Topics covered in course

Models for discrete data

Draw \(X_i \overset{IID}{\sim} F , 1 \leq i \leq N\)
Let \(R(X_i), C(X_i)\) be discrete random variables: presence or absence of a trait; labelling a trait.
Sample can be summarized in table

	\(c_1\)	\(\dots\)	\(c_J\)	Row total
\(r_1\)	\(Y_{11}\)	\(\dots\)	\(Y_{1J}\)	\(Y_{1.}\)
\(\vdots\)	\(\dots\)	\(\dots\)	\(\dots\)	\(\dots\)
\(r_I\)	\(Y_{I1}\)	\(\dots\)	\(Y_{IJ}\)	\(Y_{I.}\)
Column total	\(Y_{.1}\)	\(\dots\)	\(Y_{.J}\)	\(Y_{..}\)

Distribution described by

\[ \pi_{ij} = P_F(R=i, C=j), \qquad 1 \leq i \leq I, 1 \leq j \leq J \]

Common questions

Independence

\[ \pi_{ij} = \pi_{i.} \cdot \pi_{.j}, \qquad 1 \leq i \leq I, 1 \leq j \leq J \]

Homogeneity: when \(I=J\) and values are common

\[ \pi_{i.} = \pi_{.i}, \qquad 1 \leq i \leq I \]

Models:

\(Y \sim \text{Multinom}(N, \pi)\)

Alternative:

\(N \sim \text{Poisson}(\lambda) \to Y \sim \text{Poisson}(\lambda \cdot \pi)\)
That is, \(Y_{ij}\)’s are independent with distribution

\[ Y_{ij} \sim \text{Poisson}(\lambda \pi_{ij}) \]

Regression

## Linear regression

Response matrix: \(Y \in \mathbb{R}^{n \times q}\)
Design matrix: \(X \in \mathbb{R}^{n \times p}\)

Usual model

IID draws from

\[ Y_i | X_i \sim N(X_i^T\beta, \sigma^2) \]

Usual estimation problem

\[ \hat{\beta} = \text{argmin}_{\beta} \|Y - X\beta\|^2_2 \]

Binary regression models: \(Y \in \{0,1\}^n\)

### Usual model

For some CDF \(F\):

\[ Y_i | X_i \sim \text{Binomial}(F(X_i^T\beta)) \]

Common choices:
1. probit: \(F \sim N(0,1)\)
2. logit: \(F(x) = e^x/(1+e^x)\).

Usual estimation problem

\[ \hat{\beta} = \text{argmin}_{\beta} - 2 \log L(\beta | X, Y) = \text{argmin}_{\beta} \text{DEV}(\beta | X, Y) \]

Asymptotic distribution? Inference?

Multinomial: \(Y \in \{1, \dots, L\}^n\)

Some models we’ll see

Baseline logistic
Ordered logistic

Survival analysis

\(T\) a survival time

Basic object: survival function and hazard

\[ P_{\beta}(T > t |X) = \exp \left(-\int_0^t h_{\beta}(s;X) \; ds \right) \]

Complications: censoring, truncation.

Non-parameteric methods

Kaplan-Meier estimator: direct estimate of \(P(T > t)\) based on IID draws of censored observations from \(F\)
Log-rank test

Usual model

Cox model:

\[ \frac{h_{\beta}(s;X_1)}{h_{\beta}(s;X_0)} = \exp\left((X_1-X_0)^T\beta \right) \]

Introduction

web.stanford.edu/class/stats305b

Topics covered in course

Models for discrete data

Common questions

Models:

Alternative:

Regression

Usual model

Usual estimation problem

Binary regression models: \(Y \in \{0,1\}^n\)

Usual estimation problem

Multinomial: \(Y \in \{1, \dots, L\}^n\)

Some models we’ll see

Survival analysis

Basic object: survival function and hazard

Non-parameteric methods

Usual model

Exponential families

Examples

Tools for exponential families