Are you trying to get a more normally distributed data?. How does the current data look like? Is it unimodal and does it have skewness. ? These answers r important for an appropriate answer.

So why I am aiming to do is to basically give a more equal weight to species that are rare and abundant. I am trying to find out if there are differences in the composition of assemblages between different islands and regions. Most of the species sampled are rare (present in an average of two sites and an average abundance of 15 individuals). There are, however, six species that are abundant (1000+ individuals) and common (present in 35 of 41 sites studied). So the data transformation should ideally decrease the distance in the abundance of these species, while maintaining some of its variability. For what I've been reading, for ecological data is recommended to apply a power transformation (square or fourth root are often mentioned). My dataset has lots of zeroes, so it is left skewed, but unimodal.

You said that your data include counts, and I get the impression that you want to use those counts as a criterion variable. It’s typical for counts to differ substantially from the normal distribution, particularly when the mean of the count is small. So, in this sense, transforming your skewed and kurtotic counts data so they look more Gaussian is a little like trimming a square peg for a round hole. Consider using a square hole. 

If you use the generalized linear model, you can keep your count data as they are by doing Poisson regression or negative-binomial regression. You can go multivariate and/or multi-level with these kinds of regression, too. Atkins and colleagues published a nice tutorial on the topic using examples from addiction research: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3513584/. If you’re an R user, Gelman and Hill discuss it in their regression text: https://www.amazon.com/Analysis-Regression-Multilevel-Hierarchical-Models/dp/052168689X. McElreath also discusses it a bit in his recent Bayesian statistics text: http://xcelab.net/rm/statistical-rethinking/. But if you’re not an R user, Hilbe devoted an entire book to the subject, in which he used R, SAS, and Stata: https://www.amazon.com/Negative-Binomial-Regression-Joseph-Hilbe/dp/0521198151. And Agresti’s been writing about Poisson and negative-binomial regression for years: https://www.amazon.com/Categorical-Data-Analysis-Alan-Agresti/dp/0470463635 and https://www.amazon.com/Foundations-Linear-Generalized-Probability-Statistics/dp/1118730038.   

Thanks Solomon, I'll have a read.

Denisse,

Several have suggested that you might transform data to

achieve an approximately normal distribution. Generally speaking,

the normality assumption is only required at a point where you

are trying to make a probability statement, such as a p-value

for a test of hypothesis. For simple mathematical manipulations

such as computing distances to help you discover patterns and

structure in data, the normality assumption is not required. Even

least squares linear model estimates are best linear unbiased estimates

under assumptions of independence and homogeneous variances without

the assumption of normality. My point is to carefully review the

assumptions of the analysis you are conducting and only be concerned

about normality if it is assumed.

That said, there are other reasons to consider transforming data. One of

these, as you suggested in your question, is to alter the emphasis of

rare and abundant taxa. The family of power transforms gives you

a sliding scale tool to accomplish this. One form of the power

family was proposed by Box & Cox (https://en.wikipedia.org/wiki/Power_transform)

which formulates the power transform as a function of lamda.

When lamba=1 the data are untransformed, as lambda decreases toward 0,

the relative influence of more abundant taxa is diminished. In

the limit as lamda reaches 0, the transformation becomes a logarithmic function.

It also might be helpful to transform to ranks, which essentially levels

the influence of abundant and rare taxa. Another tool is to use a

distance/similarity measure based on percent similarity.

Bear in mind that there is no 'correct' choice among these options.

You should be guided by choosing a transformation that helps to

disclose patterns in your data that make sense according to

the science that you practice. In defense of those who suggested

transforming to achieve normality, I frequently find that the

data start to make sense when they are transformed to be approximately

normal. Hope this helps. You are invited to continue

this dialogue if it does.

regards, Elgin

Agreed with Elgin Perr, a Box-Cox transformation will be useful in this regard. But for count data and for pattern recognition, you can use discrete distributions such as Poisson.

Thanks everyone for your answers, they have been quite helpful in understanding the different types of transformations available. Now, I am trying to establish when should I consider applying a transformation to my data for a case like mine (trying to get a more even weight between rare and common species). Is there a rule of thumb for these cases? Would I need to consider a transformation when, for example, there is a ten fold difference between the lowest and the highest recorded counts? The answer I seem to read the most is "it depends". But to know when to apply them, you probably need more experience and knowledge about stats, which is an area I am not hugely familiar with. Is there a guideline for this at all? I'd really appreciate your thoughts on this.

Transformations are not only used to normalize data, but also to stabilize variance (that is, to make the variance homogeneous). The third reason in Time Series analysis is to make the seasonal effect additive. As per the choice of appropriate  transformation, Akpanta and Iwueze (2009) showed Bartlett's transformation as a power transformation and that the slope, (beta) of the regression line of the natural log of the group standard deviation against the natural log of the group means , determines the appropriate transformation. If beta is zero, no transformation, if half, square root transformation, if one, logarithmic transformation, etc

The question stated by Denisse is a quite standard one but not easy to answer. The appropriate answer depends critically on the underlying assumptions for the data and the stated problem. For a problem in species composition, to use a "compositional data approach" is becoming standard. For a general review of the state of the art you can have a look at "Modeling and Analysis of Compositional Data" (Pawlowsky-Glahn, Egozcue, Tolosana-Delgado, 2015)

http://bcs.wiley.com/he-bcs/Books?action=index&itemId=1118443063&bcsId=9510

The main assumption for compositional data is that the relevant information is contained in the ratios between the abundances of species. Therefore, does not matter about the number of individuals measured but on the proportions of the species. When this is so, one realizes that the scale of proportions is not absolute but relative. That is, observing 5 individuals compared with 10 individuals is just observing half abundance; observing 1000 individuals compared with an abundance of 1005 is almost the same. In this circumstances, standard mean values and variances are no longer valid, and a suitable transformation is mandatory in order to transform into a new scale which can be assumed near to absolute. If you are interested in compositional information (I think this is your case) the immediate recipe is: "transform your data using an isometric log-ratio transformation (ilr)", so that you represent your proportions by ilr-coordinates. The interesting result is that standard statistics can be safely applied to that ilr-coordinates (see Mateu-Figueras, G., Pawlowsky-Glahn, V. and Egozcue, J. J.:

The principle of working on coordinates. In Pawlowsky-Glahn, V. and Buccianti A. (Eds.) Compositional Data Analysis: Theory and Applications,

ISBN-10: 0-470-71135-3, Wiley, Chichester UK, 2011.

However, your data are counts that may be very low, including zeros, and large counts. Large counts, divided by the number of observations can be identified with proportions or probabilities. However, low counts cannot. If you observe 1000 individuals and you get 0 of species A, it does not mean that the actual proportion for species A is 0 but a small proportion. In these cases with low counts, the counts by themselves are not considered compositional. What is effectively compositional are the probabilities (proportions of each species) in a multinomial sampling. This is what is done in multinomial logistic regression. For more information see

Josep-Antoni Martín-Fernández et al.

Bayesian-multiplicative treatment of count zeros in compositional data sets, Statistical Modelling 2015; 15(2): 134-158

With respect to other transformations. The logit transformation is a limiting case of Box-Cox transformation. The ilr-coordinates are a multivariate generalization of the univariate logit. Componentwise logarithmic transformation produces analyses that are dependent on the total number of counts. No transformation confronts you with spurious correlation if the data is assumed compositional.

Compositional data packages are available in R: "compositions", "zCompositions","robCompositions". See also the free stand alone program CoDapack (teaching software) for exploratory analysis of compositional data.

if you want to ttransformation yor data to normalize, in first step, plot histogram of data and pay attention to shape. in next step chooose pereach variable , best function.

yor function may be ssin(x) or LOG(x).

dont afraid to test any ffunction

There is a mistake in my previous response. It was written "The logit transformation is a limiting case of Box-Cox tranformation". This is not true. What is a limiting case of Box-Cox transformation is the logarithm, not involving a ratio as argument. Excuse me.

NEVER. check out our new paper

Article TO DETERMINE SKEWNESS, MEAN AND DEVIATION WITH A NEW APPROAC...

Dear Guven Mehmet Gunver,

After checking your recommended paper, I went to my initial impression: I do not agree with your "NEVER" (transform data).

From your paper, I got the impression that you think about the data as if they were "true", and any transformation destroy something essential to the data themselves. However, data are the output of a frequently complex processes of calibration which are themselves a transformation of the initial measurements. In most cases, data sets are transformed previous to their presentations as such. The next impression is that you think that many times the transformation of the data are carried out to get a sample distribution close to the normal distribution. Well, this can be true in many cases, and I agree that transformations would not be used for approaching normality only or for suppressing skewness.

My idea (hopefully not only mine) is that transformations are essential for changing the scale of the data. Frequently, to transform the data scale into a scale which is assumed absolute or near to it. In your paper, you measure deviations by ordinary subtraction; then you construct a weighting of these differences. This means that you are implicitly transforming the absolute scale on the real numbers (Lebesgue measure). This contradicts your "NEVER".

Finally, I would like to remaind that after Fréchet (1948), it is quite clear that the mean and variance are defined on the basis of a distance in the sample space, not necessarily the Euclidean distance, i.e. on the scale in the sampling space. Median, average, geometric and harmonic means are only means corresponding to different scales assumed in the sampling space.

By the way, part of the data used in your paper can be considered compositional. This kind of data normally require a log-ratio transformation for a consistent analysis using standard tools.

Excuse me if my criticism is too hard. I think this corresponds to your "NEVER".

Fréchet, M. (1948), Les éléments Aléatoires de Nature Quelconque dans une Espace Distancié, Annales de l'Institut Henri Poincaré, vol. 10 (4), pp. 215-308,

Dear professor,

thank you very much for critisizing our paper. You are right. My "never" was wrong. I change it to "ever" ; for any data stack, the same robust transformation. Our paper would be a solution to the statistics'  biggest problem : coherence. We suggest that, any continuous data stack can be evaluated by our methodology. It means that, the data would be described by exactly the same parameters, whereever or whenever the calculation executed. You may have more idea if you check the paper linked

Article Crowdsourced research: Many hands make tight work

Here's my belated answer. I like to use PROC UNIVARIATE in SAS to examine data distributions before using parametric statistics. For non-Gaussian (e.g., skewed) distributions, I do a relevant transformation & then re-examine the results w/ PROC UNIVARIATE, to make sure that it worked.

-Bob Vadas, Jr.

I have started working with the vegan package in R which does Non-metric Multidimensional Scaling (NMDS) for this kind of community data. There is a default "auto" transformation option which seems to prefer

1. sqrt of the data (to reduce the disparity in scales between species)

2. "Wisconsin" transformation (normalise each column by the maximum value, then normalise each row by the sum of the values)

So this will tend to give slightly more weight to species with low counts (after sqrt + col normalisation) and weight sites according to total abundance (proportion of total observations). Then typically the Bray-Curtis dissimilarity measure is used for the ordination (plotting the site (dis)similarities in 2 dimensions), which also does some scaling.

This discussion is useful:

https://ordnews.colostate.narkive.com/lMWF502c/ordnews-1593-log-sqrt-and-other-transformation-with-bray-curtis-dissimilarity

For much of my own biotic-assemblage work (see

https://www.researchgate.net/profile/Robert_Vadas), I’ve depended on factor analysis (w/ a Varimax rotation) to examine assemblage patterns across time & space. This is typically for percent-abundance data. I prefer this multivariate-stat. approach b/c it weights towards more-abundant spp., as does my use of the Simpson/Levins index for spp. diversity.

Factor analysis is pretty robust, so stat. transformations aren’t critical, in contrast to bivariate-stat. analyses.

-Bob