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You are given a logistic regression equation that predicts the probability of having a disease dependent on whether or not three risk factors are present. For any given patient whose chance of disease you want to predict, you are able to find out whether or not they have the first two risk factors, but you have no information either way about the third. In a situation like this, is there any way the regression equation could reasonably be used?

Of course, you can't simply set the third term equal to zero, since that supposes the patient does not have the risk factor. Is there any other option? Say you know that the prevalence of the third risk factor in the patient's population is 25%. Could you set your third independent variable equal to 0.25 and then use the equation to estimate the patient's probability of disease?

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  • $\begingroup$ To clarify my question, let me add that in the situation I'm describing, I have no information about the third risk factor for any patients in my population. It's not the case that I have the information for some patients but not others. $\endgroup$ – LLP Dec 1 '17 at 15:26
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Solution 1: Missing value is set as a special category

This is especially adapted if the variable is already categorical: like "smokes", "doesn't smoke". You add the category "missing answer". This is when you consider a missing value is informative and you want to exploit this information: something you want to analyse.

Solution 2: simple imputation (not recommended)

The values are automatically filled. Practically they are replaced by:

  • the mean given the other known variables if continuous
  • the mode given the other known variables if categorical

This is known as "regression" in https://en.wikipedia.org/wiki/Imputation_(statistics)#Single_imputation. This may create a bias and is not recommended nowadays.

Solution 3: multiple imputation

The values are also automatically filled. The method considers the missing value has a distribution given the other variables and simulates several datasets according to this distribution. For example if the variable can have values A or B, and for a line in the dataset the probability of A/B (given the other variables) is say 30%/70%, datasets are generated by picking one of A and B with these probabilities. This require a special analysis that most stat software implement. You can read more here: https://en.wikipedia.org/wiki/Imputation_(statistics)#Multiple_imputation.

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The answer above is already very good, but I just want to clarify something about your question. It's not clear from your question whether you have any information on the third risk factor. Benoit's answer works only if the data is available for some respondents and missing for others. If you don't have any information on the variable for any of your patients, it doesn't do any good to impute missing data. You probably know that, but I just want to make sure the bases are covered!

One other option for how to deal with the missing data if you have less than 5% missings and the missing data is missing at random, is to just perform the analysis on the cases without missing data, aka listwise deletion or complete case analysis.

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  • $\begingroup$ Thanks for pointing that out. I meant that I don't have any data points at all on the third variable for my patients. I added a comment to my main question above to clarify. $\endgroup$ – LLP Dec 1 '17 at 15:29

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