* Examples for transformations done in class on November 20, 2017;

* Example 1;
* Simulated Poisson Count Data;

PROC IMPORT Out=S3A3.Poisson
    Datafile="Data\PoisData.txt"
    DBMS=DLM REPLACE;
	Getnames=YES;
	Datarow=2;
run;

* Untransformed model - Problem with heteroscedasticity;
PROC REG Data=S3A3.Poisson plots=none;
    Model Y=X;
	Plot Y*X;
	Plot Student.*Pred.;
	Plot Student.*nqq.;
run;

* Transform Y to square root and run new model;
Data S3A3.Poisson;
    Set S3A3.Poisson;
	SqrtY=sqrt(Y);
	LogY=log(Y);
run;

PROC MEANS Data=S3A3.Poisson Mean Var;
    Var Y SqrtY LogY;
	By X;
run;

PROC REG Data=S3A3.Poisson plots=none;
    Model SqrtY=X;
	Plot SqrtY*X;
	Plot Student.*Pred.;
	Plot Student.*nqq.;
run;

* In this model the heteroscedasticity has been solved but there;
* is now evidence of non-linearity;
* Try taking the square root of the covariate too;
Data S3A3.Poisson;
    Set S3A3.Poisson;
	SqrtX=sqrt(X);
run;

PROC REG Data=S3A3.Poisson plots=none;
    Model SqrtY=SqrtX;
	Plot SqrtY*SqrtX;
	Plot Student.*Pred.;
	Plot Student.*nqq.;
run;

* This seems to solve the problems;

* What if we used a different transformation, such as log?;
Data S3A3.Poisson;
    Set S3A3.Poisson;
	LogY=log(Y);
	LogX=log(X);
run;

PROC REG Data=S3A3.Poisson plots=none;
    Model LogY=X;
	Plot LogY*X;
	Plot Student.*Pred.;
	Plot Student.*nqq.;
run;

PROC REG Data=S3A3.Poisson plots=none;
    Model LogY=LogX;
	Plot LogY*LogX;
	Plot Student.*Pred.;
	Plot Student.*nqq.;
run;

* Example 2;
* Simulated data with the standard deviation proportional to the covariate;
PROC IMPORT Out=S3A3.Hetero
    Datafile="Data\HeteroData.txt"
    DBMS=DLM REPLACE;
	Getnames=YES;
	Datarow=2;
run;

* Untransformed model - clear heteroscedasticity;
PROC REG Data=S3A3.Hetero plots=none;
    Model Y=X;
	Plot Y*X;
	Plot Student.*Pred.;
	Plot Student.*nqq.;
run;

* Transformed model mentioned in class notes;
DATA S3A3.Hetero;
    Set S3A3.Hetero;
    Xprime=1/X;
	Yprime=Y/X;
run;

PROC REG Data=S3A3.Hetero plots=none;
    Model Yprime=Xprime;
	Plot Yprime*Xprime;
	Plot Student.*Pred.;
	Plot Student.*nqq.;
	PLOT H.*Xprime;
run;
* Note that this does result in some observations having high leverage!

* A Real Data Example;
PROC IMPORT OUT=S3A3.Cedar
  Datafile="Data\Cedar.txt"
  DBMS=DLM REPLACE;
  Getnames=YES;
  Datarow=2;
run;

PROC REG Data=S3A3.Cedar plots=none;
  Model Height=Dbh;
  Plot Height*Dbh;
  Plot Student.*Predicted.;
  Plot Student.*nqq.;
run;
quit;

Data S3A3.Cedar;
  Set S3A3.Cedar;
  LogDbh=log(Dbh);
  SqrtDbh=sqrt(Dbh);
  RecSqrtDbh=1/SqrtDbh;
run;

PROC REG Data=S3A3.Cedar plots=none;
  Model Height=SqrtDbh;
  PLOT Height*SqrtDbh;
  Plot Student.*Predicted.;
  Plot Student.*nqq.;
run;
quit;

PROC REG Data=S3A3.Cedar plots=none;
  Model Height=LogDbh;
  Plot Height*LogDbh;
  Plot Student.*Predicted.;
  Plot Student.*nqq.;
run;
quit;

PROC REG Data=S3A3.Cedar plots=none;
  Model Height=RecSqrtDbh;
  PLOT Height*RecSqrtDbh;
  Plot Student.*Predicted.;
  Plot Student.*nqq.;
run;
quit;

Data S3A3.Cedar;
  Set S3A3.Cedar;
  LogHeight=log(Height);
  SqrtHeight=sqrt(Height);
  RecSqrtHeight=1/SqrtHeight;
run;

PROC REG Data=S3A3.Cedar plots=none;
  Model SqrtHeight=sqrtDbh;
  Plot SqrtHeight*sqrtDbh;
  Plot Student.*Predicted.;
  Plot Student.*nqq.;
run;
quit;

PROC REG Data=S3A3.Cedar plots=none;
  Model LogHeight=sqrtDbh;
  Plot LogHeight*sqrtDbh;
  Plot Student.*Predicted.;
  Plot Student.*nqq.;
run;
quit;

PROC REG Data=S3A3.Cedar plots=none;
  Model LogHeight=LogDbh;
  Plot LogHeight*LogDbh;
  Plot Student.*Predicted.;
  Plot Student.*nqq.;
run;
quit;

PROC REG Data=S3A3.Cedar plots=none;
  Model sqrtHeight=logDbh;
  Plot sqrtHeight*logDbh;
  Plot Student.*Predicted.;
  Plot Student.*nqq.;
run;
quit;