Solved – Logistic Regression: Interpreting Continuous Variables

logisticregression

I had a couple of questions about interpreting odds ratios for continuous variables in logistic regression. I feel like these are basic questions about logistic regression (and probably about regression in general), and although I'm slightly ashamed that I don't know the answers, I'm gonna swallow my pride and ask them so I know them in the future!

Here's my situation…I'm looking at a sample of adjudicated youth who, as part of their probation, were enrolled in a job/life skills training program. I wanted to see the extent to which the age at which they were released from the program predicted employment six months post-release from the program.

(Also, keep in mind that there are other predictors in the model, but I've excluded them because they are not statistically significant and I want to keep this as clear as possible.)

Predictor:
Age of release from training program (Mean age = 17.4, SD=1.2, Range 14.3-20.5)

Outcome:
Employed or not (Employed=1, Not Employed=0)

Result:
Odds ratio 3.01 (p<.005)
(I've excluded goodness of fit stats, etc. because I'm seeking answers about the interpretation of the odds ratio only; I feel comfortable w/ the evaluation of model fit, CI's, etc.)

Putting it into words:
As age increases by one year, the odds of being employed six months post-discharge increase by three units.

Questions:

1) When I say, "As age increases by one year…" what is the starting point for age?

Does age start at zero? For example, "As age increases from 0 [i.e., the lowest age if you were to place this model on a graph]…”

Does age start at the the lowest age among the range of ages in the sample? For example, “As age increases from 14.3…”

Does age start at the mean age of the sample? For example, “As age increases from 17.4…”,

2) Would centering help me interpret this result OR is that only effective in interpreting the y-int?
If it would help, I was thinking of doing either mean centering or subtracting the lowest age in the range from all the other ages in the sample.
Any suggestions?

3) Finally, is it appropriate to say that compared to a 14-year-old youth, a 17-year-old youth is nine times more likely to be employed?
I ask because I know that logistic regression assumes a sigmoidal relationship, and I’m curious as to whether this 3 unit increase in odds remains consistent at any point along the regression line.

Thanks so much!

Aaron

Best Answer

1) Since it is an odds ratio it doesn't matter where you start. The odds for an 18 year old are 3 times those for a 17 year old. Or the odds for a 17 year old are 1/3 those of an 18 year old. Same thing. If you want to get the probability that a person of a particular age will be employed, you can use the formula with the parameter estimates (not the ORs). Or you can get the program you are using to do it for you.

2) Whether centering helps is a matter of opinion. I don't find centered models clearer, but some people do.

3) The odds are not exactly the same as "likely" (although many people speak as if they were) and the odds for a 17 year old would be 27 times those of a 14 year old.

Finally, I'd be cautious about this model. The model assumes that the OR is the same between 14 and 15, 15 and 16 and so on. That seems unlikely to me, based on what I know about the subject.

It is allowed (in principle) but it is probably not a good functional form

It is okay to do this in principle, so long as you are aware of what this means in terms of the meaning of the predictors in your model. You need to scrutinise whether this leads to a sensible relationship between age and your latent response variable. In principle it is allowable to use any functional form between your age variable and your latent response, but in practice you might get a functional form that is implausible. Let me give you some guidance on how to judge this.

Ordinarily, when we put a continuous age variable as a predictor in a logistic regression model, this means that we are proposing a linear relationship$^\dagger$ between age and the latent logistic response (i.e., the latent continuous response underlying the binary outcome). If you add binary predictors for particular age categories (e.g., teens, seniors, etc.) on top of this, then this means that the proposed relationship now allows "jumps" at the boundaries of those age categories. This means that you now have a piecewise linear function where the piecewise lines are parallel, but there are discontinuities due to jumps at the category boundaries.

Your proposal is a generalisation of the straight-line function and it is allowable as a model input in principle. However, while this type of proposed functional relationship between age and the latent logistic response is allowed, it has some obvious drawbacks that make it implausible in most cases --- particularly the lack of continuity. Ultimately, you need to decide whether or not it is sensible in the context of your problem to allow "jumps" in the effect of age at the category boundaries of your age groups. In most cases this it will be quite dubious to believe that becoming a teenager suddenly causes a jump in the latent response variable, so the type of model you a proposing is rarely going to be sensible.

A more sensible alternative to what you are proposing is to use a piecewise linear function that imposes a continuity requirement on age, but allows different slopes over different age categories (called a linear spline). (This is achieved by using interactions between the age variable and the dummy variables and imposing appropriate constraints on the parameters; full details of linear splines are outside the scope of the present answer.) This gives a reasonably flexible functional form for the age variable, allowing for differing effects in different stages of life, but also imposing a continuity requirement on the overall effect of age. While this is probably an improvement of your proposal, it is still a rather crude form for the relationship, and it would naturally lead to consideration of higher-order spline functions (e.g., a cubic spline) as a further refinement of the relationship. Ultimately, I think the line of thinking you are pursuing is likely to lead you to consider some kind of higher-order spline function for the age variable, with your age categories forming the intervals used for the splines (and their boundaries giving the x-values for the knots).

Regardless of what particular form you decide to use, it is important to interpret the effect of age holistically using all the model variables that are determined by age. Consequently, you wouldn't interpret the main age term or the dummy terms independently --- you would interpret them together as giving the overall effect of age on your latent logistic response.

$^\dagger$ Strictly speaking, this is an affine relationship if we include consideration of the intercept term. I will call it "linear" here as a shorthand for a straight-line function.

Best Answer

Related Solutions

Solved – logistic regression backwards selection

Is it OK to include a continuous predictor variable and dumthe predictors based on that continuous predictor

It is allowed (in principle) but it is probably not a good functional form

Related Question