Some basic things(homework presentation style,

basic algebra and stats)

Some guidelines for homework presentation:

Try to always talk in terms of substance rather than the symbols only.

Some guidelines for homework presentation:

Try to always talk in terms of substance rather than the symbols only.

For instance:

u=sq2 – we know that u equals s times q squared

q0=.02 – q0 is .02

Δq=p0u+q0v – we know that the delta q equals p0

times u plus q0 times v

Some guidelines for homework presentation:

Try to always talk in terms of substance rather than the symbols only.

For instance:

u=sq2 – we know that the rate of mutation of the A1 into the A2 allele equals the product of the selection coefficient s and the

squared frequency of the other, A2, allele

Some guidelines for homework presentation:

Try to always talk in terms of substance rather than the symbols only.

For instance:

u=sq2 – we know that the rate of mutation of the A1 into the A2 allele equals the product of the selection coefficient s and the

squared frequency of the other, A2, allele

q0=.02 – the frequency of the less common, value decreasing (or disease-causing?) allele in the parental generation is .02

Some guidelines for homework presentation:

Try to always talk in terms of substance rather than the symbols only.

For instance:

u=sq2 – we know that the rate of mutation of the A1 into the A2 allele equals the product of the selection coefficient s and the

squared frequency of the other, A2, allele

q0=.02 – the frequency of the less common, value decreasing (or disease-causing?) allele in the parental generation is .02

Δq=p0u+q0v – we know that the change in the frequency of the A2 allele over the generations will equal the sum of a component due

to mutation of A1 into A2, and a component due to the back mutation (of A2 into A1). Because the assignment says that

back mutation is negligible (and this is because the initial frequency of the A2 allele, q0, is very small), this can simplify

to: Δq=p0u

This is not easy (for anyone). If you think people a) fully grasp Falconer with complete ease after a single reading, or b) have no difficulties always talking in terms of substance rather than only symbols, you are wrong. It is difficult, so do not worry (you are not the only one).

This is not easy (for anyone). If you think people a) fully grasp Falconer with complete ease after a single reading, or b) have no difficulties always talking in terms of substance rather than only symbols, you are wrong. It is difficult, so do not worry (you are not the only one).

If you think you lack the basis for the course: you most probably do not. All it requires is a bit of high school math (which I will revise in the next slides), and lots of careful reading (and re-reading) of Falconer. Fully grasping the material is not easy and requires re-reading the book multiple times. E.g., I’ve read it many times and still discover new things with every new reading. It’s the nature of the material. On the up side: it keeps things fun!

This is not easy (for anyone). If you think people a) fully grasp Falconer with complete ease after a single reading, or b) have no difficulties always talking in terms of substance rather than only symbols, you are wrong. It is difficult, so do not worry (you are not the only one).

If you think you lack the basis for the course: you most probably do not. All it requires is a bit of high school math (which I will revise in the next slides), and lots of careful reading (and re-reading) of Falconer. Fully grasping the material is not easy and requires re-reading the book multiple times. E.g., I’ve read it many times and still discover new things with every new reading. It’s the nature of the material. On the up side: it keeps things fun!

In short: do not worry. If you are having trouble understanding the material, it does not necessarily mean you are falling behind. All I will ask from you is to just keep reading the book, and ask when something isn’t clear.

Some basic algebra and stats:

Some basic algebra and stats:

Binomial expansion: (p + q)n

For us, the n=2 case is particularly relevant:

(p + q)2 = p2 + 2pq + q2

It is relevant, for instance, because of the following application:

Some basic algebra and stats:

Binomial expansion: (p + q)n

For us, the n=2 case is particularly relevant:

(p + q)2 = p2 + 2pq + q2

It is relevant, for instance, because of the following application:

Genotype frequencies in the offspring generation as a function of allele frequencies in the parental generation

Maternal gametes (and their frequencies)

Paternal gametes (and their frequencies)

A1 (p) A2(q)

A1 (p) A1A1 (p2) A1A2 (pq)

A2 (q) A2A1 (qp) A2A2 (q2)

Some basic algebra and stats:

Binomial expansion: (p + q)n

For us, the n=2 case is particularly relevant:

(p + q)2 = p2 + 2pq + q2

It is relevant, for instance, because of the following application:

Here, you can arrive at thesolution by counting theresulting genotype frequenciesin the table, or via the binomialexpansion:

(p + q)2 = p2 + 2pq + q2

Genotype frequencies in the offspring generation as a function of allele frequencies in the parental generation

Maternal gametes (and their frequencies)

Paternal gametes (and their frequencies)

A1 (p) A2(q)

A1 (p) A1A1 (p2) A1A2 (pq)

A2 (q) A2A1 (qp) A2A2 (q2)

Some basic algebra and stats:

Variance

y345

10981267

my=5.5

Some basic algebra and stats:

Variance

Vy = S (yi - my)2 / N

- the mean square

- we’ll address sums of squares and mean squares more in the lext lecture!

y345

10981267

my=5.5

Some basic algebra and stats:

Covariance

x y1 32 43 54 105 96 87 18 29 610 7

mx=5.5

my=5.5

Some basic algebra and stats:

Covariance

covxy = [ S (xi – mx) (yi – my) ] / N

- the mean cross-product

x y1 32 43 54 105 96 87 18 29 610 7

mx=5.5

my=5.5

Some basic algebra and stats:

Standardized covariance (correlation coefficient or standardized regression coefficient)

Some basic algebra and stats:

Standardized covariance (correlation coefficient or standardized regression coefficient)

Correlation:

rxy = covxy / sdxsdy

sd = √var

rxy = covxy / √varX√vary

Some basic algebra and stats:

Standardized covariance (correlation coefficient or standardized regression coefficient)

Correlation:

rxy = covxy / sdxsdy

sd = √var

rxy = covxy / √varX√vary

Regression:

covOP = bOP varP → bOP = covOP / varP

OPbOP

varP

Some basic algebra and stats:

Proportions vs. counts

y345

10981267

my=5.5

Some basic algebra and stats:

Proportions vs. counts

Mean:

my = Syi / N

so, the mean value is the sum of values divided by their number.

y345

10981267

my=5.5

Some basic algebra and stats:

Proportions vs. counts

Mean:

my = Syi / N

so, the mean value is the sum of values divided by their number.

Sometimes, the number is given as a proportion (i.e., on a scale from 0 to 1). This is what we have been doing with allele (andsometimes genotype) frequencies, for instance.

If instead of the absolute number we have a proportion for each value, to obtain the mean we multiply each value by its respectiveproportion, and sum over all the values.

my = Syipi

y345

10981267

my=5.5

Some basic algebra and stats:

Proportions vs. counts

For instance:

my = Syi / N = (3+4+5+…7) / 10 = 5.5

my = Syipi = 3* 1/10 + 4 * 1/10 + … 7 * 1/10 = (3+4+5+…7) / 10

= 5.5

y345

10981267

my=5.5

Some basic algebra and stats:

Proportions vs. counts

For instance:

my = Syi / N = (3+4+5+…7) / 10 = 5.5

my = Syipi = 3* 1/10 + 4 * 1/10 + … 7 * 1/10 = (3+4+5+…7) / 10

= 5.5

→ so, same thing

y345

10981267

my=5.5

Some basic algebra and stats:

Scaling values as deviations from the meanx y1 32 43 54 105 96 87 18 29 610 7

mx=5.5

my=5.5

Some basic algebra and stats:

Scaling values as deviations from the mean

Vy = S (yi - my)2 / N

covxy = [ S (xi – mx) (yi – my) ] / N

x y1 32 43 54 105 96 87 18 29 610 7

mx=5.5

my=5.5

Some basic algebra and stats:

Scaling values as deviations from the mean

Vy = S (yi - my)2 / N

covxy = [ S (xi – mx) (yi – my) ] / N

These are the expressions for the variance and the covariance.Both require expressing each value as a deviation from themean of its respective variable (e.g., x – mx). Sometimes, however, in the derivations in the Falconer book, the valueshave already been so expressed (i.e., the mean has been subtracted at some point). To obtain the variance/covarianceof the values already scaled in this way:

x y1 32 43 54 105 96 87 18 29 610 7

mx=5.5

my=5.5

Some basic algebra and stats:

Scaling values as deviations from the mean

Vy = S (yi - my)2 / N

covxy = [ S (xi – mx) (yi – my) ] / N

These are the expressions for the variance and the covariance.Both require expressing each value as a deviation from themean of its respective variable (e.g., x – mx). Sometimes, however, in the derivations in the Falconer book, the valueshave already been so expressed (i.e., the mean has been subtracted at some point). To obtain the variance/covarianceof the values already scaled in this way:

Vy = Syi2 / N

covxy = Sxiyi / N

x y1 32 43 54 105 96 87 18 29 610 7

mx=5.5

my=5.5

Some basic algebra and stats:

Scaling values as deviations from the mean

Vy = S (yi - my)2 / N

covxy = [ S (xi – mx) (yi – my) ] / N

These are the expressions for the variance and the covariance.Both require expressing each value as a deviation from themean of its respective variable (e.g., x – mx). Sometimes, however, in the derivations in the Falconer book, the valueshave already been so expressed (i.e., the mean has been subtracted at some point). To obtain the variance/covarianceof the values already scaled in this way:

Vy = Syi2 / N or in case of proportions:Vy = Syi

2pi

covxy = Sxiyi / Ncovxy = Sxiyipi

x y1 32 43 54 105 96 87 18 29 610 7

mx=5.5

my=5.5

This is most of the basics you need for the course. If you want, you can print these slides and bring them to class.