Errors in Hypothesis Testing; Linear Regression

M. Drew LaMar
September 07, 2016


https://xkcd.com/539/

Introduction to Quantitative Biology, Fall 2016

Class announcements

  • Upcoming Booz Allen Events (http://www.boozallen.com/)
    • Diversity Brunch: September 9, 10 AM - 11:30 AM in the Cohen Career Center
    • Fall Career & Internship Fair: September 9, 12 PM - 4 PM in the Sadler Center
    • Meet the Firms Friday: September 16, 11 AM - 2 PM in Miller Hall
    • Corporate Presentation/Info Session: September 27, 6 PM - 8 PM in Miller Hall

Interval estimates


http://www.smbc-comics.com/comic/2011-11-23

Hypothesis testing

So, P = 0.04. Is that good?

https://youtu.be/7jSE3JANx14?t=4m29s

Jelly Beans


https://xkcd.com/882/

P-Values


https://xkcd.com/1478/

Hypothesis Testing


https://xkcd.com/892/

Errors in Hypothesis Testing

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Definition: Type I error is rejecting a true null hypothesis. The probability of a Type I error is given by \[ \mathrm{Pr[Reject} \ H_{0} \ | \ H_{0} \ \mathrm{is \ true}] = \alpha \]

Definition: Type II error is failing to reject a false null hypothesis. The probability of a Type II error is given by \[ \mathrm{Pr[Do \ not \ reject} \ H_{0} \ | \ H_{0} \ \mathrm{is \ false}] = \beta \]

Errors in Hypothesis Testing

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Errors in Hypothesis Testing - Power

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Definition: The power of a statistical test (denoted \( 1-\beta \)) is given by \[ \begin{align*} \mathrm{Pr[Reject} \ H_{0} \ | \ H_{0} \ \mathrm{is \ false}] & = 1-\beta \\ & = 1 - \mathrm{Pr[Type \ II \ error]} \end{align*} \]

Probability of errors in hypothesis testing

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  • \( \alpha \) is the significance level
  • \( 1-\beta \) is the power

Statistical power example
https://qubeshub.org/tools/statpowerviz/

Power analysis

Power of a statistical test is a function of
     - Significance level \( \alpha \)
     - Variability of data
     - Sample size
     - Effect size

  • Desired power is set by researcher (typically 80%)
  • Significance level set by researcher
  • Data variability and effect size can be estimated by previous studies or pilot studies
  • Sample size is then calculated to achieve desired power given previous fixed attributes

Regression

Definition: Regression is the method used to predict values of one numerical variable (response) from values of another (explanatory).

Note: Regression can be done on data from an observational or experimental study.

We will discuss 3 types:

  • Linear regression
  • Nonlinear regression
  • Logistic regression

Linear regression

Definition: Linear regression draws a straight line through the data to predict the response variable from the explanatory variable.

Slope determines rate of change of response with explanatory - humans lose 0.076 units of genetic diversity with every 10,000 km from East Africa.

Formula for the line

Definition: For the population, the regression line is

\[ Y = \alpha + \beta X, \]
where \( \alpha \) (the intercept) and \( \beta \) (the slope) are population parameters.

Definition: For a sample, the regression line is

\[ Y = a + b X, \]
where \( a \) and \( b \) are estimates of \( \alpha \) and \( \beta \), respectively.

Graph of the line

  • \( a \): intercept
  • \( b \): slope

Assumptions of linear regression

Note: At each value of \( X \), there is a population of \( Y \)-values whose mean lies on the true regression line (this is the linear assumption).

Assumptions of linear regression

  • Linearity
  • Residuals are normally distributed
  • Constant variance of residuals
  • Independent observations

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