Presentation Title Goes Here…presentation subtitle.

Data Transformation

Leilani Nora

Assistant Scientist

Crop Research Informatics Laboratory

International Rice Research Institute

:: color, composition, and layout2

Assumptions in the ANOVA

ANOVA is valid only when the following assumptions

are met:

• Additive Effects

• Independence of errors

- Assured by proper randomization

• Homogeneity of variance

• Normal distribution

:: color, composition, and layout3

Assumptions in the ANOVA

Most common violation in the ANOVA is the homogeneity of

variances. If this assumption is not met, most often the

normality and additive assumptions are also violated.

Transforming the data often homogenizes the variances and

consequently solves the problem of non-additive effect and

non-normality of variances.

:: color, composition, and layout4

Data Transformation

Commonly used transformations

• Logarithmic Transformation

• Square-root Transformation

• Arc Sine Transformation

:: color, composition, and layout5

Replace with 100-25/nFor obs = 100%

Replace with 25/nFor obs = 0%

Arcsin(sqrt(y/100))Other ranges

No transformation30 – 70

Sqrt(100-Y)80 – 100

Sqrt(Y)0 – 20

% whiteheads

% survival

Percentage data

Tiller no.

100 grain weight

Percent filled grains

Panicle number

No transformationYield and yield

components

ExampleTransformationVariable (Y)

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Log(Y+1)Some obs. < 10

Log(Y)All obs. > 10

No. of insects per plot

No. of egg masses per

plant

Whole numbers that

cover a wide range

Sqrt (Y+.5)Some obs. < 10

Sqrt (Y)All obs. > 10

No. of infested plants

per plot

No. of insects caught

in traps

No. of weeds per plot

Small whole numbers

ExampleTransformationVariable (Y)

:: color, composition, and layout7 :: color, composition, and layout8

PLOT OF LS RESIDUALS AGAINST FITTED VALUES

..............................................................

: :

: :

: * :

: * :

8. -: * :

: :

: * * :

: 2 * * * * :

: 2 * * :

0. -: * * * ** :

: * * * * * :

: ** * * ** :

: * * :

: * :

-8. -: * :

: :

: :

: :

: * :

-16. -: :

:............................................................:

: : : : : :

-5. 0. 5. 10. 15. 20.

:: color, composition, and layout9

A satisfactory residual plot should give this overall impression

:: color, composition, and layout10

A pattern or trend in the

distribution of points in the

residual plot may be

encountered indicating

transformation is needed.

:: color, composition, and layout11

PLOT OF LS RESIDUALS AGAINST FITTED VALUES

..............................................................

: :

: :

: * :

: * :

8. -: * :

: :

: * * :

: 2 * * * * :

: 2 * * :

0. -: * * * ** :

: * * * * * :

: ** * * ** :

: * * :

: * :

-8. -: * :

: :

: :

: :

: * :

-16. -: :

:............................................................:

: : : : : :

-5. 0. 5. 10. 15. 20.

before transformation

:: color, composition, and layout12

PLOT OF LS RESIDUALS AGAINST FITTED VALUES

..............................................................

: * :

: * :

: * :

: ** :

6. -: * * *:

: * * * * :

: * * * :

: * * * * * * :

: * * * :

0. -: * * * :

: * * :

: * * 2 :

: * * :

: * * * :

-6. -: * * :

: * * * * * :

: * :

: * :

: * :

-12. -: :

:............................................................:

: : : : : :

47.5 52.5 57.5 62.5 67.5 72.5

after transformation