Sampling Distributions

Figure 3.2: Four frequency distributions used to test diversity estimators. Portion of total is shown as a function of transcript abundance. The zero abundance class was eliminated from frequency calculations. The remaining net diversity (S) and number of individuals (N) are summarized for each distribution.

We used four frequency distributions to validate diversity estimators (Figure 3.2), chosen to model ecological distributions [76,89]. The four sample distributions from four families, each producing long-tailed curves, having many rare and a few common individuals [38,89] were:

1. Poisson, a discrete distribution with $x \ge 1$, $g(x)=e^{-\lambda} \lambda^x / x!, \lambda=1$, which simplifies to $(ex!)^{-1}$;

2. exponential, $e^{-1}$;

3. log-normal, with parameters log-mean=1 and log-variance=0.7; and

4. negative binomial, with parameters size=1 and probability=0.5.

Methods in R, version 1.1.1 [62], generated the validation distributions (rpois, rexp, rlnorm, and rnbinom).

To make continuous distributions discrete, R rounded down decimal values to the nearest integer (e.g., 0.999 = 0). The zero abundance, null class was eliminated from frequency calculations because it is not observable; no representatives are present, so not counted in diversity (c.f. [38]). Sampling was from $g(x)$ as $m \times n$ samples, m=20 replicates of n=500 individuals, with replacement between samples.

The true underlying frequency distribution of transcripts in a cell is unknown, but expression assay results indicate that the general form of the distribution is likely to vary across cell types [74].