Page 1

STATISTICS IN MEDICINE, VOL. 14,811-819 (1995)

THE LOG TRANSFORMATION IS SPECIAL

OLIVER N. KEENE

Department of Medical Statistics, GIaxo Research and Development Ltd., Greenford Road, Greenford,

Middlesex. UB4 OHE. U.K.

SUMMARY

The logarithmic (log) transformation is a simple yet controversial step in the analysis of positive continuous

data measured on an interval scale. Situations where a log transformation is indicated will be reviewed. This

paper contends that the log transformation should not be classed with other transformations as it has

particular advantages. Problems with using the data themselves to decide whether or not to transform will

be discussed. It is recommended that log transformed analyses should frequently be preferred to untrans-

formed analyses and that careful consideration should be given to use of a log transformation at the protocol

design stage.

1. INTRODUCTION

The use of t-tests, analysis of variance and analysis of covariance for continuous positive data on

an interval scale is widespread. One of the easiest modifications to these simple parametric

methods is the prior use of a log transformation.

Conventional wisdom dictates that the data should be analysed untransformed and the

residuals examined for outliers, deviations from Normality and other indications of departures

from the required assumptions. If this investigation indicates that the problems are severe then

transformations may be considered.’ Often it is recommended that a Box-Cox analysis be

performed.2 These procedures may lead to a log transformation, but may equally well lead to

some other transformation and depend on the actual data observed.

In clinical trials, the analysis strategy should as far as possible be specified in advance in the

protocol. Because many of the approaches to decisions on transformations are essentially

subjective, this has led to a widespread suspicion of the use of any transformation.

Trials performed by pharmaceutical companies are heavily influenced by the attitudes of

regulatory authorities. This suspicion of transformations is reflected in the FDA guideline’ for the

format and content of the statistical section of an application. This states:

‘Unnecessary data transformation should be avoided. In the event a data transforma-

tion was performed, a rationale for the choice of data transformation along with

interpretation of the estimates of treatment effects based on transformed data should be

provided.’

At the least, this provides some discouragement to a pharmaceutical company to transform their

data. It is clear that an industry statistician should not analyse the data using a number of

transformations and pick the most favourable to the company. However, a consequence of this

guideline is that the log transformation is grouped with all other types of transformation and is

given no special status.

CCC 0277-67 15/95/0808 1 1-09

0

1995 by John Wiley & Sons, Ltd.

Received March 1993

Revised April 1994

Page 2

812

0. KEENE

Table I. Gastric half-emptying time (min)

Subject Treatment A

Period

Treatment B

Period

Treatment C

Period Response Response Response

1

2

3

4

5

6

7

8

9

10

11

12

Mean

SD

1

3

3

1

2

2

3

1

2

1

3

2

84

87

85

82

83

110

215

50

92

70

97

95

96

40

2

2

1

3

1

3

2

3

3

2

1

1

62

108

85

46

70

110

86

46

50

61

40

147

76

33

3

1

2

2

3

1

1

2

1

3

2

3

58

38

96

61

46

66

42

34

80

55

78

57

59

19

1.1. Example: gastric emptying study

This was a three period, three treatment crossover study in 12 volunteers. The primary endpoint

was gastric half-emptying time (min) and the data are given in Table I. In many medical journals,

data such as these would be analysed untransformed. For the comparison of treatments A and B

a non-significant P-value (P = 0.14) would be quoted and possibly a confidence interval for the

treatment difference (95 per cent CI: - 7 to 47 min). There might be some discussion of the large

value for subject 7 on treatment 1 and even an additional analysis excluding this value.

The increase in standard deviation with the mean is suggestive of the need for a log transforma-

tion. If the data are log transformed prior to analysis, the increase of 29 percent between

treatments A and B now approaches significance (P = 0,083; 95 per cent CI: - 4 per cent to 72

percent). The large value for subject 7 is no longer a potential outlier.

The log transformed analysis is more supportive of a treatment effect than the untransformed

analysis. However, no log transformation is specified in the protocol. Given the suspicion of log

transformed analyses, it is clearly easier to convince a sceptical reviewer of the possibility of

a treatment effect if the log transformed analysis had been planned in advance.

2. WHY DO WE NEED TO TRANSFORM CONTINUOUS DATA?

There are a number of reasons why an analysis on a ratio scale may be preferred to an analysis on

the original scale.

2.1. Clinical importance relates t o a ratio scale

When the magnitude of an effect is commonly perceived in terms of percentage change between

treatments, this is usually a good indication that the clinical importance relates to a ratio scale. It

seems perverse to base the statistical analysis on absolute values when changes to small responses

are more clinically important than changes to large responses. Where baseline information is

available, a common approach is to analyse the percentage change of a variable from baseline.

Patients with small baseline values can have a greatly inflated influence on the analysis of

Page 3

THE LOG TRANSFORMATION IS SPECIAL

813

percentage change and this is generally a poor way of incorporating baseline information." A log

transformation weights observations automatically according to a ratio scale and reduces

problems associated with percentage changes from baseline.

2.2. End-point represents a ratio of variables or a reciprocal

In general, if two variables are approximately Normally distributed with similar variability, then

it is unlikely that their ratio will have an approximate Normal distribution. Kronmal' discusses

other problems associated with use of untransformed ratios as the dependent variable in

regression analyses. By applying a log transformation, the ratio of the variables is now expressed

as a difference of two variables and the assumptions required by analysis of variance or

regression analysis are usually much more realistk6 It is sometimes arbitrary which way round

a ratio is expressed and an analysis of the log of the ratio makes this irrelevant. Where a

variable is the reciprocal of another, a log transformation allows identical inferences for both

variables.

2.3. Inference depends on ratios

Often the inference to be made depends on ratios, for example, conclusions of bioequivalence

depend on ratios of treatments. An analysis based on untransformed data then requires division

of the treatment difference by an estimated treatment mean. Either this is done crudely

by ignoring the estimation error in the treatment mean or more precisely by application

of Fieller's theorem.' Derivation of a confidence interval with finite positive limits is then

only possible provided the estimates of both treatment means are significantly greater than

zero.'

A much more straightforward solution is provided by use of a log transformation, where the

treatment differences will automatically provide treatment ratios when transformed back to the

original scale.

2.4. Multiplicative models

Effects frequently act multiplicatively and variability often increases with the size of the measure-

ment.g A log transformation is explicitly recommended when the standard deviation is propor-

tional to the mean value."

Variables such as biochemical measurements typically show a skewed distribution,' which

can often be made symmetric using a log transformation. It has been argued that 'the theoretical

justification for using this transformation for most scientific observations is probably better than

that for using no transformation at all'.''

2.5. Example: pharmacokinetic studies

The need for a log transformation of AUC and Cmax in pharmacokinetic studies has been

discussed for a long time. These discussions illustrate many of the above points.

The inference to be made is for ratios between treatments. Pharmacokinetic considerations

indicate that effects act multiplicatively. In simple situations, AUC is inversely related to plasma

clearance and Cmax is inversely related to the volume of di~tribution.'~ Recent consensus

statement^'^ and regulatory guidelines' '* '' have unequivocably favoured the prior use of log

transformation. All these documents recommend that the data are not used to determine the

correct transformation.

Page 4

814

0. KEENE

Another parameter commonly determined is the half-life (tllz), which is calculated according to

the following formula:

t1/2 = ln(2)/k,

where k, is the elimination rate constant. Use of a log transformation for tllz yields identical

inferences for tllZ and k,.

3. STATISTICAL METHODS

3.1. Statistical objectives

The objectives of a clinical trial are typically to compare two or more treatments and to provide

estimates and confidence intervals for the size of the effect as well as P-values. These should be

made on a meaningful clinical scale. This is now required by EC GCP” and confidence intervals

are explicitly requested by some medical journals.’ * The usefulness of analyses should therefore

be guided by their ability to produce such estimates readily.

Analyses should take full account of the experimental design and be able to identify outlying

values and interactions where these are of interest.

3 . 2 . Families of transformations

The most well-known family of transformations is the Box-C~x:’~

z = { ( YA - 1)/1 (1 z 0)

l a y ) (1 = 0)

where I = 1 implies no transformation, A = 0 gives a log transformation, 1 = 0.5 a square root

and 1 = - 1 a reciprocal.

As a referee has pointed out, the log is the only member of the Box-Cox family of transforma-

tions for which the transform of a positive-valued variable can be truly Normal, because the

transformed variable is defined over the whole of the range from - 00 to 00.

A modified version of the log transformation may be obtained by using the transformation

z = log( y - c), where c corresponds to a lower bound for y.” Berry” has proposed that this

transform be used ‘whenever a parametric analysis is planned’. One of the motivations behind this

is that a log transformation may give undue weight to small values.

While a particular transformation may satisfy statistical criteria regarding distributional

assumptions, there is a compelling reason to favour the log transformation. Treatment means

may be directly transformed back to the original scale for all these transformations, but for

treatment differences only the simple log transformation provides a direct back transformation,

allowing treatment differences on the transformed scale to be interpreted as ratios on the original

scale. If a transformation is repeatedly and routinely used for the same type of data, then some

familiarity with the interpretation of treatment differences on the transformed scale may arise

with experience, but most of these transformations are used for a particular dataset and a different

approach will often be followed for a subsequent dataset.

3.3. Generalized linear models

While analysis of variance assumes a model where explanatory variables produce additive effects

on the response and where the error variance is constant, generalized linear models split the

Page 5

THE LOG TRANSFORMATION IS SPECIAL

815

model into systematic and random components. Transformations seek to achieve both objectives

simultaneously. An advantage of generalized linear models over simple data transformation is

that a transformation to produce additivity can be made quite independently of a transformation

to produce approximate Normality or constancy of variance.22 Extensions of this approach allow

the error component to be modelled as a function of parameters.

Their main disadvantage is the analysis of clinical trial data lies in their more complex nature

and consequent unfamiliarity to the non-statistical audience. Diagnostic tools are also less

well-developed compared with a least squares analysis of log transformed data.23

A log transformation is still special in the framework of generalized linear models in that only

the identify and log link functions allow the simple interpretation of treatment differences as

discussed above.

3.4. Non-parametric methods

Non-parametric methods (or distribution free methods) are not as susceptible as model based

methods to controversy over the justification of assumption^.^^ They are also often presented as

an alternative for situations when an untransformed parametric analysis appears unsatisfactory.2

It is clear that the Wilcoxon approach is useful for analyses of two-treatment studies, whether

crossover or parallel group. Methods for evaluating confidence intervals based on a Wilcoxon

approach are now in widespread use.”. 26

Some authors advocate use of the rank transformation, which may be useful for minimizing the

impact of outliers and in deriving P-values. Methods of deriving estimates of treatment effects

from such analyses have not been widely discussed. In general, estimation based on non-

parametric methods works well for the simple cases, but where the design is more complex, for

example involving covariates, more research is required.

In clinical studies, one of the advantages of non-parametric methods, that they are robust to

outliers, may be a disadvantage because such points may reflect a sub-population which requires

in~estigation.~’

For some common designs, for example crossovers with more than two periods, no non-

parametric method is recommended by a recent textbook.28 Assessment of interactions, such as

treatment by centre interactions, typically requires use of parametric methods.24

In the situations where a non-parametric analysis provides a good alternative to parametric

analysis the issue of use of a transformation is still important. For two period crossovers the

standard analysis29 is based on period differences, which will not in general provide the same

analysis as one based on an analysis of ratios between periods. The standard method of

calculation for confidence intervals corresponding to the Wilcoxon two sample case is based on

individual treatment difference^.^^ Again a transformation to ratios will affect the confidence

intervals.

4. LET THE DATA DECIDE?

4.1. Procedure

A common method of data analysis frequently recommended in books on statistic^,^.^^ uses

a procedure which will be called ‘Let the data decide’. This approach requires the data first to be

analysed untransformed and then an assessment of goodness-of-fit to be made. If there is evidence

of a departure from assumptions, a choice from a variety of transformations or non-parametric

methods is made.