Transforming the data

Modified from:

Gotelli and Allison 2004. Chapter 8; Sokal and Rohlf 2000 Chapter 13

What is a transformation?

It is a mathematical function that is applied to all the observations of a given variable

• Y represents the original variable, Y* is the transformed variable,

and f is a mathematical function that is applied to the data

YfY *

Most are monotonic:

• Monotonic functions do not change the rank order of the data, but they do change their relative spacing, and therefore affect the variance and shape of the probability distribution

There are two legitimate reasons to transform your data before analysis

• The patterns in the transformed data may be easier to understand and communicate than patterns in the raw data.

• They may be necessary so that the analysis is valid

They are often useful for converting curves into straight lines:

The logarithmic function is very useful when two variables are related to each other by multiplicative or exponential functions

Logarithmic (X): )log(10 XY

)log(10 XY

y = Ln(x)

0

5

10

15

20

1 100 10000 1000000

log(x)

Y

0

5

10

15

20

0 50000 100000 150000 200000

x

Y

Example:Asi’s growth (50 % each year)Year weight

1 10.0

2 15.0

3 22.5

4 33.8

5 50.6

6 75.9

7 113.9

8 170.9

9 256.3

10 384.4

11 576.7

12 865.0

Exponential: XeY 10

0.0

200.0

400.0

600.0

800.0

1000.0

0 5 10 15

year

wei

gh

t (g

)

y = 6.6667e0.4055x

1.0

100.0

10000.0

0 5 10 15

yearw

eig

ht

(g)

XY 10 )ln()ln(

Example: Species richness in the Galapagos Islands

Power: 10

XY

)log()log()log( 10 XY

0

100

200

300

400

0 2000 4000 6000 8000

Area

Ric

hn

ess

Nspecies

Power(Nspecies)

1

10

100

1000

0.1 10 1000 100000

Area

Ric

hn

ess

Nspecies

Power(Nspecies)

Statistics and transformation

Data to be analyzed using analysis of variance must meet to assumptions:

• The data must be homoscedastic: variances of treatment groups need to be approximately equal

• The residuals, or deviations from the mean must be normal random variables

Lets look an example

• A single variate of the simplest type of ANOVA (completely randomized, single classification) decomposes as follows:

• In this model the components are additive with the error term εij distributed normally

ijiijY

However…

• We might encounter a situation in which the components are multiplicative in effect, where

• If we fitted a standard ANOVA model, the observed deviations from the group means would lack normality and homoscedasticity

ijiijY

The logarithmic transformation

• We can correct this situation by transforming our model into logarithms

)log(* YY

Wherever the mean is positively correlated with the variance the logarithmic transformation is likely to remedy the situation and make the variance independent of the mean

We would obtain

• Which is additive and homoscedastic

)log()log()log()log( ijiijY

The square root transformation

• It is used most frequently with count data. Such distributions are likely to be Poisson distributed rather than normally distributed.

In the Poisson distribution the variance is the same as the mean.

Transforming the variates to square roots generally makes the variances independents of the means for these type of data.

When counts include zero values, it is desirable to code

all variates by adding 0.5.

The box-cox transformation

• Often one do not have a-priori reason for selecting a specific transformation.

• Box and Cox (1964) developed a procedure for estimating the best transformation to normality within the family of power transformation

/)1(* YY

)log(* YY

)0( for)0( for

The box-cox transformation

• The value of lambda which maximizes the log-likelihood function:

yields the best transformation to normality within the family of transformations

s2T is the variance of the transformed values (based on v degrees of freedom). The second term involves the sum of the ln of untransformed values

)ln()1(ln2

2 Yn

vs

vL T

box-cox in R (for a vector of data Y)

>library(MASS)

>lamb <- seq(0,2.5,0.5)

>boxcox(Y_~1,lamb,plotit=T)

>library(car)

>transform_Y<-box.cox(Y,lamb)

-2 -1 0 1 2

-24

.0-2

3.5

-23

.0-2

2.5

log

-Lik

elih

oo

d

95%

What do you conclude from this plot?

Read more in Sokal and Rohlf 2000 page 417

The arcsine transformation

• Also known as the angular transformation

• It is especially appropriate to percentages

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

lineal

arcsine

The arcsine transformation

YY arcsin*

It is appropriate only for data expressed as proportions

Proportion original data

Tra

nsfo

rmed

da

ta

Since the transformations discussed are NON-LINEAR, confidence limits computed in

the transformed scale and changed back to the original

scale would be

asymmetrical