Seminar notes: Confidence intervals for a variance ratio (July 17, 2006). Category: Analysis of variance, Category: Ask Professor Mean

One of the talks at the 18th Annual Applied Statistics in Agriculture Conference, sponsored by Kansas State University was "Selecting the Best Confidence Interval for a Variance Ratio (or Heritability)" by Brent Burch, Northern Arizona University. Here are my notes from that talk.

Variance ratios have many applications, including genetics, where there are both genetic effects and environmental effects. You would be interested in the variation due to genetics and due to environment as well as the ratio of those estimates of variation. Most confidence intervals (CIs) rely on a typical approximation using a weighted sum of Chi-square random variables. The best CI (confidence interval) could look at bias, expected length, or a new concept that Dr. Burch advocated, the radius of curvature.

I had forgotten what it means to say that a CI is unbiased, so I appreciated the definition: a CI is unbiased if the probability of covering the true value (usually 1 - alpha) is larger than the probability of covering any false value.

If you define a probability function for the coverage probability for any estimate (either true or false) then if the derivative is equal to zero at the true value, then the confidence interval is unbiased. The expected length of the interval is the area under the curve defined by this function.

The radius of curvature provides a measure of the size of the circle that fits the curve at a point. You can calculate this quantity using a simple formula involving the first and second derivative. The radius of curvature combines expected length and bias.

Dr. Burch showed an example where there were 165 different ways to compute a CI. The radius of curvature allowed him to select a procedure that had minimal length but not a serious degree of bias.

This webpage was written by Steve Simon and was last modified on 07/08/2008.