Distributions, Iteration, Simulation

Why R will rock your world

(if it hasn’t already)

Simulation

Sampling Calculation Iteration Data storage Summation

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

> dnorm(0) <-default mean=0, sd=1[1] 0.3989423

input

output

> pnorm(2) <-default mean=0, sd=1[1] 0.9772499

input

output

> 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

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 …

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

}

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

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)

Conditional operators

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

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

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

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

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

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

mvrnorm()

part of MASS library Allows generation of n-dimensional matrices

of multivariate normal distributions Also useful for simulating data for unrelated

random normal variablesefficient code and less work

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)

output

Back to R script

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

Back to R script

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

MZ

DZ

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

a2 = 2*(corMZ-corDZ) c2 = (2*corDZ)-corMZ) e2=1-a2-c2

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