Chapter 8 Probability & Simulation

8.1 Learning Objectives

8.1.1 Basic

  1. Understand what types of data are best modeled by different distributions
  2. Generate and plot data randomly sampled from the above distributions
  3. Test sampled distributions against a null hypothesis
  4. Define the following statistical terms:
  5. Calculate power using iteration and a sampling function

8.1.2 Intermediate

  1. Generate 3+ variables from a multivariate normal distribution and plot them

8.1.3 Advanced

  1. Calculate the minimum sample size for a specific power level and design

8.3 Distributions

Simulating data is a very powerful way to test your understanding of statistical concepts. We are going to use simulations to learn the basics of probability.

8.3.1 Uniform Distribution

The uniform distribution is the simplest distribution. All numbers in the range have an equal probability of being sampled.

Take a minute to think of things in your own research that are uniformly distributed.

8.3.1.1 Sample continuous distribution

runif(n, min=0, max=1)

Use runif() to sample from a continuous uniform distribution.

8.3.1.2 Sample discrete distribution

sample(x, size, replace = FALSE, prob = NULL)

Use sample() to sample from a discrete distribution.

You can use sample() to simulate events like rolling dice or choosing from a deck of cards. The code below simulates rolling a 6-sided die 10000 times. We set replace to TRUE so that each event is independent. See what happens if you set replace to FALSE.

Distribution of dice rolls.

Figure 8.1: Distribution of dice rolls.

You can also use sample to sample from a list of named outcomes.

##  [1] "cat"    "cat"    "cat"    "cat"    "ferret" "dog"    "bird"  
##  [8] "cat"    "dog"    "fish"

Ferrets are a much less common pet than cats and dogs, so our sample isn’t very realistic. You can set the probabilities of each item in the list with the prob argument.

##  [1] "fish" "dog"  "cat"  "dog"  "cat"  "dog"  "fish" "dog"  "cat"  "fish"

8.3.2 Binomial Distribution

The binomial distribution is useful for modeling binary data, where each observation can have one of two outcomes, like success/failure, yes/no or head/tails.

8.3.2.1 Sample distribution

rbinom(n, size, prob)

The rbinom function will generate a random binomial distribution.

  • n = number of observations
  • size = number of trials
  • prob = probability of success on each trial

Coin flips are a typical example of a binomial distribution, where we can assign heads to 1 and tails to 0.

##  [1] 1 1 1 0 1 1 0 1 0 0 1 1 1 0 0 0 1 0 0 0
##  [1] 1 1 1 0 1 0 1 1 1 0 1 1 1 0 0 1 1 1 1 1

You can generate the total number of heads in 1 set of 20 coin flips by setting size to 20 and n to 1.

## [1] 13

You can generate more sets of 20 coin flips by increasing the n.

##  [1] 10 14 11  7 11 13  6 10  9  9

You should always check your randomly generated data to check that it makes sense. For large samples, it’s easiest to do that graphically. A histogram is usually the best choice for plotting binomial data.

Run the simulation above several times, noting how the histogram changes. Try changing the values of n, size, and prob.

8.3.2.2 Exact binomial test

binom.test(x, n, p)

You can test a binomial distribution against a specific probability using the exact binomial test.

  • x = the number of successes
  • n = the number of trials
  • p = hypothesised probability of success

Here we can test a series of 10 coin flips from a fair coin and a biased coin against the hypothesised probability of 0.5 (even odds).

## 
##  Exact binomial test
## 
## data:  fair_coin and n
## number of successes = 4, number of trials = 10, p-value = 0.7539
## alternative hypothesis: true probability of success is not equal to 0.5
## 95 percent confidence interval:
##  0.1215523 0.7376219
## sample estimates:
## probability of success 
##                    0.4 
## 
## 
##  Exact binomial test
## 
## data:  biased_coin and n
## number of successes = 7, number of trials = 10, p-value = 0.3438
## alternative hypothesis: true probability of success is not equal to 0.5
## 95 percent confidence interval:
##  0.3475471 0.9332605
## sample estimates:
## probability of success 
##                    0.7

Run the code above several times, noting the p-values for the fair and biased coins. Alternatively, you can simulate coin flips online and build up a graph of results and p-values.

  • How does the p-value vary for the fair and biased coins?
  • What happens to the confidence intervals if you increase n from 10 to 100?
  • What criterion would you use to tell if the observed data indicate the coin is fair or biased?
  • How often do you conclude the fair coin is biased (false positives)?
  • How often do you conclude the biased coin is fair (false negatives)?

8.3.2.3 Statistical terms

The effect is some measure of your data. This will depend on the type of data you have and the type of statistical test you are using. For example, if you flipped a coin 100 times and it landed heads 66 times, the effect would be 66/100. You can then use the exact binomial test to compare this effect to the null effect you would expect from a fair coin (50/100) or to any other effect you choose. The effect size refers to the difference between the effect in your data and the null effect (usually a chance value).

{#p-value} The p-value of a test is the probability of seeing an effect at least as extreme as what you have, if the real effect was the value you are testing against (e.g., a null effect). So if you used a binomial test to test against a chance probability of 1/6 (e.g., the probability of rolling 1 with a 6-sided die), then a p-value of 0.17 means that you could expect to see effects at least as extreme as your data 17% of the time just by chance alone.

{#alpha} If you are using null hypothesis significance testing (NHST), then you need to decide on a cutoff value (alpha) for making a decision to reject the null hypothesis. We call p-values below the alpha cutoff significant. In psychology, alpha is traditionally set at 0.05, but there are good arguments for setting a different criterion in some circumstances.

{#false-pos}{#false-neg} The probability that a test concludes there is an effect when there is really no effect (e.g., concludes a fair coin is biased) is called the false positive rate (or Type I Error Rate). The alpha is the false positive rate we accept for a test. The probability that a test concludes there is no effect when there really is one (e.g., concludes a biased coin is fair) is called the false negative rate (or Type II Error Rate). The beta is the false negative rate we accept for a test.

The false positive rate is not the overall probability of getting a false positive, but the probability of a false positive under the null hypothesis. Similarly, the false negative rate is the probability of a false negative under the alternative hypothesis. Unless we know the probability that we are testing a null effect, we can’t say anything about the overall probability of false positives or negatives. If 100% of the hypotheses we test are false, then all significant effects are false positives, but if all of the hypotheses we test are true, then all of the positives are true positives and the overall false positive rate is 0.

{#power}{#sesoi} Power is equal to 1 minus beta (i.e., the true positive rate), and depends on the effect size, how many samples we take (n), and what we set alpha to. For any test, if you specify all but one of these values, you can calculate the last. The effect size you use in power calculations should be the smallest effect size of interest (SESOI). See (Lakens, Scheel, and Isager 2018)(https://doi.org/10.1177/2515245918770963) for a tutorial on methods for choosing an SESOI.

Let’s say you want to be able to detect at least a 15% difference from chance (50%) in a coin’s fairness, and you want your test to have a 5% chance of false positives and a 10% chance of false negatives. What are the following values?

  • alpha =
  • beta =
  • false positive rate =
  • false negative rate =
  • power =
  • SESOI =

{#conf-int} The confidence interval is a range around some value (such as a mean) that has some probability (usually 95%, but you can calculate CIs for any percentage) of containing the parameter, if you repeated the process many times.

A 95% CI does not mean that there is a 95% probability that the true mean lies within this range, but that, if you repeated the study many times and calculated the CI this same way every time, you’d expect the true mean to be inside the CI in 95% of the studies. This seems like a subtle distinction, but can lead to some misunderstandings. See (Morey et al. 2016)(https://link.springer.com/article/10.3758/s13423-015-0947-8) for more detailed discussion.

8.3.2.4 Sampling function

To estimate these rates, we need to repeat the sampling above many times. A function is ideal for repeating the exact same procedure over and over. Set the arguments of the function to variables that you might want to change. Here, we will want to estimate power for:

  • different sample sizes (n)
  • different effects (bias)
  • different hypothesised probabilities (p, defaults to 0.5)

Once you’ve created your function, test it a few times, changing the values.

## [1] 0.271253

8.3.2.5 Calculate power

Then you can use the replicate() function to run it many times and save all the output values. You can calculate the power of your analysis by checking the proportion of your simulated analyses that have a p-value less than your alpha (the probability of rejecting the null hypothesis when the null hypothesis is true).

## [1] 0.4678

1e4 is just scientific notation for a 1 followed by 4 zeros (10000). When you’re running simulations, you usually want to run a lot of them. It’s a pain to keep track of whether you’ve typed 5 or 6 zeros (100000 vs 1000000) and this will change your running time by an order of magnitude.

8.3.3 Normal Distribution

8.3.3.2 T-test

t.test(x, y, alternative, mu, paired)

Use a t-test to compare the mean of one distribution to a null hypothesis (one-sample t-test), compare the means of two samples (independent-samples t-test), or compare pairs of values (paired-samples t-test).

You can run a one-sample t-test comparing the mean of your data to mu. Here is a simulated distribution with a mean of 0.5 and an SD of 1, creating an effect size of 0.5 SD when tested against a mu of 0. Run the simulation a few times to see how often the t-test returns a significant p-value (or run it in the shiny app).

## 
##  One Sample t-test
## 
## data:  sim_norm
## t = 5.1431, df = 99, p-value = 1.367e-06
## alternative hypothesis: true mean is not equal to 0
## 95 percent confidence interval:
##  0.3027835 0.6831659
## sample estimates:
## mean of x 
## 0.4929747

Run an independent-samples t-test by comparing two lists of values.

## 
##  Welch Two Sample t-test
## 
## data:  a and b
## t = 0.043602, df = 197.5, p-value = 0.9653
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -0.2813281  0.2940499
## sample estimates:
## mean of x mean of y 
## 0.5123162 0.5059554

The paired argument defaults to FALSE, but it’s good practice to always explicitly set it so you are never confused about what type of test you are performing.

8.3.3.3 Sampling function

We can use the names() function to find out the names of all the t.test parameters and use this to just get one type of data, like the test statistic (e.g., t-value).

##  [1] "statistic"   "parameter"   "p.value"     "conf.int"    "estimate"   
##  [6] "null.value"  "stderr"      "alternative" "method"      "data.name"  
##          t 
## 0.04360244

Alternatively, use broom::tidy() to convert the output into a tidy table.

## # A tibble: 1 x 10
##   estimate estimate1 estimate2 statistic p.value parameter conf.low
##      <dbl>     <dbl>     <dbl>     <dbl>   <dbl>     <dbl>    <dbl>
## 1  0.00636     0.512     0.506    0.0436   0.965      197.   -0.281
## # … with 3 more variables: conf.high <dbl>, method <chr>,
## #   alternative <chr>

If you want to run the simulation many times and record information each time, first you need to turn your simulation into a function.

Run it a few times to check that it gives you sensible values.

## [1] 0.1002539

Now replicate the simulation 1000 times.

## [1] 0.2926

Run the code above several times. How much does the power value fluctuate? How many replications do you need to run to get a reliable estimate of power?

Compare your power estimate from simluation to a power calculation using power.t.test(). Here, delta is the difference between m1 and m2 above.

## 
##      Two-sample t test power calculation 
## 
##               n = 100
##           delta = 0.2
##              sd = 1
##       sig.level = 0.05
##           power = 0.2902664
##     alternative = two.sided
## 
## NOTE: n is number in *each* group

You can plot the distribution of p-values.

What do you think the distribution of p-values is when there is no effect (i.e., the means are identical)? Check this yourself.

Make sure the boundary argument is set to 0 for p-value histograms. See what happens with a null effect if boundary is not set.

8.3.4 Bivariate Normal

8.3.4.1 Correlation

You can test if two continuous variables are related to each other using the cor() function.

Below is one way to generate two correlated variables: a is drawn from a normal distribution, while x and y the sum of and another value drawn from a random normal distribution. We’ll learn later how to generate specific correlations in simulated data.

## [1] 0.5500246

Set n to a large number like 1e6 so that the correlations are less affected by chance. Change the value of the mean for a, x, or y. Does it change the correlation between x and y? What happens when you increase or decrease the sd for a? Can you work out any rules here?

cor() defaults to Pearson’s correlations. Set the method argument to use Kendall or Spearman correlations.

## [1] 0.529553

8.3.4.2 Sample distribution

What if we want to sample from a population with specific relationships between variables? We can sample from a bivariate normal distribution using mvrnorm() from the MASS package.

##           [,1]      [,2]
## [1,] 1.0000000 0.5081377
## [2,] 0.5081377 1.0000000

Plot your sampled variables to check everything worked like you expect. It’s easiest to convert the output of mvnorm into a tibble in order to use it in ggplot.

## Warning: `as_tibble.matrix()` requires a matrix with column names or a `.name_repair` argument. Using compatibility `.name_repair`.
## This warning is displayed once per session.

8.3.5 Multivariate Normal

You can generate more than 2 correlated variables, but it gets a little trickier to create the correlation matrix.

8.3.5.1 Sample distribution

##           [,1]      [,2]      [,3]
## [1,] 1.0000000 0.5983590 0.1529026
## [2,] 0.5983590 1.0000000 0.6891871
## [3,] 0.1529026 0.6891871 1.0000000

Alternatively, you can use the (in-development) package faux to generate any number of correlated variables. It also allows to to easily name the variables and has a function for checking the parameters of your new simulated data (check_sim_stats()).

## 
## ************
## Welcome to faux. For support and examples visit:
## http://debruine.github.io/faux/
## - Get and set global package options with: faux_options()
## ************
## Warning: All elements of `...` must be named.
## Did you want `multisim_data = c(A, B, C)`?
## # A tibble: 3 x 7
##       n var       A     B     C  mean    sd
##   <dbl> <chr> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1   200 A      1     0.54  0.1   10.5  7.27
## 2   200 B      0.54  1     0.73  20.6  9.46
## 3   200 C      0.1   0.73  1     30.7  9.64

8.3.5.2 3D Plots

You can use the plotly library to make a 3D graph.

## 
## Attaching package: 'plotly'
## The following object is masked from 'package:MASS':
## 
##     select
## The following object is masked from 'package:ggplot2':
## 
##     last_plot
## The following object is masked from 'package:stats':
## 
##     filter
## The following object is masked from 'package:graphics':
## 
##     layout

8.4 Example

This example uses the Growth Chart Data Tables from the US CDC. The data consist of height in centimeters for the z-scores of –2, -1.5, -1, -0.5, 0, 0.5, 1, 1.5, and 2 by sex (1=male; 2=female) and half-month of age (from 24.0 to 240.5 months).

8.4.1 Load & wrangle

We have to do a little data wrangling first. Have a look at the data after you import it and relabel Sex to male and female instead of 1 and 2. Also convert Agemos (age in months) to years. Relabel the column 0 as mean and calculate a new column named sd as the difference between columns 1 and 0.

## Parsed with column specification:
## cols(
##   Sex = col_character(),
##   Agemos = col_character(),
##   `-2` = col_double(),
##   `-1.5` = col_double(),
##   `-1` = col_double(),
##   `-0.5` = col_double(),
##   `0` = col_double(),
##   `0.5` = col_double(),
##   `1` = col_double(),
##   `1.5` = col_double(),
##   `2` = col_double()
## )

If you run the code above without putting dplyr:: before the select() function, you might get an error message. This is because the MASS package also has a function called select() and, since we loaded MASS after tidyverse, the MASS function becomes the default. When you loaded MASS, you should have seen a warning like “The following object is masked from ‘package:dplyr’: select”. You can use functions with the same name from different packages by specifying the package before the function name, separated by two colons.

8.4.2 Plot

Plot your new data frame to see how mean height changes with age for boys and girls.

8.4.3 Get means and SDs

Create new variables for the means and SDs for 20-year-old men and women.

## # A tibble: 2 x 4
##   sex      age  mean    sd
##   <chr>  <dbl> <dbl> <dbl>
## 1 male      20  177.  7.12
## 2 female    20  163.  6.46

8.4.4 Simulate a population

Simulate 50 random male heights and 50 random female heights using the rnorm() function and the means and SDs above. Plot the data.

Run the simulation above several times, noting how the density plot changes. Try changing the age you’re simulating.

8.4.5 Analyse simulated data

Use the sim_t_ind(n, m1, sd1, m2, sd2) function we created above to generate one simulation with a sample size of 50 in each group using the means and SDs of male and female 14-year-olds.

## [1] 0.002962042

8.4.6 Replicate simulation

Now replicate this 1e4 times using the replicate() function. This function will save the returned p-values in a list (my_reps). We can then check what proportion of those p-values are less than our alpha value. This is the power of our test.

## [1] 0.6428

8.4.7 One-tailed prediction

This design has about 65% power to detect the sex difference in height (with a 2-tailed test). Modify the sim_t_ind function for a 1-tailed prediction.

You could just set alternative equal to “greater” in the function, but it might be better to add the alternative argument to your function (giving it the same default value as t.test) and change the value of alternative in the function to alternative.

## [1] 0.761

8.4.8 Range of sample sizes

What if we want to find out what sample size will give us 80% power? We can try trial and error. We know the number should be slightly larger than 50. But you can search more systematically by repeating your power calculation for a range of sample sizes.

This might seem like overkill for a t-test, where you can easily look up sample size calculators online, but it is a valuable skill to learn for when your analyses become more complicated.

Start with a relatively low number of replications and/or more spread-out samples to estimate where you should be looking more specifically. Then you can repeat with a narrower/denser range of sample sizes and more iterations.

## `geom_smooth()` using method = 'loess' and formula 'y ~ x'

Now we can narrow down our search to values around 55 (plus or minus 5) and increase the number of replications from 1e3 to 1e4.

8.5 Exercises

Download the exercises. See the answers only after you’ve attempted all the questions.

E References

DeBruine, Lisa M, and Dale J Barr. 2019. “Understanding Mixed Effects Models Through Data Simulation,” June. PsyArXiv. https://doi.org/10.31234/osf.io/xp5cy.

Lakens, Daniël, and Aaron R Caldwell. 2019. “Simulation-Based Power-Analysis for Factorial Anova Designs,” May. PsyArXiv. https://doi.org/10.31234/osf.io/baxsf.

Lakens, Daniël, Anne M. Scheel, and Peder M. Isager. 2018. “Equivalence Testing for Psychological Research: A Tutorial.” Advances in Methods and Practices in Psychological Science 1 (2): 259–69. https://doi.org/10.1177/2515245918770963.

Morey, Richard D., Rink Hoekstra, Jeffrey N. Rouder, Michael D. Lee, and Eric-Jan Wagenmakers. 2016. “The Fallacy of Placing Confidence in Confidence Intervals.” Psychonomic Bulletin & Review 23 (1): 103–23. https://doi.org/10.3758/s13423-015-0947-8.