# Predicting/imputing the missing values of a Poisson GLM Regression in R?

I'm trying to explore ways of imputing missing values in a data set. My dataset contains the number of counts of an occurance (Unnatural, Natural and the sum Total) for Year(2001-2009), Month(1-12), Gender(M/F) and AgeGroup(4 groups).

One of the imputation techniques I'm exploring is (poisson) regression imputation.

Say my data looks like this:

```    Year Month Gender AgeGroup Unnatural Natural Total
569 2006     5   Male     15up       278     820  1098
570 2006     6   Male     15up       273     851  1124
571 2006     7   Male     15up       304     933  1237
572 2006     8   Male     15up       296    1064  1360
573 2006     9   Male     15up       298     899  1197
574 2006    10   Male     15up       271     819  1090
575 2006    11   Male     15up       251     764  1015
576 2006    12   Male     15up       345     792  1137
577 2007     1 Female        0        NA      NA    NA
578 2007     2 Female        0        NA      NA    NA
579 2007     3 Female        0        NA      NA    NA
580 2007     4 Female        0        NA      NA    NA
581 2007     5 Female        0        NA      NA    NA
...
```

After doing a basic GLM regression - 96 observations have been deleted due to them being missing.

Is there perhaps a way/package/function in R which will use the coefficients of this GLM model to 'predict' (ie. impute) the missing values for Total (even if it just stores it in a separate dataframe - I will use Excel to merge them)? I know I can use the coefficients to predict the different hierarchal rows - but this will take forever. Hopefully there's an one step function/method?

```Call:
glm(formula = Total ~ Year + Month + Gender + AgeGroup, family = poisson)

Deviance Residuals:
Min         1Q     Median         3Q        Max
-13.85467   -1.13541   -0.04279    1.07133   10.33728

Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept)   13.3433865  1.7541626   7.607 2.81e-14 ***
Year          -0.0047630  0.0008750  -5.443 5.23e-08 ***
Month          0.0134598  0.0006671  20.178  < 2e-16 ***
GenderMale     0.2265806  0.0046320  48.916  < 2e-16 ***
AgeGroup01-4  -1.4608048  0.0224708 -65.009  < 2e-16 ***
AgeGroup05-14 -1.7247276  0.0250743 -68.785  < 2e-16 ***
AgeGroup15up   2.8062812  0.0100424 279.444  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for poisson family taken to be 1)

Null deviance: 403283.7  on 767  degrees of freedom
Residual deviance:   4588.5  on 761  degrees of freedom
(96 observations deleted due to missingness)
AIC: 8986.8

Number of Fisher Scoring iterations: 4
```

First, be very careful about the assumption of missing at random. Your example looks like missingness co-occurs with Female and agegroup. You should really test whether missingness is related to any predictors (or whether any predictors are missing). If so, the responses could be skewed.

Second, the function you are seeking is likely to be predict, which can take a glm model. See ?predict.glm for more guidance. You may want to fit a cascade of models (i.e. nested models) to address missing values.

The mice package provides a function of the same name that allows each missing value to be predicted using a regression scheme based on the other values. It can cope with predictors also being missing because it uses an iterative MCMC algorithm.

I don't think poisson regression is an option, but if all of your counts are as large as the example normal regression should offer a reasonable approximation.