# Imputed binary response outcome

I am currently working on a group project with COVID data. The response variable is a binary outcome (positive/negative) from swab test; and let's only consider age groups (0-18, 18-60, 60+) for simplicity here. The goal here is to obtain some incidence rates and relative risks among the age groups by, for example, poisson regression.

In my situation, there are lots of subjects with missing outcome data. For these subjects, we are given the imputed "probability that the subject would have tested positive if they had been successfully swabbed". From what we were told, we would replace the missing entries with those probabilities. So now the data looks like:

ID age_group covid
1 1 1
2 3 1
3 2 0.678
4 2 0
5 1 0.221
6 3 1
... ... ...

I am very confused at this point since the response is no longer binary. What exactly is an appropriate method for achieving the goal?

• Strictly speaking, if your covid data is not binary, you shouldn't use a binomial model (logit/probit), however if the shape of the output behaves non-linearly you could use a logistic regression or, in best scenario, a linear regression, depending on the assumptions. May 23, 2022 at 17:44

It might be best to use the estimated probabilities to generate multiple data sets, each of which has a 0/1 outcome for each case based on your imputed probability values. Depending on further details of missing data and the ultimate modeling strategy, you might even be OK just omitting the cases with missing Covid-test results.

As described in van Buuren's Flexible Imputation of Missing Data, imputing multiple data sets with probabilistic models of the missing data values is a general approach typically superior to single imputations. A rule of thumb is to generate a number of data sets equal to the percentage of cases that have missing data.

The outcome model is fit to each of the multiple imputed data sets. That allows you to use logistic regression or count-based Poisson or negative binomial regression directly on each imputed data set. After all data sets are analyzed, results are combined in a way that directly accounts for the uncertainty of imputation.

It's not clear how the probability values were imputed in your situation. If you are comfortable with the imputation method, then you could do this for each imputed data set by taking a random binomial sample from 0/1 based on the indicated 0/1 probability for each ID with a missing test result.

You might, however, do just as well to omit the cases with missing outcomes, provided that only outcomes are missing and you have no predictor variables other than those that will be included in the ultimate model. Quoting van Buuren:

Suppose that the complete-data model is a regression with outcome Y and predictors X. If the missing data occur in Y only, complete-case analysis and multiple imputation are equivalent, so then complete-case analysis is preferred since it is easier, more efficient and more robust.

He continues, however:

Multiple imputation gains an advantage over complete-case analysis if additional predictors for Y are available that are not part of X.

For logistic regression with 0/1 outcome values in particular:

Suppose that the missing data are confined to either a dichotomous Y or to X, but not to both. Assuming that the model is correctly specified, the regression coefficients (except the intercept) from the complete-case analysis are unbiased if the probability to be missing depends only on Y and not on X.

That would allow you to estimate relative risks, although incidence rates (which depend on the intercept) might not be correct. Think carefully about whether your study design can provide unbiased estimates of incidence rates even with the imputed values.

Finally, if values of predictor variables were also imputed singly in the data set provided to you, it might make sense to go back to the original data set with missing values and do multiple imputations of both outcome and predictor values together.