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# One Mean

### Input.

• Counted numbers. To construct a confidence interval around one Poisson count. Input counted number in the top box. The two other boxes should have the value zero ('0').
• Percentages. To construct a confidence interval around a proportion or percentage. Input an observed proportion or an observed integer number in the top box and a number of cases in the number of cases box. If you input a number it will be divided by the number of cases before the analysis to obtain a proportion.
• Averages. To construct a confidence interval around a normally distributed average. Input an observed real number in the top box, a number of cases in the number of cases box and a standard deviation in the standard deviation box, so that there are values in each box.

### Explanation.

The program constructs confidence intervals around a mean, be it a proportion, a count or an average.

The Poisson analysis is for use with a counted variable, for example, the number of accidents on a busy junction. For the confidence intervals for a Poisson type variable the exact estimate would probably be the first choice. However, over a count of 500 there might be computing problems and the Pearson or normal approximation method is then recommended.

The six confidence intervals for a proportion are all discussed in an article by Newcombe (1998). The simple normal approximation method is taught in many introductory statistics courses. Newcombe urges strongly against its use in practical research. The exact Binomial estimate would probably be the first choice. However, over a number of cases of 400 there might be computing problems and the Wilson method is then recommended. Note that the continuity corrected Wilson confidence interval according to Newcombe is the same as the quadratic confidence interval according to Fleiss (1982). This last method has the disadvantage that if a proportion equals one both values of the confidence interval are smaller than one.

A discussion of how to construct a confidence interval for the normal mean can be found in any introductory statistics book, such as Blalock or Swinscow and Campbell.

Exact p-values tend to be more conservative than most approximate estimates. To make the exact p-value more like the approximate result the mid-p value is sometimes used. The mid-p is somewhere in the middle between p-values including the point probability values and p-values not including the point probability values. To determine the mid-p value SISA takes the average of these two values. SISA does not recommend the use of mid-p values.

The confidence interval for the difference between two means is calculated by SISA’s t-test procedure

The design effect is a factor which changes the confidence interval because of uncertainty caused by the design. Sources of such uncertainty might be the use of stratification or differences in response rates between groups targeted in the sampling. The design effect is mostly larger than 1. There are designs, however, which improve the certainty of estimates. In that case the design effect is smaller than 1. The program specifically requires to have the design effect, abbreviated as DEFF, and NOT the design factor, or DEFFT. The design effect is the square of the design factor, or, DEFF=DEFFT*DEFFT. In the SISA raking procedure and in the SISA weighting program the design effect is calculated for a number of situations. You can incorporate the design effect in the OneMean procedure to study the effect of the design of a study on confidence intervals around the mean.