visualization + design

Statistics for aneuploidy level `h` = 1 2 3 4 5 6 7 8 9 10

Tetraploid Genome Coverage Tables

Given a location `x` defined in the context of `h` chromosomes, the probability that position `x` is covered at least `\phi` times is `P_{h,\phi}` and given by $$ P_{h,\phi} = \left( 1 - \sum \frac{1}{k!} \left( \frac{\rho}{h}^k \right) e^{-\rho/h} \right)^h \tag{1} $$

For more details, see Wendl, M.C. and R.K. Wilson. 2008. Aspects of coverage in medical DNA sequencing. BMC Bioinformatics 9: 239.

For a given sequencing redundancy `\rho` (e.g. `\rho`-fold, as determined by the length of the haploid genome) of a tetraploid genome, the fraction of the tetraploid genome represented by at least `\phi` reads is reported by `P_{h,\phi}`. Coverage by fewer than `\phi` reads is reported as `1-P_{h,\phi}`. Coverage by exactly `\phi` reads is `P_{h,\phi} - P_{h,\phi+1}`. Entries for which fractional coverage is `\lt 10^{-5}` are not shown.

A rudimentary Monte Carlo simulation of genome coverage is also available, and is a useful supplement to the exact probabilities shown here.

CUSTOM DEPTH AND PLOIDY To create a table with a specific ploidy (e.g. 12) and haploid-equivalent (see below) depth (e.g. `200 \times`), use

http://mkweb.bcgsc.ca/coverage/?aneuploidy=12&depth=200

EXAMPLE 1

Suppose you carried out 3-fold redundant (`\rho=3`) sequencing of a haploid genome (`h=1`). 95.02% of the genome will be covered by at least one read (`P_{1,1}`) while 22.40% will be covered by exactly 3 reads (`P_{1,3} - P_{1,4}`).

EXAMPLE 2

You are sequencing a sample with a tumor content of 25% and you're interested in the depth of sequencing required to detect heterozygous mutations in the tumor. This scenario is equivalent to an aneuploidy = 8 genome—any given allele is present 8 times. If you sequence at (`\rho=200`), then 95% of the bases will be covered at a depth of at least `\phi = 14` (`P_{8,14} = 0.9494`). If you're satisfied with `\phi = 5` then you only need `\rho = 100` since now `P_{8,5} = 0.9580`.

ANALYTICAL vs STOCHASTIC

View plot that compares analytical vs stochastic results.

HAPLOID vs DIPLOID

View plot that compares 100x and 200x coverage of haploid and diploid genomes.

CODE

Download Perl scripts for analytical (to produce the tables below for any `\rho`) and stochastic coverage calculations.

sequencing redundancy for a tetraploid genome

View table for sequencing redundancy `\rho` = 1 2 3 4 5 6 7 8 9 10 20 25 50 75 100 of a tetraploid genome.

IMPORTANT The redundancy is always calculated using the size of the haploid genome. For example, if we collect 600 Gb of reads, our sequencing redundancy is `600 / 3 = 200 \times`. We've used the length of the haploid genome (3 Gb) in the calculation. If we now apply this `200 \times` sequencing to a diploid genome, our average coverage will not be `200 \times` but slightly less than `100 \times`.

sequencing redundancy 1-fold (`\rho / h = 0.2`)

`\phi`	`P_{h,\phi} - P_{h,\phi+1}`	`1-P_{h,\phi}`	`P_{h,\phi}`
0	0.9976	0.0000	1.0000
1	0.0024	0.9976	0.0024

sequencing redundancy 2-fold (`\rho / h = 0.5`)

`\phi`	`P_{h,\phi} - P_{h,\phi+1}`	`1-P_{h,\phi}`	`P_{h,\phi}`
0	0.9760	0.0000	1.0000
1	0.0239	0.9760	0.0240
2	0.0001	0.9999	0.0001

sequencing redundancy 3-fold (`\rho / h = 0.8`)

`\phi`	`P_{h,\phi} - P_{h,\phi+1}`	`1-P_{h,\phi}`	`P_{h,\phi}`
0	0.9225	0.0000	1.0000
1	0.0766	0.9225	0.0775
2	0.0009	0.9991	0.0009

sequencing redundancy 4-fold (`\rho / h = 1.0`)

`\phi`	`P_{h,\phi} - P_{h,\phi+1}`	`1-P_{h,\phi}`	`P_{h,\phi}`
0	0.8403	0.0000	1.0000
1	0.1548	0.8403	0.1597
2	0.0048	0.9951	0.0049
3	0.0000	1.0000	0.0000

sequencing redundancy 5-fold (`\rho / h = 1.2`)

`\phi`	`P_{h,\phi} - P_{h,\phi+1}`	`1-P_{h,\phi}`	`P_{h,\phi}`
0	0.7408	0.0000	1.0000
1	0.2432	0.7408	0.2592
2	0.0156	0.9841	0.0159
3	0.0003	0.9997	0.0003

sequencing redundancy 6-fold (`\rho / h = 1.5`)

`\phi`	`P_{h,\phi} - P_{h,\phi+1}`	`1-P_{h,\phi}`	`P_{h,\phi}`
0	0.6358	0.0000	1.0000
1	0.3260	0.6358	0.3642
2	0.0369	0.9618	0.0382
3	0.0013	0.9987	0.0013
4	0.0000	1.0000	0.0000

sequencing redundancy 7-fold (`\rho / h = 1.8`)

`\phi`	`P_{h,\phi} - P_{h,\phi+1}`	`1-P_{h,\phi}`	`P_{h,\phi}`
0	0.5340	0.0000	1.0000
1	0.3917	0.5340	0.4660
2	0.0700	0.9257	0.0743
3	0.0042	0.9957	0.0043
4	0.0001	0.9999	0.0001

sequencing redundancy 8-fold (`\rho / h = 2.0`)

`\phi`	`P_{h,\phi} - P_{h,\phi+1}`	`1-P_{h,\phi}`	`P_{h,\phi}`
0	0.4410	0.0000	1.0000
1	0.4345	0.4410	0.5590
2	0.1136	0.8755	0.1245
3	0.0105	0.9891	0.0109
4	0.0004	0.9996	0.0004

sequencing redundancy 9-fold (`\rho / h = 2.2`)

`\phi`	`P_{h,\phi} - P_{h,\phi+1}`	`1-P_{h,\phi}`	`P_{h,\phi}`
0	0.3595	0.0000	1.0000
1	0.4537	0.3595	0.6405
2	0.1635	0.8132	0.1868
3	0.0220	0.9767	0.0233
4	0.0013	0.9987	0.0013
5	0.0000	1.0000	0.0000

sequencing redundancy 10-fold (`\rho / h = 2.5`)

`\phi`	`P_{h,\phi} - P_{h,\phi+1}`	`1-P_{h,\phi}`	`P_{h,\phi}`
0	0.2901	0.0000	1.0000
1	0.4519	0.2901	0.7099
2	0.2147	0.7420	0.2580
3	0.0399	0.9567	0.0433
4	0.0033	0.9965	0.0035
5	0.0001	0.9999	0.0001

sequencing redundancy 20-fold (`\rho / h = 5.0`)

`\phi`	`P_{h,\phi} - P_{h,\phi+1}`	`1-P_{h,\phi}`	`P_{h,\phi}`
0	0.0267	0.0000	1.0000
1	0.1255	0.0267	0.9733
2	0.2607	0.1522	0.8478
3	0.2953	0.4129	0.5871
4	0.1938	0.7082	0.2918
5	0.0762	0.9020	0.0980
6	0.0186	0.9782	0.0218
7	0.0029	0.9968	0.0032
8	0.0003	0.9997	0.0003
9	0.0000	1.0000	0.0000

sequencing redundancy 25-fold (`\rho / h = 6.2`)

`\phi`	`P_{h,\phi} - P_{h,\phi+1}`	`1-P_{h,\phi}`	`P_{h,\phi}`
0	0.0077	0.0000	1.0000
1	0.0471	0.0077	0.9923
2	0.1365	0.0548	0.9452
3	0.2365	0.1913	0.8087
4	0.2608	0.4278	0.5722
5	0.1872	0.6886	0.3114
6	0.0887	0.8758	0.1242
7	0.0282	0.9646	0.0354
8	0.0061	0.9928	0.0072
9	0.0009	0.9990	0.0010
10	0.0001	0.9999	0.0001

sequencing redundancy 50-fold (`\rho / h = 12.5`)

`\phi`	`P_{h,\phi} - P_{h,\phi+1}`	`1-P_{h,\phi}`	`P_{h,\phi}`
0	0.0000	0.0000	1.0000
1	0.0002	0.0000	1.0000
2	0.0012	0.0002	0.9998
3	0.0048	0.0014	0.9986
4	0.0150	0.0062	0.9938
5	0.0368	0.0212	0.9788
6	0.0733	0.0580	0.9420
7	0.1201	0.1313	0.8687
8	0.1622	0.2514	0.7486
9	0.1797	0.4136	0.5864
10	0.1625	0.5933	0.4067
11	0.1194	0.7559	0.2441
12	0.0712	0.8753	0.1247
13	0.0344	0.9465	0.0535
14	0.0135	0.9808	0.0192
15	0.0043	0.9943	0.0057
16	0.0011	0.9986	0.0014
17	0.0002	0.9997	0.0003
18	0.0000	0.9999	0.0001

sequencing redundancy 75-fold (`\rho / h = 18.8`)

`\phi`	`P_{h,\phi} - P_{h,\phi+1}`	`1-P_{h,\phi}`	`P_{h,\phi}`
3	0.0000	0.0000	1.0000
4	0.0001	0.0000	1.0000
5	0.0006	0.0002	0.9998
6	0.0017	0.0007	0.9993
7	0.0046	0.0025	0.9975
8	0.0108	0.0071	0.9929
9	0.0222	0.0179	0.9821
10	0.0406	0.0401	0.9599
11	0.0663	0.0808	0.9192
12	0.0963	0.1470	0.8530
13	0.1242	0.2433	0.7567
14	0.1416	0.3676	0.6324
15	0.1419	0.5092	0.4908
16	0.1244	0.6512	0.3488
17	0.0950	0.7756	0.2244
18	0.0630	0.8706	0.1294
19	0.0363	0.9336	0.0664
20	0.0181	0.9699	0.0301
21	0.0078	0.9879	0.0121
22	0.0029	0.9958	0.0042
23	0.0010	0.9987	0.0013
24	0.0003	0.9996	0.0004
25	0.0001	0.9999	0.0001
26	0.0000	1.0000	0.0000

sequencing redundancy 100-fold (`\rho / h = 25.0`)

`\phi`	`P_{h,\phi} - P_{h,\phi+1}`	`1-P_{h,\phi}`	`P_{h,\phi}`
6	0.0000	0.0000	1.0000
7	0.0001	0.0000	1.0000
8	0.0002	0.0001	0.9999
9	0.0006	0.0003	0.9997
10	0.0015	0.0009	0.9991
11	0.0033	0.0023	0.9977
12	0.0069	0.0057	0.9943
13	0.0131	0.0125	0.9875
14	0.0231	0.0256	0.9744
15	0.0375	0.0487	0.9513
16	0.0564	0.0862	0.9138
17	0.0782	0.1427	0.8573
18	0.0996	0.2208	0.7792
19	0.1161	0.3204	0.6796
20	0.1234	0.4365	0.5635
21	0.1191	0.5599	0.4401
22	0.1041	0.6790	0.3210
23	0.0820	0.7831	0.2169
24	0.0581	0.8650	0.1350
25	0.0369	0.9231	0.0769
26	0.0211	0.9600	0.0400
27	0.0108	0.9811	0.0189
28	0.0049	0.9919	0.0081
29	0.0020	0.9969	0.0031
30	0.0008	0.9989	0.0011
31	0.0002	0.9997	0.0003
32	0.0001	0.9999	0.0001
33	0.0000	1.0000	0.0000

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news + thoughts

Propensity score weighting

Mon 17-03-2025

The needs of the many outweigh the needs of the few. —Mr. Spock (Star Trek II)

This month, we explore a related and powerful technique to address bias: propensity score weighting (PSW), which applies weights to each subject instead of matching (or discarding) them.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca

▲ Nature Methods Points of Significance column: Propensity score weighting. (read)

Kurz, C.F., Krzywinski, M. & Altman, N. (2025) Points of significance: Propensity score weighting. Nat. Methods 22:1–3.

Happy 2025 π Day—
TTCAGT: a sequence of digits

Thu 13-03-2025

Celebrate π Day (March 14th) and sequence digits like its 1999. Let's call some peaks.

▲ 2025 π DAY | TTCAGT: a sequence of digits. The digits of π are encoded into DNA sequence and visualized with Sanger sequencing. (details)

Crafting 10 Years of Statistics Explanations: Points of Significance

Sun 09-03-2025

I don’t have good luck in the match points. —Rafael Nadal, Spanish tennis player

Points of Significance is an ongoing series of short articles about statistics in Nature Methods that started in 2013. Its aim is to provide clear explanations of essential concepts in statistics for a nonspecialist audience. The articles favor heuristic explanations and make extensive use of simulated examples and graphical explanations, while maintaining mathematical rigor.

Topics range from basic, but often misunderstood, such as uncertainty and P-values, to relatively advanced, but often neglected, such as the error-in-variables problem and the curse of dimensionality. More recent articles have focused on timely topics such as modeling of epidemics, machine learning, and neural networks.

In this article, we discuss the evolution of topics and details behind some of the story arcs, our approach to crafting statistical explanations and narratives, and our use of figures and numerical simulations as props for building understanding.

▲ Crafting 10 Years of Statistics Explanations: Points of Significance. (read)

Altman, N. & Krzywinski, M. (2025) Crafting 10 Years of Statistics Explanations: Points of Significance. Annual Review of Statistics and Its Application 12:69–87.

Propensity score matching

Mon 16-09-2024

I don’t have good luck in the match points. —Rafael Nadal, Spanish tennis player

In many experimental designs, we need to keep in mind the possibility of confounding variables, which may give rise to bias in the estimate of the treatment effect.

▲ Nature Methods Points of Significance column: Propensity score matching. (read)

If the control and experimental groups aren't matched (or, roughly, similar enough), this bias can arise.

Sometimes this can be dealt with by randomizing, which on average can balance this effect out. When randomization is not possible, propensity score matching is an excellent strategy to match control and experimental groups.

Kurz, C.F., Krzywinski, M. & Altman, N. (2024) Points of significance: Propensity score matching. Nat. Methods 21:1770–1772.

Understanding p-values and significance

Tue 24-09-2024

P-values combined with estimates of effect size are used to assess the importance of experimental results. However, their interpretation can be invalidated by selection bias when testing multiple hypotheses, fitting multiple models or even informally selecting results that seem interesting after observing the data.

We offer an introduction to principled uses of p-values (targeted at the non-specialist) and identify questionable practices to be avoided.

▲ Understanding p-values and significance. (read)

Altman, N. & Krzywinski, M. (2024) Understanding p-values and significance. Laboratory Animals 58:443–446.

Depicting variability and uncertainty using intervals and error bars

Thu 05-09-2024

Variability is inherent in most biological systems due to differences among members of the population. Two types of variation are commonly observed in studies: differences among samples and the “error” in estimating a population parameter (e.g. mean) from a sample. While these concepts are fundamentally very different, the associated variation is often expressed using similar notation—an interval that represents a range of values with a lower and upper bound.

In this article we discuss how common intervals are used (and misused).

▲ Depicting variability and uncertainty using intervals and error bars. (read)

Altman, N. & Krzywinski, M. (2024) Depicting variability and uncertainty using intervals and error bars. Laboratory Animals 58:453–456.