Computing bovee_bia6_inppt01Transformations

7/26/2019 Computing bovee_bia6_inppt01Transformations

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SW388R7

Data Analysis &

Computers II

Slide 1

Computing Transformations

Transforming variales

Transformations for normality

Transformations for linearity


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Slide 2Transforming variales to satisfy assumptions

W!en a metri" variale fails to satisfy t!eassumption of normality# !omogeneity of varian"e# or

linearity# $e may e ale to "orre"t t!e defi"ien"y

y using a transformation%

We $ill "onsider t!ree transformations for normality#

!omogeneity of varian"e# and linearity t!e logarit!mi" transformation

t!e s'uare root transformation# and t!e inverse transformation

plus a fourt! t!at is useful for prolems of linearity

t!e s'uare transformation


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Slide 3Computing transformations in S(SS

In S(SS# transformations are otained y "omputing ane$ variale% S(SS fun"tions are availale for t!elogarit!mi" )*+1,- and s'uare root )S.RT-transformations% T!e inverse transformation uses aformula $!i"! divides one y t!e original value forea"! "ase%

/or ea"! of t!ese "al"ulations# t!ere may e datavalues $!i"! are not mat!emati"ally permissile%

/or e0ample# t!e log of ero is not definedmat!emati"ally# division y ero is not permitted#and t!e s'uare root of a negative numer results inan 2imaginary value% We $ill usually ad4ust t!evalues passed to t!e fun"tion to ma5e "ertain t!at

t!ese illegal operations do not o""ur%


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Slide 4T$o forms for "omputing transformations

T!ere are t$o forms for ea"! of t!e transformationsto indu"e normality# depending on $!et!er t!e

distriution is s5e$ed negatively to t!e left or

s5e$ed positively to t!e rig!t%

6ot! forms use t!e same S(SS fun"tions and formula

to "al"ulate t!e transformations%

T!e t$o forms differ in t!e value or argument passed

to t!e fun"tions and formula% T!e argument to t!e

fun"tions is an ad4ustment to t!e original value of

t!e variale to ma5e "ertain t!at all of t!e

"al"ulations are mat!emati"ally "orre"t%


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Slide 5/un"tions and formulas for transformations

Symoli"ally# if $e let 0 stand for t!e argumentpasses to t!e fun"tion or formula# t!e "al"ulations for

t!e transformations are

*ogarit!mi" transformation "ompute log *+1,)0-

S'uare root transformation "ompute s'rt

S.RT)0-

Inverse transformation "ompute inv 1 )0-

S'uare transformation "ompute s9 0 : 0

/or all transformations# t!e argument must e greater

t!an ero to guarantee t!at t!e "al"ulations are

mat!emati"ally legitimate%


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Slide 6Transformation of positively s5e$ed variales

/or positively s5e$ed variales# t!e argument is anad4ustment to t!e original value ased on t!e

minimum value for t!e variale%

If t!e minimum value for a variale is ero# t!ead4ustment re'uires t!at $e add one to ea"! value#

e%g% 0 ; 1%

If t!e minimum value for a variale is a negative

numer )e%g%#


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Slide 7>0ample of positively s5e$ed variale

Suppose our dataset "ontains t!e numer of oo5s

read )oo5s- for ? su4e"ts 1# 3# ,# ?# and 9# and t!e

distriution is positively s5e$ed%

T!e minimum value for t!e variale oo5s is ,% T!e

ad4ustment for ea"! "ase is oo5s ; 1%

T!e transformations $ould e "al"ulated as follo$s Compute log6oo5s *+1,)oo5s ; 1-

Compute s'r6oo5s S.RT)oo5s ; 1-

Compute inv6oo5s 1 )oo5s ; 1-


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Slide 8Transformation of negatively s5e$ed variales

If t!e distriution of a variale is negatively s5e$ed#

t!e ad4ustment of t!e values reverses# or refle"ts#

t!e distriution so t!at it e"omes positively s5e$ed%

T!e transformations are t!en "omputed on t!e

values in t!e positively s5e$ed distriution%

Refle"tion is "omputed y sutra"ting all of t!e

values for a variale from one plus t!e asolute

value of ma0imum value for t!e variale% T!is resultsin a positively s5e$ed distriution $it! all values

larger t!an ero%


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Slide 9>0ample of negatively s5e$ed variale

Suppose our dataset "ontains t!e numer of oo5s

read )oo5s- for ? su4e"ts 1# 3# ,# ?# and 9# and t!e

distriution is negatively s5e$ed%

T!e ma0imum value for t!e variale oo5s is ?% T!e

ad4ustment for ea"! "ase is = @ oo5s%

T!e transformations $ould e "al"ulated as follo$s Compute log6oo5s *+1,)= @ oo5s-

Compute s'r6oo5s S.RT)= @ oo5s-

Compute inv6oo5s 1 )= @ oo5s-


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Slide

!T!e S'uare Transformation for *inearity

T!e s'uare transformation is "omputed ymultiplying t!e value for t!e variale y itself%

It does not matter $!et!er t!e distriution ispositively or negatively s5e$ed%

It does matter if t!e variale !as negative values#sin"e $e $ould not e ale to distinguis! t!eir

s'uares from t!e s'uare of a "omparale positivevalue )e%g% t!e s'uare of @ is e'ual to t!e s'uare of;-% If t!e variale !as negative values# $e add t!easolute value of t!e minimum value to ea"! s"oreefore s'uaring it%


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Slide

>0ample of t!e s'uare transformation

Suppose our dataset "ontains "!ange s"ores )"!g- for

? su4e"ts t!at indi"ate t!e differen"e et$een test

s"ores at t!e end of a semester and test s"ores at

mid@term @1,# ,# 1,# 9,# and 3,%

T!e minimum s"ore is @1,% T!e asolute value of t!e

minimum s"ore is 1,%

T!e transformation $ould e "al"ulated as follo$s

Compute s'uarC!g )"!g ; 1,- : )"!g ; 1,-


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Slide

2

Normal Q-Q Plot of TOTAL TIME SPENT ON TH

Observed Value

121!"#2-2-#

E$%e&tedNormal

'

2

1

-1

-2

-'

Transformations for normality

TOTAL TIME SPENT ON THE INTERNET

1(

)(

!(

*(

"(

+(

#(

'(

2(

1(

(

H,sto-ram

.re/ue0&1

+

#

'

2

1

Std( ev 3 1+('+

Mea0 3 1(*

N 3 )'(

Both the histogram and the normality plot for TotalTime Spent on the Internet(netime) indicate that thevariable is not normally distributed.


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Slide

3

Descriptives

1(*' 1(+)

*(+*

1'(!)

!(2)

+(+

2'+("++1+('+

12

12

1(2

'(+'2 (2+

1+("1# (#)+

Mea0

Lo4er 5ou0d

6%%er 5ou0d

)+7 8o0f,de0&e

I0terval for Mea0

+7 Tr,mmed Mea0

Med,a0

Var,a0&eStd( ev,at,o0

M,0,mum

Ma$,mum

Ra0e

I0ter/uart,le Ra0e

S9e40ess

:urtos,s

TOTAL TIME SPENT

ON THE INTERNET

Stat,st,& Std( Error

Determine $!et!er refle"tion is re'uired

Skewness, in the table of Descriptive Statistics,indicates whether or not reflection (reversing thevalues) is required in the transformation.

f Skewness is positive, as it is in this problem,reflection is not required. f Skewness is negative,reflection is required.


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Slide

4

Descriptives

1(*' 1(+)

*(+*

1'(!)

!(2)

+(+

2'+("++1+('+

12

12

1(2

'(+'2 (2+

1+("1# (#)+

Mea0

Lo4er 5ou0d

6%%er 5ou0d

)+7 8o0f,de0&e

I0terval for Mea0

+7 Tr,mmed Mea0

Med,a0

Var,a0&e

Std( ev,at,o0

M,0,mum

Ma$,mum

Ra0e

I0ter/uart,le Ra0e

S9e40ess

:urtos,s

TOTAL TIME SPENT

ON THE INTERNET

Stat,st,& Std( Error

Compute t!e ad4ustment to t!e argument

n this problem, the minimum value is !, so " will beadded to each value in the formula, i.e. the argumentto the S#SS functions and formula for the inverse willbe$

netime + 1.


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Slide

5Computing t!e logarit!mi" transformation

%o compute the transformation,select the Compute& commandfrom the Transformmenu.


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Slide

6

Spe"ifying t!e transform variale name andfun"tion

First, in the Target Variablete't bo', type aname for the log transformation variable, e.g.lgnetime.

Second, scroll down the list of functions tofind *"!, which calculates logarithmicvalues use a base of "!. (%he logarithmicvalues are the power to which "! is raisedto produce the original number.)

Third, clickon the up

arrow buttonto move thehighlightedfunction tothe +umeric'pressionte't bo'.


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Slide

7Adding t!e variale name to t!e fun"tion

First, scroll down the list ofvariables to locate thevariable we want totransform. -lick on its nameso that it is highlighted.

Second, click on the right arrowbutton. S#SS will replace thehighlighted te't in the function

() with the name of the variable.


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Slide

8Adding t!e "onstant to t!e fun"tion

/ollowing the rules stated for determining the constantthat needs to be included in the function either toprevent mathematical errors, or to do reflection, weinclude the constant in the function argument. n thiscase, we add " to the netime variable.

-lick on the 01button to completethe computerequest.


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Slide

9T!e transformed variale

%he transformed variable which we

requested S#SS compute is shown in thedata editor in a column to the right of theother variables in the dataset.


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Slide

2!Computing t!e s'uare root transformation



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Slide

2

Spe"ifying t!e transform variale name andfun"tion

First, in the Target Variablete't bo', type aname for the square root transformationvariable, e.g. sqnetime.

Second, scroll down the list of functions tofind S23%, which calculates the square rootof a variable.

Third, clickon the uparrow button

to move thehighlightedfunction tothe +umeric'pressionte't bo'.


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Slide

22Adding t!e variale name to t!e fun"tion

Second, click on the right arrowbutton. S#SS will replace the

highlighted te't in the function() with the name of the variable.

First, scroll down the list ofvariables to locate thevariable we want totransform. -lick on its nameso that it is highlighted.


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Slide

23Adding t!e "onstant to t!e fun"tion

/ollowing the rules stated for determining the constantthat needs to be included in the function either toprevent mathematical errors, or to do reflection, weinclude the constant in the function argument. n thiscase, we add " to the netime variable.

-lick on the 01button to completethe computerequest.


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Slide


%he transformed variable which werequested S#SS compute is shown in thedata editor in a column to the right of theother variables in the dataset.


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Slide

25Computing t!e inverse transformation



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Slide

26

Spe"ifying t!e transform variale name andformula

First, in the TargetVariablete't bo', type aname for the inversetransformation variable,e.g. innetime.

Second, there is not a function forcomputing the inverse, so we typethe formula directly into theNumeric Expressionte't bo'.

Third, click on theOKbutton tocomplete thecompute request.


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Slide


%he transformed variable which we

requested S#SS compute is shown in thedata editor in a column to the right of theother variables in the dataset.


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Slide

28

Descriptives

1(*' 1(+)

*(+*

1'(!)

!(2)

+(+

2'+("++

1+('+

12

12

1(2

'(+'2 (2+

1+("1# (#)+

Mea0

Lo4er 5ou0d

6%%er 5ou0d

)+7 8o0f,de0&e

I0terval for Mea0

+7 Tr,mmed Mea0

Med,a0

Var,a0&e

Std( ev,at,o0

M,0,mum

Ma$,mum

Ra0e

I0ter/uart,le Ra0e

S9e40ess

:urtos,s

TOTAL TIME SPENTON THE INTERNET

Stat ,st,& Std( Error

Ad4ustment to t!e argument for t!e s'uaretransformation

n this problem, the minimum value is !, no ad4ustmentis needed for computing the square. f the minimumwas a number less than 5ero, we would add theabsolute value of the minimum (dropping the sign) asan ad4ustment to the variable.

t is mathematically correct to square a value of 5ero, so thead4ustment to the argument for the square transformation isdifferent. 6hat we need to avoid are negative numbers,since the square of a negative number produces the samevalue as the square of a positive number.


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Slide

29Computing t!e s'uare transformation

%o compute the transformation,

select the Compute& commandfrom the Transformmenu.


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Slide

3!

Spe"ifying t!e transform variale name andformula

First, in the TargetVariablete't bo', type aname for the inversetransformation variable,e.g. s7netime.

Second, there is not a function forcomputing the square, so we typethe formula directly into theNumeric Expressionte't bo'.

Third, click on theOKbutton tocomplete thecompute request.


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Slide


%he transformed variable which werequested S#SS compute is shown in thedata editor in a column to the right of theother variables in the dataset.

Computing bovee_bia6_inppt01Transformations

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