Dear Witold, 

When the intrinsic nature is continuous, for example in satisfaction , concordance, evaluated in scales,  since we do not pass from one category to another suddenly, there is a continuum, whereby the intrinsic nature is continued. Regarding the statistical treatment, it depends on the model that the analyst believes that it is the most appropriate and depends on the number of categories of their scales.

As for the best fit of the model, treating the variables as continuous or ordinal, everything depends on its theoretical support. The statistics should always reflect the reality so it is always preferable to have a model with lower quality than to have a model with much higher quality but that move away from reality. The analyst is the one that  should have the sensitivity to find out what the most appropriate treatment in each case, and should support the choice made.

Then " Any example on how theory indicates the aforementioned decision? " Yes, the logic, the adherence to reality that any statistical model should have. The world exists and statistics only tries to interpret it in the clearest way possible, but  never misrepresenting reality.

Have a nice time

Helena

Dear Witold, 

When the intrinsic nature is continuous, for example in satisfaction , concordance, evaluated in scales,  since we do not pass from one category to another suddenly, there is a continuum, whereby the intrinsic nature is continued. Regarding the statistical treatment, it depends on the model that the analyst believes that it is the most appropriate and depends on the number of categories of their scales.

As for the best fit of the model, treating the variables as continuous or ordinal, everything depends on its theoretical support. The statistics should always reflect the reality so it is always preferable to have a model with lower quality than to have a model with much higher quality but that move away from reality. The analyst is the one that  should have the sensitivity to find out what the most appropriate treatment in each case, and should support the choice made.

Then " Any example on how theory indicates the aforementioned decision? " Yes, the logic, the adherence to reality that any statistical model should have. The world exists and statistics only tries to interpret it in the clearest way possible, but  never misrepresenting reality.

Have a nice time

Helena

Witold, this is probably not addressing exactly your question, but you might find the simulation study linked below interesting.  HTH.

http://psych.colorado.edu/~willcutt/pdfs/Rhemtulla_2012.pdf

Lets pretend: as in this is an example and a brief literature search will unveil problems and answers but we will ignore all this.

I want to know the effect of nitrogen fertilization on plant growth and yield.

I can apply nitrogen at any value from one atom/molecule to billions of kilograms per hectare. The application of nitrogen is a discrete function (since I cannot split atoms), but I am unable to measure atoms except at the smallest of scales. Thus this feature of the data is lost due to human error and the data are effectively continuous.

However, there is only so much land to run the experiment. So I am limited to 12 plots. I have a "do nothing" treatment (3 reps), and then I can have three other treatments each of three reps.

If I assume that the effect of nitrogen is linear, I can select the treatments at 1 gram/ha, 1 kg/ha, and 1000 kg/ha and run a linear regression analysis.

If I cannot assume that the effect of nitrogen is linear, then I have a problem. What if growth is stimulated at 1 gram/ha, it is more stimulated at 500 g/ha, but the rate of stimulation declines to 1.2 kg/ha and is toxic at anything above 10 kg/ha? Sadly, I was not able to test the 500 g/ha and 10 kg/ha levels, so I don't know about these conditions. So an analysis of nitrogen as a categorical variable might indicate that there are treatment differences, while an analysis of nitrogen as a continuous variable may indicate otherwise.

You can develop a similar scenario with surveys using Likert scales. In taste tests of cookies, people cannot reliable distinguish between a like of 67 versus 68 on a 100 point scale. So one breaks the 100 point scale (continuous) into 3 to 10 distinct categories. The average score of 87.3 makes sense, even if a person could not give a rating of 87.3. So this is still a continuous variable, though it can be treated as categorical because that is how the data were collected.

Be careful with AIC. If your dependent variable is the one that is being modeled as continuous or categorical then my understanding is that something like AIC is not appropriate. This is just like A=mX+b versus log(A)=mX+b can't be compared with AIC and related metrics. If you have A=mX + nY +oZ +b versus A=m*log(X)+nY + b where Y is categorical in the second equation, then you are ok.

The statement that the choice should be theory driven is fine, except if we are doing cutting edge science then there is no proven theory to support our choice. In some cases we don't know enough to even start developing a theory. So, now what?

In part it is determined by the wiggle room in our treatment list. In the nitrogen example a great deal could happen between 1 gram/ha and 1 kg/ha. What if I had chosen my treatments as 1 g/ha, 2 g/ha, and 3 g/ha or 1 gram/ha + 1 molecule/ha versus 1 gram/ha + 2 molecules/ha. As the interval between treatments gets smaller it is less likely that strange things will happen. We also have to make some assumptions because we cannot have an experiment with Avogadro's number of treatments from 0 to 28 grams/ha.

As a general rule, I dislike turning a continuous variable into a categorical variable. The categorical variable contributes much less to our understanding of the system because it precludes any prediction of what might happen under other conditions. However, if we don't have enough data, then this might be a good thing.

Let me address another point: AIC and BIC cannot appropriately be used for this comparison. They are valid for comparing models on exactly the same data. Treating a variable as continuous vs. categorical changes the dataset.