Transcript meiosis_8_for_vlex

Meiosis and variation
Lesson 8
The χ2 test
Learning objectives
You should be able to:
Use the chi-squared (χ2) test to test the
significance of the difference between
observed and expected results. (The
formula for the chi-squared test will be
provided)
Previously we have discussed how we expect a
particular ratio from dihybrid inheritance of two
unlinked genes – to recap………………………
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Dihybrid inheritance of two unlinked genes
produces a phenotypic ratio of 9:3:3:1
A 9:3:4 ratio suggests recessive epistasis
A 12:3:1 ratio or a 13:3 ratio suggests
dominant epistasis
A 9:7 ratio suggests complementary action of
two genes
A good example of this is
the experiments that
Mendel did on pea plants
He used pea plants and looked at the inheritance of
two traits:
Seed colour:
Seed shape:
Yellow (G allele) or green (g allele)
Round (R allele) or wrinkled (r allele)
He crossed pure breeding peas with green, wrinkled seeds (ggrr)
With pure breeding peas with yellow, round seeds (GGRR)
What ratio would he expect, assuming normal dihybrid inheritance of
unlinked genes with no interaction or codominance?
Mendel’s Experimental Garden
Answer
Imagine we did Mendel’s
experiment and observed the
following ratio of phenotypes
169 yellow, round
54 green, round
51 yellow, wrinkled
14 green, wrinkled
These are our observed results
They look pretty close to the 9:3:3:1 ratio of
phenotypes that we expected but is it close enough
to say that this IS the ratio we are getting and
therefore that straightforward dihybrid inheritance
of unlinked genes, with no interaction, is operating
Our expected results
There are 288 individuals here
If we had 288 individuals and a perfect 9:3:3:1 ratio we would
expect the following number of offspring from each
phenotype
Can you calculate these yourself?
Yellow, round
Green, round
Yellow, wrinkled
Green, wrinkled
Our expected results
There are 288 individuals here
If we had 288 individuals and a perfect 9:3:3:1 ratio
we would expect the following number of offspring
Yellow, round 9/16 x 288 = 162
Green, round 3/16 x 288 = 54
Yellow, wrinkled 3/16 x 288 = 54
Green, wrinkled 1/16 x 288 = 18
These are our expected results
The χ2 test (chi-squared
test)
This is a statistical test that compares observed and expected
results to see if they differ significantly
We can use the chi-squared test to see if we are getting a
significant departure from these ratios we expected, in other
words if some other form of inheritance is occurring that we
did not know about
In order to do this we first need a null hypothesis
This is the assumption that there is no statistically significant
difference between the ratios that we expected or predicted
based on the kind of inheritance we thought was operating
and the ratios of phenotypes that we actually see.
We need to calculate the χ2
value from the expected and
observed ratios that we have
This is the equation we use (you’ll be
given the equation in the exam if you
need it)
Χ2 =
∑
(O-E)2
E
It’s easiest to use a table
to calculate this
The one on the following slide is for the
problem about pea seed colour and
shape that we were looking at
Copy and complete
group
Observed Expected
(O)
(E)
O-E
(O-E)2
(O-E)2
E
Yellow
round
Green
round
Yellow
wrinkled
Green
wrinkled
Χ2 =
group
Observed
(O)
Expected
(E)
O-E
(O-E)2
(O-E)2
E
Yellow
round
169
162
7
49
0.30
Green
round
54
54
0
0
0.00
Yellow
wrinkled
51
54
-3
9
0.17
Green
wrinkled
14
18
-4
16
0.88
Χ2 = 1.35
What does this mean?
We need to look up the this value in a table of
Χ2 values to see if it is significant, in other
words if the observed results differ
significantly from the expected results.
This would mean that we could reject our null
hypothesis that there is no significant
difference between the two.
Degrees of freedom
In order to use the table we need to know how many
degrees of freedom there are in the problem we are
looking at and how many groups there are
The number of groups or categories is easy to work
out. In the example we are looking at there are 4
possible phenotypes, hence 4 groups or categories
Degrees of freedom = number of groups -1
so in our example there are 3 degrees of freedom
The Χ2 table
Χ2
Number
of
classes
Degrees
of
freedom
2
1
0.00
0.10
0.45
1.32
2.71
3.84
5.41
6.64
3
2
0.02
0.58
1.39
2.77
4.61
5.99
7.82
9.21
4
3
0.12
1.21
2.37
4.11
6.25
7.82
9.84
11.34
5
4
0.3
1.92
3.36
5.39
7.78
9.49
11.67 13.28
6
5
0.55
2.67
4.35
6.63
9.24
11.07 13.39 15.09
0.99
0.75
0.50
0.25
0.10
0.05
0.02
0.01
99%
75%
50%
25%
10%
5%
2%
1%
Probability that
deviation is due
to chance alone
What value do we look
at?
We know the number of classes and the degrees of
freedom that are operating but which column, in
other words, which level of probability should we
look at?
In all statistics problems we use the 5% level of
probability. More of what this means in a bit………..
So what is the critical Χ2 value that we need to
compare our calculated Χ2 value with?
The Χ2 table
Χ2
Number
of
classes
Degrees
of
freedom
2
1
0.00
0.10
0.45
1.32
2.71
3.84
5.41
6.64
3
2
0.02
0.58
1.39
2.77
4.61
5.99
7.82
9.21
4
3
0.12
1.21
2.37
4.11
6.25
7.82
9.84
11.34
5
4
0.3
1.92
3.36
5.39
7.78
9.49
11.67 13.28
6
5
0.55
2.67
4.35
6.63
9.24
11.07 13.39 15.09
0.99
0.75
0.50
0.25
0.10
0.05
0.02
0.01
99%
75%
50%
25%
10%
5%
2%
1%
Probability that
deviation is due
to chance alone
So do we have a
significant difference?
If the Χ2 value you have calculated is greater
than the critical Χ2 value you have looked up
in the table then there is a significant
difference between the observed and
expected results
LEARN THIS!
This would mean that you could reject the null
hypothesis (that there was no significant difference
between observed and expected results)
So what happens with our
problem?
Our calculated Χ2 value is 1.35 which is less than the
critical Χ2 value of 7.32 from the table
There is therefore no significant difference between
our observed and expected results and we do not
reject our null hypothesis
We can assume that we were correct in thinking that
pod colour and shape were controlled by 2 unlinked
genes with no interaction between the two
Other uses of the Χ2 test
You can use the
chi-squared test
to compare sets
of data that are
categorical in
order to see if
the ratios differ
significantly
e.g. comparing the
proportion of brown and
yellow banded snail shells
from woods and fields to
see if the shells are more
likely to be yellow in one
area than another
For those that love maths
Just a little extra information for those that are interested. What
exactly did we mean when we said we were looking at the
5% probability level in the table of Χ2 values?
If we found a significant difference at this level of probability,
between our observed and expected results, we would write it
as p<0.05
This means that there would be a less than 5% chance of getting
the observed results we did if there was in reality no
difference between the two groups
We could re phrase this as there being a less than 5% chance of
our null hypothesis being correct
Practice problems
1. Black coat color in Cocker Spaniels is governed
by a dominant allele B and red coat color is
governed by its recessive allele b; solid pattern is
governed by the dominant allele of an
independently assorting locus S, and spotted
pattern by its recessive allele s. A solid-black
male is mated to a solid-red female and produces a
litter of 6 pups: 2 solid black, 2 solid red, 1 black
and white, and 1 red and white. Determine the
genotypes of the parents.
2) Aliens are attacking the earth and settling in the Bay Area.
As a budding geneticist you are interested to find that they
have two traits each determined by single independently
segregating genes. When the aliens have the dominant H
allele they have very elastic heads, while the recessive h
allele gives a very rigid head. Additionally, small brain size
is due to a dominant allele B and large brain size is
determined by a recessive allele b. When two parents
heterozygous for brain size and head elasticity mate they
produce: 9 small brained elastic headed offspring, 3 large
brained elastic headed offspring, and 3 small brained rigid
headed offspring. Do these genes follow the laws of
Mendelian segregation? How? Which class is missing?