coefficient of determination r

8/10/2019 Coefficient of Determination r

1/18

r

Coefficient of DeterminationUnit 3


2/18

Coefficient of Determination, r2

Once weve decided its appropriate to usea line, we need to think about assessing the

accuracy of predictions.


3/18


Suppose we wish to predict the price of homes in aparticular city. We take a random sample of 20houses to get y = price and x = size (our housingdata).

Clearly, we are going to get some variability in the price,since houses differ in price.

How much of this variability in price can be explained by thefact that price is related to size and houses differ in size?

If a lot of the variation in price can be accounted for by

house size, a prediction of price based on house size will bea big improvement over a prediction not based on housesize.

Our best guess, here, would be the average price of our

sample (y-bar).


4/18


The Coefficient of Determination, r2, is

the proportion of variation in y that can

be attributed to the approximate linear

relationship between x and y. (or that

can be explained by the linear

relationship between x and y).


5/18


r2is useful because:

it gives the proportion of the variance

(fluctuation) of one variable that is

predictable from the other variable

explains how much of the variability in

the y's can be explained by the fact that

they are related to x


6/18

Lets look at a formula

We find the total variation in y (SSTotal)

SSTotal = (yi ybar)

Is also called SSM (Sum of Squares about the

mean)

Is the variation around the meanlooks likevariance


7/18

Formula continued

We then find the Sum of Squared

Residuals (SSR)

SSR = (yii)

Is also called SSE or sum of squares of error

This is sometimes referred to as a measure of

the unexplained variation. Or the amount ofvariation in y that cannot be attributed to the

linear relationship between x and y


8/18

This gives us

r = 1(SSR / SSTotal)

If I multiply by 100, I get the percentage of

y variation attributable to the approximate

linear relationship between x and y.

The book uses the formula:

r = (SSMSSE) / SSM

Formula continued


9/18

Couple of Examples:

The variation of each observation (y) from is small. explains the variation in y very well

High r, high r


10/18

The variation of each observation (y) from isnot really small. doesnt explain the variation in y as well.

Poor r, poor r


11/18

Example

Suppose from our strong example thatr = .9 then r = .81

This means that 81% of the variation in the y

variable is accounted for by the linearrelationship between x and y

Suppose the other model:

r = -.4 then r = .16This means that only 16% of the variation in the

y variable is accounted for by the linear

relationship


12/18

Some points

Always use in context

Must interpret the r with our sentence. Do

not say:

The regression equation can predict 81% of

the data points

81% of data points lie on the LSRL

LSRL accounts for 81% of the data points


13/18

Properties of r2

Properties to note:

r2ranges in value from 0 to 1.0

The magnitude of r2is proportional to the

strength of the linear relation between x and y

The location of the r2value relative to 0 and 1.0

indicates the relative proximity of the linear relation

is to a perfect linear relation and no linear relation


14/18

Examples

An r2

value of 0.75 indicates that the linear relationis the distance between:

No linear relation between x and y

A perfect linear relation between x and y.

If the r2value between motivation to learn andclassroom achievement equals 0.16 for femalesand 0.04 for males, we can conclude that the linearrelation between these two variables is 4 times as

strong for females as it is for males. An r2value between systolic blood pressure and

age equal to 0.38 implies that 38% of the variabilityof between systolic blood pressure and age isaccounted for by their linear relation.


15/18

Standard Deviation about the LSRL

Se= (SSR / n-2)

This measures the typical amount by

which an observation deviates fromthe LSRL (analogous to sample

standard deviation)


16/18

Example


17/18

Homework

Textbook pp. 190196 # 15, 16, 31, 32, 47

Anova Table worksheet

Chapter 9 Projectdue October 12

th

!! Unit 3 TestOctober 14th!!


18/18

Example

3.36.pdf

anova tables.pdf

anova answers.pdf
http://localhost/var/www/apps/conversion/tmp/scratch_2/3.36.pdfhttp://localhost/var/www/apps/conversion/tmp/scratch_2/anova%20tables.pdfhttp://localhost/var/www/apps/conversion/tmp/scratch_2/anova%20answers.pdfhttp://localhost/var/www/apps/conversion/tmp/scratch_2/anova%20answers.pdfhttp://localhost/var/www/apps/conversion/tmp/scratch_2/anova%20tables.pdfhttp://localhost/var/www/apps/conversion/tmp/scratch_2/3.36.pdf

coefficient of determination r

Documents