1. TROUBLESOME CONCEPTS IN STATISTICS: r2 AND POWER
N. Scott Urquhart
Director, STARMAP
Department of Statistics
Colorado State University
Fort Collins, CO

2. STARMAP FUNDING Space-Time Aquatic Resources Modeling and Analysis Program
The work reported here today was developed under the STAR Research Assistance Agreement CR awarded by the U.S. Environmental Protection Agency (EPA) to Colorado State University. This presentation has not been formally reviewed by EPA.  The views expressed here are solely those of the presenter and STARMAP, the Program he represents. EPA does not endorse any products or commercial services mentioned in these presentation.
This research is funded by
U.S.EPA – Science To Achieve
Results (STAR) Program
Cooperative
Agreement
# CR

3. INTENT FOR TODAY
To discuss two topics which have given some of you a bit of confusion
r2 in regression
Power in the context of tests of hypotheses
Thanks for Ann Brock and Harriett Bassett for suggesting these topics
Approach: Visually illustrate the idea,
Then talk about the concepts illustrated
The sequences of graphs are available on the internet right now (address is at the end of this handout)
Questions are welcome

4. r2 IN REGRESSION
r2 provides a summary of the strength of a (linear) regression which reflects:
The relative size of the residual variability,
The slope of the fitted line, and
How good the observed values of the predictor variable are for prediction
Mainly the range of the Xs
Let’s see these features in action, then
Look at the formulas

5. WHAT MAKES r2 TICK?
varying one thing, leaving the remaining things fixed
r2 increases as residual variation decreases
r2 increases as the slope increases
r2 increases the range of x increases

6. WHAT IS r2?
r2 provides A measure of the fit of a line to a set of data which incorporates
The amount of residual variation,
The strength of the line (slope), and
How good the set of values of “x” are for estimating the line
Some areas of endeavor tend to overuse it!

7. HOW DOES r2 TELL US ABOUT VARIATION?
The following graph illustrates this:
The data scatter has r2 = 0.5 (approximately)
The red points have the same values, but all concentrated at X = 5.
{Strictly speaking the above formulas apply only in the case of bivariate regression.}
{Estimation formulas involve factors of n-1 and n-2.}

8. r2

9. FORMULAS FOR r2 But these have little intuitive appeal !
We’ll decompose observations into parts:
Mean
Regression
Residual

10. DECOMPOSING REGRESSION
This is really n equations
Square each of these equations and add them up across i.
The three cross product terms will each add to zero. (Try it!)

11. DECOMPOSING REGRESSION (continued)

12. POWER OF A TEST OF HYPOTHESIS
Power = Prob(“Being right”) = Prob(Rejecting false hypothesis)
Power depends on two main things
The difference in the hypothesized and true situations, and
The strength of the information for making the test
Sample size is very important factor
In regression it depends on the same factors as the ones which increase r2.
Again, see it, then talk about it
Power increases as D = m1 - m2 increases

13. POWER VARIES WITH DIFFERENCE (D = m1 - m2) and SAMPLE SIZE (n)

14. ON TESTS OF HYPOTHESES ( ON THE WAY TO POWER)
TRUE SITUATION
HYPOTHESIS FALSE
HYPOTHESIS TRUE
ACTION
FAIL TO REJECT THE NULL HYPOTHESIS
CORRECT ACTION
TYPE II ERROR
REJECT THE NULL HYPOTHESIS
TYPE I ERROR
CORRECT ACTION
Tests of hypotheses are designed to control
a = Prob (Type I Error)
While getting
Power = 1- Prob (Type II Error) as large as possible

15. ON TESTS OF HYPOTHESES (AN ASIDE)
Which is worse,
a type I error, or
a type II error?
It depends tremendously on perspective
Consider the criminal justice system
Truth: Accused is innocent (HO) or guilty (HA)
Action: Accused is acquitted or convicted
Type I error = Convict an innocent person
Type II error = Acquit a guilty person
Which is worse?
Consider the difference in view of the
Accused
Society – especially if accused is terrorist

16. COMPUTING THE CRITICAL REGION
Consider a simple case X ~ N( m, 1)
HO: m = 4 versus HA: m ¹ 4
Critical Region (CR) is
X £ l and X ³ u , so
0.025 = P(X £ l ) = P((X-4)/1 £ ( l - 4)/1) = P(Z £ -1.96)
l = 2.04, similarly,
u = 5.96

17. COMPUTING POWER Consider a simple case X ~ N( m, 1)
HO: m = 4 versus HA: m ¹ 4
Power (at m = 5) = ?
= Prob(XA in CR| m = 5)
XA ~ N( 5, 1)
Prob(XA £ 2.04) + Prob(XA ³ 5.96)
= Prob(Z £ -2.96) + Prob(Z ³ 0.96)
= =

18. POWER VARIES WITH DIFFERENCE (D = m1 - m2) and SAMPLE SIZE (n)

19. COMPUTING POWER USING A MEAN BASED ON n = 2 OBSERVATIONS
Consider a simple case:
When
the mean of two observations follows:
HO: m = 4 versus HA: m ¹ 4
Power (at m = 5) = ?
Critical Region (CR) is
£ l and ³ u , so
0.025 = P( £ l ) = P(( -4)/0.707 £ ( l - 4)/0.707) = P(Z £ -1.96)
So l = 4 – (1.96)(0.707) = 2.61, similarly, u = 5.39

20. COMPUTING POWER USING A MEAN BASED ON n = 2 OBSERVATIONS (continued)

21. POWER VARIES WITH DIFFERENCE (D = m1 - m2) and SAMPLE SIZE (n)

22. COMPUTING POWER USING A MEAN BASED ON n = 4 OBSERVATIONS (continued)
(This page is not in the handout – so it all would fit on one page)

23. POWER VARIES WITH DIFFERENCE (D = m1 - m2) and SAMPLE SIZE (n)

24. POWER VARIES WITH DIFFERENCE (D = m1 - m2) and SAMPLE SIZE (n)

25. DIRECTIONAL NOTE
As the alternative has been two-sided throughout this presentation, the power curves are symmetric about the vertical axis.
By examining only the positive side, we can see the curves twice as large.

26. YOU HAVE ACCESS TO THESE PRESENTATIONS
You can find each of the slide shows shown here today at:
Each show begins with authorship & funding slides
You are welcome to use them, and adapt them
But, please always acknowledge source and funding
You are free to reorder the graphs if it makes more sense for r2 to decrease than increase.
Urquhart is available to talk to AP Stat classes about statistics as a profession.
See content on the web site above.