First of all, if something is significant or not is your decision, and not (simply) the result of a test.

Actually, to unbiasedly interpret the test results, you should know before analysing the data if and how the response should be transformed. However, you obviousely don't know it... so the test results are not interpretable in the way you'd like to.

The typical way is to have a look at the residuals. Their distribution should not be too contradictory to the model assumptions (symmetry, variance homogeneity -> "normality"). From the look at the residuals you may be able to decide which model (the additive model with the untransformed response or the multiplicative model with the log-transformed response) is more appropriate (in light of the given data!).

Since you then used the very same data to decide about the model and to analyze the model coefficients, the p-values must be interpreted with greater care. Selecting a coefficient with p

First of all, if something is significant or not is your decision, and not (simply) the result of a test.

Actually, to unbiasedly interpret the test results, you should know before analysing the data if and how the response should be transformed. However, you obviousely don't know it... so the test results are not interpretable in the way you'd like to.

The typical way is to have a look at the residuals. Their distribution should not be too contradictory to the model assumptions (symmetry, variance homogeneity -> "normality"). From the look at the residuals you may be able to decide which model (the additive model with the untransformed response or the multiplicative model with the log-transformed response) is more appropriate (in light of the given data!).

Since you then used the very same data to decide about the model and to analyze the model coefficients, the p-values must be interpreted with greater care. Selecting a coefficient with p

There may be many questions that should be asked and answered before jumping to a statistical conclusion. I've listed some that come to my mind:

Is there any physical reason to log-transform your response variables? In the first model you constructed, since you deem it insignificant (insignificance is attributed to p-values being below some threshold, but what may that threshold be?? Why?), your model is simply y = e. In this case, if you plot 9 observations, do they look normal-distributed? In the second model using two factors (or maybe you mean "terms in your model"?) have you found the residuals and check for normality? Do you have any repetitions of these experiments, or maybe some other data points from central composite designs? Do you have reason to believe or neglect the existence of interactions?

Paul, exactness in a given set of data can in priciple not guarantee exactness in a more general setting. To concluse this one again needs distributional assumptions...

What is appropriate and what is not has to be judged for the particular case. Simulations can help to see what we can reasonably expect under specific conditions, what in turn can help to judge the appropriateness.

To my point of view, statistical analysis is not simply a classification task. We wish to build models that quantitatively relate different variables, I think I do not need to explain you what will distinguish "better" models from "worse" models. In your response you seen to distinguish appropriate and inappropriate models as if these were general features. In my view, models are like vehicles. The appropriatenes depends on what you will use it for. A ship, for instance, is not an inappropriate vehicle per se, but quite inappropriate if you wish to carry goods over land.

And yes, as always in research: failure is an option.

Statistics is a tool(box) to deal with (empirical) knowledge. The methods are objective, consistent, and as efficient (optimal?) as possible. In contrast, knowledge is subjective, depending on the available data and on the epistemic system (we may regard inter-subjective agreement on knowledge [quasi-]"objective"). Since knowledge is subject to changes, so the conlcusions will. Thus one can be sure about statements and about the available data, but not about the mental/epistemic concepts. The only link between these concepts and empirical data that I am aware of is a statistical or probabalistic model.

A probability "model" is a way (the only one?) to represent knowledge (in a quantitative way that can be subsjected to mathematical analyses), and this neccesarily has to be an (or based on) assumption(s). Seeing residuals that do not behave as "expected" based on these assumptions tells us that the underlying model is a very bad representation (for whatever reasons, there can be many). Finding no obvious, "severe" deviations tells us that the model a a good-enough representation for the data considered. It may turn out (and usually it does so - that is what is called scientific progress!) in future that the model needs to be revised.

As far as I understand this, this is exactly the way "Is this how quantum mechanics runs, how thermodynamics plays, how astronomers understand the universe" and "how engineering should be done". Yes.

In short: yes, they are!

Suppose we study the relationship between N2 fertilization and plant growth. At a certain level of the curve N2 ceases to be linear. We can use here for a anova? Taking logarithms is like playing with a magnifier. The problem arises because we want to fit a potential model y = axb or exponential y = abx and we want to explain with an ANOVA which is a linear model. This growth curve has a linear part but from a certein  value is transfomed in potential. How can truly interpret a relationship? !Of course.¡¡ ¿ Changing scale data taking logarithms we obtain the relationship fertilzation/ plant growth, as should actually be discovered.

Alberto, you seem to confuse "linear model" and "linear relationship". An ANOVA is also possible if non-linear relationships are fit. The "linear" in "linear model" refers to the coefficients that are fit - they must not be transformed (what is called "linear", too). However, the predictors can be transformed, so non-linear relationships between the predictors and the response can well be modeled, and the -linear- coefficients in this model can be subjected to an ANOVA as usual (when the assumption of normally distributed residuals is reasonable).

Dear all,

Perhaps I can weigh in with a very humble comment (which is very general since there is no additional information on what kind of response variable you used):

In my view (and here I agree with Jochen), significance has two sides: the statistical one (which is the p-value) and the real-world one. The statistical side gives you access to a number that (even when augmented with power analysis) remains at heart a probability.

It is our job as researchers to take this probabilistic statement and - taken together with background literature, other effects, a thorough assessment of the validity of your experimental design and setting, as well as a (likely large) grain of salt - to devise an interpretation of the results in light of your hypothesis.

I am assuming, when running the ANOVA you had a specific hypothesis that you wanted to test. Perhaps it may be good to ask the following questions to clarify the impact of your experiment: what happens if the ANOVA says that there are significant factors? what does that MEAN for my research? what predictions can I make? am I sure that I can claim this to be a "robust" effect?

My stats "upbringing" has taught me that focusing on statistical significance alone is not the full story - the interpretation that follows is the "real deal" and (in my view) the actual scientific contribution.

Best

Christian

PS: I also agree with Jochen that there are sometimes reasons to transform your response variable to better meet some assumptions about data distributions when applying **standard parametric** tests. I have no experience with the ODA approach favored by Paul here, but perhaps that could also be an appropriate point of attack for your data  (Paul: if you read this, could you say something about "power" of ODA?)

In this point I agree with Fausto. I know many cases where the interaction is the most important/main research question - but it is never analysed (not even considered) by the researchers.

I disagree with Fausto in the way that I would not call these people "so-called researchers", what to my opinion seems to be mean in a derogatory or deprecative way. They usually do a good job in their field of research anyway, and they are knowlegable and smart in many subjects. It is only that some basic ability of "statistical thinking" is missing, and usually this is caused by a bad and wrong training and a large ignorance among colleagues, reviewers and editors (who are, after all, similar people). This is clearly an important issue to improve this "statistical-thinking ability", but I it the wrong way to call these people "stupid" or similar (well, some people are in fact at least ignorant, but this should not be generalized). Wagging fingers and a moral sermons did not change this in the past, and they surely won't in the future. We must find more constructive ways of a qualitatively better statistical education of empirical scientists.

First,

You have three factors and how many replicats ?

Did you test the normality and the homogeneity of variance on your datas ?

What do you use as software ?

You don't provide us enough informations to help you !

Data transformation don't bring up the significance but it bring up normality so maybe it's not pertinent to work with untransformed data.

Imen and Fausto (who seems to be correct):

significance is related to a hypothesis, and the hypothesis is related to a (statistical) model. Using transformed data is a change of the model (and, hence, of the hypothesis). This obviousely results in an altered significance ("p-value"), simply because the p-values calculated on untransformed and on transformed data refer to different models and hypotheses. Even if the p-values were identical, the numerically identical values had still a different meaning, since they still refer to different models/hypotheses.(*)

Given background knowledge about the underlying processes, the kind of data, and possibly also the given data, one model may be more sensible/reasonable/adequate than another model. Thus, the choice between the possible different models/hypotheses is not as arbitrary as it seems.

* This should not be confused with the scenario that the same model/hypothesis is analyzed in different ways, e.g. a multiplicative linear model by using a linear model on the log-transformed response or by using a (gamma)model with log-link on the original response data.