# How to use a multiple regression model with missing inputs?

I'd like to be able to use a multiple linear regression model even when I don't have all the independent variables. Ideally, I'd also be able to calculate some indicator of confidence too.

Using the following example (shamelessly borrowed), we could imagine that I'd want to predict blood pressure without the "Age" factor.

| Independent Variable       | Regression Coefficient | T     | P-value |
|----------------------------|------------------------|-------|---------|
| Intercept                  | 68.15                  | 26.33 | 0.0001  |
| BMI                        | 0.58                   | 10.30 | 0.0001  |
| Age                        | 0.65                   | 20.22 | 0.0001  |
| Male gender                | 0.94                   | 1.58  | 0.1133  |
| Treatment for hypertension | 6.44                   | 9.74  | 0.0001  |


Omitting the age * 0.65 element of the regression equation would be the same as predicting for age 0, which has obvious problems. I suppose I could plug in the average age from the original dataset, but that would imply a greater precision than is true.

I'm leaning towards a "brute force" approach, whereby I'd calculate a multiple regression for each combination of factors, then select the appropriate one depending on the available data. Whilst I think this would work, it seems inelegant and I'm sure there must be a better way.

Is there a way to square this circle?

• I don't totally understand what you're asking. You fit a model on X1, X2, and X3, but now only X1 and X2 are available to make out of sample predictions? Commented Jun 23, 2020 at 16:58
• @shadowtalker That's exactly it. Commented Jun 23, 2020 at 17:39
• How is it that you don't have the age variable? Can't you ask the patient how old they are? What do you want the prediction for? Are you trying to estimate the population average for some specified sex, BMI & Tx combination? Commented Jun 23, 2020 at 17:47
• @gung-ReinstateMonica The blood pressure example is just illustrative. Without boring you all with the details, the actual problem is very niche, highly dimensional, and (unfortunately) has patchy data available. Commented Jun 24, 2020 at 7:53
• @gung-ReinstateMonica I missed you question on the purpose of the prediction... Within the fictional example, we're interested in the likely "blood pressure" of individuals, but only for the purposes of evaluating risk (i.e. not diagnostic). Accuracy is somewhat important, but characterising the confidence of the prediction is more so. Commented Jun 24, 2020 at 7:56