Model the proportion of a subset of total counts to determine the difference

Question

I've been working on solutions to my first unanswered questions and had been proposed to rather model the proportion of total count of deaths that are unnatural death counts. The reason why I want to do this, is because I want to determine the count of natural and unnatural deaths for a certain month of a year of a district for a specific gender.

Say my data looks like this: (Data goes up to 2009, where the whole of 2007 is missing and first 3 months of 2008 are missing - the counts of deaths.)

        District Gender Year Month AgeGroup Unnatural Natural Total
961  Khayelitsha Female 2001     1        0         0       6     6
965  Khayelitsha Female 2001     2        0         2       9    11
969  Khayelitsha Female 2001     3        0         3      10    13
973  Khayelitsha Female 2001     4        0         0      14    14
977  Khayelitsha Female 2001     5        0         0      16    16
981  Khayelitsha Female 2001     6        0         0      13    13
985  Khayelitsha Female 2001     7        0         3      11    14
989  Khayelitsha Female 2001     8        0         1      12    13
993  Khayelitsha Female 2001     9        0         0       6     6
997  Khayelitsha Female 2001    10        0         1      11    12
1001 Khayelitsha Female 2001    11        0         0       7     7
1005 Khayelitsha Female 2001    12        0         2       8    10
1009 Khayelitsha Female 2002     1        0         0      13    13
1013 Khayelitsha Female 2002     2        0         1      16    17
1017 Khayelitsha Female 2002     3        0         0       9     9
1021 Khayelitsha Female 2002     4        0         0      14    14
1025 Khayelitsha Female 2002     5        0         0      14    14
1029 Khayelitsha Female 2002     6        0         1      16    17
1033 Khayelitsha Female 2002     7        0         2      12    14
1037 Khayelitsha Female 2002     8        0         1       6     7
1041 Khayelitsha Female 2002     9        0         0       9     9
1045 Khayelitsha Female 2002    10        0         0       8     8
1049 Khayelitsha Female 2002    11        0         0       9     9
1053 Khayelitsha Female 2002    12        0         0       6     6

So what I want to do is: model the proportion of Total which is Unnatural so that I could use these models to impute the missing counts of total and unnatural for the missing period, and then use these to find the natural for the missing period. Main question now is just to model. I've been pretty confused if I should use SARIMA/ARIMA/ARMA models (as these counts are too small). I've also looked at examples that use state-space models and Kalman recursions - but I'm so confused what I should use?

Hope someone can help me in all my confusion.

Answer 1

You have monthly data for the period 2001-2009: 9 X 12 = 108 minus missing observations 12 +3 = 15 which gives you a total of 83 observation. That's not much, but you should try to use the data. It's important that you also show the uncertainty associated with the result: (1) plot the data you have, and check whether there is any trend and seasonal variation. F.ex plot the sample autocorrelation function in addition to illustrative figures/plots. Try to identify whether there is any systematic pattern in your material, (2) If seasonal pattern, use seasonal dummies (you impose a deterministic seasonality), (4) Use an autoregressive model (an autoregressive distributed lag model) y(t) = constant + lagged dependent variable + 11 seasonal dummies (if necessary) + geographical dummies (if necessary) + residual, and (4) use the model to (also) estimate the missing values.

P.