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I am trying to understand what statistical measures I can use to compare three groups having varying populations to understand which group is bad (highest probability of death or most vulnerable or weakest or whatever measure of strength is suitable) and which group is good.

Total Population
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Group A: 100
Group B: 150
Group C: 50
Group D: 900

Over the course of one year, I have recorded the following information:

Group    Deaths
----------------
A        40
B        60
C        20
D        360

I was thinking I could just calculate the probability of deaths by diving the numbers in the second table by the numbers in the first. This made sense based on the definition of probability. However, I am not sure if there is a better way to do this. Group D had almost a population of 900 which makes me think that the confidence in estimating the probability is higher whereas for Group C, we don't really have enough data to make any conclusions.

In this case, Group C and Group D had population sizes differing the order of hundreds yet we get the same probability of death for both groups leading us to conclude that Group C and Group D have the same properties (whatever they are). How do I deal with this problem (if it is a problem in the first place)? Is there another level of normalization (besides dividing a number by its group population)?

Note: If it helps, I have data on when each individual died (approximate time) in the following form:

Death_Date Group
------------------
2011-01-23 Group A
2011-01-23 Group A
2011-01-25 Group A
...
2011-01-25 Group C
...
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  • $\begingroup$ If your groups are indeed populations - not samples - then "confidence in estimating the probability" is absolute for all groups and data are always enough to make conclusions. $\endgroup$ – ttnphns Aug 17 '11 at 7:16
  • $\begingroup$ Thank you for your comment. Yes. Indeed they are not samples but populations. So from what I understand, you're saying that this is indeed the only way to do it and I can make conclusions based on this data alone? $\endgroup$ – Legend Aug 17 '11 at 7:50
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    $\begingroup$ What you appear to be touching on here, is Survival Analysis, which is a whole branch of statistics. You should read up on that first. $\endgroup$ – Nick Sabbe Aug 17 '11 at 7:57
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From what you are describing intuitively about having higher confidence in large populations, I think what you might like is calculating the confidence intervals on the probability of death for each group. This way toy should get a narrower confidence interval for Group D. You can chose a significance level that you want. If you want to model the time dependent effects as you mentioned in your update, then may be you want to look at survival analysis.

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  • $\begingroup$ +1 Thank you for your suggestions. Would you mind providing me with an example on how to do it in R? I am asking this because I am positive this calculation depends on the distribution of the data. For normally distributed data, it probably makes sense to use the standard formula given on the Wiki page. In other cases, I am guessing I will have to first test for normality of my data and then proceed with this? How do I proceed if my data is not normally distributed? Does R have anything to handle this? $\endgroup$ – Legend Aug 17 '11 at 20:19
  • $\begingroup$ Oh.. Sorry! I meant survival analysis. Would you also comment on my concern regarding data normality if possible? $\endgroup$ – Legend Aug 17 '11 at 20:47
  • $\begingroup$ First of all, I think fitting a general hazard function would be overkill since you basically want to compare different groups and see which one has better survivability. Or are you trying to figure something else out, like the shape of the survival function? Can you explain what it is that you are trying to find? Secondly, normality doesn't really enter the picture in general for survival analysis AFAIK. $\endgroup$ – highBandWidth Aug 17 '11 at 21:03
  • $\begingroup$ I see. I will try to explain what I am trying to achieve. These populations are from different areas and each face the same environment. Within the course of one year, some fail to survive the environment. What I am trying to figure out is if I can make some conclusions on their robustness (or perhaps their reliability) to the environment and want to figure out which populations are the strongest in terms of survival. At the end of the day, a question that I am interested in looking at is: Which population can I invest in for the next year (perhaps the most reliable)? $\endgroup$ – Legend Aug 17 '11 at 21:19
  • $\begingroup$ In that case I think you don't need to do survival analysis because you don't really care in the details of how they die (early or late etc.). You just want to see which one is better, and how confident you can be. Which one is better is given by the ML estimates (no. of survivors/total). The confidence intervals can give you an idea of how confident you are. ps. can you invest in more than one population? can you write the cost function you are trying to maximize? Is it the expected number of survivors? $\endgroup$ – highBandWidth Aug 17 '11 at 21:30

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