Excluding predictors with small effect sizes: not worth it to obtain data? Suppose I have the following completely-made-up logistic regression model:
will_die_from_cancer = 0.50 + 0.07*age - 0.05*weight - 0.000001*passed_expensive_screening_test

All predictors are significant at any reasonable threshold.
The variables age and weight are cheap to obtain and have relatively large effect sizes. passed_expensive_screening_test, on the other hand, is really hard to obtain (it's an expensive screening test, duh!) and has a small effect size (a relatively minuscule coefficient).  
Perhaps the inclusion of passed_expensive_screening_test ensures that my model is correctly specified, but its effect on the likelihoods is so small that it doesn't seem worth it to continue collecting these data, when age and weight (again, a very minimal example) seem to do the trick.
Is this (a) correct way of thinking about passed_expensive_screening_test and whether it should be included?
 A: It may well be that the screening test isn't informative enough to justify its expense. In general, however, looking at the parameter estimates is a bad way to judge the effect of including a covariate on model fit, because all the parameters can be affected by which covariates are in the model. For example, in your model, the coefficient for weight may well change when you remove passed_expensive_screening_test.
Instead, look at the fit of each of the models of interest. In this case, that would be one model including only age and weight, and another model including all three covariates. The fit of a logistic-regression model can be judged with proper scoring rules or with classification diagnostics such as the proportion of cases classified correctly.
Bear in mind that fit is overly optimistic as a measure of predictive accuracy, so if you want to know how well the models will perform when asked to predict the values of cases they haven't been trained with, you need to use a method such as cross-validation.
