LINEAR REGRESSION: On to Predictions!

Or: “How to amaze your friends and baffle your

enemies”

Imagine:The correlation

between 1 mile run time and VO2 Max is r = -1.0!

Suppose you know... that a 6:30 mile = 40 ml/kg/minand 6:00 mile = 45 ml/kg/minCould you predict what VO2 Max

a person would have who could run a 5:00 min mile???

Of Course you could, by finding the equivalent point on the line!

Mile Time vs. VO2 Max

MILE TIME

VO2MAX

4:00 4:30 5:00 5:30 6:00 6:30

40

45

50

55

60r = -1.0

Describing and Defining the LINETo describe a line on a

graph, we need to know:The slope The point where the line

intercepts the y axis

More Math?? Y=a+bX

General formula for a straight line

Calculated from the means, s, r

Y = a + b XY = the predicted value of y for a

given value of Xa = the point of the y interceptb = the slope of the line (rise over

run)X = the value of X ( Height) for

predicting Y (Shoe size)

Linear Regression

Maybe you recognize this general equation: Y = a+bX:

VO2 = 111.33 - (0.42 * HR)

Y = a + (-b * X)Y = dependent variableX = independent variable

How Accurate is the Prediction?When the correlation coefficient is

equal to 1.0, then every actual score will fall exactly on the prediction line.

THERE IS NO “ERROR” BETWEEN THE ESTIMATED PREDICTION and REALITY

Mile Time vs. VO2 Max

MILE TIME

VO2MAX

4:00 4:30 5:00 5:30 6:00 6:30

40

45

50

55

60r = -1.0

Get Real! In the REAL

WORLD, it is never so tidy

There is some deviation between the line and most points

Standard Error of EstimateThe predicted (estimated) score will not

be exact, there will be a margin of error between predicted and actual scores.

Thus we need to know the standard deviation of the prediction error.The SEE gives one a feel for the

accuracy of a prediction

Note the error from predicted?

R = .87

Prediction Line

Actual Scores

Body Composition data: Compared to UWW Skinfold 7 site Skinfold 3 site BIA Infrared Circumference

r=.87 SEE= 3.5% r= .87 SEE = 3.5% r=.80 SEE = 5.0% r= .80 SEE= 4.5% r= .75 SEE = 7.0%

UWW vs dissection: SEE = 2.0%

Let’s give it a try!Lab # 4: Predicting Shoe size

(dependent variable - Y)From Height (independent

variable - X)First derive the linear regression

equation, then try it out!