Chapter 4
More on Two-Variable Data
YMS 4.1
Transforming
Relationships
Basics
 Transforming
data
– Changing the scale of measurement used
when the data was collected
 Ch
4 Transforming
– Choose a power or logarithmic
transformation that straightens the data
– Why? We know how to analyze linear
relationships!
 Monotonic
Function
– f(t) moves in one direction as t increases
Algebraic Properties of Logarithms
 logbx
= y if and only if by = x
 Multiply/add
– Log (AB) = Log A + Log B
 Divide/subtract
– Log (A/B) = Log A – Log B
 Power
to front
– Log (x)A = A*Log x
 Linear
Growth
– Increases by a fixed amount in each
equal time period
 Exponential
– Increases by a fixed percentage of the
previous total
– y=abx
– Plot log y vs. x
– If a variable grows exponentially, its
logarithm grows linearly
log y = log abx
log y = log a + log bx
log y = log a + xlog b
Power Models
 Ladder
of Power Functions p201
y
= axp
 Take logarithm of both sides
straightens the data
log y = log (axp)
log y = log a + logxp
log y = log a + plogx
p213 #4.10-4.11
Homework: p222 #4.17 to 4.20
YMS 4.2
Cautions about Correlation
and Regression
Some Vocabulary
 Extrapolation
– Predicting outside the domain of values
of x used to obtain the line or curve
 Lurking
variable
– Is not among the explanatory or
response variables but can influence the
interpretation of relationships among
those variables
– Can dramatically change the conclusions
Reminders!
 Correlation
and regression only
describe linear relationships and
neither one is resistant!
 Using
averaged data
– Correlations based on averages are
usually too high when applied to
individuals
p230 #4.28 and 4.31
Explaining Association

Causation
– May not generalize to other
settings
– A direct causation is rarely
the complete explanation
– Is established by an
experiment where lurking
variables are controlled
x
y
 Common
Response
– The observed
association between x
and y is explained by a
lurking variable z
– An association is
created even though
there may be no direct
causal link
x
y
z
 Confounding
– Two variables whose
effects on a response
variable are
undistinguishable
– May be either
explanatory or lurking
variables
p237 #4.33 to 4.37
x
?
z
y
Establishing Causation

Strength
– There is a strong association between variables

Consistency
– Many different studies show the same results

Response
– Higher explanatory values produce a higher
response

Temporal Relationship
– Alleged cause precedes the effect in time

Coherence
– The alleged cause is plausible/logical
YMS 4.3
Relations in Categorical
Data
Two-Way Tables
 Row
variable/Column variable
 Marginal Distributions
– Found at the bottom or right margin
– Are entire rows/columns over the total
 Conditional
Distributions
– Only a cell that satisfies a certain
condition (given in the row/column)
Simpson’s Paradox
 The
reversal of the direction of a
comparison or an association when
data from several groups are
combined to form a single group
– Alaska Airlines vs. American West
– Business vs. Law School Admissions
Workshop Statistics 7-2 and 7-4