Stat 112 Notes 12
• Today:
– Transformations for fitting Curvilinear
Relationships (Chapter 5)
Log Transformation of Both X and
Y variables
• It is sometimes useful to transform both
the X and Y variables.
• A particularly common transformation is to
transform X to log(X) and Y to log(Y)
E (log Y | X )  0  1 log X
E (Y | X )  exp( 0  1 log X )
Heart Disease-Wine Consumption
Data (heartwine.JMP)
Bivariate Fit of Heart Disease Mortality By Wine Consumption
Residual Plot for Simple Linear Regression Model
Residual
10
8
6
3
2
1
0
-1
-2
-3
0
10
20
4
30
40
50
60
70
80
60
70
80
Wine Consumption
Residual Plot for Log-Log Transformed Model
2
0
10
20
30
40
50
60
70
80
Wine Consumption
Linear Fit
Transformed Fit Log to Log
3
Residual
Heart Disease Mortality
12
1
-1
-3
0
10
20
30
40
50
Wine Consumption
Evaluating Transformed Y Variable Models
The residuals for a log-log transformation model on the original Y-scale are
eˆi  Yi  Eˆ (Y | X i )
 Yi  exp(b0  b1 log X i )
The root mean square error and R2 on the original Y-scale are shown in JMP under Fit
Measured on Original Scale.
To evaluate models with transformed Y variables and compare their R2’s and root mean
square error to models with untransformed Y variables, use the root mean square error
and R2 on the original Y-scale for the transformed Y variables.
Linear Fit
Heart Disease Mortality = 7.6865549 - 0.0760809 Wine Consumption
Summary of Fit
RSquare
RSquare Adj
Root Mean Square Error
0.555872
0.528114
1.618923
Transformed Fit Log to Log
Log(Heart Disease Mortality) = 2.5555519 - 0.3555959 Log(Wine Consumption)
Fit Measured on Original Scale
Sum of Squared Error
Root Mean Square Error
RSquare
41.557487
1.6116274
0.5598656
The log-log transformation
provides slightly better predictions
than the simple linear regression
Model.
Interpreting Coefficients in Log-Log
Models
E (log Y | X )  0  1 log X
E (Y | X )  exp( 0  1 log X )
Assuming that
E (log Y | log X )  0  1 log X
satisfies the simple linear regression model assumptions, then
Median(Y | X )  exp( 0 ) exp( 1 X )
Thus,
Median(Y | log 2 X ) exp(  0 ) exp( 1 log 2 X )

 21
Median(Y | log X )
exp(  0 ) exp( 1 log X )
Thus, a doubling of X is associated with a multiplicative change of 2 1 in the
median of Y.
Transformed Fit Log to Log
Log(Heart Disease Mortality) = 2.5555519 - 0.3555959 Log(Wine Consumption)
Doubling wine consumption is associated with multiplying median heart disease
mortality by 20.356  0.781 .
Another interpretation of
coefficients in log-log models
For a 1% increase in X,
Median(Y | log1.01X ) exp(  0 ) exp( 1 log1.01X )

 exp(  0 )1.011
Median(Y | log X )
exp(  0 ) exp( 1 log X )
Because 1.011  1  .011 ,
a 1% increase in X in associated with a 1 percent increase in the median (or mean) of Y.
Transformed Fit Log to Log
Log(Heart Disease Mortality) = 2.5555519 - 0.3555959 Log(Wine Consumption)
Increasing wine consumption by 1% is associated with a -0.36% decrease in mean heart
disease mortality.
Similarly a 10% increase in X is associated with a 10 1 percent increase in mean heart
disease mortality.
Increasing wine consumption by 10% is associated with a -3.6% decrease in mean heart
disease mortality.
For large percentage changes (e.g., 50%, 100%) , this interpretation is not accurate.
Another Example of
Transformations: Y=Count of tree
seeds, X= weight of tree
Bivariate Fit of Seed Count By Seed weight (mg)
30000
25000
Seed Count
20000
15000
10000
5000
0
-5000
-1000
0
1000
2000
3000
Seed w eight (mg)
4000
5000
Bivariate Fit of Seed Count By Seed weight (mg)
30000
25000
Seed Count
20000
15000
10000
5000
0
-5000
-1000
0
1000
2000
3000
Seed w eight (mg)
Linear Fit
Transformed Fit Log to Log
Transformed Fit to Log
4000
5000
Linear Fit
Seed Count = 6751.7179 - 2.1076776 Seed weight (mg)
Transformed Fit to Log
Seed Count = 12174.621 - 1672.3962 Log(Seed weight (mg))
Summary of Fit
Summary of Fit
RSquare
RSquare Adj
Root Mean Square Error
Mean of Response
Observations (or Sum
Wgts)
0.220603
0.174756
6199.931
4398.474
19
RSquare
RSquare Adj
Root Mean Square Error
Mean of Response
Observations (or Sum
Wgts)
0.566422
0.540918
4624.247
4398.474
19
Transformed Fit Log to Log
Log(Seed Count) = 9.758665 - 0.5670124 Log(Seed weight (mg))
Fit Measured on Original Scale
Sum of Squared Error 161960739
Root Mean Square
3086.6004
Error
RSquare
0.8068273
Sum of Residuals
3142.2066
By looking at the root mean square error on the original y-scale, we see that
Both of the transformations improve upon the untransformed model and that the
transformation to log y and log x is by far the best.
Comparison of Transformations to
Polynomials for Tree Data
Bivariate Fit of Seed Count By Seed weight (mg)
30000
Transformed Fit Log to Log
25000
Log(Seed Count) = 9.758665 - 0.5670124*Log(Seed weight (mg))
Seed Count
20000
Fit Measured on Original Scale
15000
Root Mean Square Error
3086.6004
10000
5000
Polynomial Fit Degree=6
0
-5000
0
1000
2000
3000
Seed w eight (mg)
4000
5000
Seed Count = 1539.0377 + 2.453857*Seed weight (mg)
-0.0139213*(Seed weight (mg)-1116.51)^2
+1.2747e-6*(Seed weight (mg)-1116.51)^3
+1.0463e-8*(Seed weight (mg)-1116.51)^4
- 5.675e-12*(Seed weight (mg)-1116.51)^5
+ 8.269e-16*(Seed weight (mg)-1116.51)^6
Summary of Fit
Transformed Fit Log to Log
Polynomial Fit Degree=6
Root Mean Square Error
6138.581
For the tree data, the log-log transformation is
much better than polynomial regression.
Prediction using the log y/log x
transformation
• What is the predicted seed count of a tree
that weights 50 mg?
• Math trick: exp{log(y)}=y (Remember by
log, we always mean the natural log, ln),
i.e., elog10  10
Eˆ (Y | X  50)  exp{ Eˆ (log Y | X  50)} 
exp{ Eˆ (log Y | log X  log 50)}  exp{ Eˆ (log Y | log X  3.912)} 
exp{9.7587  0.5670 * 3.912}  exp{7.5406}  1882.96