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

brgray@...

*****************************************************************

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

There shouldn't be any loss of information since the log transformation

> 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.

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

relationship

> 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"

> 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

When the relation between x and y is linear such as in E(y)=a*x+b, the

> between x and y is linear, should the data transformation still be carried

> out (just to satisfy the statistical requirement)?

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: Chaosheng.Zhang@...

ChaoshengZhang@...

Web: http://www.nuigalway.ie/geography/zhang.html

=================================================

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