Elementary Statistics for Lawyers

References

Evett and Weir, 1998. Interpreting DNA evidence.

Balding, 2005. Weight-of-evidence for forensic DNA profiles.

Sampling Theory

• Moser and Kalton, 1971. Survey methods in social investigation.

• pg. 56 “But if the survey results are to be generalized in this way, then the part of the population chosen for study should be selected according to the rules of statistical theory.

If it is not, statistically rigorous inferences from the sample to the population cannot be made and must not be attempted.”

Moser and Kalton, 1971

• pg. 56 “It is entirely wrong to make an arbitrary selection of cases, to rely on volunteers or on people who happen to be at hand, and then claim that they are a proper sample of some particular population.”

Moser and Kalton, 1971

• pg. 82 “The layman tends to think of ‘random’ as being equivalent to ‘haphazard’ and to believe that as long as the investigator does not exercise conscious selection and avoids obvious dangers, randomness is ensured. This is far from correct.”

Taking Random Samples, Resources

• Master Address File (MAF), maintained by the census bureau

• Delivery Sequence File (DSF), used to update MAF twice a year, from USPS

• Address Control File (ACF), may include residences that don’t get mail delivery

• National Health Interview Survey (NHIS)

Probability Laws

• 0 Pr(x) 1

• The certain event has, Pr(x) = 1

Mutually exclusive events

• Probability of either A or B = Pr(A) + Pr(B)

A

B

Universe

Dependent Events

• Pr(J and K) = Pr(J) Pr(K|J)• Suppose in a population 75% are Chinese

and 25% are Asian Indian• In the Asian Indian population 4.8% are

8,9.3 at the TH01 locus [Pr(8,9.3|Asian Indian)]

• Pr(randomly chosen person is Asian Indian and 8,9.3 at TH01) = 0.25 0.048= 0.012

Independent Events

• Pr(J and K) = Pr(J) Pr(K)

• Frequency of TH01 15,16 in Caucasians = 1/8

• Frequency of vWA 16,17 in Caucasians = 1/9

• Frequency of people that are 15,16 at TH01 and 16,17 at vWA is (1/8) (1/9) = 1/72

Likelihood Ratios

• When there are two explanations for an event we can compare the probability of each event to assess their value.

• Suppose we have genetic evidence E, and it is found that the suspect, S, matches the evidence. The true culprit is C. The prosecution explanation is S=C, the defense explanation is, S=I, a randomly chosen person.

Likelihood Ratios

• A simple LR is Pr[G|S=C]/Pr[G|S=I]

• Suppose G is a TH01 genotype, 15,16.

• Pr[G|S=C] = 1 (ignoring lab error, false negative)

• Pr[G|S=I] = 1/8 (ignoring lab error, false positive)

Likelihood Ratios

• A false negative is exclusion of two identical samples due to a lab error.

• The prosecution hypothesis is really that both S=C and a false negative did not happen (e.g. two independent events).

• This happens with probability, 1 [1-Pr(false negative)]

• If the Pr(false negative) =0.01 then the numeratorin the LR is about 0.99 not 1.

Likelihood Ratios

• A false positive is inclusion of two different samples due to a lab error.

• The defense hypothesis is that S=I OR the lab made a false positive (mutually exclusive)

• This happens with probability Pr(I=G) + Pr(false positive).

• If Pr(false positive) = 0.0001 then the denominator of the LR would never be smaller than 0.0001.