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Genetics
Chapter 19
Terms
• Probability:
– Number of times an event happens divided by
the total number of possible events
• Statistics:
– Measure of certainty of a result
• Population genetics:
– Mendel’s laws
– How Mendel’s laws affect the alleles present
in a set population
A Match?
Three possible outcomes of comparing two
DNA profiles:
1. Inconclusive - unknown
– There is not enough data to determine
2. Exclusion – non-match
– The profiles are too different to possibly be
the same individual
3. Inclusion – if the DNA profiles match
– Probability of seeing this match at random is
calculated
Probability of a match
• Out of the three possibilities:
– Inconclusive
– Exclude
– Include
• Only the third requires statistics
• Probability of a random match is
calculated based on:
– Allele frequencies in a small sampling of the
population
– Population genetic principles
Method to use?
• There is often more than one method of
calculating the statistics of seeing a
random match
• Which statistic test used depends on:
– Experience of user
– Local legal system
– Practicality of the approach
– Available data
– Question that needs to be answered
Probability
• Number of times an event happens
divided by the total number of possible
events
• Probability exists on a continuum between
zero (0) and one (1)
• Zero probability means event is impossible
• Probability of one means event occurred
• Most of the time however we are uncertain
– Probability gives a measure of how certain
Mendel Understood Probability
Probability: The number of times an event occurs
divided by the number of trials during which that event
could have happened
The probability of rolling a 2 with one roll of one die:
1 event / 6 possible outcomes =
1/6
Mendel Understood Probability
The Multiplication Rule: The probability of two or more
independent events occurring simultaneously is the
product of their individual probabilities.
The probability of rolling two 2’s with a pair of dice:
The probability of rolling a 2 =
1/6
So rolling two 2’s =
1/6 x 1/6 =1/36
Mendel Understood Probability
The Addition Rule: The probability that an event can
occur in two or more alternative ways is the sum of the
separate probabilities of the different ways.
(Used to answer “either / or” questions only)
The probability of rolling a 2 or a 5 =
1/6 + 1/6 = 1/3
Probability
One more thing to remember:
p(a mutually exclusive event) = 1 – p(all the other events)
Likelihood Ratio
• Comparison of probabilities
• Probability that event occurred:
– The defendant committed the murder
– Hypothesis of the prosecution (Hp)
• Probability that even did not occur:
– Someone else committed the murder
– Hypothesis of the defense (Hd)
• Likelihood ratio = Hp/Hd
Likelihood Ratio
• Likelihood ratio = Hp/Hd
• More evidence that defendant committed
the murder
– Hp becomes larger
– LR grows larger – more convincing
• Less evidence
– Hd becomes larger
– LR shrinks – less convincing
Statistics
• Measure of the certainty of a result
• Provides a sense of how reliable a
measurement is
• Testing two alternative hypotheses:
– Ho = null hypothesis
– Ha or H1 = alternative hypothesis
• Represent the only two possible conditions
– This man did it
– Or did not do it
Hypothesis Testing
Six steps:
1. Formulate two competing hypotheses
2. Select the statistical model/test to use
3. Determine the level of significance
4. Collect and analyze the data
5. Does the test statistic meet the level of
significance or not?
6. Reject or accept null hypothesis (Ho)
(A) Hypothesis Testing Decisions
Truth about the population
Decision based on
sample examined
Accept H0
H0 True
H1 True
Correct
decision
Type II error
Type I error
Correct
decision
Reject H0
(Accept H1)
(B) Example
Defendant
Courtroom Verdict
Not Guilty
Guilty
Not Guilty
Guilty
Correct
decision
Wrongfully
acquitted
Wrongfully
accused
Correct
decision
Figure 19.2, J.M. Butler (2005) Forensic DNA Typing, 2nd Edition © 2005 Elsevier Academic Press
Chi-Squared Test
• Called the “Goodness of fit” test
• How well does the observed data fit with
the expected results?
• If the data is what you expect it to be, then
your observed will be very close to your
expected – X2 will be low
• If the data is radically different than
expected than your Χ2 will be high
Chi-Squared Test
Χ2 = (Observed – Expected)2
Expected
• Accept the hypothesis as correct if the
data fits with the expectations
– Χ2 is lower than acceptable limit
• Reject the hypothesis as wrong if the data
simply does not match to the expected
– Because Χ2 is higher than acceptable limit
Confidence Intervals
• How much data fits into your statistical
limit?
• Most often people use 95% confidence
interval
• This means that you have 95% chance of
being correct
• 5% chance of being wrong
• p-value = 0.05
Population Genetics
• Determining how frequent alleles and
genotypes are within a given population
• Need to know this in order to determine
the random chance of seeing one specific
DNA profile
• Determined by sampling a portion of the
entire population
– Because it would take too much time and
money to genotype everyone
Genetic Language:
Gene - Discrete “unit factors” of inheritance
Allele - Different forms of a gene (e.g. Y or y)
Genotype - Allelic composition of a trait
(e.g. YY, Yy, or yy)
Phenotype - Physical manifestation of a trait
(e.g. Yellow or green seed)
Genetic Language:
Homozygous – Individuals with two
identical copies of a gene
Same allele (yy)
Heterozygous - Individuals with two
different copies of a gene
Two different alleles (Yy)
Mendel’s 1st Law:
Principle of segregation
Hereditary traits are determined by discrete factors (now
called genes) that appear in pairs. During sexual development,
these pairs are separated (segregated) into gametes and only
one factor from each parent is passed to the offspring.
Alleles are randomly separated into gametes during
meiosis. One allele, at random, goes into the gamete
and then is passed to baby.
Mendel’s 2nd Law:
Independent Assortment
Inheritance of a pair of factors for one trait is
independent of the simultaneous inheritance of
factors for another trait
Two genes will assort independently and randomly
and be inherited completely separately.
Mendel’s Laws
1. Principle of Segregation
Two alleles, of one gene, segregate randomly
during formation of gametes
2. Independent Assortment
Two genes will assort independently and
randomly from each other
Bi-allelic Gene
• In bi-allelic gene there are only two alleles
possible
– T or t – for tall or short pea plants
– R or r – for wrinkled or round seeds
• p = frequency of the more common of the
two alleles
• q = frequency of the less common of the
two alleles
Hardy-Weinberg Equilibrium
Where the allele frequencies stay constant
from one generation to the next
• Often calculated with a bi-allelic gene
(p and q)
Therefore…
• p and q remaining constant
Hardy-Weinberg Equilibrium
1. If there are only two alleles then the
following must be true:
p+q=1
The frequency of the two alleles added
together must equal the entire population
(a frequency of 1)
Hardy-Weinberg Equilibrium
2. The genotype frequencies can also be
calculated:
p2 + 2pq + q2 = 1
The frequency of each homozygote equals
the frequency of the allele squared
The frequency of heterozygote is 2 times p
times q Product Rule
These three
genotypes
must add
to one
Product
and Addition
Rules
Hardy-Weinberg Equilibrium
1. Allele frequencies add to one:
p+q=1
2. The genotype frequencies can be
calculated from the allele frequencies:
p2 + 2pq + q2 = 1
Hardy-Weinberg Equilibrium
1. Allele frequencies add to one:
p+q+r=1
2. The genotype frequencies can be
calculated from the allele frequencies:
p2 + 2pq + 2pr + 2rq + q2 + r2 = 1
How it was derived:
A (p)
a (q)
A (p)
a (q)
AA
(pp)
Aa
(pq)
Aa
(pq)
aa
(qq)
Frequencies:
Allele A
=p
Allele a
=q
Genotype AA = p2
Genotype Aa = 2pq
Genotype aa = q2
HWE and Product Rule:
Genotype Five Bi-allelic
Polymorphisms
(ex. SNPs)
Het = 2pq = 2(.6)(.3)
= .36
Het = 2pq = 2(.5)(.3)
= .30
Het = 2pq = 2(.15)(.8) = .24
Homo = q2 = (.2)(.2)
= .04
Het = 2pq = 2(.80)(.18)= .29
HWE and Product Rule:
= .36
= .30
= .24
= .04
= .29
(.36)(.3)(.24)(.04)(.29) = 0.00031
Or 1/3,226
Therefore, the chance of this matching
the wrong person is 1/3,226
Alleles vs. Genotypes
Allele frequency = number of copies of one
allele within a certain population
• Divided by the total number of all alleles
within this same population
Genotype frequency = number of individuals
with a certain genotype within a certain
population
• Divided by the total number of individuals
within this sample population
Alleles vs. Genotypes
• Number of alleles will determine how
many different genotypes are possible
• n = number of alleles
• n(n + 1)/2 = number of genotypes
• Theoretically this is true
• However some alleles and genotypes are
so rare as to not really exist
• Want to use real numbers, rather than
theoretical numbers
Alleles vs. Genotypes
• Table 19.1
Part a shows number of alleles ever seen
• Theoretical number of genotypes possible
Part b shows number of common alleles
• Number of genotypes ever seen
• This second table represents actual
number of possibilities
– This is what Forensic Labs use for
calculations
Any Questions?
Read Chapter 20
Guest Lecturer –
Forensic Anthropology
A Match!
Three possible explanations for a match:
1. Suspect left DNA at crime scene
– Trial needs to determine whether that
information proves suspect committed crime
2. Suspect’s profile matches by chance
– This is why statistics of seeing this DNA
profile at random are calculated
3. Match is a false positive result
– This is avoided at all times by validating
technology, running controls and duplicates
Randomization Trials
• Also known as permutation tests
• Calculate a statistic to determine how
unusual is the sample
– In this case, how rare is the DNA profile?
• Calculate this by randomizing the
genotypes 1,000 to 100,000 times
• How many times did you see the same
DNA profile at random?
• Get a statistic without any population data
Hardy-Weinberg Equilibrium
1908
• Hardy – an English mathematician
• Weinberg – a German physician
• Both derived, independently, an algebra
calculation for what happens to allele
frequencies within a population
• Assuming all those false conditions