Simulation for Inference Bret Larget Departments of Botany and of Statistics University of Wisconsin Madison March 25 2008 1 12 Overview We will use simulation to assess uncertainty in predictions and in estimated regression coefficients This is in contrast to deriving formulas for standard errors One advantage is that we can assess uncertainty in quantities for which there are not formulas A second advantage is we do not need to learn how to derive formulas However we will require learning to write small programs in R 2 12 How many girls How many girls do we expect in 500 births The population probability of being a girl is 0 488 This is a relatively simple problem for which you learned a formula last semester We will compare answers with simulation and the formula Simulation of Probability Models Discrete Predictive Simulation 3 12 Formula Answer We can model the number of girls with a binomial distribution The mean is np 500 0 488 244 The standard deviation of the binomial distribution is p p np 1 p 500 0 488 0 512 11 2 With a normal approximation we expect about a 95 chance that the number of girls would be between 244 2 11 or from 222 to 266 A precise calculation of this probability is sum dbinom 222 266 500 0 488 1 0 9559977 Simulation of Probability Models Discrete Predictive Simulation 4 12 Simulation Approach We begin by writing a function in R to simulate the number of girls in 500 births from the binomial distribution The function rbinom simulates draws from the binomial distribution The arguments are in order the number of draws n and p Unfortunately these are named n size and prob girls sim function n p return rbinom n 1 size n prob p girls sim 500 0 488 1 244 Simulation of Probability Models Discrete Predictive Simulation 5 12 Simulation Approach cont Use the replicate function to repeat this many times say 1000 The object n girls stores the answers n girls replicate 1000 girls sim 500 0 488 mean n girls 1 244 276 sd n girls 1 10 58394 sum n girls 222 n girls 266 1000 1 0 962 Simulation of Probability Models Discrete Predictive Simulation 6 12 A Simpler Version For this particular example we could have done this simulation with a single R command without writing a function However the paradigm of writing a function to do one simulation and then using replicate to repeat this will be helpful for more complicated examples n girls2 rbinom 1000 500 0 488 mean n girls2 1 243 681 sd n girls2 1 11 01583 Simulation of Probability Models Discrete Predictive Simulation 7 12 Sibling Example What if we wanted a more complicated example where we modeled the number of girls from a Beta Binomial distribution with overdispersion similar to that observed Finding a formula in this situation would be a lot of work but a simulation is not too bad The Beta distribution is for continuous numbers between 0 and 1 and is defined by has two parameters and Setting 9 0 488 and 9 0 512 fits the data well Simulation of Probability Models Discrete Predictive Simulation 8 12 Sibling Example cont We use the function rbetabin ab from the VGAM library for the Beta Binomial distribution siblings read table siblings txt header T family size with siblings boy girl sum family size 1 145 with siblings sum girl 1 72 library VGAM girls sim function sizes p M return sum rbetabin ab length sizes sizes p M 1 p M g replicate 1000 girls sim sizes family size p 0 488 M 9 Simulation of Probability Models Discrete Predictive Simulation 9 12 Sibling Example cont mean g 1 70 87 sd g 1 6 801129 quantile g c 0 025 0 05 0 5 0 95 0 975 2 5 57 5 60 50 71 Simulation of Probability Models 95 97 5 82 84 Discrete Predictive Simulation 10 12 Sibling Example cont 25 Percent of Total 20 15 10 5 0 50 60 70 80 90 g Simulation of Probability Models Discrete Predictive Simulation 11 12 Detach VGAM If you use the VGAM library it will mess up the predict function This command will unload a library detach package VGAM Simulation of Probability Models Discrete Predictive Simulation 12 12
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