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Psych 5500/6500

Data Transformations

Fall, 2008

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Data Transformations

We are now going to examine an option we have if we find that our data appear to violate the assumption that the populations are normally distributed or the assumption that the populations have the same variance. This option is to transform our data to better fit the assumptions.

First we will look at some options on how to do this, and then we will turn to general issues and concerns about transforming data.

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You Might Not Need to Worry About the Assumptions

Remember that violation of the assumption of normality grows less serious as the N of our samples increase; and that violation of the assumption of homogeneity of variances is not important when the N’s of our groups are roughly equal. Thus you may not need to turn to transformations if the N’s of your groups are largish and approximately the same size.

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Transformations to Take Care of Anticipated Problems

Certain types of measurements routinely produce samples that violate one or both of the assumptions of normality and equal variances. The solutions are fairly well established and it is always better to anticipate a priori what transformation might be appropriate. Post hoc decisions to transform the data face two criticisms; 1) are you changing you data just to get the results you want?, and 2) is your population actually ok and you are changing your data (and thus the population it represents) to fix a problem that appeared in your sample just due to chance (i.e. the populations were actually ok)?

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Reaction TimesReaction times are often positively skewed.

Transformations that are recommended for reaction times (or any positively skewed data) are:

1Y

1Y b)

thenscores someon 0 equalmight Y ifor Y

1 Y a) 2

Ylog Y 1)

i transformi,

i transformi,

i10 transformi,

Try both 1 and 2 and see which works best.

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CountsIf the variable involves counting something

(e.g. occurrences of some behavior) then there could be a problem, particularly if low counts (around zero) occur. The floor effect of not being able to have a score below zero will effect the normality of the data, and if one of the groups has more of a floor effect than the other then the groups will have different variances. A recommended transformation in this case is:

i transformi, YY

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ProportionsProportions as measures suffer from several

problems, two of them are:

1. Proportions around .5 have greater variance than proportions close to 0 or 1 (due to floor and ceiling effects). Thus the variance of two groups will differ if one has proportions closer to .5 than the other.

2. Many people consider a difference in the mean proportion of two groups of .02 and .08 (a difference of .06 but also a quadrupling of the proportion) to be greater than a difference of .48 and .54 (also a difference of .06 but a much smaller change in terms of ratios).

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Proportions (cont.)

Stretching out both tails of the distribution can help with both of those problems. Transforms that can accomplish this are:

i1

transformi, YsinY

i

i transformi, Y1

YlogY

arcsine transform:

logit transform

Try them both and see which works best.

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Power Transformations

The general formula of raising Y to some power (‘pow’), as in, Ytransform=Y pow can be used to reshape the distribution in a variety of ways. We have already seen one use of this, the square root of Y, which is the same as Y0.5 (in SPSS that would be Y**.5).

Here we are venturing into the territory of post hoc transformations, where you try out various transformations until the distribution is the shape that you want.

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Power Transformation Strategies

Try various values for power, you might try the following to see which works best:

pow = 3, 2, 0.5, ‘0’, -0.5, -1, -2.

Obviously, Y1 isn’t on the list as Y1 = Y

The ‘0’ is in quote marks because raising Y to the power of 0 will result in changing all of your scores to ‘1’. In place of using pow=0, substitute Ytransform= log10 Y.

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Power Transformation Strategies

Instead of pure trial and error, you can use the general strategy of trying a value of pow>1 if you want to bring in a long negative tail, and pow<1 if you want to bring in a long positive tail. The further from ‘1’ in either direction the more the tail will be pulled in.

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Power Transformation Strategies

Some statistical programs will run through possible values for the power of Y and then report which one best turns the data into a normal curve (SPSS apparently doesn’t offer that).

It is important to note that whether you figure out a good value for p or the computer does, such a transformation would be purely post hoc.

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Rank Transformation

If you cannot transform you data in such a way that it becomes more normally distributed then an interesting option of last resort is to transform your data into rank scores. Each score is transformed into a rank score which indicates where it falls in a list of all the scores in the study for that variable. If observations are tied, then each observation receives the mean of the ranks they would have received if they weren’t tied.

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• For example, the (cardinal) data – Group 1: 3.4, 7.2, 5.1, 6.9– Group 2: 7.2, 5.1, 7.2, 8.4

• Would first be put into one list and ordered: – Y= 3.4, 5.1, 5.1, 6.9, 7.2, 7.2, 7.2, 8.4

• Scores of 5.1 come in 2nd and 3rd on the list, so they each get a rank of (2+3)/2=2.5. Scores of 7.2 come in 5th, 6th, and 7th, so they each get a rank of (5+6+7)/3=6.

• Ytransform=1, 2.5, 2.5, 4, 6, 6, 6, 8

• Transformed (to ranked) data:– Group 1: 1, 6, 2.5, 4– Group 2: 6, 2.5, 6, 8

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Upside of rank transformations:1. Control the effect of outliers (because

distance to extreme score is reduced).2. While they don’t create normal data they

generally reduce problems of thick tails.3. While they don’t ensure homogeneity of

variance they generally prevent very large differences in variance.

Downside of rank scores: you are losing information when you move from cardinal to rank scores.

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Parametric and Nonparametric Tests

Statistical procedures that are based upon certain assumptions about the population being true are called parametric tests. The t tests--in fact every test we will look at this semester--are parametric tests.

Statistical procedures that are not based upon certain assumptions about the population being true are called nonparametric tests. These tests are useful in that they can be applied to situations where assumptions certainly are not met, but nonparametric tests tend to have low power.

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Parametric Tests of Rank Data

Most nonparametric tests require rank data. To apply a parametric test to rank data is an interesting half-way step between parametric and non-parametric tests. By ranking the data you are not meeting the assumptions of the parametric tests but you are violating the assumptions in a way that tends to be ok (e.g. creating thin tails rather than fat tails).

While it is still better not to have to transform cardinal data to rank data (because you lose so much information and probably lose power) if you do it may be better to use a parametric test than a non-parametric on the data.

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AdvantagesThe advantages of using a parametric test on rank

data vs. using a nonparametric test are:1. You don’t have to learn another whole class of

statistical tests (nonparametric tests).2. In some cases the parameter test of rank data

may be just as good or even better than the nonparametric test (see Judd & McClelland, 1989, and Ruxton, 2006).

Judd, C. M. & McClelland, G. H. (1989). Data analysis: A model-comparison approach. New York: Harcourt College Publishers.

Ruxton, G. D. (2006). The unequal variance t-test is an underused alternative to Student's t-test and the Mann–Whitney U test. Behavioral Ecology, 17, 688 - 690.

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The Mann–Whitney U test is the nonparametric equivalent to a t test. The following table compares type 1 error rate of the U test to that of Welch’s t’ in a Monte Carlo study (Zimmerman and Zumbo, 1993, as adapted and cited in Ruxton, 2006) when both tests were applied to rank data.

N1 N2 σ1/ σ2 U t’

6 18 1 .052 .049

6 18 2 .085 .051

6 18 4 .104 .054

18 6 1 .049 .052

18 6 2 .030 .056

18 6 4 .023 .064

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Issues and Concerns RegardingTransformations

1) You are moving the data further away from reflecting reality in an attempt to better make if fit your tool.

2) If your original measure is more relevant to your theory than the transform, then can you generalize the analysis of the transformed data to the theory?

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Issues and Concerns (cont.)

3) Transforming your data to fit the tool may or may not be justified, but transforming your data to make the results fit your theory is definitely not justified. Comment: this is why transforms that are selected a priori due to known characteristics of the measure have an advantage over post hoc transforms.

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Issues and Concerns (cont.)

4) Post hoc transforms might end up fixing problems that are a fluke, that only appear in this particular sample. This shows one of the advantages of replicating studies. If the data in the first study suggests a particular transform would be useful, you might want to use it, but then withhold final judgment on its appropriateness until a second study confirms its utility.