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==Calculation of confidence intervals==
==Calculation of confidence intervals==
The calculation of confidence intervals for means or proportions follows the same basic procedure, whereas a slightly different approach is used for other measures such as rates and ratios. Most commonly, confidence intervals will not be calculated manually, but it is useful to know the general approach used.
The calculation of confidence intervals for means or proportions follows the same basic procedure, whereas a slightly different approach is used for other measures such as rates and ratios. Most commonly, confidence intervals will not be calculated manually, but it is useful to know the general approach used. This is based upon the estimation of the '''standard error''' of the parameter in question, which is the the '''standard deviation of the sampling distribution''' of the parameter. That is, if samples of a specified size were taken from the population repeatedly (with replacement after each sampling), the standard deviation of all of these results is the standard error. It is known that with large sample sizes, the sampling distribution will be normally distributed (known as the '''central limit theorem'''), and this distribution can be calculated by dividing the 'standard deviation' of this parameter in the population with the square root of the number of animals sampled (or by dividing the variance with the number of animals sampled, and taking the square root of the result).<br>
However, as these parameters in the '''population''' are not known, they must be approximated using data from the '''sample'''.
===Approach for means and proportions===
===Approach for means and proportions===
The general approach for the calculation of confidence intervals in these cases follows the following steps:
The general approach for the calculation of confidence intervals in these cases follows the following steps. The method of estimating the sample standard deviation for means and proportions will be described below:
* Calculate the sample standard deviation, as an approximation of the population standard deviation:
* Calculate the '''sample standard deviation''', as an approximation of the population standard deviation:
* Estimate the '''standard error of the mean''' or proportion by dividing the sample standard deviation by the square root of the number of animals sampled.
* Estimate the '''standard error of the mean''' or proportion by dividing the sample standard deviation by the square root of the number of animals sampled.
* Calculate the sample mean or proportion (as an approximation of the population mean or proportion).
* Calculate the '''sample mean''' or '''proportion''' (as an approximation of the population mean or proportion).
* Decide upon the confidence level required (usually 95%), and multiply the estimate of the standard error with the appropriate percentile point of the normal distribution (unless you are estimating the confidence interval for the mean of a small sample, in which case the t-distribution should be used instead). In the case of a 95% confidence interval, this is 1.96.
* Decide upon the confidence level required (usually 95%), and '''multiply the estimate of the standard error with the appropriate percentile point of the normal distribution''' (unless you are estimating the confidence interval for the mean of a small sample, in which case the t-distribution should be used instead). In the case of a 95% confidence interval, this is 1.96.
* Subtract the resultant number from the sample mean or proportion to give the lower confidence limit.
* Subtract the resultant number from the sample mean or proportion to give the '''lower confidence limit'''.
* Add the number to the sample mean or proportion to give the upper confidence limit.
* Add the number to the sample mean or proportion to give the '''upper confidence limit'''.
====Means====
As described [[Data description|earlier]], means are the most appropriate measure of central tendency for normally distributed continuous variables, and the standard deviation is the most appropriate measure of 'spread' in these cases. An adjusted form of the sample standard deviation is used to approximate the standard deviation in the population, and is calculated by dividing the sum of the squared differences from the sample mean by the number of animals sampled minus 1, and taking the square root of the answer.
====Proportions====
Proportions are the most appropriate method of description of [[Data description|categorical and binary]] variables. Although the concept of a 'variance' or 'standard deviation' for a proportion is difficult to comprehend, it is based upon the 'binomial distribution'. The variance can be estimated by multiplying the proportion of positive animals by the proportion of negative animals; and the square root of this will give the standard deviation.
===Approach for rates and ratios====
===Approach for rates and ratios===
Although the general concept behind confidence intervals for rates and ratios is the same as that for means and proportions, the method of calculation is different. The number of 'outcomes' (i.e. the numerator of the rate) can be considered to follow a 'Poisson distribution'. This facilitates the estimation of the standard error for this count, as the standard error of a Poisson variable is the square root of the expected value (that is, the square root of the number of outcomes, in our case). From this, confidence intervals can be estimated as above (i.e. estimate the standard error, multiply this by 1.96 [for 95% confidence limits], and add and subtract this value to/from the number of outcomes). Finally, to convert these confidence limits into a rate (rather than a count), the confidence limits themselves should be divided by the total amount of animal-time under observation (i.e. the denominator of the rate).
[[Category:Veterinary Epidemiology - Statistical Methods|D]]
[[Category:Veterinary Epidemiology - Statistical Methods|D]]