My understanding is that data transformation was sexy during the SAS/MINITAB/STATITCF... years when implementing generalized (mixed effect) (non)linear models was challenging. Now with open source software it is easier to use native error structure that are appropriate for your response variable (see O'Hara and Kotze 2010 paper, see also Xiao et al 2011 paper, and the like).

I have the tendency of not transforming my data but using GLMM. It is obvious that one would not look to transform binomial (yes/no) or multinomial data but instead use binomial error structure or multinomial logit. Along the same lines, instead of log+1 transforming count data, use poisson or negative binomial or quasipoisson error structure, instead of arcsine sqrt-tranforming proportion data, use GLMM with beta error structure, instead of transforming failure time data, use survival analysis,..

http://onlinelibrary.wiley.com/doi/10.1111/j.2041-210X.2010.00021.x/full

http://www.esajournals.org/doi/abs/10.1890/11-0538.1

Hi, 

To my knowledge there is nothing wrong to apply transformation to your variables. Only problem can be raised when you have to interpret your results (because transformed variables are not identical to original in scale etc). Here is a link to John Fox's compact guide to these problems: 

http://socserv.socsci.mcmaster.ca/jfox/Courses/Brazil-2009/slides-handout.pdf

Hope this helps

Jingming, it would be helpful if you could tell us what type of data it is and what type of analysis you expect to perform on your data. I'm not convinced that left skewed data presents much of a problem for linear models that assume normality.  Particularly if sample size is reasonably large.  

Transformations are done to achieve normality and homogeneity of variance (reducing heteroscedasticity). Before proceeding towards transformation one should check for these two criterion. If both the criterion are achieved, there is not need for transformation. For you assistance, normality and homoscedasticity criterion for the sample data are tested using Shapiro-Wilks’ test of normality and Leven’s test (mean) of variance (significance level α= 0.05) respectively.

If in your data standard deviations are very high, i.e. difference between the two units, you can apply transformation. Best transformation is the one by which above two criterion are achieved and also reduce  standard deviations.

For Box-Cox, it is based on the range of power transformations as a means for normalizing and variance equalizing of both positively- and negatively-skewed variables. Its a very effective variable transformation mechanism.

The type of transformation also depends on whether your data is expressed as numerical values (e.g. abundance values) or proportions (e.g. percent).

for ecological data such as occurances, the Hellinger Distance is often prefered. It accounts for the size effect.

Jingming  Hello

I propose that such a transformation must base on some probabilistic-statistic and information methodologies with utilization of expert estimations.

Some similar methodologies you can find in the attached paper:             

Eppelbaum, L., Eppelbaum,V. and Ben-Avraham, Z., 2003. Formalization and estimation of integrated geological investigations: Informational Approach. Geoinformatics, 14, No.3, 233-240. 

My understanding is that data transformation was sexy during the SAS/MINITAB/STATITCF... years when implementing generalized (mixed effect) (non)linear models was challenging. Now with open source software it is easier to use native error structure that are appropriate for your response variable (see O'Hara and Kotze 2010 paper, see also Xiao et al 2011 paper, and the like).

I have the tendency of not transforming my data but using GLMM. It is obvious that one would not look to transform binomial (yes/no) or multinomial data but instead use binomial error structure or multinomial logit. Along the same lines, instead of log+1 transforming count data, use poisson or negative binomial or quasipoisson error structure, instead of arcsine sqrt-tranforming proportion data, use GLMM with beta error structure, instead of transforming failure time data, use survival analysis,..

http://onlinelibrary.wiley.com/doi/10.1111/j.2041-210X.2010.00021.x/full

http://www.esajournals.org/doi/abs/10.1890/11-0538.1

I use the following two options to normalize negatively skewed data:

First option: You can directly transform your negatively skewed data. E.g. with an exponential or power tranformation (see http://www-01.ibm.com/support/docview.wss?uid=swg21479677)

Second option: first reflect the data in order to get a positive skew, and then use a data transformation for positively skewed data like log, inverse or sqrt (see http://support.sas.com/documentation/cdl/en/stsug/62259/HTML/default/viewer.htm#ugvartransform_sect15.htm)

thanks all the answers, all very helpful, make me thinking....

as @Dan Gwinn says, I would like to introduce my data first. I work on phylogenetic analysis on xylem traits and their relationship with cliamte, there are serveral traits variables and cliamte variables which related with each other in their groups. these data comes from hundreds of tree species, thus, most of the traits were left skewed, i.e., with many small values and few larger ones, such as vessel diameter, area percentage of vessels in the cross-sectional sapwood.

I believe @Orou Gaoue is right in his points, it is good to practice this in standard statistic models. however, situation is little different here, Generaly, the phylogeny will lead to great bias in such a dataset, so I decide to use phylogenetic informed statisics methods, such as, PICs correlations, phylo-ANOVA, phylo-PCA. as the normalty and homoscedasticity is required by standard correlation or ANOVA analysis, I guess this priciple applys to phylogentic informed methods. is that right? however, after some trials, it seems to met the results differ when using different transformation methods, say, log(), or power(), I don't know why?

if @kesh Kumaar Gautam is right, can I use different power index to tranform different traits in order to meet the two ceriterion? often in linear regression, I know boxcox() was use for a pair of variable(y~x), not two groups variable (traits ~ climate), in the former case, it is much easier to find the opimum power coeffeicent to use, while in the latter case, I am trying to use PowerTrans() in R with some doubts...  so the question is: will the transformation lead to wrong result? so far, i have no reason/evidence to support this question 

thanks @Koenraad Van Meerbeek for the lightening advices, but I could not conect to the links..it shows:."We're sorry, we cannot complete your request.".would you mind give me further help with your kind hand?

I think @Lev V. Eppelbaum's maybe a possible way to solve the problem, I will try to figure it soon.

looking forward to hear more from you all, thanks alot!

again, thanks @Levan Mumladze for the very helpful handout. I guess the question now is, by using phylogenetic informed methods, will the same protocol applys, or, will transformation really needed/necessary? as PICs is the difference between two sister branch species on the phylogeny tree and may be this will change the distribution of data for a large species pool. 

...?

I would say data transformation should be merely viewed as a technical shortcut to fit a model with certain properties. It is very important to acknowledge what happens to the actual model and its properties, such as the functional form and mean-variance relationship of the error, when transforming a variable. For example, when fitting a linear regression to log-transformed data (dependent variable), one is fitting an exponential model with multiplicative log-normal error structure. Back-transforming to raw scale is often relevant for interpretation and illustration.

I often hear that transformations are applied to make the data behave linearly and to homogenize the residual variances. This might be true, but I think these arguments are approaching the problem from the wrong end. This can possibly lead on the wrong track. Do we want to fit the data to the model or the model to the data?

As Orou Gaoue pointed out above, it is nowadays usually more convenient to use e.g. a GLM which is more explicit about all these properties and makes any transformation unnecessary. But still, GLMs handle some common special combinations of functions and data types, and there surely situations where transformations can be useful.

I agree too that one should use models appropriate to the data. If it is not possible (e.g. factor analysis, principal component analyses), I generally still prefer to transform the data if normality is one of the assumptions of the statistical technique.  Other than Box-Cox transformation, there is also log(x+k) transformation, where x is your variable and k is the constant added to achieve normality (Griffiths, 1980). It is, however, sometimes still difficult to achieve normality of data.

For proportions, I have used the logit transformation so that predicted values do not go out of range. While there are models that can take account of proportions data, I took the pragmatic approach of using data transformation as I couldn't with the available software fit a mixed model together with missing data imputation (Man et al., 2014).

References

Griffiths, D.A (1980). Interval estimation for the three-parameter lognormal distribution via the likelihood function. Applied Statistics, 29:58-68.

Man, W. Y. N., Worth, H., Kelly, A., Wilson, D. P., & Siba, P. (2014). Is endemic political corruption hampering provision of ART and PMTCT in developing countries? Journal of the International AIDS Society, 17(1). doi:10.7448/IAS.17.1.18568

"...the most of variables I am dealing with are left skewed..."  - What is the nature of these vars? Seems to me - very offen we found the opposite - the long right tale!

"Normality" is convenient for RESIDUALS! (NOT FOR RAW DATA!)

I like Bob O'Hara "propaganda", but what deal with 0 or 1 in logit tr?! ;-)

(I just use =arcsin(2p-1) it is resricted always!)

Always useful to use a model (with or without transformation) that is convenient to interpret, which has a clear physical / biological / ... sense, does not predict the absurdities and so on.

The logit transformation is convenient for interpretation as odds ratio on back-transformation of the regression coefficient. Not perfect because of the 0 and 1 limitation, but one can get around it with addition of a constant as one does with log-transformation of count data with zeros. It is difficult, however to interpret what back-transformation of arcsin means, but it does serve its purpose in other ways.

It is NOT difficult to interpret what back-transformation of arcsin means,

- it'll  back in original scale! 

I only hint: the "ultimate universal trasformation is not exist, but so called "canonical" link...

About arcsin trf, I mean that I like it before using wavelet/spectral/reg etc. for DVs mesured in frequency (success/attempts) & sometimes included zero obs. 

It is difficult, however to interpret what  "inverse hyperbolic sin" means... ;-)

I was lettle surprized about "long left tail"

Hi Jingming,

My general approach to data transformation is part of the data exploration which I apply before moving on to data analysis.

Data exploration (sometime called data clean-up)

1) Data transformation

2) Outlier detection and removal

3) Data normalization

Data analysis

The initial step in data transformation could be a simple analysis along the line of generating set of random numbers that belongs to a certain distribution family (generate the same number of data points as in your data set) and then plot it in a qq-plot. It will be very easy to spot which distribution family your dataset belongs to and then you can transform your data.

I really like to plot the normal transformed data in a histogram and it can be very helpful, because I came across a number of platform that contain systematic technical issues. A histogram of the normal transformed data is one way of spotting them. Often this technical issues show as humps or sharp peaks (zero readings) or on the left hand side of the histogram as a max reading value on the right.

Other ways for the identification of outliers could be PCA or boxplots….another of getting ride off your right hand tail is to apply a data normalization procedure.

After this you should be able to analysis your data.

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