Transformations to Achieve Linearity
When a
residual plot
reveals a data set to be nonlinear, it is often possible to
"transform" the raw data to make it more linear. This allows us to use
linear regression
techniques more effectively with nonlinear data.
What is a Transformation to Achieve Linearity?
Transforming a variable involves using a mathematical operation to
change its measurement scale. Broadly speaking, there are two
kinds of transformations.
 Nonlinear tranformation. A nonlinear transformation changes
(increases or decreases) linear
relationships between variables and, thus, changes the
correlation between variables. Examples of a nonlinear
transformation of variable x would be taking the
square root of x or the reciprocal of x.
In regression, a transformation to achieve linearity is a
special kind of nonlinear transformation. It is a nonlinear
transformation that increases the linear
relationship between two variables.
Methods of Transforming Variables to Achieve Linearity
There are many ways to transform variables to achieve linearity
for regression analysis. Some common methods are summarized below.
Method 
Transformation(s) 
Regression equation 
Predicted value (ŷ) 
Standard linear regression 
None 
y = b_{0} + b_{1}x 
ŷ = b_{0} + b_{1}x 
Exponential model 
Dependent variable = log(y) 
log(y) = b_{0} + b_{1}x 
ŷ = 10^{b0 + b1x} 
Quadratic model 
Dependent variable = sqrt(y) 
sqrt(y) = b_{0} + b_{1}x 
ŷ = ( b_{0} + b_{1}x )^{2} 
Reciprocal model 
Dependent variable = 1/y 
1/y = b_{0} + b_{1}x 
ŷ = 1 / ( b_{0} + b_{1}x ) 
Logarithmic model 
Independent variable = log(x) 
y= b_{0} + b_{1}log(x) 
ŷ = b_{0} + b_{1}log(x) 
Power model 
Dependent variable = log(y)
Independent variable = log(x) 
log(y)= b_{0} + b_{1}log(x) 
ŷ = 10^{b0 + b1log(x)} 
Each row shows a different nonlinear transformation method. The
second column shows the specific transformation applied to
dependent and/or independent variables. The third column shows
the regression equation used in the analysis. And the last
column shows the "back transformation" equation used to
restore the dependent variable to its original, nontransformed
measurement scale.
In practice, these methods need to be tested on the
data to which they are applied to be sure that they
increase rather than decrease the linearity
of the relationship. Testing the effect of a transformation
method involves looking at
residual
plots and correlation coefficients, as described in the
following sections.
Note: The logarithmic model and the power model
require the ability to work with
logarithms.
Use a
graphic calculator
to obtain the log of a number or to transform back from the logarithm
to the original number.
If you need it, the Stat Trek glossary has a brief
refresher on logarithms.
How to Perform a Transformation to Achieve Linearity
Transforming a data set to enhance linearity is a multistep,
trialanderror process.
 Compute the coefficient of determination (R^{2}),
based on the transformed variables.
 If the tranformed R^{2} is greater than the
rawscore R^{2}, the
transformation was successful. Congratulations!
 If not, try a different
transformation method.
The best tranformation method (exponential model, quadratic
model, reciprocal model, etc.) will depend on nature of the
original data. The only way to determine which method is best
is to try each and compare the result (i.e.,
residual
plots, correlation coefficients).
A Transformation Example
Below, the table on the left shows data for independent and dependent
variables  x and y, respectively. When we apply a linear regression
to the untransformed raw data, the
residual
plot shows a nonrandom pattern (a Ushaped curve), which
suggests that the data are nonlinear.
x 
1 
2 
3 
4 
5 
6 
7 
8 
9 
y 
2 
1 
6 
14 
15 
30 
40 
74 
75 



Suppose we repeat the analysis, using a quadratic model to transform
the dependent variable. For a quadratic model, we use the square
root of y, rather than y, as the dependent variable. Using the transformed data, our
regression equation is:
y'_{t} = b_{0} + b_{1}x
where
y_{t} = transformed dependent variable, which is equal to
the square root of y
y'_{t} = predicted value of the transformed dependent variable y_{t}
x = independent variable
b_{0} = yintercept of transformation regression line
b_{1} = slope of transformation regression line
The table below shows the transformed data we analyzed.
x 
1 
2 
3 
4 
5 
6 
7 
8 
9 
y_{t} 
1.41 
1.00 
2.45 
3.74 
3.87 
5.48 
6.32 
8.60 
8.66 



Since the transformation was based on the quadratic model
(y_{t} = the square root of y),
the transformation regression equation can be expressed in terms of the
original units of variable Y as:
y' = ( b_{0} + b_{1}x )^{2}
where
y' = predicted value of y in its orginal units
x = independent variable
b_{0} = yintercept of transformation regression line
b_{1} = slope of transformation regression line
The residual plot (above right) shows residuals based on predicted raw scores
from the transformation
regression equation. The plot suggests that the transformation
to achieve linearity was successful. The pattern of residuals is
random, suggesting that the relationship between the independent
variable (x) and the transformed dependent variable (square root of y)
is linear. And the coefficient
of determination was 0.96 with the transformed data versus only
0.88 with the raw data. The transformed data resulted in a better
model.
Test Your Understanding
Problem
In the context of
regression
analysis,
which of the following statements is true?
I. A linear transformation increases the linear relationship
between variables.
II. A logarithmic model is the most effective transformation method.
III. A residual plot reveals departures from linearity.
(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I, II, and III
Solution
The correct answer is (C). A linear transformation neither increases nor
decreases the linear relationship between variables; it preserves the
relationship. A nonlinear transformation is used to
increase the relationship between variables.
The most effective transformation method depends on the data
being transformed. In some cases, a logarithmic model may be more
effective than other methods; but it other cases it may be less
effective.
Nonrandom patterns in a
residual plot
suggest a departure from linearity in the data being plotted.