50 calves are expected to die each month.
Treatmetn with our product expects to reduce that to 0 deaths.
If we do not use a placebo, our product only, what is the sample size needed to prove statistical significance?
I also need to know how the number is justified.
I have no special training in statistics, but just from commonsense: it depends upon 1) do you wish to prove that the treatment decreased the mortality to some degree, or that the mortality after the treatment is zero? ( in the latter case the proof, strictly speaking, is never possible: even if your sample is ten million, it always remains possible that the mortality is one per twenty million!); if, on the other hand, you wish to prove some decrease of mortality than first of all you must precise: 50 calves out of how many die each month (the proportion rather than absolute number): the higher the proportion of calves dying before the treatment the less numerous must the sample be to demonstrate that the treatment is really effective (in the extreme case, if pre-treatment mortality was 100% - i.e. 50 of 50 - than even very small sample may be sufficient: three survivors from the sample of five after treatment seems rather significantly different from 50 of 50 before...).).
Hi, you may have a lookt at this software: http://www.gpower.hhu.de/
With it's help, you can calculate sample sizes, effect sizes and so on. Plus: as there is also a paper about it, it is easy to cite.
The question is not well posed. You can only show that the (expected) number of deaths per month is larger than zero, or that it is smaller than some value greater zero. The data should be Poisson-distributed (counts!). Thus, the Poisson distribution tells you how likely you would expect to get more than k deaths per month when the rate is 50/month, for instance. Under the hypothesis that the rate is 50/month, the probability of observing less than 40 deaths per month is lower than 0.05. Under the hypothesis that the rate is 1/month, an observation window of a single month is not sufficient to reject the hypothesis (even if not a single dead is observed, the probability under this hypothesis is still 0.37). In this case you would have to find how long the observation window needs to be to allow you to reject the hypothesis (a 3 month period would allow you to reject the hypothesis with 95% confidence, when during this time not a single dead was observed).
My apologies. 50% of calves die per month on average. I am not very good at statistics. What other type of information is needed to find the sample size?
The question was posed as 50 calves each month, then corrected to 50% each month. This is not clear as it does not specify the age of the calves. Does this mean that 50% of the calves die in their first month? Is there a smaller or larger age interval?
How large is the pool of calves available to treat? How many can be treated in the trial?
There is no need to have a placebo. Treat as many as you can and compare to the untreated.
Is it possible to carry on the treatment for an extended time? Is the treatment one time? Dose the treament require repeated doses?
Suppose you treat 50 calves. Expected deathes are 25 with a standard deviation of 5. There is one chance in twenty that you will have only 15 deathes. Treat 200 with expected deathes of 100 and a standard deviation of 10. There is one chance 20 that you will have 80 deaths.
Test for survival, live calves. The normal rate is 50% live after one month.
A positive treatment will result in more calves surviving. Untreated calves will have A surviving of 2A calves. Treatment will result in additional survival B. The total surviving will be A+B. The survival process is Poisson. The uncertainty of a Poisson value is the square root of the value. The additional survivors is A+B-A and its propagated uncertainty is sqrt(2A+B). The relative uncertainty in B is sqrt(2A+B)/B.
Assume you wish to determine B to a relative uncertainty of 0.1.
Start with 100 calves for A = 50. Zero deaths will be B=50. The relative uncertainty of B is 0.2.
Start with 200 calves for A = 100. Zero deaths will be B=100. The relative uncertainty of B is 0.1.
Note: a Poisson rate of zero has a 95% upper bound of 3. If the death rate is zero 3 to 4 deaths might occur, so the larger the number of test subjects, the smaller will be the effect of zero rate upper bound.
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