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 = ab x a = initial amount b = 1 + rate

Power law model y = a(x p ) 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.