it is frowned upon to collapse a continuous variable into a binary variable. You lose valuable information. your power decreases, and you run the risk of getting results that may not make sense (as you noticed yourself). I am attaching a couple of articles that discuss this issue. One discusses the general topic for both dependent and independent variables and other focuses specifically on the outcome variable.

I hope this helps!

Ariel

it is frowned upon to collapse a continuous variable into a binary variable. You lose valuable information. your power decreases, and you run the risk of getting results that may not make sense (as you noticed yourself). I am attaching a couple of articles that discuss this issue. One discusses the general topic for both dependent and independent variables and other focuses specifically on the outcome variable.

I hope this helps!

Ariel

Great article, Paul!

Thank you both for your responses. Allow me to add a final paper on the list.

It explains that in small samples and increase in correlations may be observed. Large samples a loss of power.

The practice is so common that it feels natural at times dichotomizing variables.

The paper describes also viable solutions when dichotomizing should be acceptable. When a variable is conceptually split in levels through certain thresholds. However, the authors classify that thresholds should not be the median or mean.

Thanks for this paper, Michael. I'll add it to my collection. Note the last sentence in the abstract: "The authors present the case that dichotomization is rarely defensible and often will yield misleading results." This is entirely consistent with the other papers that Paul and I provided...

You can also find very useful discussions about this topic from the Frank J Harrell's book and courses : http://biostat.mc.vanderbilt.edu/wiki/Main/RmS

I quote : "Categorizing continuous predictors into intervals is a disaster"

A good reason for creating binary predictors from continuous predictors is to overcome the problem of linearity: linear regression assumes that the relationship between Y and X is a straight line or that you can transform X and/or Y so that the relationship is linear. Note that Y = b0 + b1.F1(X) + b2.F2(X) + ... is still linear, i.e. the expression must be linear in the parameters b0, b1, ... for linear regression to be an appropriate model. If you cannot find an appropriate equation to represent to relationship between Y and the X-variables, then creating binary predictors might be a way of at least obtaining some predictions.

With a set of special circumstances it may be reasonable to change a continuous variable into a binary response. An example could be an unstable population that is mostly at low abundance but occasionally switches to outbreak proportions. Within each state the abundance is a bit variable, but there is no overlap. So one could count the number of grasshoppers per m2 and if this number is < 10 then abundance is zero, if the number is >10 then abundance is one. The difference between 1 grasshopper per m2 and two grasshoppers is not relevant to the dynamics of the outbreak and such detail introduces irrelevant variability into the model. Maybe grasshoppers are difficult to count. If the true density was 2 grasshoppers per m2, Joe will record 2.4, Lisa will record 1.4, and George will record 1.8 (on average). In a real situation, one may not have the resources to estimate these values so one cannot apply a "correction factor" to each person's data. Changing counts into a binary variable may avoid the problem. Chick survival depends on other food resources when grasshopper counts are low, but greatly boosts survival when grasshoppers are plentiful.

In general, I agree with the quote submitted by Lucas.

In your case, I don't really see alcohol use as a binary variable for predicting domestic abuse because there are too many confounding factors. Gender, weight, other drugs, past history, ... and for the population is the proportion of these factors within the sampled populations the same across all 50 states? There will be overlap in the categories abuse versus not-abuse and converting to this binary state will result in many misclassifications. So with a binary variable you will have to argue that the misclassification probability is the same across all 50 states. I would be very uncomfortable making that argument.

The answer was written 12 days ago, but I have difficulty to submit it until 10 minutes gao.

I do agree with Ariel, using binary to replace the continuous factor will lose information.  However, if the relationship between the outcome and the predictor is non-linear, or even not non-monotonic, say BMI (body Mass Index), over weight is not good for health, too skinny is also bad for health, categorizing or windowing the continuous factor into multi-level category (or using k-1 binary variables to describe k categories)  will be necessary and  get better model fitting and power rather than to use it as a continuous factor.  Akaike Information  criterion can be used to compare the model fitting: which one should be used?

Dear Yuanzhang,

If I understood correctly, you are suggesting using AIC to determine if you should use 5 versus 8 levels. I'll you BMI to try and visualize the choices (these are completely fictitious).

1) Soldier success = BMI + gender + married + training

2) BMI = dietary fiber + exercise + gender + caloric intake + heart rate

I now want to convert BMI into a categorical variable, but I don't know how many categories to break it into. Probably any integer from 3 to 10 could work, though probably closer to three. So I will change BMI into BMI3 if there are 3 states, and BMI10 if there are ten states.

In the first model, I could make seven models and use AIC to select the best one.

In the second case, I don't think this will work. It is my understanding that AIC is good for comparing models with the same dependent variable. So I can compare y=x1+x2+x3 versus y=x1+x2+x3+x1*x2+x1*x3.

It is not appropriate for comparing y=x1+x2+x3 versus z=x1+x2+x3 or log(y)=x1+x2+x3.

Do you agree, or have I missed something?

In fact, we should decide the number of levels by variety of reasons:

1.      Check sample size, avoiding a group only have a few records.

2.      After grouping, checking the association, if the association of two neighbor groups are similar you may combine them

3.      Using graph to see if non-linear relationship exists

4.       If possible, using fewer groups.

You may use AIC to verify it, but not only depend on AIC.

If you have 5 groups, you need adding 4 parameters, if you use 4 groups you needing 3 parameters. For BMI, 5 or 4 groups are probably good enough.

For categorical predictor, I think 10 is too many in general. You need generate 9 dummy variables, some of them will be similar.   

AIC should be used for same data, same outcome, only the predictors are different

This approach of generating a binary variable from a continuous variable is frowned upon, as stated in previous posts and supported by articles provided. Unless you have a very good reason for doing so, I would recommend that you leave your variable continuous.