Transcript Chapter 4 More on Two

Chapter 4
More on Two-Variable Data
“Each of us is a statistical impossibility
around which hover a million other lives
that were never destined to be born”
Loren Eiseley
4.1
Some models for scatterplots with
non-linear data (pp. 176-197)
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Exponential growth
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Growth or decay function
x
y

ab
Form:
Power function
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Form: y  axb
Logarithms
log b x  y if and only if b  x
y
 x  0, b  0, b  1
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Rules for logarithms
log  AB   log A  log B
 A
log    log A  log B
B
log A p  p  log A
In other words…
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The log of a product is the sum of the logs.
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The log of a quotient is the difference of
the logs.
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The log of a power is the power times the
log.
4.2
Interpreting Correlation and
Regression (pp. 206-214)
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Overview:
Correlation and regression need to be interpreted with
CAUTION. Two variables may be strongly associated,
but this DOES NOT MEAN that one causes the other.
High Correlation does not imply causation!
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We need to consider lurking variables and common
response.
Extrapolation
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The use of a regression line or curve to
make a prediction outside of the domain of
the values of your explanatory variable x
that you used to obtain your line or curve.
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These predictions cannot be trusted.
Lurking Variable
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A variable that affects the relationship of the
variables in the study.
NOT INCLUDED among the variables studied.
Example: strong positive association might exist
between shirt size and intelligence for teenage
boys. A lurking variable is AGE.
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Shirt size and intelligence among teenage boys
generally increases with age.
If there is a strong association between two variables x
and y, any one of the following statements could be
true:
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x causes y:
 Association DOES NOT imply causation, but causation
could exist.
Both x and y are responding to changes in some unobserved
variable or variables.
 This is called common response.
The effect of x on y is hopelessly mixed up with the effects of
other variables on y.
 This is called confounding.
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Always a potential problem in observational studies.
Can be somewhat controlled in experiments with a control group
and a treatment group.
4.3
Relations in Categorical Data
(pp. 215-226)
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Overview:
We can see relations between two or more
categorical variables by setting up tables.
So far, we have studied relationships with a
quantitative response variable.
Notation
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Prob(X) is the probability that X is true.
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Prob(X/Y) is the probability that X is true,
given that Y is true
Two-way Table
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Describes the relationship between two
categorical variables:
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Row totals and column totals give MARGINAL
DISTRIBUTIONS of the two variables
separately.
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Row variable
Column variable
DO NOT give any information about the
relationships between the variables.
Can be used in the calculation of probabilities.
Example: 200 employees of a company are classified
according to the Table below, where A, B, and C are
mutually exclusive.
Have A
20
Have B
40
Male
30
10
40
80
Totals
50
50
100
200
Female
Have C Totals
60
120
Example: (con’t)
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What is the probability that a randomly chosen
person is female?
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What is the probability that a randomly chosen
person has property A?
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Prob(F) = 120/200 = 60%
Prob(A) = 50/200 = 25%
If a randomly chosen person is female, what is
the probability that she has property B?
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Prob(B/F) = 40/50 = 80%
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Note: equals Prob(B and F)/Prob(B)
Example: (con’t)
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If a randomly chosen person has property
C, what is the probability that the
individual is male?
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Prob(M/C) = 40/100 = 40%
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Note: equals Prob(C and M)/Prob(M)
If a randomly chosen person has B or C,
what is the probability that the person is
male?
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Prob(M/B or C) = 50/150 = 33.3%
Simpson’s Paradox
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The reversal of the direction of a
comparison or an association when data
from several groups are combined to form
a single group.
Lurking variables are categorical.
An extreme form of the fact that observed
associations can be misleading when there
are lurking variables.
Example of Simpson’s Paradox
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First Half of BB Season
Hits
Times
Bat
Caldwell 60
at bat
200
avg.
.300
Wilson 29
100
.290
Batting avgs. For entire season:
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Second Half of BB Season
Hits
Times
Bat
at bat
avg.
50
200
.250
1
5
.200
Caldwell: 110/400 = .275
Wilson: 30/105 = .286
Calwell had a better avg. than Wilson in each half; however,
Caldwell ends up with a LOWER OVERALL avg. than Wilson.