I want to calculate simultaneous confidence intervals for proportions for a nominal variable with three classes. For each class I expect more than 20 samples. I read that the method of Glaz & Sison is only suitable for variables with 7 classes or more, so I decided to use the Goodman method. I would like to calculate in advance the required sample size for 95%-simultaneous confidence intervalintervals with predefined lengths. But unfortunately I have no idea how. GNU R provides a nice routine to calculate these CIs but I found none to calculate the sample size. I would appreciate any suggestions on how to calculate the sample size.
UCL=mean+/-(table value)*stnderror
Compute UCLand solve for n. This should give you an approximate value. See any elementary statistics book for examples . Best wishes David Booth BTW always increase n to take into account your approximations
Best wishes David Booth
To calculate the sample size for simultaneous confidence intervals for multinomial proportions, you can use the following formula:
n = [(z(α/2) * √(p(1-hat)q(1-hat) + p(2-hat)q(2-hat) + ... + p(k-hat)q(k-hat))) / d]^2
where:
- n is the required sample size
- z(α/2) is the critical value of the standard normal distribution corresponding to the desired level of significance α (usually 0.05 for a 95% confidence level)
- p(1-hat), p(2-hat), ..., p(k-hat) is the estimated proportions for each of the k categories of the multinomial variable, based on a pilot study or prior knowledge
- q(1-hat) = 1 - p(1-hat), q(2-hat) = 1 - p(2-hat), ..., q(k-hat) = 1 - p(k-hat)
- d is the maximum allowable half-width of the confidence intervals
The maximum allowable half-width (d) depends on the desired level of precision, and it can be calculated using the following formula:
d = z(α/2) * √(∑[p(i-hat)q(i-hat)] / (nk))
where:
- k is the number of categories of the multinomial variable
- n is the sample size
- p(i-hat) and q(i-hat) are as defined above
Note that in practice, the estimated proportions may not be available from a pilot study or prior knowledge, and therefore, they may need to be estimated from the sample data. In this case, you would need to use an iterative approach to estimate the required sample size based on the observed proportions.
It is important to consult with a statistician to ensure that the assumptions underlying this formula are met and to address any specific issues related to your study design and data.
To calculate the sample size for simultaneous confidence intervals for multinomial proportions, you can use the following formula:
n = [(Zα/2 + Zβ)² * N * P] / [d² * (N - 1) + (Zα/2 + Zβ)² * P * (1 - P)]
where:
- n is the required sample size
- Zα/2 is the critical value of the standard normal distribution corresponding to the desired level of confidence
- Zβ is the critical value of the standard normal distribution corresponding to the desired level of power
- N is the population size
- P is the proportion of the population in each category of interest
- d is the maximum allowable difference between the true proportion and the estimated proportion
This formula assumes that the sample is randomly selected and that the sampling is done with replacement. It also assumes that the population is large compared to the sample size so finite population correction is not needed.
To use this formula, you will need to specify the desired level of confidence, the desired level of power, the maximum allowable difference between the true and estimated proportions, and the population size and proportion in each category of interest. You can then use a statistical software package or a calculator to solve for the required sample size.
Keep in mind that this formula provides an estimate of the required sample size, and the actual sample size may vary depending on the specific conditions of the study. It is also important to ensure that the assumptions of the formula are met and that the sample is representative of the population of interest.
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