No, there are no guidelines. This all depends on your data (as so often in statistics). Oh, well, there might be two suggestion be taken as "guidelines":

1) the categories should make sense practically/scientifically

2) the response should be homogeneous within each category

The best solution. however, was to have a sensible model of the time-course. Sure, this is not always available. But one should first think hard before one starts putting nice quantitative values into ugly arbitrary categories...

No, there are no guidelines. This all depends on your data (as so often in statistics). Oh, well, there might be two suggestion be taken as "guidelines":

1) the categories should make sense practically/scientifically

2) the response should be homogeneous within each category

The best solution. however, was to have a sensible model of the time-course. Sure, this is not always available. But one should first think hard before one starts putting nice quantitative values into ugly arbitrary categories...

Let me add to Jochen's excellent response. Reducing continuous data into categories can result in a loss of data and power (this is especially true when you dichotomize a continuous variable).  Keeping this in mind, you should try and strike a balance between keeping the data as rich as possible (that is, keeping the categories as close to continuous as possible), and categorizing them into meaningful buckets (in this case, so that you don't lose track of the relationship between the Y and X).

Have you considered using splines for the time variable? This may solve your problem and allow you to keep the natural time-periods.

Have you considered other methods for "very curvilinear" manners, like lowess, splines, etc.?  Depending on your theories/population, it would seem difficult to meet Jochen's suggestions for average weekly work hours. If you can, great (perhaps your sample are in a particular company with only three values for this), but my first choice is usually not to categorize.

Daniel and I appear to be tracking :-)

Yes, you can use splines. I did not think of this possibility.

(still a real model would be better, because this would be science)*

*such a "real model" could, for instance, "explain" that and how things change over the course of a week (for example the concentration may decrease from monday to friday, mayby this further depends on the age of the employee), one may see long-term trends, more or less convolved with weekly changes and so on. All such things get lost when you categorise or when you use splines. These are only some out-of-the-blue examples. It takes lots of good subject knowledge and hard work to build a reasonable, appropriate and helpful model, and time series analyses are generally among the most difficult candidates...

The problem is indeed how to justify that a solution "makes sense" (practically and scientifically). I uploaded a table (above) that shows what is sensible and consistent to me. I don't know yet how reviewers would find that though. Does it suffice to say that I made categories which are symmetrical (round hours and hours in between as separate categories) and that I chose the reference category for it appears to be the optimal with respect to its effect on Y?

I am not familiar with spines or lowess. If you believe it is worth checking, could you suggest some literature on the topic?   

While you can find suggestions in the literature, there is no theoretically "right" answer to this.  From a theoretical standpoint, for every additional category you create, it uses a degree of freedom which comes directly from the error degrees of freedom.  Thus, the error mean square is larger making your design less powerful.  If the variable is not a variable of interest and only one that is used to increase the precision of the design, you might consider using it as a covariate.  If it is a variable of focus and you must break it into ranges, logic should prevail.  If you have natural breaking points or theoretical breaking points, use them.  If not, decide how many groups you wish to have, divide into equal groups.  Since your variable, time, is a ratio variable, I think that might be your best approach.  Be careful to be sure there are enough observations in each group to be meaningful.  If the variable in question is less than interval, you might consider running a frequency distribution of the variable and braking it up so that there are an equal number of observations in each group.

A problem similar to yours can be found in sampling from finite population when you want to create strata (a set of dummy variables) “segmenting” a continuous variable. A popular rule in this context is known as the Dalenius-Hodges rule. It is based on the idea of minimizing the variance of the continuous variables within strata. A description may be found in the book from Sarndal, Swensson, and Wretman (1992, pp. 462–464). The rule may be used also to find out the optimal number of dummies, that typically ranges between 4 and 6. Of course, this rule does not consider the substantive meaning of the dummies, that can be relevant in many applications.

Similar to Enrico's suggestion, you could use a regression tree to decide where to split a continuous variable (and use things like cross-validation to decide how many pieces to break the variable into), but the commenters have pointed out if you don't have a reason for doing this you should avoid it and it is better to break it based on your theories than on where an algorithm chooses.

Could you show a plot of the data (response vs. time)? This would be far more instructive than the table.

ad James: Yes, there is usually a trade-off between increased flexibility by more groups and a loss of sensitivity by a decreased number of degrees of freedom. But the value of all this depends on the sample size we talk about. If we talk about d.f.=300 vs. d.f.=294, then I would not care about these lost 6 d.f.

Another remark - only if hypothesis tests and p-values are required:

If you search the way how to analyze the data best based on the available data, then any hypothesis tests in the final analysis on this same data are biased. It is perfectly fine to find a good analysis/model based on the available data, but the same data must not be used for hypothesis tests. If there is enough data, one can split the data (randomly or stratified) in two groups, find the good analysis/model in one group and perform hypothesis tests on the other group. If there is not enough data available, well, then there is not enough data available. Testing hypotheses might then require to sample additional data.

For free literature on splines and lowess, an introduction is available on http://www-bcf.usc.edu/~gareth/ISL/ (which includes R  code to run these analyses), more advanced coverage in http://statweb.stanford.edu/~tibs/ElemStatLearn/ and see their recommendations (at the end of the chapters in ESL) for further recommendations. Simon Wood's work (http://www.maths.bath.ac.uk/~sw283/) can also be consulted. 

I added a frequency distribution above as I am not sure if a plot would be very informative. 

I was also wondering about the issue of hypothesis testing in this case. In fact the theory (and previous studies) suggest a linear relationship but that is not what I find. I am not sure how to conceptualize the paper now. I assume I have to offer some new theoretical framework which can explain my findings? That would be not theory-testing but theory-building approach, how do you think? 

you could try cox regression if you have continuous time variable

Based on the data that you have shown, it looks like you already have categories (although in a linear pattern). You could choose to further collapse those categories to make them more meaningful based on whatever your hypothesis is. For example, if you believe mothers who do not work would have different results than mothers who do, that may be a meaningful category that you could use in your analysis. If you believe that mothers who work full-time or more would have different outcomes that mothers who work less than full-time, that may be a good category to use. 

Sebrina's suggestion is very good. There are a class of models called zero-inflated models, that can be used when the variable with zeros being qualitatively different from non-zeros is on the left-side of the equation. When on the right side, one dummy variable for work v non-work and then whatever curve you want to model the work variable.

Dear Akallyski,

I also agree Mr Wright, I do not think it is a good choice to categorize a contiunous variable. In addtion, your categories of the working hours, such as, 1-9, 10, 11-19, 20, and so on, do not make sence scientifically. The better categories should be 1-5,6-10,11-15,16-20, etc. It is more practical and scientifical.

Dear Sebrina, Daniel and Wei-Sheng, 

As I already mentioned, the problem I have is that the theory suggest a linear relationship, i.e. with each increase of work hours the outcome variable also increases, but this is not what I find. The relationship is in fact very curvilinear, as you can see on the table. Combining work hours based on theory, e.g. not working, working part time, working full-time and working long work hours makes less sense practically though. According to Jochen Wilhelm "the response should be homogeneous within each category". Do you suggest I should just bury the evidence by combining heterogeneous groups together just because the theory suggests that or because it is more scientific? I am reluctant to do that.  

Hi Plamen,

I would plot them with hours in the smallest increment you have (if it is just hours you will need to jitter the points) and the outcome variable on the y axis. I think the scatter plot is my favorite statistical procedure.Sometimes this is enough to show what is going on, but a lot of packages will add a lowess curve or similar to show patterns.

Given the number of cases at 30 and 40, those seem qualitatively different. Do you have occupation variables in your data set? If not, you could presumably get data from the gov't to show which types of jobs are more likely to have these round numbers.

If all you are worried about is saying whether the theory is right or not (the linear one), then it sounds like a scatter plot is fine. But it depends why you are doing this.

Dan

In addition to what Daniel recommends, I might also hypothesize that there may be a "3rd variable" problem that may be explaining some of these results. For example, household income (and the proportion of the income that belongs to the mother), relationship/household status (single parenting vs. co-parenting), employed in a career vs. employed in a low-skill job, these could be affecting your results.  If you are not already accounting for these and have them in your dataset, you may see if adding one or more changes your results to fit with the existing theory.

I would also echo Wei-Sheng's comments that you may want to attend to the categories that your data is currently in. Right now, you have 1-9, 10, 11-19, 20, etc. I would suggest considering, at the least, collapsing the single digit categories with the ranges. (e.g.- 1-10, 11-20, 21-30, etc). I do not think it would be "burying results" to do so.

Dear Daniel, 

of course, I started my analysis with a lot of plots and they also demonstrate non-linearity. One complication is that I am using longitudinal data and I found that the cross-sectional patterns differ from the longitudinal ones, which could be due to omitted variable bias, as Sebrina suggests. 

(Sebrina) I use fixed-effects model for analysing longitudinal data, which controls for all unobservable time-constant characteristics of the individuals. From those that can differ over time, I also control for family income, number of children in the HH, family structure, work hours of the partner and occupation. 

With regard to what I am doing that for, well, I can prove the theory correct to some extent because long work hours indeed have the strongest positive effect, and by combining categories based on "theory and practice" I can easily bury the disturbing findings. I think it would be way more interesting paper if I demonstrate this curvilinear effect though, even if I am not able to explain it. I believe it is scientifically more correct as well. Don't you agree?

You can decide on no of categories by looking at the range.Usually the number of categories should not be more than 7.The reference point can be always - to start with at the lowest category,and if you want to compare the effect of long duration of work on outcome then this category can also be used as reference point.

Venkat 

First look into the values of quartiles.

Then categorize

1st class interval =  lessthan 1 st quartile

2nd class = 1stquartile to 2nd

etc