data transformations

Data Transformations


For some data sets, it may be necessary to transform variables– e.g. change units (lb to kg, ˚C to ˚F, etc.)

• This is simply a change in the scale, and such transformations are called ‘Linear’.

• Linear transformations consist of (1) multiplying all the observations by a constant, (2) adding a constant to all observations, or (3) both.

Data Transformations Multiplicative transformation example

– Y = weight in kg– Y’ = weight in lb– Y’ = 2.2Y

Additive transformation example– Measurements of nitrate (mg/l) → Y

• Y = 0.3, 0.35, 0.5, 0.42, 0.38, 0.56…

– Add 1 to each number → Y’• Y’ = 1.3, 1.35, 1.5, 1.42, 1.38, 1.56…


Additive and Multiplicative example– Body temperature measurements in ˚C (Y) were

taken for 47 women; if we convert to ˚F (Y’): • Y’ = 1.8Y + 32

Multiplicative transformations affect S in the same way that they affect the mean:– e.g., if mean Y = 22, and mean Y’ = 2.2Y

– then SY’ = 2.2SY


Additive transformations, however, don’t affect S

Original observations

Deviations Transformed observations

Deviations

0.36 -0.3 1.36 -0.3

0.40 0.1 1.40 0.1

0.42 0.3 1.42 0.3

0.38 -0.1 1.38 -0.1

Mean 0.39 1.39


Additive transformations thus effectively move probability distributions to the left or the right – but the shape of the histogram is unchanged.

Multiplicative transformations shrink or stretch the probability distribution

Nonlinear Transformations

These sorts of transformations affect data in more complex ways.

Examples:

2'

1'

'

)log('

YY

YY

YY

YY

Nonlinear Transformations

These transformations do change the essential shape of frequency distributions

They are thus used to try and make distributions more symmetric – i.e., are tools to achieve normality.

Transformations to achieve normality

If the distribution is skewed to the right (the most common problem) then each of the following transformations will help produce a more symmetric distribution.

The transformations are listed in order of how much they will pull in a right-skewed distribution. Y

Y

Y

Y

Y

1

1

)ln(

)(log10

Transformations to achieve normality

Percentage or proportion data is a special case – it often appears binomially distributed– e.g., 0-100%, 0-1

Here the appropriate transformation is:

YY arcsin'

Results

Tables and figures - must have a purpose

Results: Tables

When to use:– Present numerical values– Large amounts of information

Rules– Numbered consecutively– Must be able to stand alone– Vertical arrangement– Title goes above the table– Definitions/’explanations’ go below the table

“Bad Table”

Carbon Source

Glucose Sucrose Mannitol

Growth rate (generations/h)

0.93 0.21 0.47

Activity of ODC (mol CO2/h)

12.6 6.9 1.5

Activity of SDH (mmol fumarate/h)

137.7 19.3 50.9

Table 6. Growth rate of cell cultures and activity of ornithine decarboxylase (ODC) and succinate dehydrogenase (SDH)in Pseudomonas aeruginosa in response to various carbonsources

“Good Table”

Enzyme activity

Carbon Source

Growth rate (generations/h)

ODC (mol/CO2/h)

SDH (mmol fumarate/h)

Glucose 0.93 12.6 137.7

Sucrose 0.21 6.9 19.3

Mannitol 0.47 1.5 50.9

Table 7. Growth rate of cell cultures and activity of ornithine decarboxylase(ODC) and succinate dehydrogenase (SDH) in Pseudomonas aeruginosa inresponse to various carbon sources

Table 4. Response of male fighting fish (Betta splendens) totheir image in a mirrora

aPrior to the experiment, fish had been visually isolated from one another for 2 wk. Observation period for each fish was 30 s.

Results: Figures

Use to illustrate important points – summarize your data

Number graphs consecutively– separately from tables

Must be able to stand alone Titles go below figure or on separate “Figure

Legends” page Know when to use specific types of graphs

– Bar graph vs histogram– Scatter plot vs line graph

Bar graph (refer to page 57)

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C. rap. E. ang. H. aur.

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Problems?

Bar graph (refer to page 57)

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Cleared quadratControl quadrat

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0 to 2 4 to 6 8 to 10 12 to 14 16 to 18 20 to 22

Disance from parent plant (cm)

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Results: Graphs

Do not forget to include error bars– Is your data significant?– Are there differences

Complete figure legend

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Figure 2. Production of flowers by three species of plants in the absence of interspecific competition and under natural conditions


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Figure 2. Production of flowers by three species of plants in the absence of interspecific competition (cleared quadrats) and undernatural conditions (control quadrats). The plants were Campanularapunculoides, Epilobium angustifolium, and Hieracium aurantiacum. Plotted are means for eight randomly chosen quadrats. Each 1 x 1 m2.


Text – Data summary– Do not discuss or draw conclusions

Statistics– Incorporate statistics into the verbal text– Be careful when using the word “significant”– Refer to appropriate tables and figures

• When do you use “Figure” and when do you use “Fig.”?

As shown in Figure 1, the shoreline of Hicks Pond was generally predominated by grasses and sedges.

Observed frequencies of turtles obtaining food differed significantly from expected frequencies (x2=58.19, df=8, P<0.001; Fig. 2).

data transformations

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