Simple Linear Regression

• Least squares line• Interpreting coefficients• Prediction• Cautions• The formal model

Section 2.6, 9.1, 9.2 Professor Kari Lock MorganDuke University

Exam 2 GradesIn Class:

Lab:

Total:

• Test whether this data provides evidence that Melanoma is found significantly more often on the left side of the body: one categorical variable -> single proportion

• 2011 Hollywood movies: If the sample is the same as the population, then no need for inference!

• Standard deviation of a bootstrap distribution is the standard error

Comments on In-Class Exam

• Most common reason for points off: applying the wrong method

• The first step should ALWAYS be asking yourself: What is/are the variable(s)? Are they categorical or quantitative?

• Always plot/visualize your data. Outliers can strongly affect the results; you should either explain why they are left in, or else remove them

Comments on Lab Exam

• For any one or two variables, resample( ) gives a confidence interval

• For any two variables, reallocate( ) tests for an association between the variables

• No conditions to check!• Automatically deals with missing data!• Only two commands to remember!• No distributions to remember!

Simulation Methods

MODELING

• Can you estimate the temperature on a summer evening, just by listening to crickets chirp?

Crickets and Temperature

Response Variable, y

ExplanatoryVariable, x

We will fit a model to predict temperature based on cricket chirp rate

• A linear model predicts a response variable, y, using a linear function of explanatory variables

• Simple linear regression predicts on response variable, y, as a linear function of one explanatory variable, x

• We will create a model that predicts temperature as a linear function of cricket chirp rate

Linear Model

Regression LineGoal: Find a straight line that best fits the data in a

scatterplot

Predicted and Actual Values

• The actual response value, y, is the response value observed for a particular data point

• The predicted response value, , is the response value that would be predicted for a given x value, based on a model

• In linear regression, the predicted values fall on the regression line directly above each x value

• The best fitting line is that which makes the predicted values closest to the actual values

y

Predicted and Actual Values

yy

Residual

• The residual for each data point is

or the vertical distance from the point to the line

residual {

actual predicted yy

ˆ 63.5 61.44 2.06yy

Want to make all the residuals as small as possible.

How would you measure this?

Least Squares Regression

• Least squares regression chooses the regression line that minimizes the sum of the squared residuals

2

1

minimize ( )ˆn

i ii

y y

Least Squares Regression

• The estimated regression line is

• Slope: increase in predicted y for every unit increase in x

• Intercept: predicted y value when x = 0

Equation of the Line

0 1ˆ ˆˆi iy x

Intercept Slope

Regression in R> lm(Temperature~Chirps)

Call: lm(formula = Temperature ~ Chirps)

Coefficients: (Intercept) Chirps 37.6786 0.2307

37.69 0.23temperature Chirps

Regression Model

37.69 0.23temperature chirps

Which is a correct interpretation?a) The average temperature is 37.69b) For every extra 0.23 chirps per minute, the

predicted temperate increases by 1 degreec) Predicted temperature increases by 0.23

degrees for each extra chirp per minuted) For every extra 0.23 chirps per minute, the

predicted temperature increases by 37.69

Units

37.69 0.23temperature chirps

• It is helpful to think about units when interpreting a regression equation

y units y units x units

units

units

y

x

degrees degrees chirps per minute

degrees/chirps per min

0 1ˆ ˆˆi iy x

Prediction

• The regression equation can be used to predict y for a given value of x

• If you listen and hear crickets chirping about 140 times per minute, your best guess at the outside temperature is

37.69 0.23temperature chirps

37.69 0.23 140 69.9

Prediction37.69 0.23 140 69.9

Prediction

If the crickets are chirping about 180 times per minute, your best guess at the temperature is

(a) 60(b) 70(c) 80

37.69 0.23 180 79.09

Exam Scores 2 24.2 0.7 1Exam Exam

Calculate your residual.

Prediction 37.69 0.23temperature chirps

The intercept tells us that the predicted temperature when the crickets are not chirping at all is 37.69. Do you think this is a good prediction?

(a) Yes(b) No

Regression Caution 1

• Do not use the regression equation or line to predict outside the range of x values available in your data (do not extrapolate!)

• If none of the x values are anywhere near 0, then the intercept is meaningless!

Duke Rank and Duke Shirts

a) positively associated

b) negatively associated

c) not associated

d) other

Are the rank of Duke among schools applied to and the number of Duke shirts owned

2 4 6 8

1020

3040

DukeRank

DukeShirts

Duke Rank and Duke Shirts

a) positively associated

b) negatively associated

c) not associated

d) other

Are the rank of Duke among schools applied to and the number of Duke shirts owned

2 4 6 8

1020

3040

DukeRank

DukeShirts

2 4 6 8

1020

3040

DukeRank

DukeShirts

Regression Caution 2

• Computers will calculate a regression line for any two quantitative variables, even if they are not associated or if the association is not linear

• ALWAYS PLOT YOUR DATA!

• The regression line/equation should only be used if the association is approximately linear

Regression Caution 3

• Outliers (especially outliers in both variables) can be very influential on the regression line

• ALWAYS PLOT YOUR DATA!

http://illuminations.nctm.org/LessonDetail.aspx?ID=L455

Life Expectancy and Birth Rate

Coefficients: (Intercept) LifeExpectancy 83.4090 -0.8895

Which of the following interpretations is correct?

(a) A decrease of 0.89 in the birth rate corresponds to a 1 year increase in predicted life expectancy

(b) Increasing life expectancy by 1 year will cause the birth rate to decrease by 0.89

(c) Both

(d) Neither

Regression Caution 4

• Higher values of x may lead to higher (or lower) predicted values of y, but this does NOT mean that changing x will cause y to increase or decrease

• Causation can only be determined if the values of the explanatory variable were determined randomly (which is rarely the case for a continuous explanatory variable)

Explanatory and Response

• Unlike correlation, for linear regression it does matter which is the explanatory variable and which is the response

37.69 0.23temperature chirps

157.8 4.25chirps Temperature

r = 0

Challenge: If the correlation between x and y is 0, what would the regression line be?

• The population/true simple linear model is

• 0 and 1, are unknown parameters

• Can use familiar inference method!

0 1i i iy x òIntercept Slope

Simple Linear Model

Random error

• Confidence intervals and hypothesis tests for the slope can be done using the familiar formulas:

• Population Parameter: 1, Sample Statistic:

• Use t-distribution with n – 2 degrees of freedom

Inference for the Slope

sample statistic null value

SEt

*sample statistic t SE

1

Inference for Slope

7n

Give a 95% confidence interval for the true slope.

Is the slope significantly different from 0? (a) Yes(b) No

Confidence Interval

7n

*statistic t SE > qt(.975,5) [1] 2.570582

2.57 0.014

0.194,

0.2

66

3

0.2

We are 95% confident that the true slope, regressing temperature on cricket chirp rate, is between 0.194 and 0.266 degrees per chirp per minute.

Hypothesis Test

7n

0 1

1

: 0

: 0aH

H

0.23116.21

0.014t

> 2*pt(16.21,5,lower.tail=FALSE) [1] 1.628701e-05

There is strong evidence that the slope is significantly different from 0, and that there is an association between cricket chirp rate and temperature.

Small Samples• The t-distribution is only appropriate for large samples (definitely not n = 7)!

• We should have done inference for the slope using simulation methods...

If results are very significant, it doesn’t really matter if you get the exact p-value… you come to the same conclusion!

• Details here

• Group project on regression (modeling)

• If you want to change groups, email me TODAY OR TOMORROW. If other people in your lab section want to change, I’ll move people around.

• Need a data set with a quantitative response variable and multiple explanatory variables; explanatory variables must have at least one categorical and at least one quantitative

• Proposal due next Wednesday

Project 2