choice of the Regularization Parameter
Efficiently obtaining the entire solution path is nice, but we still have to choose a specific \(\lambda\) regularization parameter. This is critical as \(\lambda\) controls the bias-variance tradeoff.
Traditional model selection methods rely on various metrics like Mallows’ \(C_p\), AIC, BIC, and adjusted \(R^2\).
Internal statistical validation (Cross validation) is a popular modern alternative, which offers some of these benefits:
- Choice is based on predictive performance,
- Makes fewer model assumptions,
- More widely applicable.
9.6 Cross Validation Motivation
Ideally, we would like a separate validation set for choosing \(\lambda\) for a given method. Reusing training sets may encourage overfitting and using testing data to pick \(\lambda\) may underestimates the true error rate. Often, when we do not have enough data for a separate validation set, cross validation provides an alternative strategy.
9.7 \(n\)-Fold Cross Validation
We have already seen examples of using cross-validation, e.g., Chapter 13, and Chapter 20 provides more details about this internal statistical assessment strategy.
We can use either automated or manual cross-validation. In either case, the protocol involves the following iterative steps:
- Randomly split the training data into \(n\) parts (“folds”).
- Fit a model using data in \(n-1\) folds for multiple \(\lambda\text{s}\).
- Calculate some prediction quality metrics (e.g., MSE, accuracy) on the last remaining fold, see Chapter 13.
- Repeat the process and average the prediction metrics across iterations.
Common choices of \(n\) are 5, 10, and \(n\) (which corresponds to leave-one-out CV). One standard error rule is to choose \(\lambda\) corresponding to smallest model with MSE within one standard error of the minimum MSE.
9.8 LASSO 10-Fold Cross Validation
Now, let’s apply an internal statistical cross-validation to assess the quality of the LASSO and Ridge models, based on our Parkinson’s disease case-study. Recall our split of the PD data into training (yTrain, XTrain) and testing (yTest, XTest) sets.
plotCV.glmnet <- function(cv.glmnet.object, name="") {
df <- as.data.frame(cbind(x=log(cv.glmnet.object$lambda), y=cv.glmnet.object$cvm,
errorBar=cv.glmnet.object$cvsd), nzero=cv.glmnet.object$nzero)
featureNum <- cv.glmnet.object$nzero
xFeature <- log(cv.glmnet.object$lambda)
yFeature <- max(cv.glmnet.object$cvm)+max(cv.glmnet.object$cvsd)
dataFeature <- data.frame(featureNum, xFeature, yFeature)
plot_ly(data = df) %>%
# add error bars for each CV-mean at log(lambda)
add_trace(x = ~x, y = ~y, type = 'scatter', mode = 'markers',
name = 'CV MSE', error_y = ~list(array = errorBar)) %>%
# add the lambda-min and lambda 1SD vertical dash lines
add_lines(data=df, x=c(log(cv.glmnet.object$lambda.min), log(cv.glmnet.object$lambda.min)),
y=c(min(cv.glmnet.object$cvm)-max(df$errorBar), max(cv.glmnet.object$cvm)+max(df$errorBar)),
showlegend=F, line=list(dash="dash"), name="lambda.min", mode = 'lines+markers') %>%
add_lines(data=df, x=c(log(cv.glmnet.object$lambda.1se), log(cv.glmnet.object$lambda.1se)),
y=c(min(cv.glmnet.object$cvm)-max(df$errorBar), max(cv.glmnet.object$cvm)+max(df$errorBar)),
showlegend=F, line=list(dash="dash"), name="lambda.1se") %>%
# Add Number of Features Annotations on Top
add_trace(dataFeature, x = ~xFeature, y = ~yFeature, type = 'scatter', name="Number of Features",
mode = 'text', text = ~featureNum, textposition = 'middle right',
textfont = list(color = '#000000', size = 9)) %>%
# Add top x-axis (non-zero features)
# add_trace(data=df, x=~c(min(cv.glmnet.object$nzero),max(cv.glmnet.object$nzero)),
# y=~c(max(y)+max(errorBar),max(y)+max(errorBar)), showlegend=F,
# name = "Non-Zero Features", yaxis = "ax", mode = "lines+markers", type = "scatter") %>%
layout(title = paste0("Cross-Validation MSE (", name, ")"),
xaxis = list(title=paste0("log(",TeX("\\lambda"),")"), side="bottom", showgrid=TRUE), # type="log"
hovermode = "x unified", legend = list(orientation='h'), # xaxis2 = ax,
yaxis = list(title = cv.glmnet.object$name, side="left", showgrid = TRUE))
}
#### 10-fold cross validation ####
# LASSO
library("glmnet")
library(doParallel)
cl <- makePSOCKcluster(6)
registerDoParallel(cl)
set.seed(seed) # set seed
# (10-fold) cross validation for the LASSO
cvLASSO = cv.glmnet(XTrain, yTrain, alpha = 1, parallel=TRUE)
# plot(cvLASSO)
# mtext("CV LASSO: Number of Nonzero (Active) Coefficients", side=3, line=2.5)
plotCV.glmnet(cvLASSO, "LASSO")# Report MSE LASSO
predLASSO <- predict(cvLASSO, s = cvLASSO$lambda.1se, newx = XTest)
testMSE_LASSO <- mean((predLASSO - yTest)^2); testMSE_LASSO## [1] 233.183plotCV.glmnet <- function(cv.glmnet.object, name="") {
df <- as.data.frame(cbind(x=log(cv.glmnet.object$lambda), y=cv.glmnet.object$cvm,
errorBar=cv.glmnet.object$cvsd), nzero=cv.glmnet.object$nzero)
featureNum <- cv.glmnet.object$nzero
xFeature <- log(cv.glmnet.object$lambda)
yFeature <- max(cv.glmnet.object$cvm)+max(cv.glmnet.object$cvsd)
dataFeature <- data.frame(featureNum, xFeature, yFeature)
plot_ly(data = df) %>%
# add error bars for each CV-mean at log(lambda)
add_trace(x = ~x, y = ~y, type = 'scatter', mode = 'markers',
name = 'CV MSE', error_y = ~list(array = errorBar)) %>%
# add the lambda-min and lambda 1SD vertical dash lines
add_lines(data=df, x=c(log(cv.glmnet.object$lambda.min), log(cv.glmnet.object$lambda.min)),
y=c(min(cv.glmnet.object$cvm)-max(df$errorBar), max(cv.glmnet.object$cvm)+max(df$errorBar)),
showlegend=F, line=list(dash="dash"), name="lambda.min", mode = 'lines+markers') %>%
add_lines(data=df, x=c(log(cv.glmnet.object$lambda.1se), log(cv.glmnet.object$lambda.1se)),
y=c(min(cv.glmnet.object$cvm)-max(df$errorBar), max(cv.glmnet.object$cvm)+max(df$errorBar)),
showlegend=F, line=list(dash="dash"), name="lambda.1se") %>%
# Add Number of Features Annotations on Top
add_trace(dataFeature, x = ~xFeature, y = ~yFeature, type = 'scatter', name="Number of Features",
mode = 'text', text = ~featureNum, textposition = 'middle right',
textfont = list(color = '#000000', size = 9)) %>%
# Add top x-axis (non-zero features)
# add_trace(data=df, x=~c(min(cv.glmnet.object$nzero),max(cv.glmnet.object$nzero)),
# y=~c(max(y)+max(errorBar),max(y)+max(errorBar)), showlegend=F,
# name = "Non-Zero Features", yaxis = "ax", mode = "lines+markers", type = "scatter") %>%
layout(title = paste0("Cross-Validation MSE (", name, ")"),
xaxis = list(title=paste0("log(",TeX("\\lambda"),")"), side="bottom", showgrid=TRUE), # type="log"
hovermode = "x unified", legend = list(orientation='h'), # xaxis2 = ax,
yaxis = list(title = cv.glmnet.object$name, side="left", showgrid = TRUE))
}
#### 10-fold cross validation ####
# Ridge Regression
set.seed(seed) # set seed
# (10-fold) cross validation for Ridge Regression
cvRidge = cv.glmnet(XTrain, yTrain, alpha = 0, parallel=TRUE)
# plot(cvRidge)
# mtext("CV Ridge: Number of Nonzero (Active) Coefficients", side=3, line=2.5)
plotCV.glmnet(cvRidge, "Ridge")# Report MSE Ridge
predRidge <- predict(cvRidge, s = cvRidge$lambda.1se, newx = XTest)
testMSE_Ridge <- mean((predRidge - yTest)^2); testMSE_Ridge## [1] 233.183stopCluster(cl)Note that the predict() method applied to cv.gmlnet or glmnet forecasting models is effectively a function wrapper to predict.gmlnet(). According to what you would like to get as a prediction output, you can use type="..." to specify one of the following types of prediction outputs:
type="link", reports the linear predictors for “binomial”, “multinomial”, “poisson” or “cox” models; for “gaussian” models it gives the fitted values.type="response", reports the fitted probabilities for “binomial” or “multinomial”, fitted mean for “poisson” and the fitted relative-risk for “cox”; for “gaussian” type “response” is equivalent to type “link”.type="coefficients", reports the coefficients at the requested values fors. Note that for “binomial” models, results are returned only for the class corresponding to the second level of the factor response.type="class", applies only to “binomial” or “multinomial” models, and produces the class label corresponding to the maximum probability.type="nonzero", returns a list of the indices of the nonzero coefficients for each value ofs.
9.9 Stepwise OLS (ordinary least squares)
For a fair comparison, let’s also obtain an OLS stepwise model selection, see Chapter 16.
dt = as.data.frame(cbind(yTrain,XTrain))
ols_step <- lm(yTrain ~., data = dt)
ols_step <- step(ols_step, direction = 'both', k=2, trace = F)
summary(ols_step)##
## Call:
## lm(formula = yTrain ~ L_insular_cortex_ComputeArea + L_insular_cortex_Volume +
## R_insular_cortex_Volume + L_cingulate_gyrus_ComputeArea +
## R_cingulate_gyrus_Volume + L_caudate_Volume + L_putamen_ComputeArea +
## L_putamen_Volume + R_putamen_ComputeArea + Sex + Weight +
## Age + chr17_rs11868035_GT + chr17_rs11012_GT + chr17_rs393152_GT +
## chr17_rs12185268_GT, data = dt)
##
## Residuals:
## Min 1Q Median 3Q Max
## -38.326 -9.250 0.067 9.257 54.184
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.6772645 6.8299694 0.685 0.493636
## L_insular_cortex_ComputeArea -0.0082211 0.0046771 -1.758 0.079131 .
## L_insular_cortex_Volume 0.0021880 0.0012140 1.802 0.071834 .
## R_insular_cortex_Volume -0.0013032 0.0008958 -1.455 0.146080
## L_cingulate_gyrus_ComputeArea 0.0074028 0.0016421 4.508 7.40e-06 ***
## R_cingulate_gyrus_Volume -0.0014608 0.0003854 -3.790 0.000161 ***
## L_caudate_Volume -0.0044248 0.0013782 -3.211 0.001371 **
## L_putamen_ComputeArea -0.0144807 0.0053722 -2.695 0.007159 **
## L_putamen_Volume 0.0055039 0.0021430 2.568 0.010379 *
## R_putamen_ComputeArea 0.0065605 0.0023052 2.846 0.004527 **
## Sex 2.9342207 1.2369794 2.372 0.017896 *
## Weight 0.0555807 0.0346594 1.604 0.109145
## Age 0.1524618 0.0593021 2.571 0.010301 *
## chr17_rs11868035_GT -1.5710859 0.7447326 -2.110 0.035167 *
## chr17_rs11012_GT -7.7050024 1.9846968 -3.882 0.000111 ***
## chr17_rs393152_GT -4.4644593 2.3601881 -1.892 0.058867 .
## chr17_rs12185268_GT 12.9794629 2.9244435 4.438 1.02e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 14.81 on 907 degrees of freedom
## Multiple R-squared: 0.0995, Adjusted R-squared: 0.08362
## F-statistic: 6.264 on 16 and 907 DF, p-value: 2.123e-13We use direction=both for both forward and backward selection and choose the optimal one. k=2 specifies AIC and BIC criteria, and you can choose \(k\sim \log(n)\).
Then, we use the ols_step model to predict the outcome \(Y\) for some new test data.
betaHatOLS_step = ols_step$coefficients
var_step <- colnames(ols_step$model)[-1]
XTestOLS_step = cbind(rep(1, nrow(XTest)), XTest[,var_step])
predOLS_step = XTestOLS_step%*%betaHatOLS_step
testMSEOLS_step = mean((predOLS_step - yTest)^2)
# Report MSE OLS Stepwise feature selection
testMSEOLS_step## [1] 243.939Alternatively, we can predict the outcomes directly using the predict() function, and the results should be identical:
pred2 <- predict(ols_step,as.data.frame(XTest))
any(pred2 == predOLS_step)## [1] TRUE9.10 Final Models
Let’s identify the most important (predictive) features, which can then be interpreted in the context of the specific data.
# Determine final models
# Extract Coefficients
# OLS coefficient estimates
betaHatOLS = fitOLS$coefficients
# LASSO coefficient estimates
betaHatLASSO = as.double(coef(fitLASSO, s = cvLASSO$lambda.1se)) # s is lambda
# Ridge coefficient estimates
betaHatRidge = as.double(coef(fitRidge, s = cvRidge$lambda.1se))
# Test Set MSE
# calculate predicted values
XTestOLS = cbind(rep(1, nrow(XTest)), XTest) # add intercept to test data
predOLS = XTestOLS%*%betaHatOLS
predLASSO = predict(fitLASSO, s = cvLASSO$lambda.1se, newx = XTest)
predRidge = predict(fitRidge, s = cvRidge$lambda.1se, newx = XTest)
# calculate test set MSE
testMSEOLS = mean((predOLS - yTest)^2)
testMSELASSO = mean((predLASSO - yTest)^2)
testMSERidge = mean((predRidge - yTest)^2)This plot shows a rank-ordered list of the key predictors of the clinical outcome variable (total UPDRS, y <- data1$UPDRS_part_I + data1$UPDRS_part_II + data1$UPDRS_part_III).
# Plot Regression Coefficients
# create variable names for plotting
library("arm")
par(mar=c(2, 13, 1, 1)) # extra large left margin
varNames <- colnames(data1[ , !(names(data1) %in% drop_features)]); varNames; length(varNames)## [1] "L_insular_cortex_ComputeArea" "L_insular_cortex_Volume"
## [3] "R_insular_cortex_ComputeArea" "R_insular_cortex_Volume"
## [5] "L_cingulate_gyrus_ComputeArea" "L_cingulate_gyrus_Volume"
## [7] "R_cingulate_gyrus_ComputeArea" "R_cingulate_gyrus_Volume"
## [9] "L_caudate_ComputeArea" "L_caudate_Volume"
## [11] "R_caudate_ComputeArea" "R_caudate_Volume"
## [13] "L_putamen_ComputeArea" "L_putamen_Volume"
## [15] "R_putamen_ComputeArea" "R_putamen_Volume"
## [17] "Sex" "Weight"
## [19] "Age" "chr12_rs34637584_GT"
## [21] "chr17_rs11868035_GT" "chr17_rs11012_GT"
## [23] "chr17_rs393152_GT" "chr17_rs12185268_GT"
## [25] "chr17_rs199533_GT"## [1] 25# # Graph 27 regression coefficients (exclude intercept [1], betaHat indices 2:27)
# coefplot(betaHatOLS[2:27], sd = rep(0, 26), pch=0, cex.pts = 3, main = "Regression Coefficient Estimates", varnames = varNames)
# coefplot(betaHatLASSO[2:27], sd = rep(0, 26), pch=1, add = TRUE, col.pts = "red", cex.pts = 3)
# coefplot(betaHatRidge[2:27], sd = rep(0, 26), pch=2, add = TRUE, col.pts = "blue", cex.pts = 3)
# legend("bottomright", c("OLS", "LASSO", "Ridge"), col = c("black", "red", "blue"), pch = c(0, 1 , 2), bty = "o", cex = 2)
df <- as.data.frame(cbind(Feature=attributes(betaHatOLS)$names[2:26], OLS=betaHatOLS[2:26],
LASSO=betaHatLASSO[2:26], Ridge=betaHatRidge[2:26]))
data_long <- gather(df, Method, value, OLS:Ridge, factor_key=TRUE)
data_long$value <- as.numeric(data_long$value)
# Note that Plotly will automatically order your axes by the order that is present in the data
# When using character vectors - order is alphabetic; in case of factors the order is by levels.
# To override this behavior, specify categoryorder and categoryarray for the appropriate axis in the layout
formY <- list(categoryorder = "array", categoryarray = df$Feature)
plot_ly(data_long, x=~value, y=~Feature, type="scatter", mode="markers",
marker=list(size=20), color=~Method, symbol=~Method, symbols=c('circle-open','x-open','hexagon-open')) %>%
layout(yaxis = formY)# par()9.11 Model Performance
We next quantify the performance of the models.
# Test Set MSE Table
# create table as data frame
MSETable = data.frame(OLS=testMSEOLS, OLS_step=testMSEOLS_step, LASSO=testMSELASSO, Ridge=testMSERidge)
# convert to markdown
kable(MSETable, format="pandoc", caption="Test Set MSE", align=c("c", "c", "c", "c"))| OLS | OLS_step | LASSO | Ridge |
|---|---|---|---|
| 247.3367 | 243.939 | 233.183 | 233.183 |
9.12 Compare the selected variables
var_step = names(ols_step$coefficients)[-1]
var_lasso = colnames(XTrain)[which(coef(fitLASSO, s = cvLASSO$lambda.min)!=0)-1]
intersect(var_step,var_lasso)## [1] "L_insular_cortex_ComputeArea" "L_insular_cortex_Volume"
## [3] "R_insular_cortex_Volume" "L_cingulate_gyrus_ComputeArea"
## [5] "R_cingulate_gyrus_Volume" "L_caudate_Volume"
## [7] "L_putamen_ComputeArea" "L_putamen_Volume"
## [9] "R_putamen_ComputeArea" "Sex"
## [11] "Weight" "Age"
## [13] "chr17_rs11868035_GT" "chr17_rs11012_GT"
## [15] "chr17_rs393152_GT" "chr17_rs12185268_GT"coef(fitLASSO, s = cvLASSO$lambda.min)## 26 x 1 sparse Matrix of class "dgCMatrix"
## s1
## (Intercept) 1.7349137334
## L_insular_cortex_ComputeArea -0.0031461632
## L_insular_cortex_Volume 0.0007428576
## R_insular_cortex_ComputeArea .
## R_insular_cortex_Volume -0.0007386605
## L_cingulate_gyrus_ComputeArea 0.0060323275
## L_cingulate_gyrus_Volume .
## R_cingulate_gyrus_ComputeArea -0.0004655064
## R_cingulate_gyrus_Volume -0.0009788965
## L_caudate_ComputeArea .
## L_caudate_Volume -0.0031007230
## R_caudate_ComputeArea .
## R_caudate_Volume -0.0007081574
## L_putamen_ComputeArea -0.0099013352
## L_putamen_Volume 0.0038351119
## R_putamen_ComputeArea 0.0056737358
## R_putamen_Volume .
## Sex 2.6406206800
## Weight 0.0577691358
## Age 0.1642627834
## chr12_rs34637584_GT -2.1268900504
## chr17_rs11868035_GT -1.4279120502
## chr17_rs11012_GT -6.6808710457
## chr17_rs393152_GT -3.3231021738
## chr17_rs12185268_GT 10.9433767577
## chr17_rs199533_GT .Stepwise variable selection for OLS selects 12 variables, whereas LASSO selects 9 variables with the best \(\lambda\). There are 6 common variables common for both OLS and LASSO.
9.13 Summary
Traditional linear models are useful but also have their shortcomings:
- Prediction accuracy may be sub-optimal.
- Model interpretability may be challenging (especially when a large number of features are used as regressors).
- Stepwise model selection may improve the model performance and add some interpretations, but still may not be optimal.
Regularization adds a penalty term to the estimation:
- Enables exploitation of the bias-variance tradeoff.
- Provides flexibility on specifying penalties to allow for continuous variable selection.
- Allows incorporation of prior knowledge.
10 Knockoff Filtering (FDR-Controlled Feature Selection)
10.1 Simulated Knockoff Example
Variable selection that controls the false discovery rate (FDR) of salient features can be accomplished in different ways. The knockoff filtering represents one strategy for controlled variable selection. To show the usage of knockoff.filter we start with a synthetic dataset constructed so that the true coefficient vector \(\beta\) has only a few nonzero entries.
The essence of the knockoff filtering is based on the following three-step process:
- Construct the decoy features (knockoff variables), one for each real observed feature. These act as controls for assessing the importance of the real variables.
- For each feature, \(X_j\), compute the knockoff statistic, \(W_j\), which measures the importance of the variable, relative to its decoy counterpart, \(\tilde{X}_j\). This importance is measured by comparing the corresponding parameter estimates, \(\hat{\beta_{X_j}}\) and \(\hat{\beta_{\tilde{X}_j}}\), obtained via regularized linear modeling (e.g., LASSO).
- Determine the overall knockoff threshold. This is computed by rank-ordering the \(W_j\) statistics (from large to small), walking down the list of \(W_j\)’s, selecting variables \(X_j\) corresponding to positive \(W_j\)’s, and terminating this search the last time the ratio of negative to positive \(W_j\)’s is below the default FDR \(q\) value, e.g., \(q=0.10\).
Mathematically, we consider \(X_j\) to be unimportant (i.e., peripheral or extraneous) if the conditional distribution of \(Y\) given \(X_1,...,X_p\) does not depend on \(X_j\). Formally, \(X_j\) is unimportant if it is conditionally independent of \(Y\) given all other features, \(X_{-j}\):
\[ Y \perp X_j | X_{-j}.\] We want to generate a Markov Blanket of \(Y\), such that the smallest set of features \(J\) satisfies this condition. Further, to make sure we do not make too many mistakes, we search for a set \(\hat{S}\) controlling the false discovery rate (FDR):
\[ FDR(\hat{S}) = \mathrm{E} \left (\frac{\#j\in \hat{S}:\ x_j\ unimportant}{\#j\in \hat{S}} \right) \leq q\ (e.g.\ 10\%).\]
Let’s look at one simulation example.
# Problem parameters
n = 1000 # number of observations
p = 300 # number of variables
k = 30 # number of variables with nonzero coefficients
amplitude = 3.5 # signal amplitude (for noise level = 1)
# Problem data
X = matrix(rnorm(n*p), nrow=n, ncol=p)
nonzero = sample(p, k)
beta = amplitude * (1:p %in% nonzero)
y.sample <- function() X %*% beta + rnorm(n)To begin with, we will invoke the knockoff.filter using the default settings.
# install.packages("knockoff")
library(knockoff)
y = y.sample()
result = knockoff.filter(X, y)
print(result)## Call:
## knockoff.filter(X = X, y = y)
##
## Selected variables:
## [1] 4 11 12 25 27 42 47 82 87 91 96 104 112 123 127 130 148 152 157
## [20] 182 206 227 231 236 239 242 245 254 258 278 292 298The false discovery proportion (fdp) is:
fdp <- function(selected) sum(beta[selected] == 0) / max(1, length(selected))
fdp(result$selected)## [1] 0.0625This yields an approximate FDR of \(0.10\).
The default settings of the knockoff filter uses a test statistic based on LASSO – knockoff.stat.lasso_signed_max, which computes the \(W_j\) statistics that quantify the discrepancy between a real (\(X_j\)) and a decoy, knockoff (\(\tilde{X}_j\)), feature coefficient estimates:
\[W_j=\max(X_j, \tilde{X}_j) \times sgn(X_j - \tilde{X}_j). \] Effectively, the \(W_j\) statistics measures how much more important the variable \(X_j\) is relative to its decoy counterpart \(\tilde{X}_j\). The strength of the importance of \(X_j\) relative to \(\tilde{X}_j\) is measured by the magnitude of \(W_j\).
The knockoff package includes several other test statistics, with appropriate names prefixed by knockoff.stat. For instance, we can use a statistic based on forward selection (\(fs\)) and a lower target FDR of \(0.10\).
result = knockoff.filter(X, y, fdr = 0.10, statistic = stat.glmnet_coefdiff) # Old: statistic=knockoff.stat.fs)
#knockoff::stat.forward_selection Importance statistics based on forward selection
#knockoff::stat.glmnet_coefdiff Importance statistics based on a GLM with cross-validation
#knockoff::stat.glmnet_lambdadiff Importance statistics based on a GLM
#knockoff::stat.glmnet_lambdasmax GLM statistics for knockoff
#knockoff::stat.lasso_coefdiff Importance statistics based the lasso with cross-validation
#knockoff::stat.lasso_coefdiff_bin Importance statistics based on regularized logistic regression with cross-validation
#knockoff::stat.lasso_lambdadiff Importance statistics based on the lasso
#knockoff::stat.lasso_lambdadiff_bin Importance statistics based on regularized logistic regression
#knockoff::stat.lasso_lambdasmax Penalized linear regression statistics for knockoff
#knockoff::stat.lasso_lambdasmax_bin Penalized logistic regression statistics for knockoff
#knockoff::stat.random_forest Importance statistics based on random forests
# knockoff::stat.sqrt_lasso Importance statistics based on the square-root lasso
#knockoff::stat.stability_selection Importance statistics based on stability selection
#knockoff::verify_stat_depends Verify dependencies for chosen statistics)
fdp(result$selected)## [1] 0.09090909One can also define additional test statistics, complementing the ones included in the package already. For instance, if we want to implement the following test-statistics:
\[W_j= || X^t . y|| - ||\tilde{X^t} . y||.\]
We can code it as:
new_knockoff_stat <- function(X, X_ko, y) {
abs(t(X) %*% y) - abs(t(X_ko) %*% y)
}
result = knockoff.filter(X, y, statistic = new_knockoff_stat)
# print indices of selected features
print(sprintf("Number of KO-selected features: %d", length(result$selected)))## [1] "Number of KO-selected features: 22"cat("Indices of KO-selected features: ", result$selected)## Indices of KO-selected features: 4 12 25 27 47 87 91 96 123 127 130 152 157 182 206 227 231 242 245 258 292 298fdp(result$selected)## [1] 010.2 Knockoff invocation
The knockoff.filter function is a wrapper around several simpler functions that (1) construct knockoff variables (knockoff.create); (2) compute the test statistic \(W\) (various functions with prefix knockoff.stat); and (3) compute the threshold for variable selection (knockoff.threshold).
The high-level function knockoff.filter will automatically normalize the columns of the input matrix (unless this behavior is explicitly disabled). However, all other functions in this package assume that the columns of the input matrix have unitary Euclidean norm.
10.3 PD Neuroimaging-genetics Case-Study
Let’s illustrate controlled variable selection via knockoff filtering using the real PD dataset.
The goal is to determine which imaging, genetics and phenotypic covariates are associated with the clinical diagnosis of PD. The dataset is available at the DSPA case-study archive site.
10.3.1 Preparing the data
The data set consists of clinical, genetics, and demographic measurements. To evaluate our results, we will compare diagnostic predictions created by the model for the UPDRS scores and the ResearchGroup factor variable.
10.3.2 Fetching and cleaning the data
First, we download the data and read it into data frames.
data1 <- read.table('https://umich.instructure.com/files/330397/download?download_frd=1', sep=",", header=T)
# we will deal with missing values using multiple imputation later. For now, let's just ignore incomplete cases
data1.completeRowIndexes <- complete.cases(data1) # table(data1.completeRowIndexes)
prop.table(table(data1.completeRowIndexes))## data1.completeRowIndexes
## FALSE TRUE
## 0.3452381 0.6547619# attach(data1)
# View(data1[data1.completeRowIndexes, ])
data2 <- data1[data1.completeRowIndexes, ]
Dx_label <- data2$ResearchGroup; table(Dx_label)## Dx_label
## Control PD SWEDD
## 121 897 13710.3.3 Preparing the design matrix
We now construct the design matrix \(X\) and the response vector \(Y\). The features (columns of \(X\)) represent covariates that will be used to explain the response \(Y\).
# Construct preliminary design matrix.
# define response and predictors
Y <- data1$UPDRS_part_I + data1$UPDRS_part_II + data1$UPDRS_part_III
table(Y) # Show Clinically relevant classification## Y
## 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
## 54 20 25 12 8 7 11 16 16 9 21 16 13 13 22 25 21 31 25 29 29 28 20 25 28 26
## 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51
## 35 41 23 34 32 31 37 34 28 36 29 27 22 19 17 18 18 19 16 9 10 12 9 11 7 10
## 52 53 54 55 56 57 58 59 60 61 62 63 64 66 68 69 71 75 80 81 82
## 11 5 7 4 1 5 9 4 3 2 1 6 1 2 1 2 1 1 2 3 1Y <- Y[data1.completeRowIndexes]
# X = scale(ncaaData[, -20]) # Explicit Scaling is not needed, as glmnet auto standardizes predictors
# X = as.matrix(data1[, c("R_caudate_Volume", "R_putamen_Volume", "Weight", "Age", "chr17_rs12185268_GT")]) # X needs to be a matrix, not a data frame
drop_features <- c("FID_IID", "ResearchGroup", "UPDRS_part_I", "UPDRS_part_II", "UPDRS_part_III")
X <- data1[ , !(names(data1) %in% drop_features)]
X = as.matrix(X) # remove columns: index, ResearchGroup, and y=(PDRS_part_I + UPDRS_part_II + UPDRS_part_III)
X <- X[data1.completeRowIndexes, ]; dim(X)## [1] 1155 26summary(X)## L_insular_cortex_ComputeArea L_insular_cortex_Volume
## Min. : 50.03 Min. : 22.63
## 1st Qu.:2174.57 1st Qu.: 5867.23
## Median :2522.52 Median : 7362.90
## Mean :2306.89 Mean : 6710.18
## 3rd Qu.:2752.17 3rd Qu.: 8483.80
## Max. :3650.81 Max. :13499.92
## R_insular_cortex_ComputeArea R_insular_cortex_Volume
## Min. : 40.92 Min. : 11.84
## 1st Qu.:1647.69 1st Qu.:3559.74
## Median :1931.21 Median :4465.12
## Mean :1758.64 Mean :4127.87
## 3rd Qu.:2135.57 3rd Qu.:5319.13
## Max. :2791.92 Max. :8179.40
## L_cingulate_gyrus_ComputeArea L_cingulate_gyrus_Volume
## Min. : 127.8 Min. : 57.33
## 1st Qu.:2847.4 1st Qu.: 6587.07
## Median :3737.7 Median : 8965.03
## Mean :3411.3 Mean : 8265.03
## 3rd Qu.:4253.7 3rd Qu.:10815.06
## Max. :5944.2 Max. :17153.19
## R_cingulate_gyrus_ComputeArea R_cingulate_gyrus_Volume L_caudate_ComputeArea
## Min. : 104.1 Min. : 47.67 Min. : 1.782
## 1st Qu.:2829.4 1st Qu.: 6346.31 1st Qu.: 318.806
## Median :3719.4 Median : 9094.15 Median : 710.779
## Mean :3368.4 Mean : 8194.07 Mean : 657.442
## 3rd Qu.:4261.8 3rd Qu.:10832.53 3rd Qu.: 951.868
## Max. :6593.7 Max. :19761.77 Max. :1453.506
## L_caudate_Volume R_caudate_ComputeArea R_caudate_Volume
## Min. : 0.1928 Min. : 1.782 Min. : 0.193
## 1st Qu.: 264.0013 1st Qu.: 660.696 1st Qu.: 893.637
## Median : 998.2269 Median :1063.046 Median :1803.281
## Mean : 992.2892 Mean : 894.806 Mean :1548.739
## 3rd Qu.:1568.3643 3rd Qu.:1183.659 3rd Qu.:2152.509
## Max. :2746.6208 Max. :1684.563 Max. :3579.373
## L_putamen_ComputeArea L_putamen_Volume R_putamen_ComputeArea
## Min. : 6.76 Min. : 1.228 Min. : 13.93
## 1st Qu.: 775.73 1st Qu.:1234.601 1st Qu.:1255.62
## Median :1029.17 Median :1911.089 Median :1490.05
## Mean : 959.15 Mean :1864.390 Mean :1332.01
## 3rd Qu.:1260.56 3rd Qu.:2623.722 3rd Qu.:1642.41
## Max. :2129.67 Max. :4712.661 Max. :2251.41
## R_putamen_Volume Sex Weight Age
## Min. : 3.207 Min. :1.000 Min. : 43.20 Min. :31.18
## 1st Qu.:2474.041 1st Qu.:1.000 1st Qu.: 69.90 1st Qu.:53.87
## Median :3510.249 Median :1.000 Median : 80.90 Median :62.16
## Mean :3083.007 Mean :1.347 Mean : 82.06 Mean :61.25
## 3rd Qu.:3994.733 3rd Qu.:2.000 3rd Qu.: 90.70 3rd Qu.:68.83
## Max. :7096.580 Max. :2.000 Max. :135.00 Max. :83.03
## chr12_rs34637584_GT chr17_rs11868035_GT chr17_rs11012_GT chr17_rs393152_GT
## Min. :0.00000 Min. :0.0000 Min. :0.0000 Min. :0.0000
## 1st Qu.:0.00000 1st Qu.:0.0000 1st Qu.:0.0000 1st Qu.:0.0000
## Median :0.00000 Median :1.0000 Median :0.0000 Median :0.0000
## Mean :0.01212 Mean :0.6364 Mean :0.3654 Mean :0.4468
## 3rd Qu.:0.00000 3rd Qu.:1.0000 3rd Qu.:1.0000 3rd Qu.:1.0000
## Max. :1.00000 Max. :2.0000 Max. :2.0000 Max. :2.0000
## chr17_rs12185268_GT chr17_rs199533_GT time_visit
## Min. :0.0000 Min. :0.0000 Min. : 0.00
## 1st Qu.:0.0000 1st Qu.:0.0000 1st Qu.: 9.00
## Median :0.0000 Median :0.0000 Median :24.00
## Mean :0.4268 Mean :0.4052 Mean :23.83
## 3rd Qu.:1.0000 3rd Qu.:1.0000 3rd Qu.:36.00
## Max. :2.0000 Max. :2.0000 Max. :54.00mode(X) <- 'numeric'
Dx_label <- Dx_label[data1.completeRowIndexes]; length(Dx_label)## [1] 115510.3.4 Preparing the response vector
The knockoff filter is designed to control the FDR under Gaussian noise. A quick inspection of the response vector shows that it is highly non-Gaussian.
h <- hist(Y, breaks='FD', plot = F)
plot_ly(x = h$mids, y = h$density, type = "bar") %>%
layout(bargap=0.1, title="Histogram of Computed Variable Y = (UPDRS) part_I + part_II + part_III")A log-transform may help to stabilize the clinical response measurements:
# hist(log(Y), breaks='FD')
h <- hist(log(Y), breaks='FD', plot = F)
plot_ly(x = h$mids, y = h$density, type = "bar") %>%
layout(bargap=0.1, title="Histogram of log(Y)")For binary outcome variables, or ordinal categorical variables, we can employ the logistic curve to transform the polytomous outcomes into real values.
The Logistic curve is \(y=f(x)= \frac{1}{1+e^{-x}}\), where y and x represent probability and quantitative-predictor values, respectively. A slightly more general form is: \(y=f(x)= \frac{K}{1+e^{-x}}\), where the covariate \(x \in (-\infty, \infty)\) and the response \(y \in [0, K]\).
10.4 Running the knockoff filter
We now run the knockoff filter along with the Benjamini-Hochberg (BH) procedure for controlling the false-positive rate of feature selection. More details about the Knock-off filtering methods are available here.
Before running either selection procedure, remove rows with missing values, reduce the design matrix by removing predictor columns that do not appear frequently (e.g., at least three times in the sample), and remove any columns that are duplicates.
library(knockoff)
Y <- data1$UPDRS_part_I + data1$UPDRS_part_II + data1$UPDRS_part_III
table(Y) # Show Clinically relevant classification## Y
## 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
## 54 20 25 12 8 7 11 16 16 9 21 16 13 13 22 25 21 31 25 29 29 28 20 25 28 26
## 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51
## 35 41 23 34 32 31 37 34 28 36 29 27 22 19 17 18 18 19 16 9 10 12 9 11 7 10
## 52 53 54 55 56 57 58 59 60 61 62 63 64 66 68 69 71 75 80 81 82
## 11 5 7 4 1 5 9 4 3 2 1 6 1 2 1 2 1 1 2 3 1Y <- as.matrix(Y[data1.completeRowIndexes]); colnames(Y) <- "y"
mode(Y)## [1] "numeric"# X = scale(ncaaData[,-20]) # Explicit Scaling is not needed, as glmnet auto standardizes predictors
# X = as.matrix(data1[,c("R_caudate_Volume", "R_putamen_Volume","Weight", "Age", "chr17_rs12185268_GT")]) # X needs to be a matrix, not a data frame
drop_features <- c("FID_IID", "ResearchGroup", "UPDRS_part_I", "UPDRS_part_II", "UPDRS_part_III")
X <- data1[ , !(names(data1) %in% drop_features)]
X = as.matrix(X) # remove columns: index, ResearchGroup, and y=(PDRS_part_I + UPDRS_part_II + UPDRS_part_III)
X <- X[data1.completeRowIndexes,]; dim(X); mode(X)## [1] 1155 26## [1] "numeric"View(cbind(X,Y))
# Direct call to knockoff filtering looks like this:
fdr <- 0.4
set.seed(1234)
result = knockoff.filter(X, Y, fdr=fdr, knockoffs=create.second_order); print(result$selected) # Old: knockoffs='equicorrelated')## L_cingulate_gyrus_ComputeArea R_putamen_ComputeArea
## 5 15
## Weight Age
## 18 19
## chr17_rs11868035_GT
## 21# knockoff::create.fixed Fixed-X knockoffs
#knockoff::create.gaussian Model-X Gaussian knockoffs
#knockoff::create.second_order Second-order Gaussian knockoffs
#knockoff::create.solve_asdp Relaxed optimization for fixed-X and Gaussian knockoffs
#knockoff::create.solve_equi Optimization for equi-correlated fixed-X and Gaussian knockoffs
#knockoff::create.solve_sdp Optimization for fixed-X and Gaussian knockoffs
#knockoff::create_equicorrelated Create equicorrelated fixed-X knockoffs.
#knockoff::create_sdp Create SDP fixed-X knockoffs.
#knockoff::create.vectorize_matrix Vectorize a matrix into the SCS format
names(result$selected)## [1] "L_cingulate_gyrus_ComputeArea" "R_putamen_ComputeArea"
## [3] "Weight" "Age"
## [5] "chr17_rs11868035_GT"knockoff_selected <- names(result$selected)
# Run BH (Benjamini-Hochberg)
k = ncol(X)
lm.fit = lm(Y ~ X - 1) # no intercept
p.values = coef(summary(lm.fit))[,4]
cutoff = max(c(0, which(sort(p.values) <= fdr * (1:k) / k)))
BH_selected = names(which(p.values <= fdr * cutoff / k))
knockoff_selected; BH_selected## [1] "L_cingulate_gyrus_ComputeArea" "R_putamen_ComputeArea"
## [3] "Weight" "Age"
## [5] "chr17_rs11868035_GT"## [1] "XL_insular_cortex_ComputeArea" "XL_insular_cortex_Volume"
## [3] "XL_cingulate_gyrus_ComputeArea" "XL_putamen_ComputeArea"
## [5] "XL_putamen_Volume" "XR_putamen_ComputeArea"
## [7] "XSex" "XWeight"
## [9] "XAge" "Xchr17_rs11868035_GT"
## [11] "Xchr17_rs11012_GT" "Xchr17_rs393152_GT"
## [13] "Xchr17_rs12185268_GT"list(Knockoff = knockoff_selected, BHq = BH_selected)## $Knockoff
## [1] "L_cingulate_gyrus_ComputeArea" "R_putamen_ComputeArea"
## [3] "Weight" "Age"
## [5] "chr17_rs11868035_GT"
##
## $BHq
## [1] "XL_insular_cortex_ComputeArea" "XL_insular_cortex_Volume"
## [3] "XL_cingulate_gyrus_ComputeArea" "XL_putamen_ComputeArea"
## [5] "XL_putamen_Volume" "XR_putamen_ComputeArea"
## [7] "XSex" "XWeight"
## [9] "XAge" "Xchr17_rs11868035_GT"
## [11] "Xchr17_rs11012_GT" "Xchr17_rs393152_GT"
## [13] "Xchr17_rs12185268_GT"# Alternatively, for more flexible Knockoff invocation
set.seed(1234)
knockoffs = function(X) create.gaussian(X, 0, Sigma=diag(dim(X)[2])) # identify var-covar matrx Sigma of rank equal to the number of features
stats = function(X, Xk, y) stat.glmnet_coefdiff(X, Xk, y, nfolds=10) # The Output X_k is an n-by-p matrix of knockoff features
result = knockoff.filter(X, Y, fdr=fdr, knockoffs=knockoffs, statistic=stats); print(result$selected)## L_cingulate_gyrus_ComputeArea R_putamen_ComputeArea
## 5 15
## Age chr17_rs11868035_GT
## 19 21
## chr17_rs12185268_GT
## 24# Housekeeping: remove the "X" prefixes in the BH_selected list of features
for(i in 1:length(BH_selected)){
BH_selected[i] <- substring(BH_selected[i], 2)
}
intersect(BH_selected,knockoff_selected)## [1] "L_cingulate_gyrus_ComputeArea" "R_putamen_ComputeArea"
## [3] "Weight" "Age"
## [5] "chr17_rs11868035_GT"We see that there are some features that are selected by both methods suggesting they may be indeed salient.
Try to apply some of these techniques to other data from the list of our Case-Studies.