FISH AND SHELLFISH CONSUMPTION PERCENTILES


 Procedure

The vast majority of individuals surveyed had eaten no seafood in the week prior to questioning. The distribution of
consumption for those adults who actually reported consumption in the previous week is highly dependent on how the
consumption information is obtained. That is, the estimate of portion size depends on how the standard portion is
defined. It is usual for respondents to report consumption in regular simple fractions of the standard. This leads, as will
be seen, to non-normal distributional shapes and in many cases, to bimodal distributions. There are a number of
approaches to estimating percentiles.


Parametric Approach

The first is to assume the data follow a specific parametric form. The parameters of the distribution, usually a function
of the first and second moments are estimated from the data and then the percentiles are estimated from the fitted
distribution. This is the approach presented by Murry and Burmaster (1994) and Ruffle, et. al. (1994) assuming
reported weekly consumption following a lognormal distribution. One drawback of this method is that one cannot
determine confidence intervals for percentile estimates.


Distribution Free Approach

A second approach is to use distribution-free estimates of percentiles calculated directly from the order statistics (sorted
observations). No distributional assumptions are made and confidence intervals for the percentile estimates are available.

Assume n respondents report consumption of a specific seafood in the previous week. Let X,, X2, ..., X, represent the
reported consumption levels for all respondents. The order statistics associated with this sample is denoted min=X(,),
X(2), ..., XY)=max and is essentially the observations in ascending order. A natural estimator of the p-th quantile of the
true distribution, denoted C, is the p-th quantile of the distribution of the sample values. That is


 S- (1-g) +gX (1)

where j=[p(n+l)], g=[p(n+l) j] and [x] denotes the greatest integer less than or equal to x. This estimator is
asymptotically normally distributed with expectation equal to the true percentile, C, and variance p(1-p)/f(Cp ); f(.)
is the true density of the unknown distribution. Other estimators which are quite similar to Equation (1) but which
involve weighted averages of more order statistics have been proposed but offer little additional benefit (Harrell and
Davis, 1982: Kaigh and Lachenbruch, 1982).

Confidence intervals for the pth percentile, Cp, follows from the probability statement

 ((2)
 P (~X < X) i p (1-p) (2)


This probability depends on the percentage, p, the sample size n, and the order statistics u and v, but not on any
particular distribution, F(x). To construct a 100(1-a)% confidence interval for the percentile, ,p, one searches for (u,v)
pairs such that Pr( Xu) < C, < X(v)) is as close to 1-a as possible. Exact 95% confidence intervals may not always be
possible. In some situations, primarily with low sample sizes and values ofp close to zero or one, it may not be possible
to construct a confidence interval at all.