PSU.EDU

1. DataAnalysis inWF ED
3 December 2015

2. Y = a + bx

3. Y
X
(x, y)
e
a y = a + bx + e
b = y / x
Y =  + X

4. y = a + bx + e
y must be continuous.
x can be nominal, ordinal,
interval, or ratio scaled.
b is the slope of the
regression line, or y | x
a is the the point on the y
axis at which the
regression line with slope
b intercepts the y axis.

5. y = a + bx + e
y must be continuous.
x can be nominal, ordinal,
interval, or ratio scaled.
b is the slope of the
regression line, or y | x
a is the the point on the y
axis at which the
regression line with slope
b intercepts the y axis.
THE PREDICTED
VALUE OF Y
IS CONTINUOUS

6. y = a + bx + e
y must be continuous.
x can be nominal, ordinal,
interval, or ratio scaled.
b is the slope of the
regression line, or y | x
a is the the point on the y
axis at which the
regression line with slope
b intercepts the y axis.
BUT WHAT IF Y IS
NOMINAL?

7. y = a + bx + e
y must be continuous.
x can be nominal, ordinal,
interval, or ratio scaled.
b is the slope of the
regression line, or y | x
a is the the point on the y
axis at which the
regression line with slope
b intercepts the y axis.
THE PREDICTED
VALUE OF A
NOMINALLY-
SCALED Y COULD
EXCEEDTHE
LIMITS OF Y

8.  y
• Is “nominal,” i.e., is composed of unordered
categories
• Is “binary,” i.e., has two values
 Examples of binary, nominal y variables
• Graduation status — graduated; not graduated
• Poverty status — living in poverty; not living in
poverty
 y variable usually coded “0” and “1” (e.g., “1,”
if graduated, and “0” otherwise

9.  We model the probability of being in the positive
category (“1”) on the dependent variable rather
that the other category (“0”), given the
independent variables
 Examples of binary logistic regression questions
• The probability of graduating, given being male
rather than female
• The probability of living in poverty, given living in
Alabama rather than any other state in the U.S.
 Predicted probabilities must not < 0 or >1 by
definition

10.  Pr[y = 1 | x] = f(x)
 f(x) chosen is x / 1 + x because, for any value of x,
Pr[y = 1 | x] must range between 0 and 1
 So, Pr[y = 1 | x] = x / 1 + x
 In binary logistic regression,
Pr[y = 1 | x] = exb / 1 + exb can be rearranged to
Pr[y = 1 | x] = 1 / 1 + e-xb

11. Pr[y = 1 | x] = 1 / 1 + e-xby = a + bx + e

12.  Pr[y = 1 | x] = 1 / 1 + e-xb, where
• Pr[y = 1 | x] is the probability that y = 1, given
independent variables, x
• e is the exponential function; is approximately equal
to 2.71828
• b is termed the “logit,” which is the log of the odds
that y = 1 and not 0
 log(b) is called the “odds ratio,” which is the
ratio between the odds of y = 1 and the odds of
y = 0

13.  log(b) is called the “odds ratio,” which is the
ratio between the odds of y = 1 and the odds
of y = 0
 The odds ratio
• Can vary from 0 to positive infinity
• Is equal to 1 if the odds of y = 1 and y = 0 are the
same
• Is <1 if the odds of y = 1 < y = 0
• Is >1 of the odds of y = 1 > y = 0

14. In R with mtcars

15. DataAnalysis inWF ED
3 December 2015