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Osborne, Jason (2002). Notes on the use of data transformations. Practical Assessment, Research & Evaluation, 8(6). Retrieved December 20, 2014 from http://PAREonline.net/getvn.asp?v=8&n=6 . This paper has been viewed 147,147 times since 5/30/2002.
Jason W. Osborne, Ph.D
Data transformations are the
application of a
mathematical modification to the values of a variable.
There are a great variety of possible data transformations, from adding
constants to multiplying, squaring or raising to a power, converting to
logarithmic scales, inverting and reflecting, taking the square root of the
values, and even applying trigonometric transformations such as sine wave
transformations. The goal of this
paper is to begin a discussion of some of the issues involved in data
transformation as an aid to researchers who do not have extensive
mathematical
backgrounds, or who have not had extensive exposure to this issue before,
particularly focusing on the use of data transformation for normalization
of
variables.
Many statistical procedures assume that the
variables are normally
distributed. A significant violation
of the assumption of normality can seriously increase the chances of the
researcher committing either a Type I or II error (depending on the nature
of
the analysis and the non-normality). However,
Micceri (1989) points out that true normality is exceedingly rare in
education and psychology. Thus, one
reason (although not the only reason) researchers utilize data
transformations
is improving the normality of variables. Additionally,
authors such as Zimmerman (e.g., 1995, 1998) have pointed out that
non-parametric tests (where no explicit assumption of normality is made)
can
suffer as much, or more, than parametric tests when normality assumptions
are
violated, confirming the importance of normality in all statistical analyses,
not just parametric analyses.
There are multiple options for dealing with
non-normal data. First, the
researcher must make certain that the non-normality is due to a valid
reason
(real observed data points). Invalid
reasons for non-normality include things such as mistakes in data entry,
and
missing data values not declared missing. Researchers
using NCES databases such as the National Education Longitudinal Survey of
1988
will often find extreme values that are intended to be missing.
In Figure 1 we see that the Composite Achievement Test scores variable
(BY2XCOMP) ranges from about 30 to about 75, but also has a group of
missing
values assigned a value of 99. If
the researcher fails to remove these the skew for this variable is 1.46,
but
with the missing values appropriately removed, skew drops to 0.35, and thus
no
further action is needed. These
are simple to remedy through correction of the value or declaration of
missing
values. However, not all non-normality is due to data
entry error or
non-declared missing values. Two
other reasons for non-normality are the presence of outliers (scores that
are
extreme relative to the rest of the sample) and the nature of the variable
itself. There is great debate in
the literature about whether outliers should be removed or not. I am
sympathetic to Judd and McClelland's (1989) argument
that outlier removal is desirable, honest, and important. However,
not all researchers feel that way (c.f. Orr, Sackett,
and DuBois, 1991). Should a
researcher remove outliers and find substantial non-normality, or choose
not to
remove outliers, data transformation is a viable option for improving
normality
of a variable. It is beyond the
scope of this paper to fully discuss all options for data
transformation.
This paper will focus on three of the most common data transformations
utilized for improving normality discussed in texts and the
literature:
square root, logarithmic, and inverse transformations.
Readers looking for more information on data transformations might refer
to Hartwig and Dearing (1979) or Micceri (1989).
There are several ways to tell whether a
variable is substantially
non-normal. While researchers tend
to report favoring "eyeballing the data," or visual inspection
(Orr,
Sackett, and DuBois, 1991), researchers and reviewers often are more
comfortable
with a more objective assessment of normality, which can range from simple
examination of skew and kurtosis to examination of P-P plots (available
through
most statistical software packages) and inferential tests of normality,
such as
the Kolmorogov-Smirnov test (and adaptations of this test—researchers
wanting
more information on the K-S test and other similar tests should consult the
manual for their software as well as Goodman (1954), Lilliefors (1967),
Rosenthal (1968), and Wilcox (1997), probably in that order).
These can be useful to a researcher needing to know whether a
variable’s distribution is significantly different from a normal (or other)
distribution.
While
many researchers in the social sciences are
well-trained in statistical methods, not many of us have had significant
mathematical training, or if we have, it has often been long
forgotten.
This section is intended to give a brief refresher on what really happens
when one applies a data transformation.
^{0}, 100
is 10^{2}, 16 is 10^{1.2}, and so on.
Thus, log_{10}(100)=2 and log_{10}(16)=1.2.
However, base 10 is not the only option for log transformations.
Another common option is the Natural Logarithm, where the
constant e (2.7182818)
is the base. In
this case the natural log 100 is 4.605. As
the logarithm of any negative number or number less than 1 is undefined, if
a
variable contains values less than 1.0 a constant must be added to move the
minimum value of the distribution, preferably to 1.00. There
are good reasons to consider a range of bases
(Cleveland (1984) argues that base 10, 2, and
x)
is to compute 1/x. What this does is
essentially make very small numbers very large, and very large numbers very
small. This transformation has the
effect of reversing the order of your scores.
Thus, one must be careful to reflect, or reverse the distribution prior
to applying an inverse transformation. To
reflect, one multiplies a variable by -1, and then adds a constant to the
distribution to bring the minimum value back above 1.0.
Then, once the inverse transformation is complete, the ordering of the
values will be identical to the original data.
In
general, these three transformations have been presented
in the relative order of power (from weakest to most powerful).
However, it is my preference to use the minimum amount of transformation
necessary to improve normality.
Data transformations are valuable tools, with many
benefits.
However, they should be used appropriately, in an informed manner.
Too many statistical texts gloss over this issue, leaving researchers
ill-prepared to utilize these tools appropriately.
All of the transformations examined here reduce non-normality by reducing
the relative spacing of scores on the right side of the distribution more
than
the scores on the left side. However, the very act of altering the relative
distances between data
points, which is how these transformations improve normality, raises issues
in
the interpretation of the data. If
done correctly, all data points remain in the same relative order as prior
to
transformation. This allows
researchers to continue to interpret results in terms of increasing
scores.
However, this might be undesirable if the original variables were meant
to be substantively interpretable (e.g., annual income, years of age,
grade,
GPA), as the variables become more complex to interpret due to the
curvilinear
nature of the transformations. Researchers
must therefore be careful when interpreting results based on transformed
data. This issue is illustrated in Figure 2 and Table 1.
While the original variable has equal spacing
between values in Figure 2 (the X axis represents the original values), the other three
lines depict the curvilinear nature of the transformations. The quality of
the transformed variable is different from the original variable.
If a variable with those qualities were subjected to a square root
transformation, where the variable's old values were {0, 1, 2, 3, 4} the
new
values are now {0, 1, 1.41, 1.73, 2}—the intervals are no longer equal
between
successive values. The examples presented in Table 1 elaborate on this
point.
It quickly becomes evident that these transformations change the relative
distance between adjacent values that were previously equidistant (assuming
interval or ratio measurement). In
the non-transformed variable, the distance between values would be an equal
1.0
distance between each increment (1, 2, 3, etc.).
However, the action of the transformations dramatically alters this equal
spacing. For example, where the
original distance between 1 and 2 had been 1.0, now it is 0.41, 0.30, or
0.50,
depending on the transformation. Further,
while the original distance between 19 and 20 had been 1.0 in the original
data,
it is now 0.11, 0.02, or 0.00, depending on the transformation.
Thus, while the order of the variable has been retained, order is all
that has been maintained. The equal
spacing of the original variable has been eliminated.
If a variable had been measured on interval or ratio scales, it has now
been reduced to ordinal (rank) data. While
this might not be an issue in some cases, there are some statistical
procedures
that assume interval or ratio measurement scales.
For researchers with a strong mathematical or
statistical background,
the points made in this section are self-evident. However, over the
years many of my students and colleagues
have helped me to realize that to many researchers this point is not
self-evident; further, it is not explicitly discussed in many statistical
texts.
First, note that adding a constant to a
variable changes only the mean,
not the standard deviation or variance, skew, or kurtosis.
However, the size of the constant and the place on the number line that
the constant moves the distribution to can influence the effect of any
subsequent data transformations. As
alluded to above, it is my opinion that researchers seeking to utilize any
of
the above-mentioned data transformations should first move the distribution
so
its leftmost point (minimum value) is anchored at 1.0.
This is due to the differential effects of the
transformations across
the number line. All three
transformations will have the greatest effect if the distribution is
anchored at
1.0, and as the minimum value of the distribution moves away from 1.0 the
effectiveness of the transformation diminishes dramatically.
Recalling that these transformations improve
normality by compressing
one part of a distribution more than another, the data presented in Table 1
illustrates this point. For all
three transformations, the gap between 1 and 2 is much larger than between
9 and
10 (0.41, 0.30, and 0.50 vs. 0.16, 0.05, 0.01).
Across this range, the transformations are having an effect by
compressing the higher numbers much more than the lower numbers.
This does not hold once one moves off of 1.0, however.
If one had a distribution achored at 10 and ranging to 20, the gap
between 10 and 11 (0.15, 0.04, 0.01) is not that much different than the
gaps
between 19 and 20 (0.11, 0.02, 0.00). In
a more extreme example, the difference between 100 and 101 is almost the
same as
between 108 and 109. In order to demonstrate the effects of minimum
values on the efficacy of
transformations, data were drawn from the National Education Longitudinal
Survey
of 1988. The variable used
represented the number of undesirable things (offered drugs, had something
stolen, threatened with violence, etc.) that had happened to a student,
which
was created by the author for another project.
This variable ranged from 0 to 6, and was highly skewed, with 40.4%
reporting none of the events occurring, 34.9% reporting only one event, and
less
than 10% reporting more than two of the events occurring.
The initial skew was 1.58, a substantial deviation from normality, making
this variable a good candidate for transformation. The relative effects of transformations on the skew of this variable are
presented in Table 2.
As the results indicate, all three types of
transformations worked very
well on the original distribution, anchored at a minimum of 1.
However, the efficacy of the transformation quickly diminished as
constants were added to the distribution. Even
a move to a minimum of 2 dramatically diminished the effectiveness of the
transformation. Once the minimum reached 10, the skew was over 1.0
for all
three transformations, and at a minimum of 100 the skewness was approaching
the
original, non-transformed skew in all three cases. These results
highlight the importance of the minimum value
of a distribution should a researcher intend to employ data transformations
on
that variable. These results should also be considered when a
variable has a range of,
say 200-800, as with SAT or GRE scores where non-normality might be an
issue.
In cases where variables do not naturally have 0 as their minimum, it
might be useful to subtract a constant to move the distribution to a 0 or 1
minimum.
Unfortunately, many statistical texts
provide minimal instruction
on the utilization of simple data transformations for the purpose of
improving
the normality of variables, and coverage of the use of other
transformations or
for uses other than improving normality is almost non-existent.
While seasoned statisticians or mathematicians might intuitively
understand what is discussed in this paper, many social scientists might
not be
aware of some of these issues. The first recommendation from this paper is
that researchers always
examine and understand their data The second recommendation is to know the
requirements of the data
analysis technique to be used. As
Zimmerman (e.g., 1995, 1998) and others have pointed out, even
non-parametric
analyses, which are generally thought to be “assumption-free” can benefit
from examination of the data. The third recommendation is to utilize data
transformations with
care—and never unless there is a clear reason.
Data transformations can alter the fundamental nature of the data, such
as changing the measurement scale from interval or ratio to ordinal, and
creating curvilinear relationships, complicating interpretation.
As discussed above, there are many valid reasons for utilizing data
transformations, including improvement of normality, variance
stabilization,
conversion of scales to interval measurement (for more on this, see the
introductory chapters of Bond and Fox (2001), particularly pages
17-19).
The fourth recommendation is that, if
transformations are to be
utilized, researchers should ensure that they anchor the variable at a
place
where the transformation will have the optimal effect (in the case of these
three, I argue that anchor point should be 1.0). Beyond that, there
are many other issues that researchers need to familiarize themselves
with.
In particular, there are several peculiar types of variables that benefit
from attention. For example,
proportion and percentage variables (e.g., percent of students in a school
passing end-of-grade tests) and count variables of the type I presented
above
(number of events happening) tend to violate several assumptions of
analyses and
produce highly-skewed distributions. While
beyond the scope of this paper, these types of variables are becoming
increasingly common in education and the social sciences, and need to be
dealt with appropriately. The reader interested in these issues
should refer to sources such as Bartlett (1947) or Zubin (1935), or other, more
modern sources that deal with these issues, such as Hopkins (2002, available at
http://www.sportsci.org/resource/stats/index.html).
In brief, when using count variables researchers should use the square root of
the counts in the analyses, which takes care of
count data issues in most cases. Proportions
require an arcsine-root
transformation. In order to apply
this transformation, values must be between 0 and 1.
A square root of the values is taken, and the inverse sine (arcsine) of
that number is the resulting value. However,
in order to use this variable in an analysis, each observation must be
weighted by
the number in the denominator of the proportion
Baker,
G. A. (1934). Transformation of
non-normal frequency distributions into normal distributions.
Bond,
T. G., & Fox, C. M. (2001). Bartlett,
M. S., (1947). The use of
transformation. Cleveland,
W. S. (1984). Graphical methods for
data presentation: Full scale
breaks, dot charts, and multibased logging. Cohen,
J., & Cohen, P. (1983). Finney,
D. J. (1948). Transformation of
frequency distributions. Goodman, L. A. (1954). Kolmogorov-Smirnov
tests for psychological research. Lilliefors, H. W. (1968). On the Kolmogorov-Smirnov
test for normality with mean and variance unknown. Judd, C. M., & McClelland, G.H. (1989). Orr,
J. M., Sackett, P. R., & DuBois, C. L. Z. (1991). Outlier
detection and treatment in I/O psychology:
A survey of researcher beliefs and an empirical illustration.
Pedhazur,
E. J. (1997). Rosenthal, R. (1968). An application of the Kolmogorov-Smirnov
test for normality with estimated
Wilcox, R. R. (1997). Some practical reasons
for reconsidering the Kolmogorov-Smirnov test. Zimmerman,
D. W. (1998). Invalidation of
parametric and nonparametric statistical tests by concurrent violation of
two
assumptions. Zubin,
J. (1935). Note on a transformation function for proportions and
percentages.
The author would like to express his gratitude to his former students at the University of Oklahoma for providing him with the impetus to write this paper.
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Descriptors: Statistical Distributions; Data Analysis; Nonnormal Distributions; *Parametric Analysis; Skew Curves; Normal Curve |