The correlation coefficient, r,

tells us about strength (scatter)

and direction of the linear

relationship between two

quantitative variables.

In addition, we would like to have a numerical description ( model ) of

how both variables vary together. For instance, is one variable increasing

faster than the other one? And we would like to make predictions based

on that numerical description. The relationship above looks linear . . .But which line best

describes our data?

The regression line

The least-squares regression line is the unique line such

that the sum of the squares of the vertical distances of the

data points to the line is the smallest possible.

ˆ y 0.125x 41.4

Year Powerboats Dead Manatees

1977 447 13

1978 460 21

1979 481 24

1980 498 16

1981 513 24

1982 512 20

1983 526 15

1984 559 34

1985 585 33

1986 614 33

1987 645 39

1988 675 43

1989 711 50

1990 719 47

And these equations are available in R through the function lm(y~x) ("lm" means "linear model"). Try lm on the manatee data… (manatee.csv)

The equation completely describes the regression line.

To plot the regression line you only need to choose two x values, put them into the prediction equation, calculate y, and draw the line that goes through those two points... or let R do it for you with the abline function (abline(lm(y~x)))

Hint: The regression line always passes through the mean of x and y.

The points you use for drawing the regression line are computed from the equation. .125*450-41.4 = 14.85.125*700-41.4= 46.1So plot the points (450,14.85) & (700,46.1)

ˆ y 0.125x 41.4

X

X

The distinction between explanatory and response variables is crucial in

regression. If you exchange y for x in calculating the regression line, you

will get a different line.

Regression examines the distance of all points from the line in the y

direction only.

Hubble telescope data about

galaxies moving away from earth:

These two lines are the two

regression lines calculated either

correctly (x = distance, y = velocity,

solid line) or incorrectly (x =

velocity, y = distance, dotted line).

Year Powerboats Dead Manatees

1977 447 13

1978 460 21

1979 481 24

1980 498 16

1981 513 24

1982 512 20

1983 526 15

1984 559 34

1985 585 33

1986 614 33

1987 645 39

1988 675 43

1989 711 50

1990 719 47

There is a positive linear relationship between the number of powerboats registered and the number of manatee deaths.

(in 1000’s)

The least squares regression line has the equation:

1.214.415.62ˆ 4.41)500(125.0ˆ =−=⇒−= yyRoughly 21 manatees - do this with R using the predict function (see help(predict))

Thus if we were to limit the number of powerboat registrations to 500,000, what

could we expect for the number of manatee deaths?

ˆ y 0.125x 41.4

ˆ y 0.125x 41.4

• The least-squares regression line of y on x is the line that minimizes the sum of the squares of the vertical distances of the data points to the line.

• The equation of the l-s line is usually represented as = b0 + b1 x where

= the predicted value of y

b0 = the intercept (predicted value of y when x=0)

b1 = the slope of the prediction line

• The correlation coefficient, r, is related to the l-s regression line as follows: the square of r (r2) is equal to the fraction of the variation in the values of the response variable y that is explained by the least squares regression of y on x. (See next slide)

yy

r=0.994, r-square=0.988

r=0.921, r-square=0.848

Here are two plots of height (response) against age (explanatory) of some children. Notice how r2 relates to the variation in heights...

• Homework:– Read pages 8-10 in the Reading & Problems 2.1 on

Linear Regression– note the R functions used here:model1=lm(y~x)plot(x,y) ; abline(model1)plot(model1)coef(model1) ; resid(model1) ; fitted(model1)plot(fitted(model1),resid(model1))– Read at least one of the online sources for simple

linear regression ( I like the second one…)http://www.stat.yale.edu/Courses/1997-98/101/linreg.htmhttp://www.statisticalpractice.com/http://onlinestatbook.com/rvls/http://www.sportsci.org/resource/stats/index.html

• Homework(cont.)– FPG (mg/ml) - fasting plasma glucose (measured at

home) HbA (% - measured in doctor's office). Can you predict FPG by HbA? Plot, compute the correlation coefficient, compute and plot the regression line and get a residual plot. Are there any unusual cases? Influential Points? Outliers?