29 Lecture 20 - 2019
29.1 Measurement error
There is always some error in measurement. Sigma in our above models captures error on outcomes. What about error on predictors? What if error isn’t constant?
29.1.1 Example: waffle divorces
Error on outcome (divorce rate) is heterogeneous because smaller states have larger error.
A -> M -> D -> D_obs, A -> D, N -> D_obs
D is the true, unobserved divorce rate. D_obs is the observed divorce rate.
Approach:
- Treat true divorce rate as an unknown parameter
- The observed rate is sampled from a Gaussian distribution
D_obs ~ Normal (D true, D standard error)
This gives us shrinkage considering the standard error as well as the modeled relationship between divorce rate and median age.
29.2 Error on predictor
29.2.1 Example: marriage rate
Consider error on marriage rate:
Likelihood for observed rate, for each state
M_obs ~ Normal (M true, M standard error)
M true ~ Normal (0, 1)
Note: this is a simplification that misses that M and A are associated.
29.2.2 Sources of error
Data come from some uncertain procedure but we often discard the uncertainty when we get to the analysis/model.
For example, taking a series of measurements but then modeling their group-level averages instead of the raw value. We toss away information about variation, sample size, etc.
29.3 Missing data
Usual approach is to only consider the complete cases. This is either
actively done by the user or silently, by software eg lm. This discards a lot
of information and is not harmless.
Options include multiple imputation (frequentist approach based on..), Bayesian imputation. Do not: replace missing values with the mean. The model takes these as the true value.
Missing values can be a confound.
29.3.1 Types of missingness
29.3.1.1 MCAR: Missing completely at random
When the response is completely independent of the missingness, the missingness variable is not a confound and you do not need to do imputation, but if do it will add precision.
For example, brain neocortex proportion. If there are missing values, we can consider B obs = function(B, R_B) where R_B is the missingness of B. There are two paths from B -> K, the direct path from B obs -> B -> K and the indirect path from B obs -> B <- U -> M -> K. There is no back door.
When would we ever really get a completely random missingness?
29.3.1.2 MAR: Missing at random
Missingness more likely for specific values of another variable.
For example, brain neocortex proportion and body mass M. Species that are larger or smaller are more likely to have missing values. Maybe there’s an size related observation bias. Now, M -> R_B creates a backdoor. To close the backdoor, we can condition on M. Then the missingness R_B value is ignorable, but we must do the imputation.
29.3.1.3 MNAR: Missing not at random
Missingness is more likely for specific values of the response (or for specific values of an unobserved variable that influences both the response and the missingness).
For example, if brain neocortex proportion is higher in monkeys more similar to humans and we are more likely to study these monkeys. So our missing values in B are in species with lower B. Alternatively, an unobserved variable may influence both the missingness and B. We can’t close this backdoor. The only thing you can do is model the error.
29.3.2 Imputing
29.3.2.1 Example: milk energy MAR
Each unobserved value becomes a parameter. These values are imputed, estimating from the observed values and the model structure.
The result is increased precision but, in this example, it doesn’t consider the relationship between B and body mass. Solution is to use a multivariate normal to impute given the relationship between B and body mass.