Simple Regression 1

Simple Regression I

Simple Regression 2

Correlation tells us how strongly Y and X are related … but regression estimates the form of this relationship

We’ll begin with simple regression, which assumes the form:

Regression Analysis

ii XbbY 10ˆ

Simple Regression 3

Y is the variable we want to predict

We believe X influences how Y behaves

Ŷi is the estimated value of Y at Xi

b0 is the Y-intercept in the equation

b1 is the slope of the regression line

Regression Notation

Simple Regression 4

Our goal: Find the straight line that best fits the data we’ve collected

The best equation will be the one that minimizes the error in fit

The equation is:

The fit error is thus:

Fitting the Regression Line

ii XbbY 10ˆ

iii YYe ˆ

Simple Regression 5

Obtaining the line

0

2

4

6

8

10

12

14

0 1 2 3 4 5 6 7

- Errors

+ Errors

Simple Regression 6

The fit error for the ith point on the scatterplot diagram is:

We would like the sum of the + errors to be the same as the sum of the – errors.

However, there are many lines that can make this happen.

Balancing out the errors

iii YYe ˆ

Simple Regression 7

Zero Error Lines

Simple Regression 8

So, which of these solutions is the best one?

Select the line with the minimum sum of squared error terms. This is called least-squares regression.

The “Least Squares” Line

Simple Regression 9

Intercept:

Slope:

* note COVAR here is Excel’s functional calculation which is the population covariance not the sample covariance

The Least Squares Estimators

1*

)(

),(1

n

n

xVar

yxCOVAR

SS

SSb

x

xy

XbYb 10

Simple Regression 10

Some values can be calculated directly using the means, variances, and covariances.

For one-variable (simple) regression, can add a trendline to a chart.

Can use the Data Analysis Tool, Regression Can use the Excel function LINEST.

Getting the Estimates in Excel

Simple Regression 11

Regression with mail data

100 200 300 400 500 600 700 8000

5

10

15

20

25

f(x) = 0.0297026689096403 x + 0.191221309475372

X

Y

Uses Excel’s Trend Line function

Simple Regression 12

Output from Data Analysis Tool

Simple Regression 13

Output from LINESTThe LINEST function must be entered as an array formula. For the example, highlight the cells E3:F7, type the formula “=LINEST(Orders,Weight,1,1)”, then CTRL-SHFT-ENTER.

Simple Regression 14

Remember the variables are X = weight in pounds and Y = orders in 1000s

The estimated intercept (b0) tells us that if there was no mail, we still have a minimum of (.1912)(1000) or 191.2 orders per day.

The estimated slope (b1) tells us that each pound of mail tends to bring with it (.0297)(1000) or 29.7 orders.

Interpretation of Results

Simple Regression 15

There are two standard ways to judge:

1. How much of the variation in the Y values (orders) can be attributed to the different values of X (weight of mail)?

2. In general, how small (or large) are the errors in fit?

How Good Is Our New Model?

Simple Regression 16

The Coefficient of Determination:

The R2 value is:

◦ Always between 0 and 1

◦ Is the percentage of variation explained by the model.

◦ The square of correlation (for simple regression)

R2 – A Universal Measure of Fit

Yin variationThe

iprelationsh Y-X by the explained Yin variationThe2 R

Simple Regression 17

ANOVA table: Total variation in the Y values is SST = 449.76

The amount of unexplained variation isSSE = 12.12

The difference is thus the variation explained by the regression equation orSSR = 449.76 – 12.12 = 437.64

The ratio of explained to total is how we get R2 = 437.64/449.76 = .973

How is R2 computed?

Simple Regression 18

For every observation i, its error is given by:

To find the “typical error,” use this formula:

This is the “Standard Error”, also the √MSE.

Size of the Typical Error (S)

iii YYe ˆ

2

2

n

eS

n

ii

Simple Regression 19

The typical error (called the standard error of prediction) for our regression model is: S = .7258

This means that we typically misestimate the actual number of orders per day by (.7258)(1000) = 725.8

That may sound like a lot, but you have to consider that we have between 5 and 20 thousand orders each day, average (13.22)*(1000) = 13200, then the percentage error is only 725.8 / 13200 = 5.5%.

S in our example

Simple Regression 20

Sales Data

Simple Regression 21

Sales Data Manual

Simple Regression 22

Sales Data Graphical

Simple Regression 23

Sales Data Tools

Simple Regression 24

Sales Data LINEST