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Transformations: an introduction
Reasons for using transformations
There are many reasons for transformation. The list here is not
comprehensive.
- Convenience
- Reducing skewness
- Equal spreads
- Linear relationships
- Additive relationships
If you are looking at just one variable, 1, 2 and 3 are relevant, while
if you are looking at two or more variables, 4 and 5 are more important.
However, transformations that achieve 4 and 5 very often achieve 2 and 3.
- Convenience A transformed scale may be as natural as the original
scale and more convenient for a specific purpose (e.g. percentages
rather than original data, sines rather than degrees).
One important example is standardisation, whereby values are adjusted for
differing level and spread. In general
value - level
standardised value = -------------.
spread
Standardised values have level 0 and spread 1 and have no units: hence
standardisation is useful for comparing variables expressed in different
units. Most commonly a standard score is calculated using the mean and
standard deviation (sd) of a variable:
x - mean of x
z = -------------.
sd of x
Standardisation makes no difference to the shape of a distribution.
- Reducing skewness A transformation may be used to reduce skewness. A
distribution that is symmetric or nearly so is often easier to handle and
interpret than a skewed distribution. More specifically, a normal or
Gaussian distribution is often regarded as ideal as it is assumed by many
statistical methods.
To reduce right skewness, take roots or logarithms or reciprocals (roots
are weakest). This is the commonest problem in practice.
To reduce left skewness, take squares or cubes or higher powers.
-
Equal spreads A transformation may be used to produce approximately
equal spreads, despite marked variations in level, which again makes data
easier to handle and interpret. Each data set or subset having about the
same spread or variability is a condition called homoscedasticity: its
opposite is called heteroscedasticity. (The spelling -sked- rather than -sced- is also used.) - Linear relationships When looking at relationships between variables,
it is often far easier to think about patterns that are approximately
linear than about patterns that are highly curved. This is vitally
important when using linear regression, which amounts to fitting such
patterns to data. (In Stata, regress is the basic command for
regression.)
For example, a plot of logarithms of a series of values against time has
the property that periods with constant rates of change (growth or
decline) plot as straight lines.
- Additive relationships Relationships are often easier to analyse when
additive rather than (say) multiplicative. So
y = a + bx
in which two terms a and bx are added is easier to deal with than
y = ax^b
in which two terms a and x^b are multiplied. Additivity is a vital issue
in analysis of variance (in Stata, anova, oneway, etc.).
In practice, a transformation often works, serendipitously, to do several
of these at once, particularly to reduce skewness, to produce nearly
equal spreads and to produce a nearly linear or additive relationship.
But this is not guaranteed.
Review of most common transformations
The most useful transformations in introductory data analysis are the
reciprocal, logarithm, cube root, square root, and square. In what
follows, even when it is not emphasised, it is supposed that
transformations are used only over ranges on which they yield (finite)
real numbers as results.
Reciprocal
The reciprocal, x to 1/x, with its sibling the negative reciprocal, x to
-1/x, is a very strong transformation with a drastic effect on
distribution shape. It can not be applied to zero values. Although it
can be applied to negative values, it is not useful unless all values are
positive. The reciprocal of a ratio may often be interpreted as easily as
the ratio itself: e.g.
population density (people per unit area) becomes area per person;
persons per doctor becomes doctors per person;
rates of erosion become time to erode a unit depth.
(In practice, we might want to multiply or divide the results of taking
the reciprocal by some constant, such as 1000 or 10000, to get numbers
that are easy to manage, but that itself has no effect on skewness or
linearity.)
The reciprocal reverses order among values of the same sign: largest
becomes smallest, etc. The negative reciprocal preserves order among
values of the same sign.
Logarithm
The logarithm, x to log base 10 of x, or x to log base e of x (ln x), or
x to log base 2 of x, is a strong transformation with a major effect on
distribution shape. It is commonly used for reducing right skewness and
is often appropriate for measured variables. It can not be applied to
zero or negative values. One unit on a logarithmic scale means a
multiplication by the base of logarithms being used. Exponential growth
or decline
y = a exp(bx)
is made linear by
ln y = ln a + bx
so that the response variable y should be logged. (Here exp() means
raising to the power e, approximately 2.71828, that is the base of
natural logarithms.)
An aside on this exponential growth or decline equation: put x = 0, and
y = a exp(0) = a,
so that a is the amount or count when x = 0. If a and b > 0, then y grows
at a faster and faster rate (e.g. compound interest or unchecked
population growth), whereas if a > 0 and b < 0, y declines at a slower
and slower rate (e.g. radioactive decay).
Power functions y = ax^b are made linear by log y = log a + b log x so
that both variables y and x should be logged.
An aside on such power functions: put x = 0, and for b > 0,
y = ax^b = 0,
so the power function for positive b goes through the origin, which often
makes physical or biological or economic sense. Think: does zero for x
imply zero for y? This kind of power function is a shape that fits many
data sets rather well.Consider ratios y = p / q where p and q are both positive in practice.
Examples are
males / females;
dependants / workers;
downstream length / downvalley length.
Then y is somewhere between 0 and infinity, or in the last case, between
1 and infinity. If p = q, then y = 1. Such definitions often lead to
skewed data, because there is a clear lower limit and no clear upper
limit. The logarithm, however, namely
log y = log p / q = log p - log q,
is somewhere between -infinity and infinity and p = q means that log y =
- Hence the logarithm of such a ratio is likely to be more symmetrically
distributed.
Cube root
The cube root, x to x^(1/3). This is a fairly strong transformation with
a substantial effect on distribution shape: it is weaker than the
logarithm. It is also used for reducing right skewness, and has the
advantage that it can be applied to zero and negative values. Note that
the cube root of a volume has the units of a length. It is commonly
applied to rainfall data.
Applicability to negative values requires a special note. Consider
(2)(2)(2) = 8 and (-2)(-2)(-2) = -8. These examples show that the cube
root of a negative number has negative sign and the same absolute value
as the cube root of the equivalent positive number. A similar property is
possessed by any other root whose power is the reciprocal of an odd
positive integer (powers 1/3, 1/5, 1/7, etc.).
This property is a little delicate. For example, change the power just a
smidgen from 1/3, and we can no longer define the result as a product of
precisely three terms. However, the property is there to be exploited if
useful.
Square root
The square root, x to x^(1/2) = sqrt(x), is a transformation with a
moderate effect on distribution shape: it is weaker than the logarithm
and the cube root. It is also used for reducing right skewness, and also
has the advantage that it can be applied to zero values. Note that the
square root of an area has the units of a length. It is commonly applied
to counted data, especially if the values are mostly rather small.
Square
The square, x to x^2, has a moderate effect on distribution shape and it
could be used to reduce left skewness. In practice, the main reason for
using it is to fit a response by a quadratic function y = a + b x + c
x^2. Quadratics have a turning point, either a maximum or a minimum,
although the turning point in a function fitted to data might be far
beyond the limits of the observations. The distance of a body from an
origin is a quadratic if that body is moving under constant acceleration,
which gives a very clear physical justification for using a quadratic.
Otherwise quadratics are typically used solely because they can mimic a
relationship within the data region. Outside that region they may behave
very poorly, because they take on arbitrarily large values for extreme
values of x, and unless the intercept a is constrained to be 0, they may
behave unrealistically close to the origin.
Squaring usually makes sense only if the variable concerned is zero or
positive, given that (-x)^2 and x^2 are identical.
Which transformation?
The main criterion in choosing a transformation is: what works with the
data? As examples above indicate, it is important to consider as well two
questions.
What makes physical (biological, economic, whatever) sense, for example
in terms of limiting behaviour as values get very small or very large?
This question often leads to the use of logarithms.
Can we keep dimensions and units simple and convenient? If possible, we
prefer measurement scales that are easy to think about. The cube root of
a volume and the square root of an area both have the dimensions of
length, so far from complicating matters, such transformations may
simplify them. Reciprocals usually have simple units, as mentioned
earlier. Often, however, somewhat complicated units are a sacrifice that
has to be made.
Psychological comments - for the puzzled
The main motive for transformation is greater ease of description.
Although transformed scales may seem less natural, this is largely a
psychological objection. Greater experience with transformation tends to
reduce this feeling, simply because transformation so often works so
well. In fact, many familiar measured scales are really transformed
scales: decibels, pH and the Richter scale of earthquake magnitude are
all logarithmic.
However, transformations cause debate even among experienced data
analysts. Some use them routinely, others much less. Various views,
extreme or not so extreme, are slightly caricatured here to stimulate
reflection or discussion. For what it is worth, I consider all these
views defensible, or at least understandable.
"This seems like a kind of cheating. You don't like how the data are, so
you decide to change them."
"I see that this is a clever trick that works nicely. But how do I know
when this trick will work with some other data, or if another trick is
needed, or if no transformation is needed?"
"Transformations are needed because there is no guarantee that the world
works on the scales it happens to be measured on."
"Transformations are most appropriate when they match a scientific view
of how a variable behaves."
Often it helps to transform results back again, using the reverse or inverse transformation:
reciprocal t = 1 / x reciprocal x = 1 / t
log base 10 t = log_10 x 10 to the power x = 10^t
log base e t = log_e x = ln x e to the power x = exp(t)
log base 2 t = log_2 x 2 to the power x = 2^t
cube root t = x^(1/3) cube x = t^3square root t = x^(1/2) square x = t^2
How to do transformations in Stata
Basic first steps
-
Draw a graph of the data to see how far patterns in data match the
simplest ideal patterns. Try dotplot or scatter as appropriate. -
See what range the data cover. Transformations will have little effect
if the range is small. - Think carefully about data sets including zero or negative values.
Some transformations are not defined mathematically for some values, and
often they make little or no scientific sense. For example, I would never
transform temperatures in degrees Celsius or Fahrenheit for these reasons
(unless to Kelvin).
Standard scores (mean 0 and sd 1) in a new variable are obtained by
. egen stdpopi = std(popi)
whereas the basic transformations can all be put in new variables by
generate:
. gen recener = 1/energy
. gen logeener = ln(energy)
. gen l10ener = log10(energy)
. gen curtener = energy^(1/3)
. gen sqrtener = sqrt(energy)
. gen sqener = energy^2
. gen logitp = logit(p) if p is a proportion . gen logitp = logit(p / 100) if p is a percent . gen frootp = sqrt(p) - sqrt(1-p) if p is a proportion . gen frootp = sqrt(p) - sqrt(100-p) if p is a percent
Cube roots of negative numbers require special care. Stata uses a general
routine to calculate powers and does not look for special cases of
powers. Whenever negative values are present, a more general recipe for
cube roots is sign(x) * (abs(x)^(1/3)). Similar comments apply to fifth,
seventh, roots etc.
Note any messages about missing values carefully: unless you had missing
values in the original variable, they indicate an attempt to apply a
transformation when it is not defined. (Do you have zero or negative
values, for example?)
It is not always necessary to create a transformed variable before
working with it. In particular, many graph commands allow the options yscale(log) and xscale(log). This is very useful because the graph is
labelled using the original values, but it does not leave behind a
log-transformed variable in memory.
Other commands
Stata offers various other commands designed to help you choose a
transformation. ladder, gladder and qladder try several transformations
of a variable with the aim of showing how far they produce a more nearly
normal (Gaussian) distribution. In practice such commands can be
helpful, or they can be confusing at an introductory level: for examples,
they can suggest a transform at odds with what your scientific knowledge
would indicate. boxcox and lnskew0 are more advanced commands that should
be used only after studying textbook explanations of what they do. Box
and Cox (1964) is the key original reference.
For some statistical people any debate about transformation is largely
side-stepped by the advent of generalised linear models. In such models,
estimation is carried out on a transformed scale using a specified link
function, but results are reported on the original scale of the response.
The Stata command is glm.
Transformations for proportions and percents (more advanced)
Data that are proportions (between 0 and 1) or percents (between 0 and
100) often benefit from special transformations. The most common is the logit (or logistic) transformation, which is
logit p = log (p / (1 - p)) for proportions
OR logit p = log (p / (100 - p)) for percents
where p is a proportion or percent.
This transformation treats very small and very large values
symmetrically, pulling out the tails and pulling in the middle around 0.5
or 50%. The plot of p against logit p is thus a flattened S-shape.
Strictly, logit p cannot be determined for the extreme values of 0 and 1
(100%): if they occur in data, there needs to be some adjustment.
One justification for this logit transformation might be sketched in
terms of a diffusion process such as the spread of literacy. The push
from zero to a few percent might take a fair time; once literacy starts
spreading its increase becomes more rapid and then in turn slows; and
finally the last few percent may be very slow in converting to literacy,
as we are left with the isolated and the awkward, who are the slowest to
pick up any new thing. The resulting curve is thus a flattened S-shape
against time, which in turn is made more nearly linear by taking logits
of literacy. More formally, the same idea might be justified by imagining
that adoption (infection, whatever) is proportional to the number of
contacts between those who do and those who do not, which will rise and
then fall quadratically. More generally, there are many relationships in
which predicted values cannot logically be less than 0 or more than 1
(100%). Using logits is one way of ensuring this: otherwise models may
produce absurd predictions.
The logit (looking only at the case of proportions)
logit p = log (p / (1 - p))
can be rewritten
logit p = log p - log (1 - p)
and in this form it can be seen as a member of a set of folded
transformations
transform of p = something done to p - something done to (1 - p).
This way of writing it brings out the symmetrical way in which very high
and very low values are treated. (If p is small, 1 - p is large, and vice
versa.) The logit is occasionally called the folded log. The simplest
other such transformation is the folded root (that means square root)
folded root of p = root of p - root of (1 - p).
As with square roots and logarithms generally, the folded root has the
advantage that it can be applied without adjustment to data values of 0
and 1 (100%). The folded root is a weaker transformation than the logit.
In practice it is used far less frequently.
Two other transformations for proportions and percents met in the older
literature (and still used occasionally) are the angular and the probit.
The angular is
arcsin(root of p)
or the angle whose sine is the square root of p. In practice, it behaves
very like
p^0.41 - (1 - p)^0.41,
which in turn is close to
p^0.5 - (1 - p)^0.5,
which is another way of writing the folded root (Tukey 1960). The probit
is a transformation with a mathematical connection to the normal
(Gaussian) distribution, which is not only very similar in behaviour to
the logit, but also more awkward to work with. As a result, it is now
less seen, except in more advanced applications, where it retains several
advantages.
Transformations as a family (more advanced)
The main transformations mentioned previously, with the exception of the
logarithm, namely the reciprocal, cube root, square root and square, are
all powers. The powers concerned are
reciprocal -1
cube root 1/3
square root 1/2
square 2
Note that the sequence of explanation was not capricious, but in
numerical order of power. Therefore, these transformations are all
members of a family. In addition, contrary to what may appear at first
sight, the logarithm really belongs in the family too. Knowing this is
important to appreciating that the transformations used in practice are
not just a bag of tricks, but a series of tools of different sizes or
strengths, like a set of screwdrivers or drill bits. We could thus fill
out this sequence, the ladder of transformations as it is sometimes
known, with more powers, as for example in
reciprocal square -2
reciprocal -1
(yields one) 0
cube root 1/3
square root 1/2
identity 1
square 2
cube 3
fourth power 4
Among the additions here, the identity transformation, say x^1 = x, is
the transformation that is, in a sense, no transformation. The graph of x
against x is naturally a straight line and so the power of 1 divides
transformations whose graph is convex upwards (powers less than 1) from
transformations whose graph is concave upwards (powers greater than 1).
Powers less than 1 squeeze high values together and stretch low values
apart, and powers more than 1 do the opposite.
The transformation x^0, on the other hand, is degenerate, as it always
yields 1 as a result. However, we will now see that in a strong sense log
x (meaning, strictly, the natural logarithm or ln x) really belongs in
the family at the position of power 0.
If you know calculus, you will know that the sequence of powers
..., x^-3, x^-2, x^-1, x^0, x^1, x^2, ...
has as integrals, apart from additive constants,
..., -x^-2 / 2, -x^-1, ln x, x, x^2 / 2, x^3 / 3, ...
and the mapping can be reversed by differentiation. So integrating x^(p -
1) yields x^p / p, unless p is 0, in which case it yields ln x. Thus we
can define a family
t_p(x) = x^p if p != 0,
= ln x if p == 0.
The notion of choosing from a family when we choose a power or logarithm
is a key idea. It follows that we can usually choose a different member
of the family if the transformation turns out to be too weak, or too
strong, for our purpose and our data.
Many discussions of transformations focus on slightly different families,
for a variety of mathematical and statistical reasons. The canonical
reference here is Box and Cox (1964), although note also earlier work by
Tukey (1957). Most commonly, the definition is changed to
t_p(x) = (x^p - 1) / p if p != 0,
= ln x if p == 0.
This t(x, p) has various properties which point up family resemblances.-
ln x is the limit as p -> 0 of (x^p - 1) / p.
-
At x = 1, t_p(x) = 0, for all p.
-
The first derivative (rate of change) of t_p(x) is x^(p - 1) if p != 0
and 1 / x if p == 0. At x = 1, this is always 1. - The second derivative of t_p(x) is (p - 1) x^(p - 2) if p != 0 and -1
/ x^2 if p == 0. At x = 1, this is always (p - 1).
Another small change of definition has some similar consequences, but
also some other advantages. Consider
t_p(x) = [(x + 1)^p - 1] / p if p != 0,
= ln(x + 1) if p == 0.
This t(x, p) has various properties which also point up family
resemblances.
-
If p = 1, t_p(x) = x.
-
At x = 0, t_p(x) = 0, for all p. So all curves start at the origin.
-
The first derivative (rate of change) of t_p(x) is (x + 1)^(p - 1) if
p != 0 and 1 / (x + 1) if p == 0. At x = 0, this is always 1. So the
curves have the same slope at the origin. - The second derivative of t_p(x) is (p - 1) (x + 1)^(p - 2) if p != 0
and -1 / (x + 1)^2 if p == 0. At x = 0, this is always (p - 1).
The most useful consequence, however, is that this definition can be
extended more easily to variables that can be both positive and negative,
as will now be seen.
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