## The Product Rule

The lack of population structuring with allele frequencies in Hardy-Weinberg equilibrium and linkage equilibrium (see Chapter 20)justifies the assumption that genotypes are independent at unlinked loci. With the assumption of independence, it then becomes possible to equate the overall match probability with the product of the locus-specific match probabilities. This combination of locus-specific match probabilities is referred to as the product rule. In other words, the match probability for the STR locus D13S317 can be combined with additional STR loci such as TH01 and D18S51 to decrease the odds of a random match to an unrelated individual.

Table 21.2 calculates the match probabilities for all 13 CODIS loci with the same STR profile using the three different U.S. population databases contained in Appendix II. Since the STR profile used for these examples is from a Caucasian and is present in the Appendix II allele frequency database, the Caucasian population is expected to provide the most common frequency estimate. This turns out to be the case where the calculated STR profile frequency estimates are 1.20 X 10-15 using the Caucasian allele frequencies and 6.04X 10-17 and 5.57 X 10-17 using the African-American and Hispanics allele frequencies, respectively (Table 21.2). As noted by Weir (2003), focusing on a suspect's racial group in the calculations has an element of conservativeness.

Table 21.2 Random match probabilities with all 13 CODIS loci

Often the rarity of a calculated DNA profile goes beyond one in billions (109) or trillions (1012) to numbers that are not frequently used because they are so large. A list of some big number names is contained in Table 21.3 to aid in verbal descriptions of rare DNA profiles. For example, the inverted value of 1.20x 10-15 is 1 in 8.37x 1014 or 0.84x 1015 (one in 0.84 quadrillion).

A1

A2

Allele 1 freq (p)

Allele 2 freq (q)

Expected genotype freq

From U.S. Caucasian (N

= 302); Appendix II

- sample in database

D13S317

11

14

0.33940

0.04801

2pq

0.0326

TH01

6

6

0.23179

P2

0.0537

D18S51

14

16

0.13742

0.13907

2pq

0.0382

D21S11

28

30

0.15894

0.27815

2pq

0.0884

D3S1358

16

17

0.25331

0.21523

2pq

0.1090

D5S818

12

13

0.38411

0.14073

2pq

0.1081

D7S820

9

9

0.17715

p2

0.0314

D8S1179

12

14

0.18543

0.16556

2pq

0.0614

CSF1PO

10

10

0.21689

p2

0.0470

FGA

21

22

0.18543

0.21854

2pq

0.0810

D16S539

9

11

0.11258

0.32119

2pq

0.0723

TPOX

8

8

0.53477

p2

0.2860

VWA

17

18

0.28146

0.20033

2pq

0.1128

AMEL

X

Y

Combined Frequency

1 in

1.20E-15

Combined Frequency

From African-American (N = 258); Appendix II - sample not in database; wrong population

 D13S317 11 14 0.3062 0.03488 2pq 0.0214 TH01 6 6 0.12403 p2 0.0154 D18S51 14 16 0.07198 0.15759 2pq 0.0227 D21S11 28 30 0.25775 0.17442 2pq 0.0899 D3S1358 16 17 0.33527 0.20543 2pq 0.1377

Expected genotype freq

From African-American (N = 258); Appendix II - sample not in database; wrong population

DBS818

D7S820

D8S1179

CSF1PO

D16SB39 TPOX VWA AMEL

17 X

13 9

14 10 22 11

0.3B271 0.108B3 0.14147 0.2B681 0.11628 0.19B74 0.37209 0.2422B

0.23837

0.30039

0.19B74 0.31783

0.1BB04

2pq p2

2pq p2 2pq 2pq p2 2pq

1 in

0.1682 0.0118 0.08B0 0.0660 0.04BB 0.1244 0.138B 0.07B1

6.04E-17

Product Rule Combined Frequency

From Hispanic (N = 140); Appendix II - sample not in database; wrong population

D13S317 11

CSF1PO FGA

D16SB39 TPOX VWA AMEL

14 28

TH01 D18SB1 D21S11 D3S13B8 16 DBS818 12 D7S820

D8S1179 12

17 X

14 6 16 30

13 9

14 10 22 11

18 Y

0.23B71 0.21429 0.13929 0.09643 0.28B71 0.3B000 0.11071 0.14286 0.23214 0.16786 0.13929 0.47143 0.21786

0.04643

0.13B71 0.26071 0.203B7 0.12B00

0.2B000

0.1B00G 0.26071

0.17143

2pq p2

0.0219

0.04B9

2pq 0.0378

2pq 0.0B03

2pq 0.1163

2pq 0.087B

0.0123

2pq 0.0714

0.0B39

2pq 0.0B04

2pq 0.0726

p2 0.2222

2pq 0.0747

5.57E-17

Product Rule Combined Frequency

Table 21.3

Names of big numbers with their corresponding scientific notation. From http://www.gomath.com/ htdocs/ToGoSheet/Algebra /bignumber.html.

 101 Ten 102 Hundred 103 Thousand 106 Million 109 Billion 1012 Trillion 1015 Quadrillion 1018 Quintillion 1021 Sextillion 1024 Septillion 1027 Octillion 1030 Nonillion 1033 Decillion 1036 Undecillion 1039 Duodecillion 1042 Tredecillion 1045 Quattuordecillion 1048 Quindecillion 1051 Sexdecillion 1054 Septendecillion 1057 Octodecillion 1060 Novemdecillion 1063 Vigintillion 10100 Google

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### Responses

• Donnamira
How to use the product rule in forensics?
1 year ago
• ROXY
How is the product rule applied in forensic?
12 days ago