Description of Statistical
Analysis:
Maximum Likelihood Ratio test
Figure 4a suggest that the loss of Sir3 function appears to have a major effect on expression of genes within 6-8 kb of telomeres, whereas the effects of histone H4 depletion appear to extend to nearly 20 kb (Figure 4a, table 1). In other words, the fraction of genes derepressed is reduced with distance from the telomere in both mutants, but the decline appears to be much sharper for Sir3 than for histone H4. We tested this observation statistically by using a maximum likelihood approach to test the hypothesis that the rate of decrease in the two mutants is the same.
Genes within 40 kb of a telomere throughout the genome were pooled and ordered according to their distance from a telomere. Genes with sequence homologues whose mRNA levels could not be distinguished with certainty (genes with an Affymetrix "_f" qualifier) were excluded from the analysis. We examined the genes closest to a telomere in consecutive intervals of 50 genes, and calculated the fraction of genes derepressed in each mutant in each interval. The data for the first two 50-gene intervals is listed below:
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Strain: |
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Histone H4 |
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Histone H4 |
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Sir3 |
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Sir3 |
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We wished to test whether the rate of decline (56% to 52% vs 16% to 2%) differed significantly between the two mutants. Since we are comparing rates of decline from different initial frequencies, we employed the following simple maximum likelihood test. Let a and ac be the fraction of genes derepressed in the histone H4 depletion data in the first and second 50-gene intervals, respectively. The parameter c is thus the proportional decrease in genes depressed as one moves from the first interval to the second. Similarly, b and bd are the fraction of genes derepressed in the histone H4 depletion data in the first and second 50-gene intervals, respectively. The parameter d is thus the proportional decrease in genes derepressed as one moves from the first interval to the second
We wish to test the hypothesis that c=d. To test this, we calculate the maximum likelihood estimates (MLE) y' = (a', b',c', d') where the four parameters are permitted to vary freely and y'' = (a'', b'', c'', c'') where the parameters are maximized subject to the constraint that c=d. The log of the ratio of the two likelihoods tends to a chi-squared distribution with one degree of freedom (see below).
We numerically calculated the MLEs, using the Mathematica software package. The resulting likelihood ratio is statistically significant (p < 0.02), allowing us to reject the hypothesis that the rate of decline is the same in the two mutants.
Similarly tests were performed using different intervals sizes (for example, the 50 most telomere-proximal genes versus the next 100 or 150 genes) with similar results.