Some basic things (homework presentation style, basic algebra and stats)

1. Some basic things(homework presentation style,basic algebra and stats)

2. Some guidelines for homework presentation: Try to always talk in terms of substance rather than the symbols only.

3. 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 knowthat the delta q equals p0 timesu plus q0 timesv

4. 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

5. Some guidelines for homework presentation: Try to always talk in terms of substancerather 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

6. 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

7. 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).

8. 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! 

9. 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.

10. Some basic algebra and stats:

11. 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:

12. 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:

13. 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 the • solution by counting the • resulting genotype frequencies • in the table, or via the binomial • expansion: • (p + q)2 = p2 + 2pq + q2

14. Some basic algebra and stats: • Variance

15. 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!

16. Some basic algebra and stats: • Covariance

17. Some basic algebra and stats: • Covariance • covxy = [ S (xi – mx) (yi – my) ] / N • - the mean cross-product

18. Some basic algebra and stats: • Standardized covariance • (correlation coefficient or standardized regression coefficient)

19. Some basic algebra and stats: • Standardized covariance • (correlation coefficient or standardized regression coefficient) • Correlation: • rxy = covxy / sdxsdy • sd = √var • rxy = covxy / √varX√vary

20. 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 = bOPvarP→ bOP = covOP / varP varP bOP P O

21. Some basic algebra and stats: • Proportions vs. counts

22. 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.

23. 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 (and • sometimes 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 respective • proportion, and sum over all the values. • my = Syipi

24. 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

25. 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

26. Some basic algebra and stats: • Scaling values as deviations from the mean

27. 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

28. 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 the • mean of its respective variable (e.g., x – mx). Sometimes, • however, in the derivations in the Falconer book, the values • have already been so expressed (i.e., the mean has been • subtracted at some point). To obtain the variance/covariance • of the values already scaled in this way:

29. 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 the • mean of its respective variable (e.g., x – mx). Sometimes, • however, in the derivations in the Falconer book, the values • have already been so expressed (i.e., the mean has been • subtracted at some point). To obtain the variance/covariance • of the values already scaled in this way: • Vy = Syi2 / N • covxy = Sxiyi / N

30. 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 the • mean of its respective variable (e.g., x – mx). Sometimes, • however, in the derivations in the Falconer book, the values • have already been so expressed (i.e., the mean has been • subtracted at some point). To obtain the variance/covariance • of the values already scaled in this way: • Vy = Syi2 / N or in case of proportions: Vy = Syi2pi • covxy = Sxiyi / N covxy = Sxiyipi

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