Chapter 3Linear Regression and

Correlation

Descriptive Analysis &Presentation of Two Quantitative Data

Chapter Objectives

• To be able to present two-variables data in tabular and graphic form

• Display the relationship between two quantitative variables graphically using a scatter diagram.

• Calculate and interpret the linear correlation coefficient.

• Discuss basic idea of fitting the scatter diagram with a best-fitted line called a linear regression line.

• Create and interpret the linear regression line.

Terminology

• Data for a single variable is univariate data

• Many or most real world models have more than one variable … multivariate data

• In this chapter we will study the relations between two variables … bivariate data

Bivariate Data

• In many studies, we measure more than one variable for each individual

• Some examples are– Rainfall amounts and plant growth– Exercise and cholesterol levels for a group

of people– Height and weight for a group of people

Types of Relations

When we have two variables, they could be related in one of several different ways– They could be unrelated– One variable (the input or explanatory or predictor

variable) could be used to explain the other (the output or response or dependent variable)

– One variable could be thought of as causing the other variable to change

Note: When two variables are related to each other, one variable may not cause the change of the other variable. Relation does not always mean causation.

Lurking Variable

• Sometimes it is not clear which variable is the explanatory variable and which is the response variable

• Sometimes the two variables are related without either one being an explanatory variable

• Sometimes the two variables are both affected by a third variable, a lurking variable, that had not been included in the study

Example 1

• An example of a lurking variable• A researcher studies a group of

elementary school children– Y = the student’s height– X = the student’s shoe size

• It is not reasonable to claim that shoe size causes height to change

• The lurking variable of age affects both of these two variables

More Examples

• Some other examples• Rainfall amounts and plant growth

– Explanatory variable – rainfall – Response variable – plant growth– Possible lurking variable – amount of sunlight

• Exercise and cholesterol levels– Explanatory variable – amount of exercise– Response variable – cholesterol level– Possible lurking variable – diet

Three combinations of variable types:

1. Both variables are qualitative (attribute)

2. One variable is qualitative (attribute) and the other is quantitative (numerical)

3. Both variables are quantitative (both numerical)

Types of Bivariate Data

TV Radio NPMale 280 175 305Female 115 275 170

Two Qualitative Variables

• When bivariate data results from two qualitative (attribute or categorical) variables, the data is often arranged on a cross-tabulation or contingency table

Example: A survey was conducted to investigate the relationship between preferences for television, radio, or newspaper for national news, and gender. The results are given in the table below:

Row Totals

760560

Col. Totals 395 450 475 1320

TV Radio NP

Male 280 175 305Female 115 275 170

Marginal Totals• This table, may be extended to display the marginal

totals (or marginals). The total of the marginal totals is the grand total:

Note: Contingency tables often show percentages (relative frequencies). These percentages are based on

the entire sample or on the subsample (row or column) classifications.

• The previous contingency table may be converted to percentages of the grand total by dividing each frequency by the grand total and multiplying by 100

Percentages Based on the Grand Total(Entire Sample)

– For example, 175 becomes 13.3%

TV Radio NP Row TotalsMale 21.2 13.3 23.1 57.6Female 8.7 20.8 12.9 42.4Col. Totals 29.9 34.1 36.0 100.0

1751320

100 133

.

• These same statistics (numerical values describing sample results) can be shown in a (side-by-side) bar graph:

Illustration

0

5

10

15

20

25

TV Radio NP

Male

Female

Percentages Based on Grand Total

Percent

Media

Percentages Based on Row (Column) Totals• The entries in a contingency table may also be expressed as

percentages of the row (column) totals by dividing each row (column) entry by that row’s (column’s) total and multiplying by 100. The entries in the contingency table below are expressed as percentages of the column totals:

Note:These statistics may also be displayed in a side-by-side bar graph

One Qualitative & One Quantitative Variable

1. When bivariate data results from one qualitative and one quantitative variable, the quantitative values are viewed as separate samples

2. Each set is identified by levels of the qualitative variable

3. Each sample is described using summary statistics, and the results are displayed for side-by-side comparison

4. Statistics for comparison: measures of central tendency, measures of variation, 5-number summary

5. Graphs for comparison: side-by-side stemplot and boxplot

Example Example: A random sample of households from three different

parts of the country was obtained and their electric bill for June was recorded. The data is given in the

table below:

• The part of the country is a qualitative variable with three levels of response. The electric bill is a quantitative variable. The electric bills may be compared with numerical and graphical techniques.

Comparison Using Box-and-Whisker Plots

Northeast Midwest West20

30

40

50

60

70

ElectricBill

The Monthly Electric Bill

The electric bills in the Northeast tend to be more spread out than those in the Midwest. The bills in the West tend to be higher than both those in the Northeast and Midwest.

Descriptive Statistics for Two Quantitative Variables

Scatter Diagrams and correlation coefficient

Two Quantitative Variables

• The most useful graph to show the relationship between two quantitative variables is the scatter diagram

• Each individual is represented by a point in the diagram– The explanatory (X) variable is plotted on the

horizontal scale– The response (Y) variable is plotted on the

vertical scale

Example: In a study involving children’s fear related to being hospitalized, the age and the score each child made on the Child Medical Fear Scale (CMFS) are given in the table below:

Age (x ) 8 9 9 10 11 9 8 9 8 11CMFS (y ) 31 25 40 27 35 29 25 34 44 19

Age (x ) 7 6 6 8 9 12 15 13 10 10CMFS (y ) 28 47 42 37 35 16 12 23 26 36

Example

Construct a scatter diagram for this data

• age = input variable, CMFS = output variable

Solution

Child Medical Fear Scale

1514131211109876

50

40

30

20

10

CMFS

Age

Another Example

• An example of a scatter diagram

Note: the vertical scale is truncated to illustrate the detail relation!

Types of Relations

• There are several different types of relations between two variables– A relationship is linear when, plotted on a scatter

diagram, the points follow the general pattern of a line– A relationship is nonlinear when, plotted on a scatter

diagram, the points follow a general pattern, but it is not a line

– A relationship has no correlation when, plotted on a scatter diagram, the points do not show any pattern

Linear Correlations

• Linear relations or linear correlations have points that cluster around a line

• Linear relations can be either positive (the points slants upwards to the right) or negative (the points slant downwards to the right)

Positive Correlations

• For positive (linear) correlation– Above average values of one variable are

associated with above average values of the other (above/above, the points trend right and upwards)

– Below average values of one variable are associated with below average values of the other (below/below, the points trend left and downwards)

• As x increases, y also increases:Example: Positive Correlation

55504540353025201510

60

50

40

30

20

Output

Input

Negative Correlations

• For negative (linear) correlation– Above average values of one variable are

associated with below average values of the other (above/below, the points trend right and downwards)

– Below average values of one variable are associated with above average values of the other (below/above, the points trend left and upwards)

• As x increases, y decreases:Example: Negative Correlation

Output

Input

55504540353025201510

95

85

75

65

55

Nonlinear Correlations

• Nonlinear relations have points that have a trend, but not around a line

• The trend has some bend in it

No Correlations

• When two variables are not related– There is no linear trend– There is no nonlinear trend

• Changes in values for one variable do not seem to have any relation with changes in the other

• As x increases, there is no definite shift in y:Example: No Correlation

302010

55

45

35

Output

Input

Distinction between Nonlinear & No Correlation

Nonlinear relations and no relations are very different– Nonlinear relations are definitely patterns …

just not patterns that look like lines– No relations are when no patterns appear at

all

Example

• Examples of nonlinear relations– “Age” and “Height” for people (including both

children and adults)– “Temperature” and “Comfort level” for people

• Examples of no relations– “Temperature” and “Closing price of the Dow

Jones Industrials Index” (probably)– “Age” and “Last digit of telephone number” for

adults

Please Note Perfect positive correlation: all the points lie along a

line with positive slope Perfect negative correlation: all the points lie along a

line with negative slope

If the points lie along a horizontal or vertical line: no correlation

If the points exhibit some other nonlinear pattern: nonlinear relationship

Need some way to measure the strength of correlation

Linear Correlation Coefficient

Measure of Linear Correlation

• The linear correlation coefficient is a measure of the strength of linear relation between two quantitative variables

• The sample correlation coefficient “r” is

1

n

s)yy(

s)xx(

r y

i

x

i

Note: are the sample means and sample variancesof the two variables X and Y.

yx SSYX ,,,

Properties of Linear Correlation Coefficients

Some properties of the linear correlation coefficient– r is a unitless measure (so that r would be the same

for a data set whether x and y are measured in feet, inches, meters etc.)

– r is always between –1 and +1. r = -1 : perfect negative correlation r = +1: perfect positive correlation

– Positive values of r correspond to positive relations– Negative values of r correspond to negative relations

Various Expressions for r

There are other equivalent expressions for the linear correlation r as shown below:

22 )()(

))((

yyxx

yyxxr

yx SSn

yyxxr

)1(

))((

However, it is much easier to compute r using the short-cut formula shown on the next slide.

Short-Cut Formula for r

SS “sum of squ ares for ( )x x” x

x

n 2

2

SS “sum of squ ares for ( )y y” y

y

n 2

2

SS “sum of squares for ( )xy xy” xyx y

n

rxy

x y SS

SS SS( )

( ) ( )

Example: The table below presents the weight (in thousands of pounds) x and the gasoline mileage (miles per gallon) y for ten different automobiles. Find the linear correlation coefficient:

Example

x y x2 y2 xy

x y x2 y2 xy

Completing the Calculation for r

SS( )

( ).y y

y

n 2

22

1066530910

1116 9

SS( ) .( . )( )

.xy xyx y

n 1010 9

34 1 30910

42 79

rxy

x y

SS

SS SS

( )

( ) ( )

.

( . )( . )0.

42 79

7 449 1116 947

SS( ) .

( . ).x x

x

n 2

22

123 7334 110

7 449

Please Note r is usually rounded to the nearest hundredth

r close to 0: little or no linear correlation

As the magnitude of r increases, towards -1 or +1, there is an increasingly stronger linear correlation between the two variables

We’ll also learn to obtain the linear correlation coefficient from the graphing calculator.

Positive Correlation Coefficients

Strong Positiver = .8

Moderate Positiver = .5

Very Weakr = .1

• Examples of positive correlation

• In general, if the correlation is visible to the eye, then it is likely to be strong

Negative Correlation Coefficients

Strong Negativer = –.8

Moderate Negativer = –.5

Very Weakr = –.1

• Examples of negative correlation

• In general, if the correlation is visible to the eye, then it is likely to be strong

Nonlinear versus No Correlation

Nonlinear Relation No Relation

• Nonlinear correlation and no correlation

• Both sets of variables have r = 0.1, but the difference is that the nonlinear relation shows a clear pattern

Interpret the Linear Correlation Coefficients

• Correlation is not causation!• Just because two variables are correlated does

not mean that one causes the other to change• There is a strong correlation between shoe sizes

and vocabulary sizes for grade school children– Clearly larger shoe sizes do not cause larger

vocabularies– Clearly larger vocabularies do not cause larger shoe

sizes

• Often lurking variables result in confounding

How to Determine a Linear Correlation?

• How large does the correlation coefficient have to be before we can say that there is a relation?

• We’re not quite ready to answer that question

Summary

• Correlation between two variables can be described with both visual and numeric methods

• Visual methods– Scatter diagrams– Analogous to histograms for single variables

• Numeric methods– Linear correlation coefficient– Analogous to mean and variance for single variables

• Care should be taken in the interpretation of linear correlation (nonlinearity and causation)

Linear Regression Line

Learning Objectives

• Find the regression line to fit the data and use the line to make predictions

• Interpret the slope and the y-intercept of the regression line

• Compute the sum of squared residuals

Regression Analysis

• Regression analysis finds the equation of the line that best describes the relationship between two variables

• One use of this equation: to make predictions

Best Fitted Line• If we have two variables X and Y which tend to be

linearly correlated, we often would like to model the relation with a line that best fits to the data.

• Draw a line through the scatter diagram

• We want to find the line that “best” describes the linear relationship … the regression line

Residuals

• One difference between math and stat is that statistics assumes that the measurements are not exact, that there is an error or residual

• The formula for the residual is alwaysResidual = Observed – Predicted

• This relationship is not just for this chapter … it is the general way of defining error in statistics

What is a Residual?

• Here shows a residual on the scatter diagram The regression line

The x value of interest

The observed value y

The residual

The predicted value y

Example

• For example, say that we want to predict a value of y for a specific value of x– Assume that we are using y = 10 x + 25 as our model– To predict the value of y when x = 3, the model gives

us y = 10 3 + 25 = 55, or a predicted value of 55– Assume the actual value of y for x = 3 is equal to 50– The actual value is 50, the predicted value is 55, so

the residual (or error) is 50 – 55 = –5

Method of Least Squares• We want to minimize the prediction errors or residuals, but we need

to define what this means• We use the method of least-squares which involves the following 3

steps:1. We consider a possible linear model to fit the data2. We calculate the residual for each point3. We add up the squares of the residuals ( We square all of the residuals

to avoid the cancellation of positive residuals and negative residuals, since some observed values are under predicted, some of the observed valued are over predicted by the proposed linear model.)

• The line that has the smallest overall residuals ( i.e. the sum of all the squares of the residuals) is called the least-squares regression line or simply the regression line which is the best-fitted line to the data.

Method of Least Squares

• Assume the equation of the best-fitting line:

Where (called, y hat) denotes the predicted value of

• Least squares method:

Find the constants b0 and b1 such that the sum

of the overall prediction errors is as small as possible

210

2 ))(()ˆ( xbbyyy

xbby 10ˆ

y y

b b x 0 1y

Illustration• Observed and predicted values of y:

y y

x

y

y

( , )x y

) ( ,x y

y

Linear Regression Line

• The equation for the regression line is given by

– denotes the predicted value for the response variable.

– b1 is the slope of the least-squares regression line– b0 is the y-intercept of the least-squares regression

lineNote: Different textbooks may use different notations for the slope

and the intercept.

Y

xbby 10ˆ

Find the Equation of a Linear Regression Line

• The equation is determined by:

b0: y-intercept

b1: slope

bx x y y

x x

xyx1 2

( )( )

( )

( )( )

SSSS

)( 1

10 xby

n

xbyb

• Values that satisfy the least squares criterion:

Example: A recent article measured the job satisfaction of subjects with a 14-question survey. The data below represents the job satisfaction scores, y, and the salaries, x, for a sample of similar individuals:

1) Draw a scatter diagram for this data

2) Find the equation of the line of best fit (i.e., regression line)

Example

• Preliminary calculations needed to find b1 and b0:

Finding b1 & b0

x y x2 xy

x y x2 xy

Linear Regression Line

SS( )( )( )

.xy xyx y

n

4009

234 1338

118 75

bxyx1

118 75229 5

5174 SSSS

( )( )

..

0.

b

y b x

n01 133 5174 234

814902

(0. )( )

.

SS( ) .x x

x

n

2

22

7074234

8229 5

Equation of the line of best fit: . 0. x 149 517y Solution 1)

Scatter Diagram

21 23 25 27 29 31 33 35 37

12

13

14

15

16

17

18

19

20

21

22

JobSatisfaction

Salary

Job Satisfaction SurveySolution 2)

Please Note Keep at least three extra decimal places while doing the calculations

to ensure an accurate answer

When rounding off the calculated values of b0 and b1, always keep at least two significant digits in the final answer

The slope b1 represents the predicted change in y per unit increase in x

The y-intercept is the value of y where the line of best fit intersects the y-axis. That is, it is the predicted value of y when x is zero.

( , )x y The line of best fit will always pass through the point

Please Note

• Finding the values of b1 and b0 is a very tedious process

• We should also know to use Graphing calculator for this

• Finding the coefficients b1 and b0 is only the first step of a regression analysis– We need to interpret the slope b1

– We need to interpret the y-intercept b0

Making Predictions1. One of the main purposes for obtaining a regression

equation is for making predictions

y2. For a given value of x, we can predict a value of

3. The regression equation should be used only to cover the sample domain on the input variable. You can estimate values outside the domain interval, but use caution and use values close to the domain interval.

4. Use current data. A sample taken in 1987 should not be used to make predictions in 1999.

Interpret the Slope

• Interpreting the slope b1

– The slope is sometimes defined as as

– The slope is also sometimes defined as as

• The slope relates changes in y to changes in x

RunRise

xinChangeyinChange

Interpret the Slope

• For example, if b1 = 4

– If x increases by 1, then y will increase by 4– If x decreases by 1, then y will decrease by 4– A positive linear relationship

• For example, if b1 = –7

– If x increases by 1, then y will decrease by 7– If x decreases by 1, then y will increase by 7– A negative linear relationship

Example

• For example, say that a researcher studies the population in a town (which is the y or response variable) in each year (which is the x or predictor variable)– To simplify the calculations, years are measured from 1900 (i.e.

x = 55 is the year 1955)

• The model used isy = 300 x + 12,000

• A slope of 300 means that the model predicts that, on the average, the population increases by 300 per year.

• An intercept of 12,000 means that the model predicts that the town had a population of 12,000 in the year 1900 (i.e. when x = 0)

Interpret the y-intercept

• Interpreting the y-intercept b0

• Sometimes b0 has an interpretation, and sometimes not– If 0 is a reasonable value for x, then b0 can be

interpreted as the value of y when x is 0– If 0 is not a reasonable value for x, then b0 does not

have an interpretation• In general, we should not use the model for

values of x that are much larger or much smaller than the observed values of x included (that is, it may be invalid to predict y for x values lying outside the range of the observed x.)

Summary

• Summarize two quantitative data– Scatter diagrams– Correlation coefficients

• Linear models of correlation– Least-squares regression line– Prediction

Obtain Linear Correlation Coefficient and Regression Line Equation from TI Calculator

1. Turn on the diagnostic tool: CATALOG[2nd 0] DiagnosticOn ENTER ENTER

2. Enter the data: STAT EDIT. Enter the x-variable data into L1 and the corresponding y-variable data into L2

3. Obtain regression line and the linear correlation r: STAT CALC 4:LinReg(ax+b) ENTER L1, L2, Y1 (Notice: to enter Y1, use VARS Y-VARS 1:Function 1:Y1 ENTER). (The screen will also show r2. Just ignore it.)

4. Display the scatter diagram and the fitted regression line: Zoom 9:ZoomStat TRACE (press up or down arrow keys to

move the cursor to the regression line. Now, you can trace the points along the line by pressing the right or left arrow keys. While the cursor is on the regression line, you can also enter a number, the screen will show the predicted value of y for the x value you just entered.)