Daniel,

So you have count data. Actually the most direct "correct" approach would be a Poisson-model (aka log-linear regression), a generalized linear model (GLM). This is a multiplicative model, only that the expected value is transformed instead of the response.

In count data, the variance and the mean value are related, therefore residuals won't be normal distributed. Note that I am talking about the residuals, not about the data itself. There is no need that the data is normal distributed! It's only and always about the residuals. So you need to specify a model and calculate the residuals. Your model should include the named predictors (temperature, pH, salinity, organic matter, etc). You can combine metric and nominal predictors. You can also investigate possible interactions between the predictors.

However, the counts you have are quite large, so a bounding problem (values close to 0) will be unlikely. The relation between variance and mean will (more or less) vanish for log-transformed data, so there is a chance that residuals of an ordinary linear model (LM) on the log-counts (a multiplicative model) will work well, too. If you are not confortable with GLMs then you might have a try with an LM and a log-transformed response.

The procedures are multiples,, and depends of research question

"Analysis" can mean everything, so ther can't be a good answer anyway...

But: What is the aim of your research?

If is is just to get some "significant p values", well... then it doen't matter at all what you do. The simplest procedure resulting in the targeted result is good enough (surely taking care that the procedure is appropriate and the assumptions are met, though). If the result is hard to achieve, one should select a method that is as powerful as possible. But that's all.

IMHO, research should aim to build useful (quantitative) models. The effect sizes in such models are parameters, so this is essentially a parametric analysis. Generalized linear models (GLMs) are very flexible here, being able to model normal, binomial, and Poisson-errors, and there are further extensions available.

To Alex's comment: Transforming, especially by "log", is not so difficult in interpret. The log turns products into sums (sums are fitted and this means, on the original scale, products; this is a multiplicative error model). The effects are log-fold-changes. This is quite reasonable for many scenarios. In a linear model with log-transformed response, the intercept on the original scale is just exp(b0), and the "i-th"coefficient bi indicated a fold-change of exp(bi). The expected response for condition "i" is exp(b0+bi) or exp(b0)*exp(bi) (intercept multiplied with fold-change). I don't find this difficult to interpret. For many biological systems, for instance gene-regulation, signalling cascades and so on, the multiplicative connection actually makes a lot of sense.

Sure, I agree. One should first *think*... too many people hope that statistics is a simpler substitute for comprehension...

Jochen is absolutely right. There is no universally correct response to this question, it depends on your research questions and the details of your design and the distribution s of the quantities you measure.

It is important to remember that non parametric tests are by no means assumption free. Depending on the conrext their assumptions can be at least as heroic as those for parametric approaches.

You should also remember that non parametrics and transformations are not the only options. There are broad clssses of models available to deal with data of very different types and distributions.

Daniel, can you give us an example of what do you want to work with?

Then we' d probably be able to help you. Otherwise, if it is a general purpose question, I think it depends on the kind of transformation. For example some times there exist singularities due to the transformation and you have to deal with them.

Thanks for all your valuable input. To make this question more clear, I added a simple example in the question description.

Daniel,

So you have count data. Actually the most direct "correct" approach would be a Poisson-model (aka log-linear regression), a generalized linear model (GLM). This is a multiplicative model, only that the expected value is transformed instead of the response.

In count data, the variance and the mean value are related, therefore residuals won't be normal distributed. Note that I am talking about the residuals, not about the data itself. There is no need that the data is normal distributed! It's only and always about the residuals. So you need to specify a model and calculate the residuals. Your model should include the named predictors (temperature, pH, salinity, organic matter, etc). You can combine metric and nominal predictors. You can also investigate possible interactions between the predictors.

However, the counts you have are quite large, so a bounding problem (values close to 0) will be unlikely. The relation between variance and mean will (more or less) vanish for log-transformed data, so there is a chance that residuals of an ordinary linear model (LM) on the log-counts (a multiplicative model) will work well, too. If you are not confortable with GLMs then you might have a try with an LM and a log-transformed response.

You should probably consider a generalised linear mixed model, depending on the sources of extra Poisson heterogeneity you have. Best thing is to find a friendly local statistician (few of usd actually bite).

Technically your method/s of analyzation should be predetermined, not chosen after your data is collected & analyzed.

@Kelly That is a bit too black and white for me. It is very much the approach used in regulatory clinical studies, where the emphasis is on ensuring that people have not chosen an analysis specifically to get evidence that a potential product is effective. Regulatory work is one of the few situations where maintenance of Type I error actually makes sense.

In most scientific situations we are principally concerned not with adopting decision rules which perform well in some hypothetical long run, but with making inferences conditional on the data we have observed See for example, Hacking, Logic of Statistical Inference, CUP, Chapter 7 for a discussion of before and after trials betting.

Those inferences will depend on a careful and nuanced analysis, which is sensitive to the particular features of the experiments and the measurements that we have. That can't be achieved with an a priori specification of the analysis.

This is one of those areas where the competing philosophies of statistics can actually lead to differences in practice. Fortunately, most statisticians, even those quite comfortable with Neyman Pearson inference, mitigate its Stalinist tendencies in practice :)

I would like to agree with Kelly and argue also for predetermining (at least part of) the methods that you will use to analyse the data. Particularly because in many disciplines parametric tests are the norm and often adopted without given proper thought to the appropriateness of the selected test. In the paper that I attach, we argue for the use of non-parametric tests, and we show that this can lead to different insights on what the data actually tell you. The topic is somewhat unrelated, but the principles could be the same.

There are no generic answers to this question. As Jochen says, it depends on your problem.

Ecologists frequently use non parametric permutation tests with no more thought as to whether or not exchangeability is plausible than others give to distributional assumptions.

In many situations, non parametric tests do not cope well with the complex dependencies that exist in data (especially in the ecological arena where there are often spatio-temporal dependencies). That is not to say that you can't develop effective permutation or other non parametric strategies for highly dependent data, but it can be very challenging to do so correctly.

Getting the dependencies and the variance structure right is usually more important than the distributional minutiae - although of course that is not always the case.

That is why you have to explore your data, and look at diagnostics before settling on a final analysis methods.

Some sort of predetermining the analysis is necessary - you have to have an analysis strategy. That strategy has to be relevant to your research questions, but it also has to be sensitive to the nature of the data.

Depends. For make the transfomation is need knon the kind of distribuition. There are many test for assessment of adherence a diferent distribution. If your dates is distributed log-normal, the transfomatios is a good idea, but if don't is not good idea. There are many king of statistic distribution. if your date is not normal or log-normal distribued the nonparametric test is recommended.

the test of anderson-Darling. Kolmogorov-Smirnov, Ryan-Joulier and Shapiro-Wilk are test for assessement the statistic distribution. and this test can answer the need or not the transformation.

is possible to make many Kind of transformatios: square root, sine, cosine, 1-sine, 1/x etc... that cam transform your dates in normal distribued.

Good question and good suggestions!. Thinking about the original question...how we decide to use one or another test? I agree with Jochen, the important thing is the residual distribution, so we ussuallly start using parametric test. When we don´t get normality, we use transformations... but If we don't get a good approximation, we think in non-parametric test...In this point, the decisión implies a crucial change in the test power (reducing it). However, I also agree with Mervyn: non-parametric no means assumptions free. But non-parametric test are more robust to assumption violation. Finally, nothing substitutes a good reseach plan and design to fill the requirements of independency and variance homogeneity.

Daniel - Jochen is quite correct about using a model that assumes a Poisson distribution. These models (i.e., models with Poisson distribution, response variable is a count) are one type of 'generalized linear model'. A number of packages are available to construct and test generalized linear models. The most sophisticated of these packages can handle models whose independent (i.e., predictor) variables are a mix of fixed and random effects, include repeated measures, are nested within each other, and so on. I use the SAS procedure GLIMMIX to construct and analyze these models, but there are also R packages that have these capabilities. However, PLEASE do get some statistical help if you go this route, because there are many potential pitfalls in specifying the model, failure to find a good solution (e.g., positive definite Hessian matrix) or any solution at all is not uncommon, etc., etc. I would also like to add that if the residuals of count data are over-dispersed (resulting in a poor fit to Poisson) a common fix-up is to assume a negative binomial distribution. Good luck!

Norma is right. You almost certainly have extra poisson variability.

If the problem is really simple (no complex hierarchical error structures) then you can just use a poisson model and inflare the parameter estimate covariance matrix using a heterogeneity factor. Any package will do that for you automatically. You may see that described as a quasi likelihood or quasi poisson approach.

if you do have more complex error structures then you need to model the situation, and use a generalised linear mixed model. But you do need help because it is an area full of pitfalls.

If you can't get local statistical advice send me more details off line and I will specify an analysis using R.

@Daniel, Finally, you can perform both of the analyses and compare accuracy of results.

Dear collegues,

Nice, and crucial debate. When we do not get normality, we use transformations. We need to study the data distribution previously to do not a lot of different transformations. If we do not get an adequate transformation, so we will use the appropriate non-parametric test. As Julio said, the decisión involves a fundamental change in the test power (reducing it). But non-parametric tests are more robust to violate the assumption of normality.

Remember that transforming the data also transforms the hypothesis. For example a log-transformation changes a linear additive model into one with multiplicative proportional effects. Other transformations (e.g. Log X+1) might not relate to any interpretable hypotheses. A second point is that many are put off non-parametric tests because the early versions were based on ranks (actually a type of data transformation) and so lost much information. Modern methods (e.g PERMANOVA) do not require reduction to ranks. However, one major limitation remains; non-parametric methods do not estimate any parameters (and that might be what you really want to do).

Ian

I am not so sure that transforming the data also transforms the hypothesis. It is a matter of research. For example if your data were lognormal, when you transform with z=log(Y) you obtain data with normal distribution. Let z be normal with homogeneus variance and let H0z) be the usual null hypotesis. If you do not reject this hypotesis you can conclude safely that H0y) have not to been rejected, because the mean of Y is a funtion of E(z) and Var(z), that are homogeneus.

If you reject H0z) then you could safely reject H0y) because the mean of Y is a function of E(z) that changes for some of the groups. I conjeture that a similar argumentation may be equally formuled for any other bijective transformation of y.

@ Mervyn, I agree my reply was a little shy on detail but from not having any detail on the method of collection or exact hypotheses, it was just my first reaction. I agree statistical tests can be beneficial "after the fact" but in general I have been taught that can create bias & leave conclusions open to argument

Guillermo

Here is a quote from my favourite textbook on statistics "Experimental Design and Data Analysis for Biologists (Quinn & Keough, 2002).

"A transformation is really changing your response variable and therefore your formal null hypothesis. You might hypothesize that growth of plants varies with density, and formalize that as the H0 that the mean growth of plants at high density equals the mean growth at low denisty. If you are forced to log-transform your data the null hypothesis becomes "mean log-growth does not vary with density", or you might say that in the first case, growth is defined as mg of weight gained, whereas after log-transforming, growth is the log-mg weight gained."

Another difference is that without transformation the hypothesis is about the arithmetic mean but with log-transformation it is really about the geometric mean. These are different null hypotheses.

Another more obvious case is with regression models. A linear model of log-transformed data (both variables) is a power function on the untransformed scales. These relationships have different shapes and their parameters mean different things (even when back-transformed) so hypotheses about those parameters must be different.

Yes, and there are (many?) cases where the transformed response and the coefficients of models on the transformed response are more sensible than from the untransformed response.

sensible in what way? Probably not biologically sensible, since the question was originally about microbes? which biological process relies on or results in log-mg weight, e.g.? I must admit I never thought about transforming the hypothesis by transforming the data, but if that's the case, then that's serious for study design etc.

I think the nature of the data should be considered and the method/s of analysis fully considered before any data is collected, especially in areas where spatial and temporal factors exist. An idea of the data you will be collecting and the questions you are answering is basic in scientific design. The probability that your data will need to be transformed should be thought about and answered before you actually see your data otherwise you have only proven that you know how to transform data into a desired result-- making any conclusions biased and unreliable to the scientific community at large-- in my humble, undergrad opinion.

I agree completely with Jochen, particularly in this case of environmental bacterial counts, so perhaps I can convince Suzanne that it is sensible. Consider the effect of a 1 degree rise in temperature on the bacterial population. Is that expected to increase the density by a fixed number of cfus (regardless of the baseline density that might vary over several orders of magnitude depending on the effects of other factors) or would it result in a proportional increase (e.g doubling)? In a river the bacterial density probably reflects the growth rate since the last disturbance so perhaps a direct measure of growth rate (or doubling time) might be an appropriate measure - and that is in effect what the log transformation is giving us.

For a contrary example, consider a salmon farm suspected of increasing nutrient levels in surrounding waters. The feed rate is constant so the effect on nutrient levels in the water is expected to be a constant increment regardless of the baseline level determined by seasonal and other factors, not a proportional increase, so log transformation would not be sensible in that case.

What I caution against is the practise of framing a hypothesis in terms of measurements on one scale, subsequently finding that residuals are not normal, transforming the data, re-running the analysis and then claiming that you have tested the original hypothesis and made optimal estimates of parameters by back-transforming the results of analysis on transformed data.

Thank you Ian for your support. That is exactly what I meant.

@Susanne: models are rarely set up to interpret expected values of the response but rather to estimate and interpret effect sizes. For a log-transformed response, log-mg seems wiered, but actually the model is made to estimate *differences* in "log-mg"s, what is straight foreward interpretable as log fold-change.

Dear Ian and Jochen

I accord with you that when we transformed the data, we obtain our conclusion over the parameters and hypotesis in the transformed variable but I know that in the example of the log transformation it can be easily translated to the original scale too.

Suppose that we have k treatments in a completely random design and our variable Y is lognormal (then Z=log(Y) is normal) and that Z has homogeneus variance (sigma*sigma) on the treatments. Then we can use over Z the anova to test: H0z) E(Z1)= E(Z2)=.. =E(Zk). Suppose that we reject H0z), then there are one or more E(Zi) E(Zj) for i,j=1,2, ...k. Then, by the properties of a lognormal variable (http://en.wikipedia.org/wiki/Lognormal), E(Zi) E(Zj) => E(Yi) = exp(E(Zi)+sigma*sigma/2) is different of E(Yj)= exp(E(Z,)+sigma*sigma/2).

Suppose that we do not reject H0z, then we can not reject H0y) either.

Gulliermo, yes! And, as far as I understood, this also applies to any kind of monotone transformation. (please correct me when I am wrong)

Just to stress this again: although transformations may reshape the data to have acceptable properties for testing (and that the conclusions apply for the untransformed response as well) the key point is that the transformed response (and the effect sizes estimated on the transformed response) can be biologically/physiologically/... more reasonable/sensible than the "original" response. A change of 5pg/ml in the protein concentration may be estimated as such but may have no biological relevance, where as the change, estimated as something like "3-fold induction" may be biologically much more relevant/sensible. [this applies also for the confidence or prediction intervals]

So I think for most purposes the estimation is more important than the hypothesis testing, and then a relevant or sensible scale of the estimated effects is important.

I would say that usually the researcher prefers to have its conclusions, about the treatments, in the same scale that he used to measure the variable. When he discovers that a transformation gives results easier to interpret and measure, then he begin to measure the variable in the new scale. For example, the acidity is measured as pH (pH=-Log10([aH+]) (http://en.wikipedia.org/wiki/PH),

it is a long time ago since that someone measured acidity in hydronium concentration ([aH+]).

I would add a note of caution: bacterial counts often have upper and lower limits of detectability. At the upper limit, the assay 'whites out' and cannot read above a certain concentration. And at the lower end, the absence of colony forming units (CFU) in a sample does not mean that the true count is zero.

I have found that taking the log of the values usually works well (I use log10, as this maps well onto real life)

In a situation like this, I recommend using interval regression, a generalisation of Tobit regression, which allows you to specify that some values are not exact, but fall within a specified range - in this case, between the upper limit and infinity. And in the case of zero observed counts, these can be modelled as being less than 0 on a log scale (meaning that the concentration is less than 1 CFU).

Interval regression is available in Stata, and probably in R. Don't know about other packages.

In R you can do interval regression with survreg (package survival). The event type for an interval censored response is 3. And as is typical for R, there are always sereval ways to solve a problem, so there is also a specialized package icenReg. There is also a package called intReg that does interval censored regression, but that seems to be outdated.