You should really think about what you want out of your analysis. In principle it is not a good idea to force a continuous function into a categorical variable. When converting a continuous variable (time) into a categorical variable, you are likely to distort your model. The distortion arises as a mismatch between cycles in the continuous variable interacting with the cut-off values in the categorical variable. Visualize a sine wave function going from 0 to +1. Now sample this in different ways, and calculate an average value for each category. Option 1) Each category level is a full cycle. Average=0.5. Option 2) Each category is a half cycle. Average=0.5,0.5. Option 3) Each category is a quarter cycle. Average=0.25, 0.75, 0.75, 0.25, and this pattern repeats. Option 4) Each category is 1.25 cycles. Average=0.45, 0.55, 0.65, 0.55, and this pattern repeats. Thus the way you build categories influences the answer that you get. Everything gets more difficult if you do not know what the cycles are, and worse yet if the cycles change in periodicity from month to month or year to year. It is even possible to conceive a scenario where the results are nothing more than an artifact of how the categorical variable was developed from continuous data.

If December 12 is one of the categories, try Julian day (1, 2...365). Day is a fairly natural biological cycle, and I would expect fewer problems. It also gives you enough "categories" that you can treat them as continuous in nonlinear models.

Variables like risk need to stay as categorical. The example given is called a Likert scale in social sciences research. Wikipedia gives a nice overview, especially with the references cited therein. If you have access, use one of the large bibliographic databases (e.g. Web of Science, AGRICOLA, CABI) and search for Likert. You could also try the newer approach that Paul advocates. However, even if the analysis methodology that you choose is independent of the underlying distribution of the data, you will still need to understand the issues associated with Likert scale data in order to discuss the results in terms that reviewers will accept.

You should really think about what you want out of your analysis. In principle it is not a good idea to force a continuous function into a categorical variable. When converting a continuous variable (time) into a categorical variable, you are likely to distort your model. The distortion arises as a mismatch between cycles in the continuous variable interacting with the cut-off values in the categorical variable. Visualize a sine wave function going from 0 to +1. Now sample this in different ways, and calculate an average value for each category. Option 1) Each category level is a full cycle. Average=0.5. Option 2) Each category is a half cycle. Average=0.5,0.5. Option 3) Each category is a quarter cycle. Average=0.25, 0.75, 0.75, 0.25, and this pattern repeats. Option 4) Each category is 1.25 cycles. Average=0.45, 0.55, 0.65, 0.55, and this pattern repeats. Thus the way you build categories influences the answer that you get. Everything gets more difficult if you do not know what the cycles are, and worse yet if the cycles change in periodicity from month to month or year to year. It is even possible to conceive a scenario where the results are nothing more than an artifact of how the categorical variable was developed from continuous data.

If December 12 is one of the categories, try Julian day (1, 2...365). Day is a fairly natural biological cycle, and I would expect fewer problems. It also gives you enough "categories" that you can treat them as continuous in nonlinear models.

Variables like risk need to stay as categorical. The example given is called a Likert scale in social sciences research. Wikipedia gives a nice overview, especially with the references cited therein. If you have access, use one of the large bibliographic databases (e.g. Web of Science, AGRICOLA, CABI) and search for Likert. You could also try the newer approach that Paul advocates. However, even if the analysis methodology that you choose is independent of the underlying distribution of the data, you will still need to understand the issues associated with Likert scale data in order to discuss the results in terms that reviewers will accept.

@ Paul,

I agree with your discussion about the problems with Likert scale data. You could add to the discussion a problem that even for a single individual, a bunch of external factors lead up to a specific answer on a specific day. However, there are no good alternatives.

Discussion, not related to original question: Ordinal data versus categorical data.

I think of ordinal data as data where position makes a difference. Feeding behavior in aphids might be an example. The aphid starts in the "non feeding" behavior. Its mouthparts touch the plant, and then penetrate the cuticle. The mouthparts can then go either towards xylem or towards phloem. The mouthparts cannot do both, and they cannot go from xylem to phloem without some intervening behaviors. Such data is a mix of ordinal (the order in which events occur), categorical (the types of events), integer (counts of the number of times each event occurs), and real (duration of each behavior).

So the question is whether a Likert scale is ordinal.

I have dose mortality data. For each dose from 1 ug/kg to 100 ug/kg I have a corresponding mortality from 0 to 100%. Clearly I cannot get to the 10 ug/kg treatment without being exposed (no matter how briefly) to the 5 ug/kg treatment. So dose is ordinal? Likewise mortality is a unidirectional variable (monotonically increasing). Is it too ordinal? I think the common response is no.

So how is the Likert scale ordinal?

Yes, the scale might be measuring a learned response. However, I can like chocolate a great deal without ever having to go through the "hate it", and "not dislike it" categories (at least I cannot recall a time that I ever hated chocolate).

I think the likert scale is categorical. I could always rank things from 0 to 100 with all real values between. This level of resolution is much beyond the resolving power of most subjects. So we break the scale into categories and change the presentation. A score of 0 to 10 is "hate" while 90 to 100 is "love." This is an artificial categorization. A natural categorization would be coding males as 0 and females as 1 (or vice versa). At least in many systems there are no other choices, and nothing in between.

So why is Likert scale data ordinal and mortality data not? I don't really see any difference between the two, and mortality data are not usually described as ordinal.

Dear Rami,

In my view the simplest way and first try should follow the above suggestion by Rami Ben Haj-Kacem.

Best wishes,

Frederico

Paul Yarnold's answers give you the full scope of the issue here and do a fabulous job of laying out the complexity of your question.  But at the base of the question is, what do you want to learn? 

Because you mentioned logistic regression, I would infer you are looking for a change in likelihood for some dichotomous outcome.  Treating your predictor on a continuum will provide information about how the likelihood changes as the predictor increases in level.  But that estimate assumes (even if you do not) that the space between intervals is the same.  One approach might be to divide up your data relative to your interval responses (i.e., just look at cases that responded 1 or 2, then just look at cases that responded 2 or 3, etc) to see how the pieces look.  You may find that you can learn empirically how the intervals behave. 

  I cannot emphasize enough - statistics are only a tool, which must serve the needs of the research question.

Hi Ranjeeta! I agree with Julia. Perhaps you should clarify the research question for us. From what I understand, you're worried about the coding of month and risk. I would use dummy variables for both. What statistical program are you using?  The problem with having so many dummy variables, as previously mentioned, is the lack of stability of the regression model, especially if you don't have a large sample size. You can also group the months by quarters. 

@Paul

So I want to hierarchically classify all types of data.

The first decision is ordinal versus categorical. In this case the categorical data are things where mathematical operations do not apply. For example color. Red+blue = ??? Numerically I can convert red to 700 nm, and blue is 450 nm, so the sum is 1150 nm, which is infrared. Purple may be what we see, and it is possible to model the result mathematically, but adding and subtracting colors doesn't seem to work with any of the standard mathematical principles (inequalities, distributive laws of addition, and so forth). One could convert the colors to their wavelength equivalent, but in our perception of red there is nothing that constrains a relationship Red>Blue.

If you have ordinal data then there are subtypes of ordinal data. The most common are integers and real numbers. There is also a type of categorical variable where a specific ordering is clearly present. The Likert scale variables go here. Dummy variables would also go here: e.g. nitrogen fertilizer at low, medium, and high levels. Your example of surrogate variables would also go here. That example was enlightening as I would usually use gender as an example for the preceding paragraph.

Did I get this right?

Dear  Ranjeeta Jain 

In logistic regression when you have a nominal independent variable, you have to create dummy variable and use "categorical" button in SPSS. You must select first or last code as reference. But in your question it seems that the variable "Risk" id dependent. If so you should perform multinomial logistic regression. Here is a useful link. Good luck

Basically data can be grouped into: nominal, ordinal, interval and ratio, the first two are categorical and last two are numerical. Many users often treat ordinal as nominal, which is usually lost power or over fitting. In the regression, the parameter –effect- of a continuous factor is a slope: the change of dependent variable per 1 unit change on continuous predictor, but for the categorical predictor, if it has k levels, k-1 parameters are needed, they are the difference on the dependent variable between two categories of the predictor. For classic linear regression, the dependent variable is itself, which follows a normal distribution. For binary outcome, the dependent variable for logistic regression is log (odds).

Logistic regression is used to predict a binary response from a binary predictor, used for predicting the outcome of a categorical dependent variable based on one or more predictor variables. That is, it is used in estimating the parameters of a qualitative response model. The probabilities describing the possible outcomes of a single trial are modeled, as a function of the explanatory (predictor) variables, using a logistic function.

However if the outcome is not binary, it is categorical with multi levels, multinomial logistic regression is used when the dependent variable in question is nominal (equivalently categorical, meaning that it falls into any one of a set of categories which cannot be ordered in any meaningful way) and for which there are more than two categories.  Such as study which blood type does a person have, given the results of various diagnostic tests,  which country will a firm locate an office in, given the characteristics of the firm and of the various candidate countries? Suppose there are 4 categories, 3 logistic function are used as dependent variables, log (pi1/pi4), log(pi2/pi4), log(pi3/pi4) are common used. You will get 3 interceptions, three sets of parameter estimations.

As predicator, a nominal variable is one that has two or more categories, but there is no intrinsic ordering to the categories.  For example, gender is a categorical variable having two categories (male and female) and there is no intrinsic ordering to the categories.  Hair color is also a categorical variable having a number of categories (blonde, brown, brunette, red, etc.) and again, there is no agreed way to order these from highest to lowest.  A purely categorical variable is one that simply allows you to assign categories but you cannot clearly order the variables.  If the variable has a clear ordering, then that variable would be an ordinal variable, as described below. For a regression, if the level of such variable is not so high, you just use them as dummy variable in the regression. But if the level is too high, such as states of USA, there will be 49 parameters in the modeling. It might be too high. Usually you may combine them as regions: NW, NE, SW, SE, etc.

However, sometimes the categorical predictor might be ordinal: an ordinal variable is similar to a nominal variable.  The difference between the two is that there is a clear ordering of the variables.  For example, suppose you have a variable, economic status, with three categories (poor, low, medium and high).  In addition to being able to classify people into these four categories, you can order the categories as low, medium and high. If we know or we guess the outcome is monotonic on it,   we can assign scores 1, 2, 3 and 4 to these four levels of income), or score as 1,2,3 and 5, if you think the difference between medium and high is large than the difference poor and low, low and medium. Then use score as number rather than dummy in the recreation.  One parameter is estimated rather than three for dummy approach. You may uses Akaike information criterion (AIC) to compare, dummy approach, 1,2,3,4 approach and 1,2,3,5 approach, the model showing small AIC is better.  

Ordinal models constrain the relationship between regressors and outcomes in specific way: suppose that it has a positive effect. As increases the lowest outcome category decreases in probability from 1 to 0 while the highest increases from 0 to 1. The probabilities for other categories are bell-shaped with modes that increase for higher categories.

As the month factor, there are twelve months. You should base on your outcome to decide which way you will use in the regression:

If the outcome (dependent variable) is monotonic, you use month as number: say Pass rate in driving ~ months on the training

If it has season effect: Say

Store Sale benefit~   categorize month into  Winter including Thanksgiving,  Christmas , new year,  or not

Or :Temperature ~  Spring, summer, fall winter

You can make plot between dependent variable vs month, to decide how many categories needed, and how to categorize months.  Using nominal or ordinal approach.

Paul, I love your answer, but I have a question - if you don't use dummy variables, how do you handle gender?  Or do you not have to?  In my work, I have to live with the variables that come to me from the feds, and they do not administer the (infinitely more detailed and informative) multidimensional gender identity scale, but just ask the respondents to indicate male/female.  Is it just the life of a secondary data analysis expert that has to roll with the punches and use the dummy variable covariate?

Nice reference - I will include it in my next quant. class.  Thnx!

Ok, so the problem is that I don't use ordinal, and I don't use nominal, and ratio in my world is not a distinct category of numbers. I had gotten to the point where I think I understood Paul. I didn't care if I agreed, but I just understood. 

In discussing things with Paul, I realize that I use a mix of computer definitions and mathematical definitions. Alpha variables are those that only have a name: e.g. "red." Numeric variables are all numbers (an alphanumeric keyboard has both numbers and letters). [Yes, there are hexadecimal numbers, but all elements obey standard mathematical operations despite what they look like.] Numbers can be real or integer. I would accept complex and irrational as some other categories, though I don't usually see this type of number in my data.

One could make a bunch of subcategories for real numbers, but it is more important to understand that ratios have a specific distribution and in some cases will need to be transformed to fit into standard statistical models. Ratio was never a separate class of number.

I can accept interval as a different class, but the definition that I first encounter is that interval is part of a set. If I have the numbers {2,4,8,16,32,64} then the interval with 8 and 32 will by definition also contain 16. I can't think of any examples in data analysis.

There are subcategories for Integers. Integers are all whole numbers from -infinity to + infinity. Natural numbers are integers from 0 to + infinity. Within the natural numbers are the cardinal numbers and ordinal numbers. Cardinal numbers are used for counting. Ordinal numbers supply order: e.g. the 3rd fish. While this came from Wikipedia, it is basically what my children are being taught in elementary school.

Wikipedia has a different page if you search for "ordinal number." Here, part of the definition is that order matters. Thus 1+a is not the same as a+1. The size of 1 depends on where you start. While I can see this as a distinct class of number, I have yet to see this type of data. So I am not sure this is a useful definition in a statistical/data analysis context.

I am ok with Paul's definition of Ordinal. I just don't see much use for the interval category, and the ratio category is bad -- unless these too have a definition that is not something that I expect. Unfortunately, I didn't understand most of Yuanzhang's comments.