• Goal of Master's project:
  • Regression methods for unbiased or consistent estimators and valid confidence intervals in a high-dimensional setting (\(p \gg N\))
  • There can be confounding
  • Response variable is discrete or right-censored
• Goal of project with Dr. Hanna:
  • Efficiently analyze large ICES medical data set using ML techniques
  • Predict the impact on the hazard rate or five-year survival rates of patients treated with a specific treatment
  • Time-dependent covariates

1. Double machine learning (DML)
2. Pooled Logistic Regression (PLR)
3. Post-selection inference (PSI) with LASSO

• For DML:
  • There are \(N\) indepdentently drawn observations: \(\{(y_i,d_i,{\mathbf{x}_i})\}^N_{i=1}\)
  • \(y_i\) is the response variable (can be continuous, discrete, or right-censored survival time)
  • \({\mathbf{x}_i}\) is a \(p\)-dimensional feature vector (may include an intercept)
  • \(d_i\) is "treatment" variable of interest (may be included in \({\mathbf{x}_i}\))
  • \({\mathbf{X}}\) is the \(N \times p\) design matrix

• (Generalized) linear model of the form shown in \(\eqref{eq:dml}\)
• Treatment effect on (function) of response is \(\alpha_0\)
• Confounding relationship for continuous or binary treatment \(d_i\) \(\eqref{eq:confound}\)
• Simulations will use \(N=100\) and \(p=100\).
• Coefficient impacts must be "sparse" \(\eqref{eq:sparse}\) or "approximately sparse" \(\eqref{eq:asparse}\)

\[ \begin{align} g_1(E[y_i|d_i,{\mathbf{x}_i}] ) &= \alpha_0 d_i + {\boldsymbol\beta}^T {\mathbf{x}_i}+ e_i \label{eq:dml} \\ g_2(E[d_i|{\mathbf{x}_i}]) &= {\boldsymbol\phi}^T {\mathbf{x}_i}+ u_i \label{eq:confound} \\ \| {\boldsymbol\beta}_0 \|_0 &= s < N \label{eq:sparse} \\ | {\boldsymbol\beta}_0 |_{(j)} &\leq A^{-\gamma}, \hspace{2mm} A>0,\hspace{2mm} \gamma>1/2 , \hspace{2mm} j=1,\dots,p \label{eq:asparse} \end{align} \]

• Inference is possible with DML on a treatment parameter due to construction of "orthogonal estimating equations" [1]
  • This counteracts bias from both regularization and selection
  • \(w_i\): measurements, \(\eta\): nuissance parameters
• For classical Gaussian case equations \(\eqref{eq:ee}\) and \(\eqref{eq:oc}\) are the Frish-Waugh-Lovell (FWL) conditions

\[ \begin{align} \mathbb{E}[\psi(w_i,\alpha_0,\eta)] &= 0 &\textbf{Estimating equation} \label{eq:ee} \\ \frac{\mathbb{E}[\psi(w_i,\alpha_0,\eta)]}{\partial \eta} &= 0 &\textbf{Orthogonality condition} \label{eq:oc} \\ \end{align} \]

• Relationship of FWL to orthogonal estimating equations
• Changes in the value of the nuissance parameters (\(\eta\)) evaluated at the true \(\alpha_0\) do not impact equations
• Hence, equations \(\eqref{eq:fwl1}\) and \(\eqref{eq:fwl2}\) can be estimated with LASSO

\[ \begin{align} d_i &= {\mathbf{x}_i}^T {\boldsymbol\phi}_0^d + u_i^d \hspace{2.2cm} E(u_i^d{\mathbf{x}_i})=\boldsymbol 0_p \label{eq:fwl1} \\ y_i &= {\mathbf{x}_i}^T {\boldsymbol\phi}_0^y + u_i^y \hspace{2.2cm} E(u_i^y{\mathbf{x}_i})=\boldsymbol 0_p \label{eq:fwl2} \\ u_i^y &= \alpha_0 u_i^d + v_i \hspace{2.5cm} E(u_i^d v_i )=0 \label{eq:fwl3} \\ E[v_iu_i^d] &= E([\underbrace{u_i^y-\alpha_0 u_i^d]u_i^d}_{\psi(w_i,\alpha_0,\eta)}) = 0 \nonumber \\ w_i &= [y_i,d_i,x_i], \hspace{3mm} \eta = [{\boldsymbol\phi}^d_0,{\boldsymbol\phi}^y_0] \nonumber \end{align} \]

• How does DML select \(\lambda\) for the LASSO estimator \(\eqref{eq:lasso}\) [2] ?
  • "Oracle-rate" convergence of estimator: \(\lambda=\sigma \cdot 2 \sqrt{2\log(pn)/n}\) [3]
    • Hard to aproximate \(\sigma\)
• Solution: square-root LASSO \(\eqref{eq:srlasso}\) [4] [5]
  • Rate-optimal \(\lambda\) independent of variance: \(\lambda=\sqrt{2\log(pn)/n}\)

\[ \begin{align} \text{LASSO}: \hspace{1cm} &\min_{\mathbf{w}} \hspace{3mm} \frac{1}{n} \sum_{i=1}^n (y_i - \mathbf{w}^T {\mathbf{x}_i})^2 + \lambda \| \mathbf{w} \|_1 \label{eq:lasso} \\ \sqrt{\text{LASSO}}: \hspace{1cm} &\min_{\mathbf{w}} \hspace{3mm} \sqrt{\frac{1}{n} \sum_{i=1}^n (y_i - \mathbf{w}^T {\mathbf{x}_i})^2} + \lambda \| \mathbf{w} \|_1 \label{eq:srlasso} \end{align} \]

• If \(y_i\) is Gaussian, my simulations suggest DML works well (or the best)

• For the GLM case, there are two approaches [6]
  • An "instrument" \(z_{0i}=z_0(d_i,{\mathbf{x}_i})\) with stimating equations: \(\eqref{eq:glm1}\) \(\eqref{eq:glm2}\) and orthogonality condition: \(\eqref{eq:glm3}\)
  • A "double selection procedure" which uses two "Post-LASSOs" and "honest confidence intervals"

\[ \begin{align} E[\{y_i - g^{-1}(\alpha_0 d_i + {\boldsymbol\beta}^T {\mathbf{x}_i}) z_{0i} \}] &= 0 \label{eq:glm1} \\ \frac{\partial}{\partial \alpha}E[\{y_i - g^{-1}(\alpha d_i + {\boldsymbol\beta}^T {\mathbf{x}_i}) z_{0i} \}] |_{\alpha=\alpha_0} &\neq 0 \label{eq:glm2} \\ \frac{\partial}{\partial {\boldsymbol\beta}}E[\{y_i - g^{-1}(\alpha d_i + {\boldsymbol\beta}^T {\mathbf{x}_i}) z_{0i} \}] |_{{\boldsymbol\beta}={\boldsymbol\beta}_0} &= \boldsymbol 0 \label{eq:glm3} \end{align} \]

• My simulations suggests it works well too

• Strengths of DML:
  • Specifically focuses on a treatment variable
  • More likely to "work"
• Issues with DML:
  • Estimation procedure for GLMs less theoretically grounded (\(\lambda\))
  • Simulations need to demonstrate how effective it is with PLR for censored and time-dependent survival data

• Logistic regression can be used to time-dependent Cox PH model [7] [8]
• Interval data is "pooled" into a long format, with each interval given a factor (risk set)

## [1] "Survival format"

##   id start stop event x
## 1  a     0    2     1 2
## 2  b     0    3     1 1
## 3  c     0    4     0 3
## 4  c     4    6     0 4
## 5  c     6   10     1 6
## 6  d     0    5     1 3

## [1] "PLR format"

##    id t0 t1 x event interval
## 1   a  0  2 2     1        1
## 2   b  0  2 1     0        1
## 3   d  0  2 3     0        1
## 4   c  0  2 3     0        1
## 5   b  2  3 1     1        2
## 6   d  2  3 3     0        2
## 7   c  2  3 3     0        2
## 8   d  3  5 3     1        3
## 9   c  3  5 4     0        3
## 10  c  5 10 6     1        4

• For example, using the Stanford heart transplant data: survival::heart, where Transplant is a time-varying covariate
• Results between two approaches are quite similar

• Or the Chronic Granulotomous Disease data dataset: survival::cgd

• Inference procedures for models that have already been selected:
  • LASSO, forward-stepwise regression, etc
• Without adjustment, or additional assumptions, inference on selected models has optimistic p-values
• Furthermore, the "LASSO" returns no p-values, SEs, CIs, etc

• Post-selection inference (PSI) is a rapidly growing and recent field
  • Lockhart, Taylor, Tibs & Tibs. A significance test for the lasso. Annals of Statistics 2014
  • I Lee, Sun, Sun, Taylor (2016) Exact post-selection inference, with the lasso. Annals of Statistics 2016
  • I Fithian, Sun, Taylor (2015) Optimal inference after model selection. arXiv. Submitted
  • I Tibshirani, Ryan, Taylor, Lockhart, Tibs (2016) Exact Post-selection Inference for Sequential Regression Procedures. To appear, JASA
  • I Tian, X. and Taylor, J. (2015) Selective inference with a randomized response. arXiv
  • I Fithian, Taylor, Tibs, Tibs (2015) Selective Sequential Model Selection. arXiv Dec 2015

• Principal idea: "any procedure for which the selection events can be characterized by a set of linear inequalities in \({\mathbf{y}}\): \(A{\mathbf{y}}\leq b\)" [9]
  • Known as "polyhedral constraints on \(y\)"
  • Includes algorithms like LASSO (fixed-\(\lambda\)), LAR, or forward-stepwise (FS)
• In FS, the variable with the largest (absolute) inner product with \(y\) is selected (compared to the last step)
  • Suppose \(y_i = \beta_0 + \beta_1 x_{i1} + \beta_2 x_{i2}\), \(x_{i0}=1\) \(\forall i\)
  • If \({\mathbf{x}_2}\) is selected in the first stage, and is positive, then:

\[ \begin{align*} \langle {\mathbf{x}_2}, {\mathbf{y}}\rangle / \langle {\mathbf{x}_2}, {\mathbf{x}_2}\rangle \geq \pm \langle {\mathbf{x}_0}{\mathbf{y}}\rangle / \langle {\mathbf{x}_0}, {\mathbf{x}_0}\rangle \\ \langle {\mathbf{x}_2}, {\mathbf{y}}\rangle / \langle {\mathbf{x}_2}, {\mathbf{x}_2}\rangle \geq \pm \langle {\mathbf{x}_1}{\mathbf{y}}\rangle / \langle {\mathbf{x}_1}, {\mathbf{x}_1}\rangle \end{align*} \]

• Which will give the following set of constraints:

\[ \begin{align*} \Bigg( {\mathbf{x}_2}^T-{\mathbf{x}_0}^T\frac{\langle {\mathbf{x}_2}, {\mathbf{x}_2}\rangle}{\langle {\mathbf{x}_0}, {\mathbf{x}_0}\rangle} \Bigg){\mathbf{y}}\geq 0 \\ \Bigg( {\mathbf{x}_2}^T+{\mathbf{x}_0}^T\frac{\langle {\mathbf{x}_2}, {\mathbf{x}_2}\rangle}{\langle {\mathbf{x}_0}, {\mathbf{x}_0}\rangle} \Bigg){\mathbf{y}}\geq 0 \\ \Bigg( {\mathbf{x}_2}^T-{\mathbf{x}_1}^T\frac{\langle {\mathbf{x}_2}, {\mathbf{x}_2}\rangle}{\langle {\mathbf{x}_1}, {\mathbf{x}_1}\rangle} \Bigg){\mathbf{y}}\geq 0 \\ \Bigg( {\mathbf{x}_2}^T+{\mathbf{x}_1}^T\frac{\langle {\mathbf{x}_2}, {\mathbf{x}_2}\rangle}{\langle {\mathbf{x}_1}, {\mathbf{x}_1}\rangle} \Bigg){\mathbf{y}}\geq 0 \\ \underbrace{A}_{4 \times N} {\mathbf{y}}\geq 0 \end{align*} \]

• For the Gaussian FS-case, the polyhedral lemma would be:
  • Before selection, \(\langle {\mathbf{x}_2}, {\mathbf{y}}\rangle \in \mathbb{R}\)
  • After selection, inner product at least as large as the second-place finisher and \({\mathbf{y}}\)
  • Conditioning on results of competition means inference can be calculated with a truncated Gaussian distribution

• For the LASSO case, we observe \(M\), the index of non-zero coefficients
  • Peform inference on \({\boldsymbol\beta}_M \in \mathbb{R}^M\), assuming \({\boldsymbol\beta}_{M^C} \simeq 0\)
• The approach has been adapted for logistic and Cox models too [10]
• For GLM works by using "one-step estimator" \(\eqref{eq:ose}\)
  • \(s_M\): is the sign of the selected coefficients
  • \(I_M\): Fisher Information matrix \(\Big( {\mathbf{X}}^T_M W(\hat{{\boldsymbol\beta}}_M) {\mathbf{X}}_M \Big)\)
  • Polyhedral constraints \(\eqref{eq:glm_poly}\)

\[ \begin{align} \bar{{\boldsymbol\beta}}_M &= \hat{{\boldsymbol\beta}}_M + \lambda \cdot I_M(\bar{{\boldsymbol\beta}}_M)^{-1}s_M \label{eq:ose} \\ \Big\{\text{diag}(s_M)&\Big[ \bar{{\boldsymbol\beta}}_M - I_M(\hat{{\boldsymbol\beta}}_M)^{-1} \lambda s_M \Big] \geq 0 \Big\} \label{eq:glm_poly} \end{align} \]

• My simulation studies for Cox models shows higher variance but similar coverage

• Stengths of PSI:
  • Developed tools and packages (selectiveInference)
  • Top-notch researchers contributing to this field
  • Highly scalable to large data sets (\(N\) or \(p\))
  • (Appears) to work with CV-selected \(\lambda\) too
• Issues with PSI:
  • Not designed for analysis of treatment variable: what if \(d_i\) isn't selected?
  • Time-varying covariates? (No conceptual problem)

• One week for time-dependent covariates and Cox-LASSO
  • Coordinate descent for Cox IRLS algorithm
• Two weeks for DML approach
  • One week for theory (\(c(\alpha)\) statistics, score equations, etc)
  • One week for simulating time-dependent survival data and testing DML