Dear all,
One week ago, I asked a question about the requirement of normal
distribution in statistics. The problem is that in most cases, the data sets
we are dealing with do not follow the normal disttibution. If normality is
required, data transformation needs to be carried out prior to (parametric)
statistical analyses. On the other hand, data transformation may cause some
other problems.
Thanks to Isobel Clark, Brian Gray and Ruben Roa for their replies and
comments. Please find the following the original question and replies.
---------------------
Dear (geo_)statisticians in the list:
I'm quite often confused with the requirement of "normal distribution" in
(geo)statistics. My question is: When the normal distribution requirement
MUST be satisfied? Specifically, in which of the following analyses, the
variables MUST follow the normal distribution? If not, what would happen?
Uni-variate analyses:
Outlier detection, Mean calculation, etc.?
Bi-variate analyses:
Correlation; Regression; etc.?
Multi-variate analyses:
Principal component, Cluster, Regression, Factor,
Discriminant, etc.?
Spatial statistics:
Spatial autocorrelation, Spatial outlier, etc.?
Geostatistics:
Variogram, Kriging, Simulation, etc.?
Regards,
Chaosheng Zhang
---------------------------------
Short answer is yes to everything.
middle length answer is that Normality is not required
for anything except where it is a basic assumption -
such as in simulation. It is necessary that the
distribution be well behaved (not skewed) and conform
to the Central Limit Theorem.
Having said that, I can't tell you where the join is
between 'well behaved' and not. It is usually fairly
obvious from probability plots and semi-variograms
when things start to get hairy.
Isobel Clark
http://geoecosse.bizland.com/BYOGeostats.htm
-----------------------------
Isobel,
Thanks for the reply. I feel this problem deserves more discussion.
I have found the message from: Gregoire Dubois, Date: Mon Mar 5, 2001,
Subject: AI-GEOSTATS: SUMMARY: Nscore transform & kriging of log normal
data sets (The original author couldn't be found, and s/he should be in the
list): "Most of geostatistics is "distribution free", i.e., the derivation
of the simple kriging, ordinary kriging and universal kriging equations do
not depend on a distributional assumption (contrary to what is sometimes
claimed)."
An example may be the "Indicator Kriging": It is impossible for the "0"s and
"1"s to follow the normal distribution.
The reason why I care about this issue is that there are at least two
problems related to data transformation (in order to follow the normal
distribution):
(1) The measurement scale is reduced. The orignal ratio/interval scale may
be reduced to the lower level of ordinal, even close to nominal, which
results in loss of raw information.
(2) Artificial relationship is introduced. We know that the lognormal
distribution is widely accepted. In correlation analysis, if the
log-transformed data are used, the correlation becomes the "log-log"
relationship, not the oginal linear relationship. In bivariate regression
analysis, the original function is:
y = a x + b
However, for the log-transformed data, the function becomes:
log(y) = a log(x) + b
or y = exp (a log(x) + b)
In many cases, it is not clear if the relationship should be linear or
"log-linear". However, the artificially introduced "log-linear" relationship
need to be proved.
The most difficult situation is that if scientifically the relationship
between x and y is linear, should the data transformation still be carried
out (just to satisfy the statistical requirement)?
Cheers,
Chaosheng
---------------------
nice point. classical statisticians are slowly eating away at how to
estimate variance structures under a marginal binary assumption using
relatively simple generalized mixed models. no, I don't know how they will
handle (if ever) the problem associated with possible underestimation of
spatial variance components after--or at the same time
as/iteratively--estimating mean components. and, of course, the focus is
typically estimation rather than prediction. regardless, the variance
component estimation question *is* approached under a marginal binary
assumption--and spatial trend in the mean or equivalent is typically the
primary part of such models. of course, with trend in the mean we face
another interesting problem--namely, that the spatial variance components
under a marginal binary assumption are a function of the mean/go to zero as
the mean goes to 0 or 1. cheers, brian
****************************************************************
Brian Gray
USGS Upper Midwest Environmental Sciences Center
575 Lester Avenue, Onalaska, WI 54650
ph 608-783-7550 ext 19, FAX 608-783-8058
[EMAIL PROTECTED]
*****************************************************************
Hi Chaosheng:
A few points about log transforms. See below.
> The reason why I care about this issue is that there are at least two
> problems related to data transformation (in order to follow the normal
> distribution):
>
> (1) The measurement scale is reduced. The orignal ratio/interval scale may
> be reduced to the lower level of ordinal, even close to nominal, which
> results in loss of raw information.
There shouldn't be any loss of information since the log transformation
is a one-to-one mapping. The sample variance is much smaller in the log
scale but the log transform is often used precisely for that purpose.
> (2) Artificial relationship is introduced. We know that the lognormal
> distribution is widely accepted. In correlation analysis, if the
> log-transformed data are used, the correlation becomes the "log-log"
> relationship, not the oginal linear relationship. In bivariate regression
> analysis, the original function is:
> y = a x + b
> However, for the log-transformed data, the function becomes:
> log(y) = a log(x) + b
> or y = exp (a log(x) + b)
> In many cases, it is not clear if the relationship should be linear or
> "log-linear". However, the artificially introduced "log-linear"
relationship
> need to be proved.
Rather, when you apply the log transform you a-priori assume the
existence of what you call the 'artificial relation', and this
assumption refer to the algebraic form of the error term rather than to
the relation between y and x. Say E(y)=f(x) is the model for y versus x,
where E is the expectation operator. If you assume an additive error
structure y_i=f(x_i)+e_i, where i indexes observation, and you consider
the e_i's as iid normal random variates, then there is no reason to
apply the log tranform. On the other hand if you assume a multiplicative
error structure such as y_i=f(x_i)*e_i, and you assume that the e_i are
iid lognormal random variates, then the log transform yields
ln(y_i)=lnf(x_i)+ln(e_i) and now the ln(e_i) are iid normal random
variates (with a much smaller variance than the e_i). The theory of the
lognormal is well developed so that there isn't actually much need to
transform the data to make it normal (e.g. Crow and Shimizu, 1988,
Lognormal distributions, Dekker Inc, NY).
> The most difficult situation is that if scientifically the relationship
> between x and y is linear, should the data transformation still be carried
> out (just to satisfy the statistical requirement)?
When the relation between x and y is linear such as in E(y)=a*x+b, the
errors are assumed to be additive (people usually do not believe that
y_i=(a*x_i+b)*e_i but rather y_i=(a*x_i+b)+e_i), so applying a log
transform to such case does not satisfy statistical requirements. On the
contrary, it goes against statistical advice.
The multiplicative error structure arises in models of the form
E(y)=a*x^b or E(y)=a*exp(b*x), or in general, in all multiplicative
processes.
As there is a central limit theoren for additive processes leading to
the normal, there is a central limit theorem for multiplicative
processes leading to the lognormal.
In the case of geostatistics, as pointed out by other people, the
kriging equations do not require distributional assumptions (though the
fitting of the model variogram to the moment-based Matheron variogram
does). If the frequency distribution of the regionalised variable looks
lognormal, it means that there is un underlying mechanism which is
multiplicative, but still i don't see why the variable should be
transformed for geostatistical analysis, except perhaps for fitting the
variogram.
Cheers
Ruben
=================================================
Dr. Chaosheng Zhang
Lecturer in GIS
Department of Geography
National University of Ireland
Galway
IRELAND
Tel: +353-91-524411 ext. 2375
Fax: +353-91-525700
Email: [EMAIL PROTECTED]
[EMAIL PROTECTED]
Web: http://www.nuigalway.ie/geography/zhang.html
=================================================
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