1. Distributions, Iteration, Simulation Why R will rock your world (if it hasn’t already)

2. Simulation Sampling Calculation Iteration Data storage Summation

3. General structure of common functions on distributions There are many distributions in R (e.g. norm is Gaussian) For every statistical distribution in R, there are 4 related functions d: density p: probability distribution q: quantile r: random number generation e.g. rnorm(10) draws ten values randomly from a standard normal distribution Like most R functions, one can alter the defaults by specifying additional parameters. What parameters can be altered varies by distrubution

4. > dnorm(0) <-default mean=0, sd=1 [1] 0.3989423 input output > pnorm(2) <-default mean=0, sd=1 [1] 0.9772499 input output

5. > qnorm(.40) [1] -0.2533471 input output > rnorm(100, mean=0,sd=1) [1] 0.66075674 -0.55118652 0.08139252 0.46958450 -0.05435657 -0.86266560 1.07616374 [8] 0.16302857 -0.89804740 -0.78564903 0.29835536 -0.71735186 -1.51407253 0.89670212 [15] 1.18914054 0.48275543 -0.07937962 -0.79158265 0.66515914 0.91851769 -0.46559128 [22] 0.73002690 0.28342742 -1.15366595 1.22958479 0.50383209 -0.42909720 -0.69166003 [29] 0.16479583 1.15306943 -0.34162134 -1.10417420 0.74769236 1.09840855 -0.51784452 [36] 0.70605966 0.28660515 -0.62594167 -0.20681508 -0.69174153 -0.85814328 -0.23921311 [43] -0.13501255 -0.26683858 -1.67594580 1.19625453 -1.46812625 0.65995661 -0.79434250 [50] 0.51520368 -0.63391224 -0.16388674 -1.19557850 0.57594851 -0.32403622 -0.69371174 [57] -1.12080737 -0.28248934 1.40289924 0.82057604 -0.78505396 -0.05197960 -1.04549759 [64] -0.01334230 1.46494433 -0.18540372 -1.80950468 -0.18717730 -0.20780014 -0.07233845 [71] 0.11554796 0.83534676 -1.28519697 0.41796046 -1.48320795 -0.37269167 1.33504531 [78] 2.89549729 1.19711458 -0.11064494 1.23430822 0.42402993 -0.70746642 -1.01444881 [85] 1.19138047 -0.24913786 -0.29479432 0.21643338 0.50766542 0.69286785 -1.58078913 [92] -0.10119364 -0.44138704 -0.81407226 -1.57577525 2.83165289 -0.55277727 -1.50489032 [99] 0.09719564 0.75148485

6. Simulating random variables in R rnorm(): uni normal mvrnorm(): multi normal rbinom(): binomial runif(): uniform rpois(): poisson rchisq():  2 rnbinom(): neg. binomial rlogis(): logistic rbeta(): beta rgamma(): gamma rgeom(): geometric rlnorm(): log normal rweibull(): Weibull rt(): t rf(): F …

7. loops for() and while() are the most commonly used others (e.g. repeat) are less useful for(i in [vector]){ do something } while([some condition is true]){ do something else }

8. for loop { for (i in 10:1){ + cat("Matt is here for ", i, " more days \n") + } cat(“Matt is gone”) } Matt is here for 10 more days Matt is here for 9 more days Matt is here for 8 more days Matt is here for 7 more days Matt is here for 6 more days Matt is here for 5 more days Matt is here for 4 more days Matt is here for 3 more days Matt is here for 2 more days Matt is here for 1 more days Matt is gone

9. The Elementary Conditional if if([condition is true]){ do this } if([condition is true]){ do this }else{ do this instead } ifelse([condition is true],do this,do this instead)

10. Conditional operators == : is equal to (NOT THE SAME as ‘=‘) ! : not != : not equal inequalities (>, =,<=) & : and by element (if a vector) | : or by element && : and total object || : y total object xor(x,y): x or y true by not both

12. Other useful functions: sample(): sample elements from a vector, either with or without replacement and weights subset(): extract subset of a data frame based on logical criteria

13. Time to get your hands dirty Open up the R file sim.R Different scripts separated by lines of “#” Two approaches to the CLT Exercise #1 Simple ACE Simulator Simple Factor model/IRT simulator Exercise #2 “solutions” to exercises at the bottom

14. Exercise #1: Create a simple genetic drift simulator for a biallelic locus The frequency of an allele at t+1 is dependent on its frequency at t. The binomial distribution might come in handy There are many ways to model this phenomenon, more or less close to reality (source: wikipedia) N=100 N=1000

15. Simulating Path Diagrams Simulating data based on a path model is usually fairly easy Latent variables are sources of variance, and usually standard normal path coefficients represent strength of effect from causal variable to effect variable. Drawing simulations using path diagrams (or something similar) can help formalize the structure of your simulation

16. ACE Model 2 measured variables 6 sources of variance 3 levels of correlation 1,.5, or 0 Standard normal latent variables Effect of latent variable on phenotype = factor score * path coefficient Phenotype #1 Phenotype #2

17. mvrnorm() part of MASS library Allows generation of n-dimensional matrices of multivariate normal distributions Also useful for simulating data for unrelated random normal variables  efficient code and less work

18. mvrnorm() == simplicity mvrnorm(n,mu,Sigma,…) n = number of samples mu = vector of means for some number of variables Sigma = covariance matrix for these variables Example: mu<- rep(0,3) Sigma<-matrix(c(1,.5,.25,.5,1,.25,.25,.25,1),3,3) ex<-mvrnorm(n=500,mu=mu,Sigma=Sigma) pairs(ex)

19. output

20. Back to R script

21. Factor Models Similar principles In a simple model, each observed variable has 2 sources of variance (factor + “error”) Psychometric models often require binary/ordinal data

22. Back to R script

23. Exercise #2: Generate a simulator for measuring the accuracy of Falconer estimation in predicting variance components MZ DZ

24. Hints: Simulation: The EEAsim6.R script contains a twin simulator that will speed things up. source(“EEAsim6.R”) will load it. function is twinsim() “zyg” variable encodes zygosity; MZ=0, DZ=1 Variables required here are numsubs, a2, c2,and e2 Calculation: cor(x,y): calculate the Pearson correlation between x and y Falconer Estimates a 2 = 2*(cor MZ -cor DZ ) c 2 = (2*cor DZ )-cor MZ ) e 2 =1-a 2 -c 2 Iteration: a “for” loop will work just fine data storage: perhaps a nsim x nestimates matrix? visualization: prior graph used plot and boxplot, do whatever you want