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(c) Use of the Correction Factor

The two methods of calculation produce the expected uncertainty of nsample bases, E(H_nb):

 

E(H_nb) =  H_g - e(n)       (bits per base). (17)

When Hs(L) is calculated from a small sample, it is too small by the amount e(n), on the average. To correct R_sequence(L), we use:

 

R_sequence(L)  =  H_g  -  [Hs(L)  +  e(n)]       (bits per base). (18)

That is, the uncertainty of the pattern is increased because there is only a small sample. Substituting equations (17) and (18) into (5) gives equation (6). H_g could also be corrected but the correction is negligible if H_g is calculated from a large sample of the organism's sequence.

The curve for E(H_nb) as a function of the number of example sites, n, (Fig. 5) has several important general properties. As the number of example sites increases, E(H_nb) approaches H_g(= 2 bits/base in the figures) since the error e(n) becomes smaller. As the number of examples drops, E(H_nb) also drops (the error increases), until at only one example E(H_nb) is zero. With only one example, the uncertainty of what the sequence is, Hs(L), is also zero. At this point, R_sequence is forced to zero (from equation 6): one cannot measure an information content from only one example.

  

Figure 5: E(H_nb) vs number of sites, n.

These data are for an equiprobable genomic composition. The curve is less than 1% lower for the composition of E. coli. Each bar represents one standard deviation above and below the curve.

The sampling error correction results in an interesting effect. If R_sequence could be measured for an infinite number of HincII sites (this would look something like Fig. 1a), the two peaks would be 2 bits/base. When the correction is made for a small sample, the peaks are less than 2 bits/base (Figs. 1b and 1c). This appears odd if we know exactly what HincII recognizes. However, given only six examples, we would not be so sure what the "real" pattern is. The sampling error correction prevents us from assuming that we have more knowledge than can be obtained from the sequences alone. That is, the value e(n) represents our uncertainty of the pattern, owing to a small sample size. In the extreme case of one sequence, we have no knowledge of what the pattern at the site is, even though we see a sequence. Because of the correction, R_sequence will be underestimated at truly conserved positions when only a few sites are known. R_sequence for six HincII sites in Fig. 1c is estimated to be 8 bits even though we "know" (by looking at more than six examples) that HincII recognizes 10 bits.