Transcript 4.1 Transforming Relationships

4.1 Transforming
Relationships
Transforming (reexpressing)
- Applying a function such as the logarithmic
or square root to a quantitative variable
- Because we may want to transform the
explanatory variable, x, or the response
variable, y, this variable will be referred to
as t.
• Monotonic function: f(t) moves in one direction
as it argument ‘t’ increases
– Monotonic increasing function: preserves the
order of the data. That is, if a > b, then f(a) > f(b).
– Monotonic decreasing function: reverses the
order of data. That is, if a < b, then f(a) < f(b)
Power functions
• For positive powers are monotonic
increasing for t > 0
• For negative powers are monotonic
decreasing for t < 0
• Linear Growth – increases by a fixed
amount in each equal time period. Linear
transformations can not straighten a
curved relationship.
– Fahrenheit to Celsius
– Miles to kilometers
• Exponential growth – increases by a fixed
percentage of the previous total
– Bacteria
– Return on investments when compounded
Exponential growth model
• y = abx
a = initial amount
b = 1 + rate
Power law model
y = a(xp)
Power law models become linear when we
apply logarithmic transformation to both
variables…
Log y = log a + p log x
The power, p, is now the slope of the
straight line that links log y to log x.