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I know that in theory for regression both the Y and factors should be continuous variables. However, I have some factors that are discrete but show both correlation and would fit a regression model.

I am looking at energy consumption and my factors are the number of calls, the data transferred, temperature, customers, number of buildings. The only continuous variable is the temperature.

Any advice if my multiple regression model is still applicable even though I have factors temperature and subscribers?

How about correlation? Would I be able to say that there is in fact say correlation between the number of customers and energy consumption?

Thanks for your help!

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    $\begingroup$ What are you referring to where you note (at the beginning) that regression requires that "factors should be continuous"? I am not aware of any such assumption. Indeed, because ANOVA is an application of (least squares) regression, and all of its factors are discrete, it would be terrible indeed to discover that it is not valid! $\endgroup$
    – whuber ♦
    Commented Mar 13, 2013 at 22:48
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    $\begingroup$ To reiterate @whuber 's point: There is nothing that says you can't do a regression with independent variables that are all discrete; if you actually read that in a book, throw the book away. More likely, you misinterpreted something. In addition, you can do regression when the dependent variable is discrete: e.g. various kinds of logistic regression, Poisson regression etc. $\endgroup$
    – Peter Flom
    Commented Mar 13, 2013 at 22:56

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The word "factor" should be used more carefully, because for some statisticians and some software packages "factor" can mean categorical variable (e.g. different types of treatments, sex, countries of origin, etc.) A "continuous factor" would sound like "corner of a circle" and confuse the heck out of the people. You may, in future, be able to more clearly express your idea if you just describe it as a "discrete independent variable."

Both continuous (number so fine that you can't name the exact point) and discrete (consists of whole numbers) variables are considered as interval/ratio. They are treated the same way when used as an independent variable in linear regression analysis. The way to discern an interval/ratio variable is to ask if every unit increment in the variable indicates the same amount of increment in the context you wish you measure. For instance, the jump from 35 to 36 degrees is the same as the jump from 43 to 44; it's the same amount of temperature difference. Likewise, the jump from 100 to 101 subscribers is the same as the jump from 1009 to 1010 subscribers. As long as this is true, your regression coefficient of that independent variable will make sense, because you can legitimately interpret it as the slope of the regression line.

General confusion appears when you mix in ordinal data, such as those 5-point "how satisfied are you?" questions. They are expressed in whole number, very easily to be confused with discrete data. However, each jump in the scale does not necessarily mean the same thing. E.g. a jump from "4: happy" to "5: very happy" is not necessarily the same as a jump from "1: very unhappy" to "2: unhappy." In that case, the variable should not be put into the regression as is, but treated differently (search "dummy variable in regression" to learn more.)

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    $\begingroup$ Re the end of the first paragraph et seq. Ironically, "discreet" is not the opposite of "continuous:" it is an antonym for "blatant." $\endgroup$
    – whuber ♦
    Commented Mar 14, 2013 at 16:13
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    $\begingroup$ It would help to make that revision systematically throughout your answer, lest your readers get the impression you think whole numbers are circumspect (and are delivered in plain brown wrappers) but ordinal data are not :-). $\endgroup$
    – whuber ♦
    Commented Mar 14, 2013 at 18:52
  • $\begingroup$ @whuber Control+F the whole thing. But wouldn't we all want our data to be a bit more discreet? Again, thanks. $\endgroup$
    – Penguin_Knight
    Commented Mar 14, 2013 at 19:20
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    $\begingroup$ Great! Now let's discuss the substance of your answer. I do not see how "continuous" = "interval/ratio." In fact, that point of view seems counterproductive, because it inhibits people from looking for nonlinear re-expressions of their "continuous" data. Next, "discrete" is a huge category. Distinguished within it are counted data, which have an inherent meaning and statistical properties, but otherwise there seems to be no universal equation of the form "discrete" = "interval/ratio" or anything like it. The whole nominal/ordered/interval/ratio classification is really not much help. $\endgroup$
    – whuber ♦
    Commented Mar 14, 2013 at 19:55
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    $\begingroup$ @whuber +1 -- some important points in there $\endgroup$
    – Glen_b
    Commented Mar 14, 2013 at 22:11
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To further understand the similarities between continuous/discrete interval and ratio variables, consider measurement precision. A continuous variable can only be measured to a certain level of precision, and as such, in reality, can only take a discrete set of values. (ie- if you are measuring with a tool of precision 0.1, the only values you will receive are 0.1,0.2,0.3, etc.)

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