My previous blog post discussed two new options in the JMP 10 Distribution platform for customizing the summary statistics and default quantiles reports. This blog post discusses another new JMP 10 feature for the Distribution platform for continuous data: Custom Quantiles. This feature gives the user even more customization options for studying quantiles. Custom Quantiles, which can be found under the Display Options menu, allows a user to enter any number and value of custom quantiles. It reports two different types of quantile estimates as well as their confidence intervals.
To demonstrate the new Custom Quantiles features, let us examine the weights of the students found in Big Class.jmp in JMP’s Sample Data folder. In particular, we want to study the 5%, 50% (median) and 95% weight quantiles for this set of students as well as their 95% confidence intervals. Figure 1 shows the Custom Quantiles dialog with our quantiles of interest entered into the list, which is the default entry method.
Figure 2 shows the Custom Quantiles report towards the bottom. The first type of quantile estimate is the same type of empirical quantile estimate as those found in the default quantiles report. Nonparametric confidence intervals based on order statistics are calculated for these empirical quantiles. With data sets that contain few rows, a significant drawback of these types of confidence intervals is that their actual probability of containing the true population quantile can be much lower than the specified confidence level. To address this limitation, we also added smoothed empirical likelihood quantile estimates that are based on a kernel density estimate. An advantage of these quantile estimates is that their confidence intervals tend to contain the true quantile with the promised confidence level.
Looking at our Custom Quantiles report for the weights, notice that the empirical quantile estimates and the smoothed empirical likelihood quantile estimates are fairly similar. Examining the first part of the report with the empirical quantile estimates, you see that the actual coverage of the confidence intervals for these 95% and 5% quantile estimates do not meet the 95% confidence level. These confidence intervals span the entire range of the data with only an 87% confidence level. Because the data sample size is relatively small at 40 observations, it is impossible to create these confidence intervals at a 95% confidence level. In addition, the confidence interval for the median, the 50% quantile estimate, is a little conservative with a confidence level of 96%. Now examine the second part of the report with the smoothed empirical likelihood quantile estimates. You see that we do get true 95% confidence intervals for all of these quantile estimates.
The following are references to learn more about these quantiles and their confidence intervals:
Hahn, G. J. and Meeker, W. Q. (1991), Statistical Intervals: A Guide for Practitioners, New York: John Wiley & Sons.
Chen, Song Xi and Hall, Peter (1993) Smoothed Empirical Likelihood Confidence Intervals for Quantiles. The Annals of Statistics, Vol. 21, No. 3, 1166-1181.