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I work in a medical company and we often analyze the days patients stay in hospital longer than needed. For example we have these data:

patient      days in hospital     days not necessary
      1                  20                      2
      2                   5                      0
      3                  13                      1
      4                   9                      3
      5                  22                      0
..

we now compute the average days per patient that were not necessary for treatment in hospital. In this example:

(2/20 + 0/5 + 1/13 + 3/9 + 0/22) / 5

where 5 is the number of patients in this example.

Then we compute a 95% confidence interval for this value (in SPSS). The result is usually an interval of the kind 

- 0.9 < \mu < 3.6

Because a negative value makes no sense we change this interval to

0 < \mu < 3.6

For me the whole process feels kind of wrong but I cannot specify it exactly. What would be a better way to do this, or does this make no sense at all?

Edit:

Some more insight on the data: We are authorized by health insurance companies and get data about patients directly from the hospitals. Our experts, usually doctors, look at each patient's case and make a report about possible unnecessary days of treatment. Reasons can be that an outpatient treatment would be enough or various others reasons. Sometimes patients also want to stay a day or two longer because they feel it is necessary. Of course health insurances have a interest in getting people out of hospital as soon as possible because that's a huge cost factor. Anyway, this is not relevant here. In this report we make a CI like the above example. Of course I cannot provide any real data nor anything that is close to it. The sample size is usually between 40 and 140 patients. There can be no negative days. I think we get negative values for the interval because it's so close to zero and the variance of the assumed normal distribution makes it so.

days not necessary is always a fraction of the total days. So for patient 1 in the above example 2 out of 20 days were not necessary. The CI should say in which area the expected value of not-necessary-days/total-days per patient is.

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  • $\begingroup$ To supply a good answer we need a reasonable model of these data and, if it's to be more than a pure guess, that requires some understanding of the data. How are the "necessary" days in the hospital determined? What are the likely causes of "unnecessary" days? Do you ever estimate a negative number of unnecessary days? That is, do you pay attention to the possibility that people might have been discharged too early? Since you're getting negative values, that suggests you may be using small datasets, which is strange: are you computing a large number of confidence intervals? $\endgroup$
    – whuber ♦
    Commented Feb 12, 2014 at 15:04
  • $\begingroup$ Saying that the unnecessary days are bounded at 0 seems flawed. (Speaking as an M.D. all that stuff about experts is tangential gobbleygook.) The only measure bounded at 0 is the total number of days. That being said you are not analyzing the data with a Poisson model so response is not a good fit to your modeling assumptions. The confidence interval with a bound less than 0 is a symptom of the use of (inappropriate) Normal theory models. $\endgroup$
    – DWin
    Commented May 30, 2014 at 3:41

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Depending upon whether the treatment period is known in advance at least at the moment when it has ended. Then I would suggest the treatment period to be an independent variable and the unnecessary as dependent with negative binomial distribution of r=1 and p estimated from the data. The CI would be obtained from the model.

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  • $\begingroup$ all cases are of patients that are already out of hospital. We analyze their data when their treatment ist over. $\endgroup$
    – spore234
    Commented Feb 13, 2014 at 8:25

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