Data Transformations

4 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 4 Multiplicative transformation example –Y = weight in kg –Y’ = weight in lb –Y’ = 2.2Y 4 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…

Data Transformations 4 Additive and Multiplicative example –Body temperature measurements in ˚ C (Y) were taken for 47 women; if we convert to ˚ F (Y’): Y’ = 1.8Y 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 S Y’ = 2.2S Y

Data Transformations 4 Additive transformations, however, don’t affect S Original observations DeviationsTransformed observations Deviations Mean

Data Transformations 4 Additive transformations thus effectively move probability distributions to the left or the right – but the shape of the histogram is unchanged. 4 Multiplicative transformations shrink or stretch the probability distribution

Nonlinear Transformations 4 These sorts of transformations affect data in more complex ways. 4 Examples:

Nonlinear Transformations 4 These transformations do change the essential shape of frequency distributions 4 They are thus used to try and make distributions more symmetric – i.e., are tools to achieve normality.

Transformations to achieve normality 4 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. 4 The transformations are listed in order of how much they will pull in a right- skewed distribution.

Transformations to achieve normality 4 Percentage or proportion data is a special case – it often appears binomially distributed –e.g., 0-100%, Here the appropriate transformation is:

Results 4 Tables and figures - must have a purpose

Results: Tables 4 When to use: –Present numerical values –Large amounts of information 4 Rules –Numbered consecutively –Must be able to stand alone –Vertical arrangement –Title goes above the table –Definitions/’explanations’ go below the table

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

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

Table 4. Response of male fighting fish (Betta splendens) to their image in a mirror a a Prior to the experiment, fish had been visually isolated from one another for 2 wk. Observation period for each fish was 30 s.

Results: Figures 4 Use to illustrate important points –summarize your data 4 Number graphs consecutively –separately from tables 4 Must be able to stand alone 4 Titles go below figure or on separate “Figure Legends” page 4 Know when to use specific types of graphs –Bar graph vs histogram –Scatter plot vs line graph

Bar graph (refer to page 57) Problems?

Bar graph (refer to page 57) Cleared quadrat Control quadrat

Results: Graphs 4 Do not forget to include error bars –Is your data significant? –Are there differences 4 Complete figure legend

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

Figure 2. Production of flowers by three species of plants in the absence of interspecific competition (cleared quadrats) and under natural conditions (control quadrats). The plants were Campanula rapunculoides, Epilobium angustifolium, and Hieracium aurantiacum. Plotted are means for eight randomly chosen quadrats. Each 1 x 1 m 2. Cleared quadrat Control quadrat

4 Text –Data summary –Do not discuss or draw conclusions 4 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 (x 2 =58.19, df=8, P<0.001; Fig. 2).