Open Journal of Statistics, 2012, 2, 389-400
http://dx.doi.org/10.4236/ojs.2012.24047 Published Online October 2012 (http://www.SciRP.org/journal/ojs)
Linear Maximum Likelihood Regression Analysis for
Untransformed Log-Normally Distributed Data
Sara M. Gustavsson, Sandra Johannesson, Gerd Sallsten, Eva M. Andersson
Department of Occupational and Environmental Medicine, Sahlgrenska University Hospital,
Academy at University of Gothenburg, Gothenburg, Sweden
Received June 25, 2012; revised July 27, 2012; accepted August 10, 2012
ABSTRACT
Medical research data are often skewed and heteroscedastic. It has therefore become practice to log-transform data in
regression analysis, in order to stabilize the variance. Regression analysis on log-transformed data estimates the relative
effect, whereas it is often the absolute effect of a predictor that is of interest. We propose a maximum likelihood
(ML)-based approach to estimate a linear regression model on log-normal, heteroscedastic data. The new method was
evaluated with a large simulation study. Log-normal observations were generated according to the simulation models
and parameters were estimated using the new ML method, ordinary least-squares regression (LS) and weighed
least-squares regression (WLS). All three methods produced unbiased estimates of parameters and expected response,
and ML and WLS yielded smaller standard errors than LS. The approximate normality of the Wald statistic, used for
tests of the ML estimates, in most situations produced correct type I error risk. Only ML and WLS produced correct
confidence intervals for the estimated expected value. ML had the highest power for tests regarding β1.
Keywords: Heteroscedasticity; Maximum Likelihood Estimation; Linear Regression Model; Log-Normal Distribution;
Weighed Least-Squares Regression
1. Introduction
Measurements in occupational and environmental re-
search, e.g. exposure and biomarkers, often have a skewed
distribution with a median smaller than the mean and
only positive values. It is also common with hetero-sce-
dasticity where the variance increases with the expected
value. Such data can often be described by a log-normal
or quasi-log-normal distribution [1].
Associations (for example between exposure and
health effects/biomarkers or between personal exposure
and background variables) are often analyzed using re-
gression models. Ordinary least squares (LS) regression
analysis is a commonly used regression method, and it is
based on the assumption of a constant variance and a
normal distribution for the stochastic term. One way of
handling non-normal data is with nonparametric median
regression (or Least-Absolute-Value regression), see e.g.
[2] where no assumptions are made about the distribution
of the response variable. The parameter estimates are
then found by minimizing the sum of absolute value of
the residuals (whereas LS minimizes the sum of squared
residuals). However, as a nonparametric method it re-
quires larger samples and it may have multiple solutions
[3].
In situations where the response variable has a skewed
distribution and an increasing variance, it has become the
practice to log-transform the response variable. Regres-
sion analysis on log-transformed data have for instance
been used to establish reference models [4], to find suit-
able biomarkers [5], to determine suitable surrogates for
exposure [6,7], and to estimate the cost function in health
economics [8]. A model in which the response variable is
log-transformed, ln(Y), will estimate the relative effect of
each predictor, whereas in many cases it is the absolute
effect that is desired [9]. It must also be considered that
e.g. a t-test for comparing the expected values of two
groups based on the mean of ln(Y) is not equal to a test
based on the mean of the original log-normal data Y,
since the expected value of Y is a function of both μ and
σ, whereas the expected value of ln(Y) is a function of
only μ. If the two σ-parameters are not equal, a t-test
based on ln(Y) may not give the correct type I-error re-
garding the difference between E[Y1] and E[Y2], [10,11].
In many cases a linear relationship between the re-
sponse and the predictor (e.g. between personal exposure
and background variables) is a reasonable assumption;
for example it is realistic that the personal exposure in-
creases linearly with time spent in a certain environment
(e.g. time spent in traffic). This linearity will be lost in a
C
opyright © 2012 SciRes. OJS

S. M. GUSTAVSSON ET AL. 390
log-transformation. On the other hand, if the log-normal
distribution is ignored in order to preserve the linearity,
tests based on the assumption of a constant variance may
give misleading results, [12].
There is a need for methods that handle log-normally
distributed data in linear regression models, based on
moderate sample sizes. In order to estimate the linear
association (and the absolute effect), but still take into
account the log-normal distribution with a non-constant
variance, we propose a maximum likelihood (ML) based
method for regression analysis. In this paper we have
evaluated this new method using large scale simulations,
which allowed us to analyze the bias, variance and dis-
tribution of the regression coefficients resulting from the
new method, as well as comparing it to LS- and weighted-
least-squares (WLS) regression analysis. A data set on
personal exposure to 1,3-butadiene in five Swedish cities
was used to illustrate the three methods.
2. Data
We considered the situation where the response variable
Y is assumed to follow a log-normal distribution (i.e. ln(Y)
is normally distributed) and where the expected value is
assumed to be a linear function of the predictors,


01
EYX 1Ypp .X

   

In regression analy-
sis, some X-variables can be included because of known
(or suspected) association with Y, in order to decrease the
variance or lower the risk of confounder effects [5].
Since Y follows a log-normal distribution, ln(Y) can be
expressed using the following model


22 ,
pZi
X e  

2
eN
01
1
ln ln
ip
YX  
where iZ

~0,

. The k
model above. Linear regression analysis on the log-
transformed data would yield an estimate of the relative
effect of Xk, rather than the absolute effect. This is illus-
trated in Figure 1.
A log-transformation is not suitable in situations when
the aim is to estimate the absolute effect of X on Y (rather
than the relative one). In a licentiate thesis [13] it was
suggested that the linear regression should be estimated
from untransformed log-normal data using maximum
likelihood methods. In this article, the properties of these
maximum likelihood estimates will be evaluated.
2.1. Simulation Study
The properties of a new method for statistical analysis
can be derived theoretically and/or by simulations and
examples. In a simulation study, a model for the variable
of interest (here personal exposure) is used to generate
samples of observations, 1n and then the pa-
rameters under investigation (here regression parameters)
are estimated from each sample. This is repeated to ob-
tain a distinctive distribution for the estimates. Our
simulation study allowed us to assess the bias and stan-
dard errors of the parameters estimated resulting from the
new maximum likelihood-based regression analysis, and
compare these results to those of LS and WLS.
,,yy
We used simulation models in which the response,
personal exposure to Particulate Matter smaller than 2.5
μm (PM2.5), was assumed to be a linear function of back-
ground variables (residential outdoor level of PM2.5,
smoking and time spent in home). Two different simula-
tion models were used to generate data. Model A had
only one predictor, the personal exposure to PM2.5-parti-
cles (μg/m3), Y, was assumed to be a linear function of
the residential outdoor concentration of PM2.5 (μg/m3),
ConcOut. Model B had three predictors and no intercept,
ˆ

is an estimate of the
absolute effect of Xk on Y. The log-transformation results
in a distortion of the linearity, as can be seen in the
Figure 1. A linear regression where Y|X follows a log-normal distribution. The absolute effect is 0.9 (left), the log-
transformation stabilizes the variance but distort the linear relation (middle) and the estimation based on log(Y) result in an
xponential function with a relative effect of 29% (right). e

S. M. GUSTAVSSON ET AL. 391
Y was a linear function of the number of cigarettes per
day, Smoke, number of hours spent in their own home,
Home, and residential outdoor concentration of PM2.5
(μg/m3), ConcOut. Since ConcOut > 0, its regression
coefficient can be interpreted as a stochastic intercept.
Datasets were simulated by generating normally distrib-
uted observations from Model A:
 
2
0.354 2,
ii
e

22
, 0.354  

lnln 4.8030.574Y ConcOut
where iZ . Samples from Model B
were simulated according to
~0eN

2
ln
ln 2.0920.761
0.4502,
i
i
Y
Smoke ConcOut
e


0.218 Home 

22
, 0.450. where iZ The parameters in the
simulation models,

01
~0eN 
,,
Z

 and 123

,,,
Z


ˆ
E
,
were estimated from real measurement data (Johannes-
son et al. [14]).
The number of repetitions needed in the simulation
study was estimated. In order to obtain a 95% confidence
interval for



that is smaller than 2·0.0005, 4 mil-
lion samples were needed. For the predictors, discrete
values were used: ConcOut = {2, 8, 14}, Smoke = {0, 7,
14} and Home = {8, 16, 24}. Sample sizes n = 108 and n
= 216 were used in the simulations and the data sets were
balanced with regard to the predictors. For Model A a
second set is also created with a data structure similar to
the observed one in the original dataset [14], which was
slightly unbalanced.
2.2. Application to Exposure Data
The properties of the three regression methods (LS, WLS
and ML) were illustrated using a set of data on personal
exposure of 1,3-butadiene from five Swedish cities. 1,3-
butadiene is an alkene and has been listed as a known
carcinogen by the International Agency for Research on
Cancer (IARC). Traffic and exposure to tobacco smoke
are considered to be two sources for personal exposure to
1,3-butadiene [15,16]. Wood burning has also been
showed to increase personal exposure [17]. The dataset
was collected in a study of exposure to carcinogens in
urban air in five Swedish cities; Gothenburg, Umea,
Malmo, Stockholm and Lindesberg, see [18], and con-
sisted of 268 measurements of personal 1,3-butadiene
exposures. Background data were collected by a ques-
tionnaire.
3. Methods
In our investigation, the outcome variable Y was as-
sumed to be log-normal with an expected value that was
a linear function of the predictors;
101,1 1,
,,pipp YX Xxx

 




From the simulated data, the parameters of the regres-
sion model were estimated using ordinary least-squares
and weighted-least-squares estimation as well as maxi-
mum-likelihood estimation, as described below.
3.1. Least-Squares Estimates
As mentioned before, the inference in ordinary least-
squares regression method, LS, is made under the as-
sumption that


2
~iid ,YX N 
22 ,iYiYi
,Yi Y
where

 
2
ˆMSE  
ˆ
. The standard deviation is assumed constant
for all Y and . The covariance matrix for the
Y
vector
L
S

is estimated as LS


1
ˆ
cov MSE



X
X 
ˆ
,
where X is a n × (p + 1) matrix. The standard errors,
SE(
L
S

), are estimated from the diagonal elements of


ˆ
cov
L
S
 .
3.2. Weighted-Least-Squares Estimation
For data that cannot be considered homoscedastic, there
are several estimation procedures, for instance the White
heteroscedasticity consistent estimator (see e.g. [19], p.
199). Since Y is assumed to follow a log-normal distri-
bution, the nature of the heteroscedasticity is known;


2
22
e1
Z

,i Yi Y


 
2
. The variance is a function of the
expected value. Thus weighted-least-squares regression
analysis, WLS, is appropriate, in which each observation
is weighted using i
iY
W



22
,
ˆ
iiLS Y
WY
. The weights can be esti-
mated from the LS regression; i




WLS
ˆ

. The co-
variance matrix for the vector is estimated as



1
WLS
ˆ
cov MSEW

X'WX 
,, yy
01
,,, ,
.
3.3. Maximum Likelihood Estimation in
Regression
The maximum likelihood estimator (MLE) of some pa-
rameter θ is the value at which the likelihood function
L(θ) attains its maximum as a function of θ, with the
sample 1n held fixed. The log likelihood is often
more convenient to work with and since the logarithm
transformation is monotone, the function log L(θ|y) can
be maximized instead. For a continuous variable the like-
lihood function is the same as the probability density
function, pdf. The MLE of θ can be found by differenti-
ating log L(θ|y) with respect to θ. When θ is a scalar it is
enough that the likelihood function is differentiable to
get a direct estimate of θ. In a regression analysis, the
aim is to estimate the parameter vector
p
Z
 






θ
12,
,,
n
yy y
.
Let be a log-normal sample where

S. M. GUSTAVSSON ET AL. 392
ˆi


01ii
xEY

 

2
~ ,
i
izZ
YN
.
Then

ln



where
2
01
σ2
ziZ
 ln
ix
 
and the log likelihood function is
 
 

2
2ln .
Z i
i
y 

2
01
2
lnlnln 2π
2
1ln ln
2
Z
ii
i
Z
n
Ln
yx



 

The derivatives were previously calculated by Yurgens
[13] and are given below with some corrections:

2
,
2
Z
ii
jj
j










2
ln 1ln ln
ik
i
kjijj
Z
x
Lyx
x
 









2
22
ln
11ln
2
ny ,
2
Z
i










l
mk
ik im
ij j
ij
Zjijj
L
xx x
x







 

 


2
,
4
3
Z
ln1 ln ln
Z
ii
ij
ZZ
Ln yx
jj
n






 












2
3
ln 2ln ln
ik
il
Zk jijj Z
x L
x
 

  

 
,
ii ll
yx









2
.
4
iijj
n
yx









,,, ,
2
22 4
ln3 ln ln
ij
ZZ Z
Ln
 
 




The maximum likelihood estimates of
01
p
Z





01
ˆˆ ˆ
,,,
 θ can be found by iterations, for
example the Newton-Raphson algorithm, see [20]. The
covariance matrix for the vector ˆ
Z





ˆi
 θ is
estimated by the inverse of the observed Fisher informa-
tion matrix (see e.g, [21]), where the elements are the
negative second derivative of the log-likelihood. One of
the known properties of MLE is, under some regularity
conditions, its asymptotic normality; when the sample
size increases the MLE of θ tend to a normal distribution
with expected value θ and a covariance matrix equal to
the inverse of the Fisher information matrix (see e.g,
[21]).
3.4. Descriptive Statistics and Inference
In the simulation study, samples of log-normal observa-
tions were generated and three different methods were
used to estimate the regression coefficients. The results
from the three methods were compared using expected
value (mean), standard error, bias, skewness, the 95%
central range and correlation of the regression estimates.
The standard error of

was denoted SE





ˆi
and


ˆi
se
the sample specific standard error for

,

, was
the estimate of i
ˆ
SE




. The bias of an estimator,
ii
ˆ
E




 , was used as a measure of the systematic
error. The s kewness of an estimator was estimated as


 
3
ˆˆˆ
EESE
 









. For log-normal data, the




22
e2e1

 skewness is

, while a symmetric
distribution has γ = 0. It has been suggested that a sample
with skewness less than 0.5 should be considered as ap-
proximately symmetric while skewness above 1 may be
considered highly skewed. The 95% central range, CR95,
was defined as the difference between the 2.5th and
97.5th percentiles. The sample-specific standard error of
ˆYX
μ
is

 
2
2
1
ˆˆˆ
ˆcov ,,
p
iiij ij
YX
iij
se μxse xx


 



ˆˆ
,
ij
cov where x0 = 1 and


ˆˆ
cov ,
ij
is the sample specific
estimation of





*0,
. 

The inference properties of the three methods were
evaluated by comparing the results from tests of the null
hypothesis H0:β = β*, where T

  in which βT is
the value specified in the simulation model. LS and WLS
estimates was tested with the t-test,






ˆˆˆ
tse  

 , which follows a Student’s t-dis-

tribution. The ML estimates were tested using a Wald-
type of test statistic, which utilized the large sample
normality of the ML-estimator and the observed Fisher
information. Under H0, the Wald statistic






ˆˆˆ
Wse

 


tribution and
asymptotically follows a N(0, 1)
dis


2
ˆ
W

asymptotically follows a chi-
, see e.
n used
on
4. Results
e presented in four sections; the distribution
square distributiong. [21], Section 9.4. Other pos-
sible large-sample test procedures for ML estimates (not
used in this study) are the score statistic and the full like-
lihood ratio statistic, see e.g. [21]. We chose the Wald
statistic because of its computational advantages.
The properties of the test statistics above, whe
log-normal data, were evaluated by the risk of type I
error and the power. The true probability of a type I error
for a specified nominal α-value (denoted α*) was esti-
mated as the proportion of test statistic values beyond the
respective critical value. The power results were based
on simulations in which the tested parameter (e.g. β0)
was varied according to H1 whereas the other parameter
(e.g. β1) was held constant.
The results ar

S. M. GUSTAVSSON ET AL.
Copyright Res. OJS
393
of ˆi

and ˆ
Z

, predictions


ˆYX
 , inference and an
appation toxposure data foutadiene. lic er 1,3-b
4.1. The Distribution of the Estimates
all three
ˆ
SE 1. Of the three methods, ML gave the smallest i




,
11% smaller than that of LS. For LS,


0
ˆ
se

overesti-
mated 0
ˆ
SE




 by over 35% for the balanced data and
about 18% for the unbalanced. For all three methods the
standard error increased when unbalanced data were used.
LS had the largest CR95, while ML has the smallest (the
differences were around 12% between LS and ML and
For data generated according to Model A,
methods produced unbiased estimates of the β-coeffi-
cients (the absolute effect of the X-variables), see Table
Table 1. The expected value (E[–]), errors (SE[–], E[se(–)]) and skewness (γ[–]) for the estimates of β, and the ex-
n = 108 n = 216
standard
pected value and standard errors for the estimate of σZa.
LS WLS ML LS WLS ML
Balanced data
β0 = 4.803 ˆ
E




 4.803 4.803 4.806 4.803 4.803 4.804
ˆ
SE




 0.486 0.441 0.430 0.343 0.312 0.304


ˆ
Ese



ˆ
0.656 0.438 0.424 0.465 0.311 0.302




95
CR
0.063 0.151 0.143 0.045 0.107 0.100
ˆ





β1 = 0.ˆ
E
1.904 1.730 1.685 1.346 1.223 1.190
574





ˆ
SE
0.574 0.574 0.574 0.574 0.574 0.574




 0.072 0.066 0.064 0.051 0.047 0.045


ˆ
Ese



ˆ
0.070 0.066 0.064 0.050 0.047 0.045




95
CR
0.125 0.054 0.053 0.089 0.039 0.039
ˆ





σZ = 0.ˆ
0.282 0.260 0.252 0.199 0.183 0.178
354
Z
E  

ˆ
- 0.351 0.350 - 0.353 0.352
Z
SE  

E
- 0.028 0.024 - 0.020 0.017
Unbalanced
β0 = 4.803 ˆ




 4.803 4.803 4.807 4.803 4.803 4.805
ˆ
SE





0.564 0.488 0.475 0.399 0.345 0.335


ˆ
Ese



0.665 0.484 0.469 0.472 0.344 0.334
ˆ




−0.030 0.156 0.149 −0.020 0.110 0.105
95
CR
ˆ





β1 = 0.574 ˆ
E
2.215 1.911 1.860 1.565 1.352 1.315





ˆ
SE
0.574 0.574 0.573 0.574 0.574 0.574





0.089 0.078 0.075 0.063 0.055
0.053


ˆ
Ese



0.082 0.077 0.075 0.058 0.055 0.053
ˆ




0.184 0.024 0.026 0.130 0.017 0.018
95
CR
ˆ





σZˆ
E
0.347 0.305 0.296 0.246 0.215 0.209
= 0.354




 - 0.351 0.350 - 0.353 0.352
ˆ
SE



 - 0.028 0.024 - 0.020 0.017

aData were ge Model A, 4 million iterations. For reference, the skewness of the normal distribution is γ = 0, whereas the log-normal data y1···yn
nerated from
from Model A has skewness γ = 1.15.
© 2012 Sci

S. M. GUSTAVSSON ET AL. 394
3% between WLS and ML). There was a strong associa-
tion between the LS- and ML-estimates; the correlation
between the estimates was 0.88 and 0.90 for 0
ˆ

and
1
ˆ

respectively. The association was weaker fohigh
lues. The WLS- and ML-estimates were even more
similar, with a correlation of 0.97 for both the β0- and
β1-estimates, and with no weakening of association for
higher values. For σZ, both WLS and ML showed only
small biases that decreased with increasing sample size,
Table 1. ML gave the smallest standard error and
seemed robust to unbalanced data. Since there is no gen-
erally accepted method for estimating σZ with LS, no
such results were presented.
Data were also generated fr
r
va
om versions of Model A in
which one of the β-parameters was set to 0. All three
methods produced unbiased estimates of β and the stan-
dard errors were much smaller than the results shown in
Table 1 (both for ˆ SE



and

ˆ
Ese



); for the
zero-parameter the sten 45 and
76% smaller than the corresponding value in Table 1,
and for the other parameter the standard error was be-
tween 26% and 52% smaller. For a situation where the
intercept is zero (β0 = 0), ML gave the smallest ˆ SE
andard error was betwe





for both parameters, 50% smaller than that of r
both ML and WLS,

LS. Fo

0
ˆ
se

was a good estimator for
0
ˆ
SE



, but for LS


0
ˆ
se

overestimated 0
ˆ
SE





by aboutuation or X has 80%. For a sitwhere the predict
no effect on the response Y (β1 = 0), all three methods
produced approximately the same 1
ˆ
SE



and


1
ˆ
se

was a good estimator for SE1
ˆ


 σZ-es
(and their standard error) werndent of the values
of β0 and β1 and had the same values as in Table 1.
For data with a large variation (large σZ) we det


. The
epe
timates
e ind
ected
th
rding to Model B. ML
pr
e occasional estimation problem for ML. The ML
method is based on iterations whereas LS and WLS have
analytical expressions for the parameter estimates, and
ML was more sensitive to large outliers. Situations in
which ML produces unreasonable results can sometimes
be avoided by excluding the extreme observations, but all
three methods will then tend to underestimate both the
intercept and the standard errors.
Data were also generated acco
ovided the smallest variation (ˆi
SE



was between
18% and 44% smaller than that of LSere was a very ). Th
small underestimation of σZ, but the bias decreases with
increasing sample size, as expected according to the
properties of MLE. As before, ML had the smaller


ˆ
Z
SE

, Table 2. For both ML and WLS,


ˆ
se

as a
imator for ˆ SE good est



. For LS,

ˆ
se

1

underes-
timated 1
ˆ
SE



by% while s 7

3
ˆ
e % - 8

ˆi
SE
did over-
estimate


 . The i
by 12% - 13%, Table 2ˆ SE





depends on the value of βi, the range and values of the
X-variables and the value of σZ. A separate simulation
ucted in 1, σZ =
ke and
was cond which β0 = 0, β1 = β2 = β3 =
0.450 and where all predictors (ConcOu t, Smo
Home) hade values between 2 and 14 and the results
showed that for this situation all the standard errors were
the same;




LS 1LS2LS3
ˆˆˆ
0.196SE SESE  ,




WLS 1WLS2WLS3
ˆˆˆ
0.169SE SESE  
and




ML 1ML2ML 3
ˆˆˆ0.161SESESE
 
Predictions
The results in Tables 1 and 2 illustrated that ˆ
.
4.2.
the

-
values were approximately symmetrical and hence the
o same would apply tˆYX
μ
for fixed x. All three meth-
d estimates of the β-parameters and ods provided unbiase
thus the point estimates of μY|X will be unbiased. For ML
and WLS,


ˆYX
se μ did adequately estimate ˆYX SE μ



,
while LS produced too large standard errors for
x
x

and too small for
x
x. This will for most values of x
result in ernfidence intervals for Ld
produce slighwer confidence intervals th
roneous coS. ML di
tly narroan WLS,
ressio
see Figure 2.
For a simple regn model (with intercept and one
X-variable) it can be shown that ˆYX SE μ


has a
minimum at 0,1 01
ˆˆ
xSESE




 


x
, in Figure 2 at
≈ 5

 0,10 1
ˆˆ
Corr ,






. As mentioned above, this
minimum was well estimated with both d WLS, ML an
icated a minimum at x = 8while LS ind. The incorrect


ˆ se μ fo
YX the underestimated corre-
lation (0,1< 0,1) and overestimating the standard error
(
r LS is a result of
rρ


00
ˆˆ
Ese SE








 ). For a regression model with no
and several X-variables, intercept ˆYX SE μ


is always
minimized for


12 0 p
xx x

 . For a multiple
reh an intercept, the minimum gression model wit


ˆ
μYX SE can be foun

d by solving the equation system

11
ˆ Var
0,1, ,.
pp
YX
i
ip
x






,,xXx
4.3. Inference
Hypothesis testing regarding separate regression pa-
rameters using the t-test (LS and WLS) or the Wald test
ted. Data were generated according to (ML) was evalua
versions of Model A where one parameter was set to zero
(βi = 0, I = {0, 1}). The null hypothesis H0:βi = 0 was
tested against both one- and two-sided alternatives in a
situation where H0 was true, Table 3.

S. M. GUSTAVSSON ET AL. 395
Table 2. The expected value (E[–]), standard errors (SE[–[se(–)]) , skewness (γ[–]) and CR95 for the estimates of βa. ], E
n = 108 n = 216
LS WLS ML LS ML WLS
β1 = 2.092 ˆ
E



2.092 2.092 2.087 2.092 2.092 2.090
ˆ
SE



0.243 0.208
0.200 0.172 0.147 0.141


ˆ
Ese



ˆ
0.224 0.205 0.196 0.159 0.146 0.140




95 ˆ
CR
0.213 0.125
0.121 0.150 0.090 0.085





β2 = 0.761 ˆ
E
0.953 0.816 0.783 0.674 0.577 0.554



ˆ
SE
0.761 0.761 0.760 0.761 0.761 0.760



0.206 0.146 0.139 0.146 0.103 0.099


ˆ
Ese



ˆ
0.205 0.144 0.137 0.146 0.103 0.098




95 ˆ
CR
0.064 0.133 0.124 0.043 0.094 0.087





β3 = 0.218 ˆ
E
0.812 0.574 0.547 0.573 0.406 0.386



ˆ
SE
0.218 0.218 0.220 0.218 0.218 0.219



0.116 0.068 0.065 0.082 0.048 0.046


ˆ
Ese



ˆ
0.131 0.066 0.063 0.093 0.047 0.045




− −
95 ˆ
CR
0.102 0.282 0.252 0.074 0.198 0.177





σZ = 0.450 ˆ
0.458 0.265 0.253 0.323 0.187 0.179
Z
E 


ˆ
- 0.444 0.443 - 0.447 0.446
Z
SE 


- 0.038 0.031 - 0.028 0.022
aData were generated from Model B, 4 million iterations. For reference, thess of theal distribution is γ, whereas t-normal dy···yn
from Model B has skewness γ = 1.53.
skewne norm = 0he logata 1
Figure 2. The ˆ 

ESμ
X
x0 = and


ˆ


se μ E
X
x0 = for
s, aped from Model A
ed ons with sample size
n = 108.
0
LS-estimates produced an α* which was much smaller
than α (α* < 0.2α, regardless of H1). This was a result of
on of ˆ SE
. the three method
Results were basplied to data generat 4 million iteration
For a model with no intercept (β = 0), tests of the
the overestimati0




 by 0

ˆ
se

. For tes
be
ts of
WLS- and ML-estimates, α* was approximately equal to
α (within 10% of the true value). A slight skewness could
observed; α* ≤ α for H1: β0 > 0 and α* ≥ α for H1:β0 <
0, while α* for H1:β0 ≠ 0 was slightly higher or the same
as α. The source of this skewness was a positive correla-
ˆ
tion between 0

and


0 se ˆ; smalues of 0
ˆ
all v


gave more extreme (nFor a model egative) test-scores.
eom Mosis
where X has no effect on Y (β1 = 0), all three tests pro-
duced α*-values close to α (within 20% of the true value).
For ML, α* was slightly higher than for LS and WLS.
The risk of type I error was approximately the same even
when the sample size was doubled (n = 216).
Data were genrated frdel A and the hypothe
H0:βi = βT was tested (βT was the parameter value speci-
fied in Model A). The results for WLS and ML (data not
shown) were that α* was within 30% and 16% of α, re-

S. M. GUSTAVSSON ET AL. 396
spectively for nominal size 0.05 and 0.10. For nominal
size 0.01 the relative deviation could be up to 70%. As
before, tests of the LS-estimates of β prod
0
uch too small (α* ≤ 0.1α). Both WLS- and ML-pro-
duced a slight skewness regarding tests of both β0 and β1,
similar to that in Table 3.
Data were also generated according to Model B (data
uced an α*
m
ˆi
SE
not shown) and the hypothesis H0:βi = βT was tested. For
all three methods, α* was closest to α for nominal size
0.10. The relative deviation was largest for nominal size
0.01; 0.7α ≤ α* ≤ 2.2α. Again, a skewness (as above)
could be seen in the tests of all three parameters, caused
by the negative skewness of the test statistics. For tests of
the LS-estimate of β1, α* was consistently too large
(1.04α ≤ α* ≤ 2.20α), whereas α* was consistently too
small in tests of β3 (0.30α ≤ α* ≤ 0.76α). These erroneous
risks were, again, the result of under- and overestimation,
respectively, of





ided and one-sided tests regarding β0 and β1a.
α = 0.050 α = 0.100
, see Table 2. Tests regarding
the LS-estimates of β2 followed the same pattern as the
tests of all the WLS- and ML-estimates. The risk of type
I error was approximately the same even when the sam-
ple size was doubled (n = 216).
The power of the tests, under the alternative hypothe-
sis H1:β1 > 0, was estimated for both α = 0.05 and α* =
0.05, see Table 4. When α = 0.05, ML had the highest
power while LS had the lowest. For LS the type I error
Table 3. Risk for type I error (α*) for two-s
α = 0.010
Alternative hypothesis H1: βi ≠ 0 βi < 0 βi > 0 βi ≠ 0 βi < 0 βi > 0 βi ≠ 0 βi < 0 βi > 0
βi n Test
β0 108 LS t-test 0.000 0.000 0.000 0.001 0.001 0.003 0.004 0.007 0.016
108 WLS t-test 0.010 0.012 0.008 0.051 0.055 0.047 0.102 0.106 0.096
108 M0.106 0.100
216 S t-test 0.000 0.000 0.001 0.003 0.003 0.007 0.014
L Wald 0.012 0.013 0.010 0.054 0.056 0.050 0.106
L0.000 0.001
216 WLS t-test
ML W
0.010
0.011
0.011
0.012
0.009
0.009
0.050
0.052 0.053 0.049
0.053 0.047 0.100
0.102 0.103
0.104 0.097
0.099 216 ald
β1 108 LS t-test 0.010 0.010 0.010 0.050 0.050 0.050 0.100 0.101 0.100
108 WLS t-test 0.011 0.011 0.010 0.052 0.051 0.051 0.102 0.102 0.101
108 ML Wald 0.012 0.012 0.011 0.055 0.053 0.053 0.106 0.104 0.103
216 LS t-test 0.010 0.010 0.010 0.050 0.050 0.050 0.100 0.100 0.100
216 WLS t-test 0.010 0.010 0.010 0.051 0.050 0.051 0.101 0.100 0.101
216 ML Wald 0.011 0.011 0.011 0.052 0.051 0.052 0.103 0.101 0.102
aDrerated odel e βi =stimre billioons nc
oweestslteryis H1:he risk of type I ert a.
NominTrue:
ata we genefrom MA wher 0 and eations aased on 4 mn iteratiwith balaed data.
Table 4. The pr for t with anative hpothesβ > 0 w
1re theror is se to 0.05
al:
Cr.659 1.659 70itical value: 11.659 1.645 1.6 1.673
β--tesald) LS LS-teald
50 0.050
0.04
.06 .533 .562 .528 .551
0
1 LS (ttest) WLS (tt) ML (W (t-test)W (tst) ML (W)
0.00 0.050 0.052 0.055 0.050 0.0
0.02 0.150 0.151 0.160 0.150 0.149 0.154
0.321 0.324 0.344 0.321 0.320 0.334
00.530 00 0.530 00
0.08 0.721 0.723 0.753 0.721 0.720 0.744
0.10.857 0.859 0.882 0.857 0.856 0.876
0.12 0.936 0.938 0.952 0.936 0.936 0.949
0.14 0.975 0.976 0.983 0.975 0.975 0.982
0.16 0.991 0.992 0.995 0.991 0.991 0.994
0.18 0.997 0.997 0.998 0.997 0.997 0.998
0.20 0.999 0.999 1.000 0.999 0.999 1.000
0.22 1.000 1.000 1.000 1.000 1.000 1.000
aere generated from with difflues but cons 4.804. Estimioased on 1 mrations with baata and sam-
ple size n = 108.
Data w Model Aerent β1 vatant β0 =atns are billion itelanced d

S. M. GUSTAVSSON ET AL. 397
ras correct butLS and ML the critical v
were adjusted to g = 0.05. adjustment
still had the highestr while WLS now had a l
pr compared to LS. The relative difference bet
thominal and trufor β1 = 0
decreases with β1.
similar to the geometric mean expo-
sure (0.345 against 0.386). Thus the data can be consid-
mod wood-fire or been in a residence
ods shwignificanrence betwe
the LS LS-regr models oothenburg
differe indesbe the ML-reon model
also shwrence betwalmo and
Lindesber= 0.041). the ML-ren model
showed a significantly loposure fosmokers
In
ation is often used to stabilize the variance,
but this distorts the linear relationship and does not give
absolute effect of a predictor.
isk w for Walues
ive α*After ML
poweower
oweween
e ne power was largest and
4.4. Application: 1,3-Butadiene Exposure in Five
Swedish Cities
The data on 1,3-butadiene were found to be highly
skewed (γ = 2.764) whereas the log-transformed values
were approximately symmetrical (γ = 0.279). The me-
dian exposure was
ered log-normal. Five predictors were included in the
el: “Have you lit a
heated with wood burning?” (Wood burning, yes/no),
“Are you a smoker?” (Smoker, yes/no), “Have you been
in an indoor environment where people where smoking?”
(ETS, yes/no), “Proportion of time spent in traffic?”
(Traffic), and “City of residence”: Umeå (Ume), Stock-
holm (Sthlm), Malmö (Malmo), Gothenburg (Gbg), and
reference category Lindersberg. These predictors had
been shown to be significant in previous studies on other
datasets [15-17].
Neither Wood burning nor Traffic were significant in
any of the regression models (for ML Traffic was bor-
derline significant, p = 0.059), Figure 3. All three meth-
oed a st diffeen the cities; in
- and Wessionnly G
d from Lrg, butgressi
oed a significant diffeeen M
g (p Only
wer ex
gressio
r non-
with no ETS (p = 0.013). There were clear differences in
range of the confidence intervals; with one exception ML
had the narrowest intervals and LS the widest.
A second analysis with only the non-smokers (n = 225)
was performed and then Traffic became significant in the
ML-regression model (p = 0.039). Otherwise the same
predictors were significant in all three analyses, as for the
full dataset.
5. Discussion
this study a new maximum likelihood-based method
for estimating a linear regression model for log-normal
heteroscedastic data was evaluated and compared with
two least squares methods. For log-normal data, the
log-transform
an estimate of the
Our simulation study demonstrated that the new
maximum likelihood method (ML) provides unbiased
estimates of the regression parameters βi, and so do the
least squares method (LS) and the weighted least squares
method (WLS).
Figure 3. The LS-, WLS- and ML-estimations of the predictor effects, βi, and their 95% confidence intervals (CI) for
regression analysis with 1,3-butadiene as the response.

S. M. GUSTAVSSON ET AL. 398
One reason for proposing a new regression method is
the need for a method to estimate a linear regression in
the presence of heteroscedasticity. The effect of ignoring
the increasing variance is demonstrated in the use of LS,
where the sror, ˆi SE tandard er




, was up to 78% larger
than thate examples that were investigated.
Since the heteroscedasticity of the data was ignored
when using LS, the confidence interval was too wide.
The results of our example with 1,3-butadiene data were
consistent with those in the simulation study; LS overall
had the widest confidence intervals and ML the narrow-
est for the predictor effects.
For all three methods, estimates of the expected re-
sponse,
of ML, for th
ˆYX
μ
, were unbiased. The confidence interval
for ˆYX
μ
is based on

ˆYX se μ, which depends on the
value of tx0. For a model with one predictor,
both ML produced an almost correct confi-
dence in the narrowest interval at
he predictor,
and WLS
terval, with
00,1
xS
0 1
ˆˆ
ESE


 

, whereas LS had its nar-
at
 
rowest co  nfidence interval 0
x
X which therefore
nfidence intervals.
For ML and WLS the sample-specific standard error,
does produce erroneous co

ˆi
se

, was a good estimator of the true standard error,
hereas for LS,

w

0
ˆ
se

greatly overestimated ˆ SE 0




.
When investigating the true risk of type I error, α*, for
tests regarding the regression parameters, the t-tests of
the LS-estimates of the intercepts produced an α* much
smaller than the nominal α. This was a consequence of
the too large

0
ˆ
se

, che distribution of the
t-statistics to have fewer observations in the tails than
expected. For both the Wald-test (ML) and the t-test used
for the WLS-estimates, the α* was approximately equal
to α (for nominal size {0.01, 0.05, 0.10} the largest rela-
tive deviations were α* = {0.021, 0.072, 0.125}). For all
three methods, the relative deviation (
ausing t
*

 ) was largest
at nominal size 0.01, indicating a skewness of the test
st
both reg
inal
atistics in the tails. Regarding the power (in tests of β1),
ML was superior to LS,arding the true and
nompower.
The precision of an estimated expected value (ˆYX
μ
)
depends both on the regression method and on the varia-
tion of the predictors ,,
1
p
X
X. If a predic
he estimated effect of that
predictor will have a large standor. In some situa-
tions a better estimation of the exposure might
tor has a
very smn t
ard err
be achieve
with
variatio
egressihe expo
models can also be used to estimate the exposure for a
whole population, provided that the distribution of the
background factors is known.
When investigating an exposure-disease associations,
the measured exposure sometimes include a stochastic
error (exposuremeas u r e d = exposuretrue + error1), which is
unrelated to the stochastic error of the (linear) expo-
sure-response model (response = δ0 + δ1·exposure + er-
ror2). The issue with regression analysis where the ex-
planatory variable has measurement errors is raised in
[23]; given the true exposure, the measured exposure has
a stochastic variation and different approaches to esti-
mating the true exposure are compared. In situations with
measurement errors, the individual- and group-based
exposure assessment approaches are often compared, see
e.g. [24,25]. In an individual-based approach, each per-
son’s measured exposure is used which often leads to a
bias of δ1 towards null. In a group-based approach, each
person is assigned an exposure based on group affiliation,
which is the same as estimating the exposure from a re-
gression model with only one explanatory variable,
Group. If several relevant explanatory variables are used
ased ap
re-re-
di
all variation, the
a model that excludes predictors with very small
n, as has been investigated in [22] for a situation
where land-use ron is used to estimate t-
sure. Apart from effect estimation, regression analysis
can be used to estimate the expected value of the re-
sponse variable given certain values of the background
factors (e.g. to estimate the personal exposure as a func-
tion of easily measured background factors). Regression
an individual-bproach. A group-based design of
ten leads to errors in the exposure which are of Berk-
son-type (exposuretrue = exposuregroup + error3), or ap-
proximately Berkson-type. As discussed in [24], in a true
Berkson-error-model, the erro r3 is independent of expo-
suregroup, whereas the approximate Berkson-error may
depend of the group size. The approximate Berkson-error
is, however, independent of error2. A group-based ap-
proach often leads to less bias in the δ1 estimate but a
larger standard error, see e.g. [24,26]. A group-based
approach with a log-linear exposure-response model can
have a substantial b
to estimate the exposure (rather than only Group) the
standard error will decrease and, thus, be more similar to
ias in the estimated exposu
sponse effect [27].
The evaluation of the new ML method, and the com-
parison to other regression methods, was conducted by a
simulation study and exemplified on a dataset. The
simulation study allowed us to assess, with great preci-
sion, the bias, standard error and the distribution of the
parameter estimates, as well as the risk of type I error
and the power of the methods. Because of the large
number of iterations, our results on bias, standard error
and the inference can be considered to be valid, for
log-normal data where the relationship is linear. The
simulation study allowed us to compare the parameters
of the simulation model (“true β values”) to the estimates
from each method and also to see how these estimates
can vary. This would not have been possible had we only
used empirical data sets, in which the “true” values are
seldom known. Also, to get a reliable estimate of the
stribution of the estimated parameters, a large number

S. M. GUSTAVSSON ET AL. 399
of data sets would be needed.
In the simulation study, data were generated from a
log-normal heteroscedastic distribution with certain pa-
rameter values. However, empirical data might be only
approximately log-normal and thus exhibit a larger vari-
ance than the one in a perfect log-normal distribution.
Thus the results from the simulation study might not hold
completely for real data sets of e.g. exposure data. Also,
in the simulation study the predictors were assumed to be
measured without measurement errors which might be
unrealistic for some predictors. Therefore, evaluation of
the new ML method should also be made on several em-
pirical data sets.
Two specific simulation models were used in this
study where the parameters were estimated from an em-
pirical dataset and therefore the simulation models can be
considered to be fairly realistic. The results on bias and
standard error are likely to be valid also in models with
other parameter values. The parameter estimates are un-
biased both for a simple model (with intercept and one
explanatory variable) and for a model with three ex-
planatory variables and no intercept, and in both situa-
tions ML gives the smallest standard errors. This sug-
gests that the results can be generalized to other datasets.
The ML estimates are derived true iterations and in
some situations (sometimes caused by unsuitable start
values or large variance in the data) the iterations don’t
converges. This may lead to unrealistic estimations. In
those cases, censoring data larger than the 95% quantile
will stabilize the results but will cause underestimations
of the intercept and standard errors. In those situations
WLS will generally give the best result.
In this study the methods have been evaluated for rela-
tively large samples (n = {108, 216}) and the conclusion
was that the asymptotic properties of maximum likely-
hood estimates are valid for these sample sizes. However,
for smaller samples, further evaluation is needed, espe-
cially regarding the distribution of the estimates. In other
studies regarding smaller samples of untransformed log-
normal data, likelihood-based approaches has been sug-
gested for constructing confidence intervals for the mean
and mean response [28,29].
This study illustrated that the new maximum likely-
hood-based regression method has good properties and
can be used for linear regression on log-normal hetero-
scedastic data. It provides an estimate of the absolute
effect of a predictor, while taking the increasing variance
into account in an optimal way. The study also demon-
strated that the results from the weighted least squares
regression (where the increasing variance is accounted
for) were very similar to those from the ML method. Al-
though ML gave slightly narrower confidence intervals
and had higher power regarding β1, both WLS and ML
can be recommended for linear regression models for
log-normal heteroscedastic data.
6. Acknowledgements
We would like to thank Urban Hjort for sharing his
MATLAB codes for the regression analysis.
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