April 6th, 2021

Today’s Class

Likely to spillover to Thursday

  1. Quick Review
    • FDR
    • Loss
  2. Regression Basics
    • Single redux
    • Multivariate
    • Interactions
    • Factors
  3. Logistic Regression
  4. Deviance
    • Out-of-sample

Quick Review

FDR Roundup

We started with the notion that a given \(\alpha\), (pvalue cutoffs) can lead to a big FDR: \(\alpha \rightarrow q(\alpha)\).

BH reverse that. They fix FDR, and find the relevant \(\alpha\). The algorithm is the key to doing that. \(q \rightarrow \alpha^*(q)\)

Loss

  • Loss is a function both of our prediction and the true outcome
  • More importantly, the driving feature of loss is our experience of making a certain error. Do we lose money? Time? Prestige?
  • Our choice of procedure is driven by this loss.
  • \(l_p(Y,\hat{Y}) = l_p(Y-\hat{Y}) = l_p(e) = \left( \frac1n \sum_{i=1}^n |e|^p\right)^{\frac{1}{p}}\)
    • E.g. \(l_2(e) = \sqrt(\frac1n \sum_{i=1}^n e^2)\).

Regression

Motivation

What is driving sales? Brand differences? Price changes? Ads?

Motivation

Blue points are based on ongoing promotional activity.
It looks like ads are important.

Motivation

Fit a line for sales by brand controlling for promotional activity.

\[ log(Sales) \approx \alpha + \gamma Brand + \beta Ads \]

\(\alpha+\gamma_b\) are like our baseline sales. But we can bring in \(\beta\) more sales with some promotional activity.

Regression

  • Regression through linear models
    • Implementation in R
  • Complications:
    • Interaction
    • Factors
  • Logistic Regression
  • Estimation: Maximum likelihood, Minimum Deviance

This should be mostly review, but perhaps with a different emphasis.

Linear Models

Many problems involve a response or outcome (y),
And a bunch of covariates or predictors (x) to be used for regression.

A general tactic is to deal in averages and lines.

\[E[y|x] = f(x'\beta)\]

Where \(x = [1,x_1,x_2,x_3,...,x_p]\) is our vector of covariates. (Our number of covariates is \(p\) again)
\(\beta = [\beta_0,\beta_1,\beta_2,...,\beta_p]\) are the corresponding coefficients.
The product \(x'\beta = \beta_0+\beta_1 x_1 + \beta_2 x_2+\cdots+\beta_p x_p\).

For simplicity we denote \(x_0 = 1\) to estimate intercepts

Marginals and Conditional Distributions