Simulating Data with R

Introduction

Prerequisite Tutorials

Before working through this tutorial, you should be familiar with the content of the following:

• Getting Started with R — R objects, basic syntax, RStudio orientation
• Handling Tables in R — data frames, dplyr verbs, tidyr reshaping
• Basic Inferential Statistics — null hypothesis testing, p-values, effect sizes (recommended for the power analysis section)
• Mixed-Effects Models — random intercepts and slopes (required for the power analysis section)

Learning Objectives

By the end of this tutorial you will be able to:

1. Explain why data simulation is a core research skill, not just a workaround for missing data
2. Use set.seed() correctly to make simulations reproducible
3. Simulate data from the most common statistical distributions used in linguistics (normal, binomial, Poisson, uniform, negative binomial, Beta, Gamma)
4. Build realistic simulated datasets for corpus studies, psycholinguistic experiments, and Likert-scale surveys
5. Simulate multivariate data with correlated predictors using the multivariate normal distribution
6. Simulate sociolinguistic variation using variable rule (Varbrul) style models
7. Simulate missing data and attrition patterns (MCAR, MAR, MNAR)
8. Simulate time-series and longitudinal data with autocorrelation and growth curve components
9. Generate synthetic textual data using template construction and bigram language models
10. Conduct a simulation-based power analysis to determine required sample size for a mixed-effects model

Citation

Schweinberger, Martin. 2026. Simulating Data with R. Brisbane: The Language Technology and Data Analysis Laboratory (LADAL). url: https://ladal.edu.au/tutorials/simulate/simulate.html (Version 2026.05.01).

Data simulation is one of the most underused and underappreciated skills in quantitative linguistics. It tends to be introduced only in the context of power analysis — “how many participants do I need?” — but its range of applications is far broader. You simulate data when you want to verify that your statistical model recovers known parameters, when you want to share a reproducible example without distributing confidential data, when you want to test how robust your analysis pipeline is to edge cases before real data arrives, or when you want to teach a concept using a dataset whose properties you control completely.

This tutorial covers simulation from first principles: how R generates pseudo-random numbers, how to sample from the distributions that matter most in linguistics, how to assemble those draws into realistic datasets, and how to build the more advanced simulation workflows — correlated predictors, sociolinguistic variation, missing data, longitudinal trajectories, synthetic text, and power analysis — that will serve you across a wide range of research scenarios.

Loading Data Instead of Simulating?

If you have real data and want to learn how to import it into R, see the companion tutorial: Loading and Saving Data in R.

---

Setup

Installing Packages

# Run once — comment out after installation
install.packages("dplyr")       # data manipulation
install.packages("tidyr")       # data reshaping
install.packages("stringr")     # string manipulation
install.packages("purrr")       # functional programming (map/walk)
install.packages("ggplot2")     # visualisation
install.packages("MASS")        # multivariate normal (mvrnorm)
install.packages("lme4")        # mixed-effects models (for power analysis)
install.packages("broom.mixed") # tidy mixed-effects model output
install.packages("faux")        # correlated data simulation
install.packages("truncnorm")   # truncated normal distribution

Loading Packages

library(dplyr)
library(tidyr)
library(stringr)
library(purrr)
library(ggplot2)
library(MASS)
library(lme4)

---

Why Simulate?

Section Overview

What you will learn: The six main use cases for data simulation in linguistic research, and why simulation belongs in every researcher’s methodological toolkit

Data simulation is not just a workaround for when you lack real data. It is a core methodological tool with at least six distinct applications:

1. Verifying your model recovers known parameters. If you simulate data with a known effect of size β = −0.08 and then fit a mixed-effects model, you should recover approximately −0.08. If your model returns something very different, the problem is with your model specification, not your data. This parameter recovery check is one of the most powerful debugging tools available.

2. Sharing reproducible examples. You can provide a self-contained, runnable script that demonstrates your analysis pipeline without distributing confidential participant data or proprietary corpus content. Every reader can run your exact analysis on your exact (simulated) data.

3. Teaching statistical concepts. Simulation gives you complete control over the data-generating process. You can show students exactly what happens when an assumption is violated, when sample size is too small, or when a confound is present — without needing to find a real dataset that happens to illustrate the point.

4. Power analysis. By simulating many datasets under a hypothesised effect size and measuring how often your model detects the effect, you can estimate the sample size needed to achieve a desired power level. This is especially valuable for complex models (mixed-effects, SEM) where analytical power formulae do not exist.

5. Stress-testing pipelines. Before your real data arrives, you can check whether your analysis code handles edge cases: completely missing conditions, zero-variance predictors, extreme outliers, unbalanced designs, or floor/ceiling effects. Better to discover these problems in simulation than in your final dataset.

6. Bayesian prior predictive checks. In Bayesian analysis, you simulate data from your prior distributions before seeing any data, to check whether your priors are sensible — do the prior-predicted datasets look anything like plausible real data?

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Reproducibility and set.seed()

Section Overview

What you will learn: What pseudo-random number generation is; how set.seed() makes simulations exactly reproducible; and the rules for using seeds responsibly in research

R’s random number generation is pseudo-random: starting from a fixed seed value, the same sequence of “random” numbers is generated every time. The seed initialises a deterministic algorithm (by default the Mersenne Twister) that produces a sequence indistinguishable from true randomness for practical purposes, but that is perfectly reproducible when you know the starting point.

# Without a seed: different results every time
sample(1:100, 5)

[1] 85  1 36 34 53

sample(1:100, 5)

[1] 27 78 59 35 28

# With a seed: identical results every time
set.seed(2026)
sample(1:100, 5)

[1] 93 97 38 45 91

set.seed(2026)   # reset to the same seed
sample(1:100, 5) # same result as above

[1] 93 97 38 45 91

Rules for set.seed() in Research

1. Always set a seed at the start of any script that uses random number generation.
2. Set the seed once, at the top of the script — not before every individual random call (which hides stochasticity and can produce artificially good results).
3. Document the seed in your methods section so others can reproduce your exact simulation.
4. Test with multiple seeds to confirm your findings are not artefacts of one particular seed value.
5. Use a memorable but arbitrary seed — the year of the study, a postal code, or a fixed arbitrary number. Never choose the seed after seeing which value gives you “nice” results — that is a form of researcher degrees of freedom.

# Recommended practice: one seed at the top of the script
set.seed(2026)

# From this point on, all random calls are reproducible
x <- rnorm(100)
y <- sample(letters, 10)
z <- rbinom(50, size = 1, prob = 0.3)

head(x, 5); y; head(z, 5)

[1]  0.52059 -1.07969  0.13924 -0.08475 -0.66664

 [1] "x" "c" "k" "v" "p" "f" "u" "a" "d" "m"

[1] 1 0 0 0 0

---

Simulating from Statistical Distributions

Section Overview

What you will learn: R’s distribution function naming convention; how to simulate from the normal, binomial, Poisson, uniform, negative binomial, Beta, and Gamma distributions; and which distribution is appropriate for which type of linguistic data

R provides a consistent family of functions for every major distribution. The naming convention uses a prefix (r, d, p, q) followed by the distribution name:

R distribution function naming convention

Prefix	Function	Example
r	Random samples	rnorm(n, mean, sd)
d	Density (PDF/PMF)	dnorm(x, mean, sd)
p	Cumulative probability (CDF)	pnorm(q, mean, sd)
q	Quantile (inverse CDF)	qnorm(p, mean, sd)

The following table summarises which distributions are most useful for linguistic data:

Distribution guide for linguistic data simulation

Distribution	R function	Typical linguistic use
Normal	rnorm()	Pitch, formants, log-transformed RT, log word frequency
Log-normal	exp(rnorm())	Raw reaction times, response latencies
Binomial	rbinom()	Accuracy (correct/incorrect), binary grammaticality judgements
Poisson	rpois()	Error counts, disfluency counts, word occurrences in short texts
Negative binomial	rnbinom()	Overdispersed counts: word frequencies, collocation counts
Uniform	runif()	Ages, random positions, presentation order
Beta	rbeta()	Proportions bounded in (0, 1): accent ratings, similarity scores
Gamma	rgamma()	Skewed positive continuous: inter-pausal intervals, vowel duration

Normal Distribution

The normal (Gaussian) distribution is appropriate for continuous variables that are symmetric around a mean. In linguistics, most continuous variables (pitch in Hz, formant values, word length) are approximately normal or can be normalised by log-transformation.

set.seed(2026)

# Simulate log-transformed reaction times
# Mean ~6.4 ≈ exp(6.4) ≈ 600 ms; SD = 0.3 on the log scale
log_rt <- rnorm(n = 500, mean = 6.4, sd = 0.3)
rt_ms  <- exp(log_rt)   # back-transform to milliseconds

data.frame(log_rt = log_rt, rt_ms = rt_ms) |>
  ggplot(aes(x = rt_ms)) +
  geom_histogram(aes(y = after_stat(density)),
                 bins = 30, fill = "steelblue", color = "white", alpha = 0.8) +
  geom_density(color = "firebrick", linewidth = 1) +
  theme_bw() +
  labs(title = "Simulated reaction times (log-normal distribution)",
       subtitle = "n = 500 | Mean ≈ 600 ms",
       x = "Reaction time (ms)", y = "Density")

cat(sprintf("Mean RT: %.1f ms | SD: %.1f ms | Range: %.0f–%.0f ms\n",
            mean(rt_ms), sd(rt_ms), min(rt_ms), max(rt_ms)))

Mean RT: 637.9 ms | SD: 198.2 ms | Range: 280–1545 ms

Binomial Distribution

The binomial distribution models binary outcomes (correct/incorrect, present/absent, target form/alternative). rbinom(n, size, prob) draws n samples from a Binomial(size, prob) distribution.

set.seed(2026)

# Simulate accuracy in a lexical decision task
# 100 participants, 80 trials each, average accuracy 85%
n_participants <- 100
n_trials       <- 80
accuracy_prob  <- 0.85

n_correct <- rbinom(n = n_participants, size = n_trials, prob = accuracy_prob)
accuracy  <- n_correct / n_trials

data.frame(accuracy = accuracy) |>
  ggplot(aes(x = accuracy)) +
  geom_histogram(binwidth = 0.02, fill = "steelblue", color = "white", alpha = 0.8) +
  geom_vline(xintercept = mean(accuracy), color = "firebrick",
             linetype = "dashed", linewidth = 1) +
  theme_bw() +
  labs(title = "Simulated accuracy in a lexical decision task",
       subtitle = sprintf("n = %d | %d trials | P(correct) = %.2f | Mean = %.3f",
                          n_participants, n_trials, accuracy_prob, mean(accuracy)),
       x = "Accuracy", y = "Count")

Poisson Distribution

The Poisson distribution models event counts when events occur independently at a constant average rate λ. In linguistics: number of errors per utterance, disfluencies per minute, or occurrences of a word per short document. A key property is that the mean equals the variance — if your data has variance much larger than the mean (overdispersion), use the negative binomial instead.

set.seed(2026)

# Simulate disfluency counts per minute for 200 speakers
corrections <- rpois(n = 200, lambda = 1.8)

data.frame(corrections = corrections) |>
  ggplot(aes(x = corrections)) +
  geom_bar(fill = "steelblue", color = "white", alpha = 0.8) +
  theme_bw() +
  labs(title = "Simulated disfluency counts per minute (Poisson)",
       subtitle = sprintf("n = 200 | λ = 1.8 | Mean = %.2f | Variance = %.2f",
                          mean(corrections), var(corrections)),
       x = "Disfluencies per minute", y = "Count")

Negative Binomial Distribution

The negative binomial extends Poisson to handle overdispersion — variance exceeding the mean — which is the norm for linguistic count data (word frequencies, collocation counts, error counts across diverse speakers).

set.seed(2026)

# Compare Poisson vs. Negative Binomial: same mean, different variance
pois_counts <- rpois(n = 500, lambda = 3.0)
nb_counts   <- rnbinom(n = 500, mu = 3.0, size = 0.8)  # size controls overdispersion

cat(sprintf("Poisson:    Mean = %.2f | Variance = %.2f\n",
            mean(pois_counts), var(pois_counts)))

Poisson:    Mean = 2.90 | Variance = 2.73

cat(sprintf("Neg. Binom: Mean = %.2f | Variance = %.2f\n",
            mean(nb_counts), var(nb_counts)))

Neg. Binom: Mean = 2.79 | Variance = 13.41

data.frame(
  count = c(pois_counts, nb_counts),
  dist  = rep(c("Poisson (λ=3)", "Neg. Binomial (μ=3, size=0.8)"), each = 500)
) |>
  ggplot(aes(x = count, fill = dist)) +
  geom_bar(position = "dodge", alpha = 0.8, color = "white") +
  scale_fill_manual(values = c("steelblue", "firebrick")) +
  theme_bw() + theme(legend.position = "top") +
  labs(title = "Poisson vs. Negative Binomial: same mean, very different variance",
       x = "Count", y = "Frequency", fill = "")

Uniform Distribution

The uniform distribution generates values equally likely across an interval. Useful for participant ages, random stimulus onset times, or positions in a text.

set.seed(2026)

# Participant ages: continuous uniform then rounded
ages <- round(runif(n = 200, min = 18, max = 65))
cat(sprintf("Ages — Range: %d–%d | Mean: %.1f\n",
            min(ages), max(ages), mean(ages)))

Ages — Range: 18–65 | Mean: 39.4

# Categorical sampling with unequal probabilities
proficiency <- sample(
  x       = c("Beginner", "Intermediate", "Advanced"),
  size    = 200,
  replace = TRUE,
  prob    = c(0.25, 0.45, 0.30)
)
print(table(proficiency))

proficiency
    Advanced     Beginner Intermediate 
          73           48           79 

Beta Distribution

The Beta distribution models proportions bounded strictly between 0 and 1. It is ideal for accent rating scales (after rescaling to (0,1)), similarity judgements, or any response that is inherently proportional.

set.seed(2026)

# Simulate accent ratings (0 = native-like, 1 = heavily accented)
# Beta(2, 5): right-skewed, most ratings near the native-like end
# Beta(5, 2): left-skewed, most ratings toward heavily-accented end
# Beta(2, 2): symmetric, centred at 0.5

ratings_native  <- rbeta(200, shape1 = 2, shape2 = 5)
ratings_l2      <- rbeta(200, shape1 = 5, shape2 = 2)

data.frame(
  rating = c(ratings_native, ratings_l2),
  group  = rep(c("L1 speakers", "Advanced L2 speakers"), each = 200)
) |>
  ggplot(aes(x = rating, fill = group)) +
  geom_histogram(binwidth = 0.05, position = "identity", alpha = 0.6, color = "white") +
  scale_fill_manual(values = c("steelblue", "firebrick")) +
  theme_bw() + theme(legend.position = "top") +
  labs(title = "Simulated accent ratings (Beta distribution)",
       subtitle = "L1: Beta(2,5) — mostly native-like | L2: Beta(5,2) — more accented",
       x = "Accent rating (0 = native-like, 1 = heavily accented)", y = "Count",
       fill = "")

Gamma Distribution

The Gamma distribution models positive, right-skewed continuous variables — particularly durations. In linguistics: vowel duration, inter-pausal interval length, or fixation durations in eye-tracking.

set.seed(2026)

# Simulate inter-pausal intervals in spontaneous speech (ms)
# shape = 2, rate = 0.01 → mean = shape/rate = 200 ms
pauses <- rgamma(n = 500, shape = 2, rate = 0.01)

data.frame(pause_ms = pauses) |>
  ggplot(aes(x = pause_ms)) +
  geom_histogram(aes(y = after_stat(density)),
                 bins = 40, fill = "steelblue", color = "white", alpha = 0.8) +
  geom_density(color = "firebrick", linewidth = 1) +
  theme_bw() +
  labs(title = "Simulated inter-pausal intervals (Gamma distribution)",
       subtitle = sprintf("n = 500 | Mean = %.0f ms | SD = %.0f ms",
                          mean(pauses), sd(pauses)),
       x = "Inter-pausal interval (ms)", y = "Density")

✎ Check Your Understanding — Question 1

A researcher simulates 500 reaction times with rnorm(500, mean = 600, sd = 80). Her supervisor asks whether this is appropriate. What is the main problem?

1. rnorm() cannot simulate reaction times — use rpois() instead
2. Reaction times are always positive and approximately log-normally distributed; the normal distribution allows negative values and is symmetric, which is unrealistic. Simulate on the log scale: exp(rnorm(500, mean = log(600), sd = 0.15))
3. 500 observations is too small for any simulation
4. Reaction times are normally distributed in large samples by the Central Limit Theorem, so this is fine

Answer

b) Reaction times are bounded at zero and right-skewed. The normal distribution with mean 600 and SD 80 produces some negative values and is symmetric — neither property matches real RT data. The standard approach is to simulate on the log scale and back-transform: rt <- exp(rnorm(n, mean = log(600), sd = 0.15)). Option (d) confuses the CLT (which applies to sample means) with the distribution of individual observations.

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Simulating Realistic Linguistic Datasets

Section Overview

What you will learn: How to assemble individual random draws into complete, realistic datasets for three common linguistic research designs: corpus frequency analysis, psycholinguistic experiments, and Likert-scale surveys

Simulating a Corpus Frequency Dataset

Corpus frequency data follow a Zipfian distribution: a small number of word types are very frequent and the vast majority are extremely rare. The frequency of the word at rank r is proportional to 1/r^α, where α ≈ 1.0 for English.

set.seed(2026)

n_types    <- 500
alpha      <- 1.0
ranks      <- 1:n_types
freq_probs <- (1 / ranks^alpha) / sum(1 / ranks^alpha)

n_tokens   <- 50000
word_freqs <- round(freq_probs * n_tokens)
word_freqs[word_freqs == 0] <- 1

corpus_freq_df <- data.frame(
  rank     = ranks,
  word     = paste0("word_", stringr::str_pad(ranks, 3, pad = "0")),
  freq     = word_freqs,
  log_rank = log(ranks),
  log_freq = log(word_freqs)
)

ggplot(corpus_freq_df, aes(x = log_rank, y = log_freq)) +
  geom_point(alpha = 0.3, size = 1, color = "steelblue") +
  geom_smooth(method = "lm", se = FALSE, color = "firebrick", linewidth = 1) +
  theme_bw() +
  labs(title = "Zipf plot: simulated corpus word frequencies",
       subtitle = sprintf("Vocabulary: %d types | Corpus: %d tokens | α = %.1f",
                          n_types, sum(word_freqs), alpha),
       x = "log(rank)", y = "log(frequency)")

head(corpus_freq_df, 5); tail(corpus_freq_df, 5)

  rank     word freq log_rank log_freq
1    1 word_001 7361   0.0000    8.904
2    2 word_002 3680   0.6931    8.211
3    3 word_003 2454   1.0986    7.805
4    4 word_004 1840   1.3863    7.518
5    5 word_005 1472   1.6094    7.294

    rank     word freq log_rank log_freq
496  496 word_496   15    6.207    2.708
497  497 word_497   15    6.209    2.708
498  498 word_498   15    6.211    2.708
499  499 word_499   15    6.213    2.708
500  500 word_500   15    6.215    2.708

Simulating a Psycholinguistic Experiment

A realistic psycholinguistic dataset requires multiple participants (a random effect), multiple items (another random effect), fixed effects of experimental conditions, and residual noise. Here we simulate a priming experiment with two conditions (Primed vs. Unprimed) and a continuous word frequency covariate.

set.seed(2026)

n_participants <- 40
n_items        <- 30
n_obs          <- n_participants * n_items

# Fixed effects on the log-RT scale
intercept      <- 6.40    # grand mean (≈ 600 ms)
beta_priming   <- -0.08   # priming advantage (~8% faster)
beta_frequency <- -0.05   # frequency effect

# Random effect SDs
sd_participant <- 0.15
sd_item        <- 0.10
sd_residual    <- 0.20

participant_ids <- paste0("P", stringr::str_pad(1:n_participants, 2, pad = "0"))
item_ids        <- paste0("I", stringr::str_pad(1:n_items, 2, pad = "0"))
re_participant  <- rnorm(n_participants, 0, sd_participant)
re_item         <- rnorm(n_items, 0, sd_item)
log_freq_item   <- rnorm(n_items, mean = 4.0, sd = 1.5)
conditions      <- rep(c(0, 1), times = n_items / 2)

sim_exp <- expand.grid(Participant = participant_ids, Item = item_ids) |>
  dplyr::mutate(
    Condition = rep(conditions, times = n_participants),
    LogFreq   = log_freq_item[match(Item, item_ids)],
    RE_part   = re_participant[match(Participant, participant_ids)],
    RE_item   = re_item[match(Item, item_ids)],
    Epsilon   = rnorm(n_obs, 0, sd_residual),
    LogRT     = intercept + beta_priming * Condition +
                beta_frequency * LogFreq + RE_part + RE_item + Epsilon,
    RT        = exp(LogRT),
    Condition = factor(Condition, labels = c("Unprimed", "Primed"))
  ) |>
  dplyr::select(Participant, Item, Condition, LogFreq, RT, LogRT)

# Sanity check
sim_exp |>
  dplyr::group_by(Condition) |>
  dplyr::summarise(Mean_RT = round(mean(RT), 1),
                   SD_RT   = round(sd(RT), 1), .groups = "drop")

# A tibble: 2 × 3
  Condition Mean_RT SD_RT
  <fct>       <dbl> <dbl>
1 Unprimed     485.  131.
2 Primed       466.  131.

ggplot(sim_exp, aes(x = Condition, y = RT, fill = Condition)) +
  geom_violin(alpha = 0.6, color = NA) +
  geom_boxplot(width = 0.2, fill = "white", outlier.alpha = 0.3) +
  scale_fill_manual(values = c("steelblue", "firebrick")) +
  theme_bw() + theme(legend.position = "none") +
  labs(title = "Simulated reaction times by priming condition",
       subtitle = "β(priming) = −0.08 on log scale",
       x = "Condition", y = "RT (ms)")

Simulating a Survey Dataset

Attitude surveys generate Likert-scale data. Here we simulate a five-item language attitude survey with two groups (L1 English vs. L1 Other):

set.seed(2026)

n_respondents  <- 120
n_l1_english   <- 60

item_means_eng   <- c(4.1, 3.8, 4.3, 3.5, 4.0)
item_means_other <- c(3.2, 3.0, 3.5, 2.8, 3.1)
item_sd          <- 0.8

# Generate continuous underlying score; round and clip to [1,5]
sim_likert <- function(n, means, sd) {
  purrr::map_dfc(means, function(m) {
    raw <- rnorm(n, mean = m, sd = sd)
    pmin(pmax(round(raw), 1), 5)
  }) |> purrr::set_names(paste0("Item", seq_along(means)))
}

survey_df <- dplyr::bind_rows(
  sim_likert(n_l1_english, item_means_eng,   item_sd),
  sim_likert(n_respondents - n_l1_english,
             item_means_other, item_sd)
) |>
  dplyr::mutate(
    Respondent = paste0("R", stringr::str_pad(1:n_respondents, 3, pad = "0")),
    Group      = c(rep("L1 English", n_l1_english),
                   rep("L1 Other",   n_respondents - n_l1_english)),
    Total      = rowSums(dplyr::across(dplyr::starts_with("Item")))
  )

ggplot(survey_df, aes(x = Total, fill = Group)) +
  geom_histogram(binwidth = 1, position = "dodge", color = "white", alpha = 0.8) +
  scale_fill_manual(values = c("steelblue", "firebrick")) +
  theme_bw() + theme(legend.position = "top") +
  labs(title = "Simulated language attitude survey: total scores by group",
       x = "Total score (5–25)", y = "Count", fill = "")

---

Simulating Multivariate and Correlated Data

Section Overview

What you will learn: Why correlated predictors matter; how to simulate multivariate normal data with a specified covariance structure using MASS::mvrnorm(); how to verify the correlation structure of your simulated data; and how to incorporate correlated predictors into a realistic regression dataset

Why Correlation Among Predictors Matters

In real linguistic datasets, predictors are almost never independent. Word frequency and word length are negatively correlated. Proficiency and years of study are positively correlated. Age of acquisition and imageability covary with frequency. If you simulate predictors independently and then fit a model, you may obtain spuriously low standard errors and inflated confidence in your results. Realistic simulation requires building in the correlations that exist in your target population.

The multivariate normal distribution is the standard tool for generating a set of continuous variables with a specified mean vector and covariance (or correlation) matrix.

Specifying a Correlation Matrix

set.seed(2026)

# Suppose we have three word-level predictors:
#   LogFreq   = log word frequency (from corpus)
#   LogLength = log word length (in characters)
#   AoA       = age of acquisition rating
#
# Realistic correlations:
#   LogFreq  ↔ LogLength: r = -0.35  (frequent words tend to be shorter)
#   LogFreq  ↔ AoA:       r = -0.50  (frequent words acquired earlier)
#   LogLength↔ AoA:       r =  0.30  (longer words tend to be acquired later)

# Build the correlation matrix
cor_mat <- matrix(
  c(
     1.00, -0.35, -0.50,
    -0.35,  1.00,  0.30,
    -0.50,  0.30,  1.00
  ),
  nrow = 3, ncol = 3,
  dimnames = list(
    c("LogFreq", "LogLength", "AoA"),
    c("LogFreq", "LogLength", "AoA")
  )
)

# Convert correlation matrix to covariance matrix
# by scaling with desired standard deviations
sds     <- c(1.5, 0.4, 2.0)   # SD of each predictor
cov_mat <- diag(sds) %*% cor_mat %*% diag(sds)

cat("Covariance matrix:\n")

Covariance matrix:

round(cov_mat, 3)

      [,1]  [,2]  [,3]
[1,]  2.25 -0.21 -1.50
[2,] -0.21  0.16  0.24
[3,] -1.50  0.24  4.00

Generating Correlated Predictors with mvrnorm()

set.seed(2026)

n_words <- 300   # 300 word stimuli

# Mean vector: log_freq ~ N(4.5, ...), log_len ~ N(1.6, ...), AoA ~ N(6.0, ...)
mu <- c(LogFreq = 4.5, LogLength = 1.6, AoA = 6.0)

# Draw from the multivariate normal
word_predictors <- MASS::mvrnorm(n = n_words, mu = mu, Sigma = cov_mat) |>
  as.data.frame()

# Verify: empirical correlations should approximate the specified ones
cat("Empirical correlations:\n")

Empirical correlations:

round(cor(word_predictors), 3)

          LogFreq LogLength    AoA
LogFreq     1.000    -0.425 -0.479
LogLength  -0.425     1.000  0.253
AoA        -0.479     0.253  1.000

# Scatter plot: LogFreq vs. AoA (should show r ≈ −0.50)
ggplot(word_predictors, aes(x = LogFreq, y = AoA)) +
  geom_point(alpha = 0.4, color = "steelblue", size = 1.5) +
  geom_smooth(method = "lm", color = "firebrick", se = TRUE) +
  theme_bw() +
  labs(title = "Simulated word stimuli: LogFreq vs. AoA",
       subtitle = sprintf("Built-in r = −0.50 | Empirical r = %.2f",
                          cor(word_predictors$LogFreq, word_predictors$AoA)),
       x = "Log word frequency", y = "Age of acquisition (rating)")

Building a Full Dataset with Correlated Predictors

set.seed(2026)

# True regression coefficients for a naming latency study (log-RT)
intercept      <- 6.50
b_logfreq      <- -0.06   # higher freq → faster
b_loglength    <- 0.12    # longer words → slower
b_aoa          <- 0.04    # later-acquired → slower
sigma_residual <- 0.18

naming_dat <- word_predictors |>
  dplyr::mutate(
    word_id  = paste0("W", stringr::str_pad(1:n_words, 3, pad = "0")),
    LogRT    = intercept +
               b_logfreq   * LogFreq +
               b_loglength * LogLength +
               b_aoa       * AoA +
               rnorm(n_words, 0, sigma_residual),
    RT       = exp(LogRT)
  )

# Fit the model — should recover the true coefficients
m_naming <- lm(LogRT ~ LogFreq + LogLength + AoA, data = naming_dat)
round(coef(m_naming), 4)

(Intercept)     LogFreq   LogLength         AoA 
     6.7487     -0.0194      0.1146     -0.0304 

cat("\nTrue vs. recovered coefficients:\n")

True vs. recovered coefficients:

true_coef <- c("(Intercept)" = intercept, LogFreq = b_logfreq,
               LogLength = b_loglength, AoA = b_aoa)
comparison <- data.frame(
  True     = round(true_coef, 4),
  Recovered = round(coef(m_naming), 4),
  Difference = round(coef(m_naming) - true_coef, 4)
)
comparison

             True Recovered Difference
(Intercept)  6.50    6.7487     0.2487
LogFreq     -0.06   -0.0194     0.0406
LogLength    0.12    0.1146    -0.0054
AoA          0.04   -0.0304    -0.0704

The faux Package for Correlated Factorial Designs

For factorial designs with within- and between-subject factors and correlated random effects, the faux package provides a high-level interface that is often easier than specifying covariance matrices manually:

library(faux)
# Simulate a 2×2 mixed design: between = Group, within = Condition
dat <- sim_design(
  within  = list(Condition = c("Primed", "Unprimed")),
  between = list(Group = c("L1", "L2")),
  n       = 40,
  mu      = list(L1 = c(600, 650), L2 = c(700, 780)),
  sd      = 80,
  r       = 0.6,   # within-subject correlation
  plot    = FALSE
)

✎ Check Your Understanding — Question 2

You want to simulate two word-level predictors — log frequency and log length — with a correlation of −0.35. You specify the following covariance matrix: Sigma = matrix(c(1, -0.35, -0.35, 1), 2, 2). Is this correct?

1. Yes — a correlation matrix is a covariance matrix when all variables have unit variance
2. No — you must first convert to a covariance matrix by multiplying by the variable SDs
3. No — correlation matrices cannot be used in mvrnorm()
4. Yes, but only if both variables have the same mean

Answer

a) Yes — a correlation matrix is a covariance matrix when all variables have unit variance.

When all standard deviations are 1, the covariance matrix equals the correlation matrix. If your variables have different SDs, you need to scale the off-diagonals: Sigma = diag(sds) %*% cor_mat %*% diag(sds). Option (b) describes what you must do when SDs differ from 1; it is incorrect when SDs are 1. Option (c) is wrong — mvrnorm() accepts any valid positive-definite matrix as Sigma.

---

Simulating Sociolinguistic Variation

Section Overview

What you will learn: What variable rule (Varbrul) modelling is; how to simulate sociolinguistic variation datasets with factor groups controlling variant choice probabilities; how to incorporate both social (speaker-level) and linguistic (token-level) constraints; and how to verify that the simulated data shows the expected patterns

Variable Rules and Varbrul Logic

Sociolinguistic variation research models the probability of a speaker selecting one linguistic variant over another (e.g. going vs. goin’, t-deletion vs. retention, copula deletion vs. retention in AAVE). The classic Varbrul framework (Cedergren & Sankoff 1974; Sankoff & Labov 1979) models variant probability as a function of:

• Social factor groups: speaker sex, age, social class, ethnicity
• Linguistic factor groups: phonological environment, grammatical category, preceding segment, following segment

In modern practice, the equivalent is a mixed-effects logistic regression, where each factor group is a fixed effect and each speaker is a random effect. Simulating such data requires:

1. Defining a base probability (input probability) for the variant
2. Specifying factor weights that raise or lower this probability for different factor levels
3. Sampling speakers with social characteristics from the target population
4. Sampling tokens per speaker with the appropriate conditional probability

Simulating a (t,d)-Deletion Dataset

(t,d)-deletion is the variable absence of final /t/ or /d/ in consonant clusters (e.g. past → pas’, wind → win’). Deletion is favoured by:

• Following segment: preceding a consonant (vs. a vowel)
• Grammatical status: monomorphemic words (vs. past tense)
• Social factors: working class (vs. middle class), less formal style

set.seed(2026)

# ---- Speaker population ----
n_speakers <- 60
speaker_df <- data.frame(
  speaker_id  = paste0("S", stringr::str_pad(1:n_speakers, 2, pad = "0")),
  sex         = sample(c("M", "F"), n_speakers, replace = TRUE, prob = c(0.5, 0.5)),
  class       = sample(c("WC", "MC"), n_speakers, replace = TRUE, prob = c(0.55, 0.45)),
  style       = sample(c("Informal", "Formal"), n_speakers, replace = TRUE, prob = c(0.6, 0.4)),
  re_speaker  = rnorm(n_speakers, 0, 0.4),   # by-speaker random intercept
  stringsAsFactors = FALSE
)

# ---- Logistic regression coefficients (log-odds of deletion) ----
intercept_logit    <- -0.5    # overall ~38% deletion base rate
b_class_wc         <-  0.6    # WC: more deletion (ref = MC)
b_style_informal   <-  0.4    # informal: more deletion (ref = formal)
b_sex_m            <-  0.3    # male: slightly more deletion (ref = F)
b_following_cons   <-  0.8    # following consonant: more deletion
b_monomorpheme     <-  0.5    # monomorphemic: more deletion

# ---- Token population ----
n_tokens_per_speaker <- 30

token_df <- tidyr::expand_grid(
  speaker_id = speaker_df$speaker_id,
  token_n    = 1:n_tokens_per_speaker
) |>
  dplyr::left_join(speaker_df, by = "speaker_id") |>
  dplyr::mutate(
    following_segment = sample(c("C", "V"), dplyr::n(), replace = TRUE,
                               prob = c(0.55, 0.45)),
    morpheme_type     = sample(c("mono", "past"), dplyr::n(), replace = TRUE,
                               prob = c(0.60, 0.40)),
    # Linear predictor (log-odds)
    eta = intercept_logit +
          b_class_wc       * (class == "WC") +
          b_style_informal * (style == "Informal") +
          b_sex_m          * (sex == "M") +
          b_following_cons * (following_segment == "C") +
          b_monomorpheme   * (morpheme_type == "mono") +
          re_speaker,
    # Convert to probability via logistic function
    p_delete = 1 / (1 + exp(-eta)),
    # Bernoulli draw
    deleted  = rbinom(dplyr::n(), size = 1, prob = p_delete)
  )

cat("Dataset dimensions:", nrow(token_df), "×", ncol(token_df), "\n")

Dataset dimensions: 1800 × 11 

cat("Overall deletion rate:", round(mean(token_df$deleted), 3), "\n\n")

Overall deletion rate: 0.7 

# ---- Check factor group effects ----
token_df |>
  dplyr::group_by(class, style) |>
  dplyr::summarise(deletion_rate = round(mean(deleted), 3),
                   n = dplyr::n(), .groups = "drop")

# A tibble: 4 × 4
  class style    deletion_rate     n
  <chr> <chr>            <dbl> <int>
1 MC    Formal           0.633   180
2 MC    Informal         0.632   690
3 WC    Formal           0.745   420
4 WC    Informal         0.778   510

# Visualise deletion rate by class and following segment
token_df |>
  dplyr::group_by(class, following_segment) |>
  dplyr::summarise(deletion_rate = mean(deleted), .groups = "drop") |>
  ggplot(aes(x = following_segment, y = deletion_rate, fill = class)) +
  geom_col(position = "dodge", alpha = 0.85, color = "white") +
  scale_fill_manual(values = c("steelblue", "firebrick"),
                    labels = c("Middle class", "Working class")) +
  scale_y_continuous(labels = scales::percent_format()) +
  theme_bw() + theme(legend.position = "top") +
  labs(title = "Simulated (t,d)-deletion rates by social class and following segment",
       subtitle = "Working class and pre-consonant environments favour deletion",
       x = "Following segment", y = "Deletion rate (%)", fill = "Social class")

# Fit a mixed-effects logistic regression to recover the built-in coefficients
m_td <- lme4::glmer(
  deleted ~ class + style + sex + following_segment + morpheme_type + (1 | speaker_id),
  data   = token_df,
  family = binomial(link = "logit")
)

# Compare true vs. recovered fixed effects
true_coefs <- c("(Intercept)" = -0.5, classWC = 0.6, styleInformal = 0.4,
                sexM = 0.3, following_segmentC = 0.8, morpheme_typemono = 0.5)

round(data.frame(
  True      = true_coefs,
  Recovered = fixef(m_td)[names(true_coefs)]
), 3)

                   True Recovered
(Intercept)        -0.5     0.887
classWC             0.6     0.712
styleInformal       0.4     0.158
sexM                0.3     0.288
following_segmentC  0.8        NA
morpheme_typemono   0.5        NA

Extending to Multi-Variant Variables

When there are more than two variants (e.g. three realisations of (ING): full -ing, reduced -in’, or syllabic nasal), use a multinomial logistic model. The simulation approach is the same: compute a log-odds for each category, apply softmax to obtain probabilities, then use sample() or rmultinom():

set.seed(2026)

# Three variants: "ing" / "in" / "n" (syllabic nasal)
# Base log-odds relative to "ing" (reference)
b_in_intercept <- -0.3   # log-odds of "in" vs "ing" at baseline
b_n_intercept  <- -2.0   # log-odds of "n" vs "ing" at baseline (rare)

# Effect of informal style on each non-reference category
b_in_informal  <-  0.8
b_n_informal   <-  0.5

n_obs   <- 400
style_v <- sample(c("Informal", "Formal"), n_obs, replace = TRUE)

# Compute multinomial probabilities via softmax
logit_in <- b_in_intercept + b_in_informal * (style_v == "Informal")
logit_n  <- b_n_intercept  + b_n_informal  * (style_v == "Informal")
logit_ing <- rep(0, n_obs)   # reference category

denom    <- exp(logit_ing) + exp(logit_in) + exp(logit_n)
p_ing    <- exp(logit_ing) / denom
p_in     <- exp(logit_in)  / denom
p_n      <- exp(logit_n)   / denom

# Sample variant for each token
variants <- purrr::map_chr(1:n_obs, function(i) {
  sample(c("ing", "in", "n"), 1, prob = c(p_ing[i], p_in[i], p_n[i]))
})

# Proportions by style
table(style_v, variants) |> prop.table(margin = 1) |> round(3)

          variants
style_v       in   ing     n
  Formal   0.380 0.567 0.053
  Informal 0.573 0.359 0.068

✎ Check Your Understanding — Question 3

In a Varbrul-style simulation, you set a factor weight of +0.8 on the log-odds scale for the “following consonant” environment. A colleague says this means deletion is 80% more likely in that environment. Are they correct?

1. Yes — a log-odds increase of 0.8 corresponds to an 80% increase in probability
2. No — a log-odds increase of 0.8 corresponds to an odds ratio of exp(0.8) ≈ 2.23, meaning the odds of deletion are approximately 2.23× higher (not the probability)
3. No — a log-odds increase of 0.8 means deletion is 8% more likely
4. Yes — on the log-odds scale, coefficients map directly to percentage changes

Answer

b) On the logistic scale, coefficients are log-odds ratios. An increase of 0.8 means the odds of deletion are multiplied by exp(0.8) ≈ 2.23. The actual change in probability depends on the baseline probability: if deletion is 40% likely at baseline, the odds are 0.40/0.60 ≈ 0.67; after multiplying by 2.23, the odds become ~1.48, corresponding to a probability of 1.48/(1+1.48) ≈ 60%. The probability increase is ~20 percentage points, not 80% or 8%.

---

Simulating Missing Data and Attrition

Section Overview

What you will learn: The three mechanisms of missingness (MCAR, MAR, MNAR); how to introduce each type of missing data into a simulated dataset; and how to verify that your chosen missingness mechanism produces the expected patterns

Why Missingness Matters

Missing data is ubiquitous in linguistic research: participants drop out of longitudinal studies, items are not answered in surveys, transcription gaps occur in corpus data, and equipment failures create holes in acoustic measurements. The mechanism by which data go missing determines which analysis strategies are valid:

Missing data mechanisms and their implications

Mechanism	Description	Implication
MCAR — Missing Completely at Random	Probability of missingness is independent of both observed and unobserved values	Complete-case analysis unbiased (though inefficient)
MAR — Missing at Random	Probability of missingness depends on observed variables but not on the missing value itself	Multiple imputation or FIML valid
MNAR — Missing Not at Random	Probability of missingness depends on the unobserved (missing) value	Standard imputation biased; requires specialised models

Simulating MCAR

Data missing completely at random: each observation has the same probability of being missing, independent of everything else. This is the rarest and most benign missingness mechanism in practice.

set.seed(2026)

# Start with complete data
n       <- 200
base_df <- data.frame(
  participant = paste0("P", stringr::str_pad(1:n, 3, pad = "0")),
  proficiency = rnorm(n, mean = 50, sd = 10),
  accuracy    = rbeta(n, 7, 2),          # ~0.78 average accuracy
  age         = round(runif(n, 18, 60))
)

# MCAR: each accuracy value independently missing with p = 0.15
mcar_mask <- rbinom(n, size = 1, prob = 0.15) == 1
base_mcar <- base_df
base_mcar$accuracy[mcar_mask] <- NA

cat(sprintf("MCAR: %.1f%% of accuracy values missing\n",
            100 * mean(is.na(base_mcar$accuracy))))

MCAR: 13.0% of accuracy values missing

# Under MCAR, missingness should be unrelated to proficiency
t.test(proficiency ~ is.na(accuracy), data = base_mcar)$p.value |>
  (\(p) cat(sprintf("t-test: proficiency ~ missing? p = %.3f (expect > 0.05)\n", p)))()

t-test: proficiency ~ missing? p = 0.064 (expect > 0.05)

Simulating MAR

Data missing at random: the probability of missingness depends on observed variables (here: proficiency score). Lower-proficiency participants are more likely to have missing accuracy scores — perhaps they found the task more difficult and gave up. This is missingness in the observed data that we can partially explain.

set.seed(2026)

# MAR: lower proficiency → higher probability of missing accuracy
# Use logistic function: prob_missing = logistic(1.5 - 0.05 * proficiency)
base_mar <- base_df
prof_c   <- scale(base_df$proficiency)[, 1]   # standardise for cleaner coefficients
p_miss_mar <- plogis(0.8 - 1.2 * prof_c)      # low proficiency → high p(missing)

mar_mask <- rbinom(n, 1, p_miss_mar) == 1
base_mar$accuracy[mar_mask] <- NA

cat(sprintf("MAR: %.1f%% of accuracy values missing\n",
            100 * mean(is.na(base_mar$accuracy))))

MAR: 71.5% of accuracy values missing

# Under MAR, missingness IS related to proficiency
t.test(proficiency ~ is.na(accuracy), data = base_mar)$p.value |>
  (\(p) cat(sprintf("t-test: proficiency ~ missing? p = %.4f (expect < 0.05)\n", p)))()

t-test: proficiency ~ missing? p = 0.0000 (expect < 0.05)

# Visualise: missing more common at low proficiency
base_mar |>
  dplyr::mutate(missing = is.na(accuracy)) |>
  ggplot(aes(x = proficiency, fill = missing)) +
  geom_histogram(binwidth = 3, position = "identity", alpha = 0.6, color = "white") +
  scale_fill_manual(values = c("steelblue", "firebrick"),
                    labels = c("Observed", "Missing")) +
  theme_bw() + theme(legend.position = "top") +
  labs(title = "MAR: missing accuracy values concentrated at low proficiency",
       x = "Proficiency score", y = "Count", fill = "Accuracy")

Simulating MNAR

Data missing not at random: the probability of missingness depends on the missing value itself. Here, participants with very low accuracy are more likely to have their score missing — perhaps they did not report results they perceived as embarrassing. This is the most dangerous mechanism because the missingness pattern cannot be fully captured from the observed data.

set.seed(2026)

# MNAR: lower accuracy → higher probability of being missing
base_mnar <- base_df
acc_c     <- scale(base_df$accuracy)[, 1]
p_miss_mnar <- plogis(0.5 - 1.8 * acc_c)   # low accuracy → high p(missing)

mnar_mask <- rbinom(n, 1, p_miss_mnar) == 1
base_mnar$accuracy[mnar_mask] <- NA

cat(sprintf("MNAR: %.1f%% of accuracy values missing\n",
            100 * mean(is.na(base_mnar$accuracy))))

MNAR: 57.0% of accuracy values missing

# Crucially: the *observed* mean accuracy is biased upward
cat(sprintf("True mean accuracy:    %.3f\n", mean(base_df$accuracy)))

True mean accuracy:    0.766

cat(sprintf("Observed mean (MNAR):  %.3f (upward bias!)\n",
            mean(base_mnar$accuracy, na.rm = TRUE)))

Observed mean (MNAR):  0.862 (upward bias!)

# Visualise the missingness vs. the (unobserved) true accuracy
data.frame(
  true_accuracy = base_df$accuracy,
  missing       = mnar_mask
) |>
  ggplot(aes(x = true_accuracy, fill = missing)) +
  geom_histogram(binwidth = 0.03, position = "identity", alpha = 0.6, color = "white") +
  scale_fill_manual(values = c("steelblue", "firebrick"),
                    labels = c("Observed", "Missing")) +
  theme_bw() + theme(legend.position = "top") +
  labs(title = "MNAR: low-accuracy observations are systematically missing",
       subtitle = "Complete-case analysis would overestimate true mean accuracy",
       x = "True accuracy", y = "Count", fill = "Accuracy")

Simulating Longitudinal Attrition

In longitudinal studies (panel surveys, SLA follow-up designs), participants drop out over time. Attrition is rarely MCAR — participants who are less engaged, less proficient, or who have relocated are more likely to drop out.

set.seed(2026)

# 80 participants, 4 measurement occasions (T1–T4)
n_part       <- 80
n_occasions  <- 4

# Participant characteristics
part_df <- data.frame(
  id          = paste0("P", stringr::str_pad(1:n_part, 2, pad = "0")),
  proficiency = rnorm(n_part, 50, 10),
  motivation  = rnorm(n_part, 5, 1.5),   # motivation score 1–10 (approx)
  stringsAsFactors = FALSE
)

# Simulate attrition: each participant has a base dropout probability per occasion
# that increases over time and is higher for lower-motivation participants
attrition_df <- tidyr::expand_grid(id = part_df$id, occasion = 1:n_occasions) |>
  dplyr::left_join(part_df, by = "id") |>
  dplyr::mutate(
    # Probability of DROPPING OUT by this occasion (cumulative)
    p_dropout = plogis(-3.0 + 0.6 * occasion - 0.3 * motivation),
    dropout   = rbinom(dplyr::n(), 1, p_dropout) == 1
  ) |>
  # Once a participant drops out, all subsequent occasions also missing
  dplyr::group_by(id) |>
  dplyr::mutate(
    first_dropout = ifelse(any(dropout), min(occasion[dropout]), Inf),
    retained      = occasion < first_dropout
  ) |>
  dplyr::ungroup()

# Retention by occasion
retention <- attrition_df |>
  dplyr::group_by(occasion) |>
  dplyr::summarise(n_retained = sum(retained), pct = round(100 * mean(retained), 1))

print(retention)

# A tibble: 4 × 3
  occasion n_retained   pct
     <int>      <int> <dbl>
1        1         79  98.8
2        2         76  95  
3        3         68  85  
4        4         64  80  

ggplot(retention, aes(x = factor(occasion), y = pct)) +
  geom_col(fill = "steelblue", alpha = 0.85, color = "white") +
  geom_text(aes(label = paste0(pct, "%")), vjust = -0.4, size = 4) +
  ylim(0, 110) +
  theme_bw() +
  labs(title = "Simulated longitudinal attrition over 4 occasions",
       subtitle = "Higher attrition among low-motivation participants (MAR)",
       x = "Measurement occasion", y = "% retained")

---

Simulating Time-Series and Longitudinal Data

Section Overview

What you will learn: How to simulate autocorrelated time series using AR(1) processes; how to model individual growth curves (random slopes over time); how to build a realistic longitudinal SLA dataset with individual learning trajectories; and how to simulate acoustic time-series data

Autocorrelated Residuals: The AR(1) Process

Many linguistic measurements taken repeatedly over time are autocorrelated — knowing the current value helps predict the next one. Speaking rate fluctuates in a smooth, wave-like way over an utterance rather than jumping randomly from one word to the next. Pitch contours are smooth. Corpus word frequencies in rolling windows are correlated across adjacent windows.

An AR(1) (first-order autoregressive) process generates a series where each value is a linear function of the previous value plus independent noise:

y_t = φ · y_{t−1} + ε_t, ε_t ~ N(0, σ²)

When |φ| < 1, the series is stationary. When φ > 0, the series has positive autocorrelation (smooth trends). When φ < 0, the series alternates around the mean.

set.seed(2026)

# Simulate AR(1) time series with different autocorrelation strengths
sim_ar1 <- function(n, phi, sigma = 1) {
  y    <- numeric(n)
  y[1] <- rnorm(1, 0, sigma / sqrt(1 - phi^2))   # stationary starting value
  for (t in 2:n) y[t] <- phi * y[t - 1] + rnorm(1, 0, sigma)
  y
}

n_steps <- 200
ts_df <- data.frame(
  t      = rep(1:n_steps, 3),
  y      = c(sim_ar1(n_steps, phi = 0.0),
             sim_ar1(n_steps, phi = 0.7),
             sim_ar1(n_steps, phi = 0.95)),
  phi    = rep(c("φ = 0.0 (iid)", "φ = 0.7", "φ = 0.95 (near unit root)"), each = n_steps)
)

ggplot(ts_df, aes(x = t, y = y, color = phi)) +
  geom_line(linewidth = 0.6, alpha = 0.9) +
  facet_wrap(~ phi, ncol = 1, scales = "free_y") +
  scale_color_manual(values = c("steelblue", "firebrick", "darkgreen")) +
  theme_bw() + theme(legend.position = "none") +
  labs(title = "AR(1) time series with varying autocorrelation strength",
       x = "Time step", y = "Value")

Individual Growth Curves

Growth curve analysis models how an outcome changes over time, allowing each participant to have their own intercept (starting level) and slope (rate of change). This is the standard framework for longitudinal SLA research.

set.seed(2026)

# 30 L2 learners measured at 5 time points (weeks 0, 4, 8, 12, 16)
n_learners    <- 30
time_points   <- c(0, 4, 8, 12, 16)

# Population-level (fixed) growth parameters
gamma_00      <- 50.0    # grand mean intercept (Week 0 proficiency)
gamma_10      <-  2.5    # grand mean slope (points per week)

# Between-learner variation (random effects)
sd_intercept  <- 8.0     # SD of individual intercepts
sd_slope      <- 0.8     # SD of individual slopes
r_int_slope   <- -0.35   # Correlation: higher starting level → slower growth
sigma_res     <- 3.5     # Residual SD

# Covariance matrix for random effects
Sigma_re <- matrix(
  c(sd_intercept^2,
    r_int_slope * sd_intercept * sd_slope,
    r_int_slope * sd_intercept * sd_slope,
    sd_slope^2),
  nrow = 2
)

# Draw random effects for each learner
re_mat <- MASS::mvrnorm(n = n_learners, mu = c(0, 0), Sigma = Sigma_re)
colnames(re_mat) <- c("b0i", "b1i")

# Simulate the full dataset
growth_df <- tidyr::expand_grid(
  learner_id = paste0("L", stringr::str_pad(1:n_learners, 2, pad = "0")),
  week       = time_points
) |>
  dplyr::mutate(
    b0i     = re_mat[match(learner_id,
              paste0("L", stringr::str_pad(1:n_learners, 2, pad = "0"))), "b0i"],
    b1i     = re_mat[match(learner_id,
              paste0("L", stringr::str_pad(1:n_learners, 2, pad = "0"))), "b1i"],
    epsilon = rnorm(dplyr::n(), 0, sigma_res),
    score   = gamma_00 + b0i + (gamma_10 + b1i) * week + epsilon
  )

# Plot individual growth trajectories
growth_df |>
  ggplot(aes(x = week, y = score, group = learner_id)) +
  geom_line(alpha = 0.3, color = "steelblue", linewidth = 0.6) +
  geom_smooth(aes(group = NULL), method = "lm", color = "firebrick",
              linewidth = 1.5, se = TRUE) +
  theme_bw() +
  labs(title = "Simulated L2 learner growth curves",
       subtitle = sprintf("n = %d learners | Grand mean slope = %.1f pts/week",
                          n_learners, gamma_10),
       x = "Week", y = "Proficiency score")

# Recover parameters with a mixed-effects model
m_growth <- lme4::lmer(
  score ~ week + (1 + week | learner_id),
  data    = growth_df,
  REML    = TRUE
)

cat("Fixed effects (true: intercept = 50.0, slope = 2.5):\n")

Fixed effects (true: intercept = 50.0, slope = 2.5):

round(lme4::fixef(m_growth), 3)

(Intercept)        week 
     51.078       2.554 

Simulating a Pitch Contour

Intonation research often analyses pitch (F0) contours across utterances. A realistic contour combines a polynomial trend (declination across the utterance) with smooth AR(1) variation and measurement noise:

set.seed(2026)

# Simulate an F0 contour for a declarative utterance
# 20 measurement points across a 2-second utterance
n_frames     <- 60
frame_times  <- seq(0, 1, length.out = n_frames)

# Population-level contour: high onset (~180 Hz), smooth declination, final rise
f0_trend <- 180 - 30 * frame_times - 15 * frame_times^2 + 20 * frame_times^3

# Speaker-to-speaker variation
n_speakers_pitch <- 5
speaker_f0_df    <- purrr::map_dfr(1:n_speakers_pitch, function(s) {
  speaker_offset <- rnorm(1, 0, 15)          # mean F0 difference
  speaker_range  <- rnorm(1, 1, 0.1)         # pitch range multiplier
  ar_noise       <- sim_ar1(n_frames, phi = 0.85, sigma = 5)  # smooth microvariation
  measurement_noise <- rnorm(n_frames, 0, 2)  # frame-level noise

  data.frame(
    speaker  = paste0("Sp", s),
    frame    = 1:n_frames,
    time_s   = frame_times,
    f0_hz    = (f0_trend + speaker_offset) * speaker_range + ar_noise + measurement_noise
  )
})

ggplot(speaker_f0_df, aes(x = time_s, y = f0_hz, color = speaker)) +
  geom_line(linewidth = 0.8, alpha = 0.85) +
  theme_bw() + theme(legend.position = "top") +
  labs(title = "Simulated F0 contours for 5 speakers (declarative utterance)",
       subtitle = "AR(1) variation + speaker-level mean and range differences",
       x = "Time (s)", y = "F0 (Hz)", color = "Speaker")

✎ Check Your Understanding — Question 4

In a growth curve simulation, you set the correlation between random intercepts and random slopes to r = −0.35. What does this mean substantively in an SLA context?

1. Learners who start at a higher proficiency level tend to show slower growth (negative intercept–slope correlation)
2. Learners who start at a higher proficiency level tend to show faster growth
3. Learners with more variable scores tend to start higher
4. The slope and intercept are negatively correlated because time is measured in weeks, not months

Answer

a) A negative intercept–slope correlation means that learners who begin the study with relatively high proficiency (large positive random intercept) tend to gain proficiency more slowly (small positive or negative random slope), while those who start lower tend to progress faster. This is a plausible ceiling-effect pattern in SLA: advanced learners have less room to improve. Option (d) is incorrect — the sign of the correlation is independent of the unit of time measurement.

---

Simulation-Based Power Analysis

Section Overview

What you will learn: What statistical power is and why simulation is preferable to analytical power formulae for complex designs; how to conduct a simulation-based power analysis for a mixed-effects logistic regression; how to plot power curves; and how to interpret the results for study planning

Why Simulate Power?

Statistical power is the probability of correctly rejecting a false null hypothesis given a true effect of a specified size. For simple designs (two-sample t-test, one-way ANOVA), analytical power formulae exist (implemented in the pwr package). For complex designs — mixed-effects models, multilevel designs, non-normal outcomes — no closed-form formulae exist. Simulation is the only general solution.

The simulation approach to power analysis proceeds as follows:

1. Specify the assumed true effect size and all model parameters (fixed effects, random effect SDs)
2. Simulate a large number of datasets (e.g. 500–2000) under these assumptions
3. Fit the intended model to each dataset and extract the p-value for the effect of interest
4. Power = proportion of simulated datasets in which p < α (e.g. p < .05)
5. Repeat for a range of sample sizes and plot the power curve

Power Analysis for a Mixed-Effects Logistic Regression

Research question: Does priming condition (primed vs. unprimed) affect the probability of target form use in an elicitation task? We expect the priming effect to be approximately OR = 2.0 (log-odds = 0.693). Each participant produces 20 responses.

set.seed(2026)

# ---- True parameters ----
beta_0          <- -0.5    # log-odds of target use at baseline (p ≈ 0.38)
beta_priming    <-  0.693  # log-OR = log(2) → OR ≈ 2
sd_participant  <-  0.6    # by-participant random intercept SD
n_items         <- 20      # items per participant per condition

# ---- Single simulation function ----
# Returns p-value for the priming fixed effect
sim_power_once <- function(n_participants, beta_0, beta_priming, sd_part, n_items) {
  part_ids  <- paste0("P", seq_len(n_participants))
  re_part   <- rnorm(n_participants, 0, sd_part)

  dat <- tidyr::expand_grid(
    participant = part_ids,
    condition   = c("Unprimed", "Primed"),
    item        = seq_len(n_items)
  ) |>
    dplyr::mutate(
      cond_num   = as.integer(condition == "Primed"),
      re         = re_part[match(participant, part_ids)],
      eta        = beta_0 + beta_priming * cond_num + re,
      p_target   = plogis(eta),
      response   = rbinom(dplyr::n(), 1, p_target)
    )

  fit <- tryCatch(
    lme4::glmer(response ~ condition + (1 | participant),
                data = dat, family = binomial),
    error = function(e) NULL
  )

  if (is.null(fit)) return(NA_real_)
  summary(fit)$coefficients["conditionUnprimed", "Pr(>|z|)"]
}

set.seed(2026)

# ---- Power curve: vary n from 20 to 80 in steps of 10 ----
n_sim           <- 00   # simulations per n (use 1000+ for publication)
n_range         <- seq(20, 80, by = 10)
alpha_level     <- 0.05

power_results <- purrr::map_dfr(n_range, function(n_part) {
  p_values <- replicate(
    n_sim,
    sim_power_once(n_part, beta_0, beta_priming, sd_participant, n_items)
  )
  data.frame(
    n_participants = n_part,
    power          = mean(p_values < alpha_level, na.rm = TRUE),
    se             = sqrt(mean(p_values < alpha_level, na.rm = TRUE) *
                          (1 - mean(p_values < alpha_level, na.rm = TRUE)) / n_sim)
  )
})

print(power_results)

  n_participants power  se
1             20   NaN NaN
2             30   NaN NaN
3             40   NaN NaN
4             50   NaN NaN
5             60   NaN NaN
6             70   NaN NaN
7             80   NaN NaN

ggplot(power_results, aes(x = n_participants, y = power)) +
  geom_ribbon(aes(ymin = power - 1.96 * se, ymax = power + 1.96 * se),
              fill = "steelblue", alpha = 0.2) +
  geom_line(color = "steelblue", linewidth = 1.2) +
  geom_point(color = "steelblue", size = 3) +
  geom_hline(yintercept = 0.80, linetype = "dashed", color = "firebrick") +
  geom_hline(yintercept = 0.90, linetype = "dotted", color = "darkgreen") +
  scale_y_continuous(labels = scales::percent_format(), limits = c(0, 1)) +
  theme_bw() +
  labs(title = "Simulation-based power curve: priming effect (OR = 2.0)",
       subtitle = sprintf("Mixed-effects logistic regression | %d items/participant | %d simulations per n",
                          n_items, n_sim),
       x = "Number of participants", y = "Power (%)")

# Find minimum n achieving 80% power
min_n_80 <- power_results |>
  dplyr::filter(power >= 0.80) |>
  dplyr::slice(1) |>
  dplyr::pull(n_participants)

cat(sprintf(
  "\nConclusion: To detect OR = 2.0 with ≥80%% power at α = .05,\nyou need approximately %d participants (20 items each).\n",
  min_n_80
))

Conclusion: To detect OR = 2.0 with ≥80% power at α = .05,
you need approximately 20 participants (20 items each).

Recommendations for Simulation-Based Power Analysis

• Use at least 500 simulations per sample size for final power estimates (200 shown here for speed; use more for publication)
• Test multiple seeds and report the range of power estimates to show stability
• Vary the assumed effect size (optimistic, realistic, conservative) and report power under each
• Report the full power curve, not just a single n — readers can then choose n for their own desired power level
• If convergence failures exceed 5% of simulations, reconsider the random effects structure

✎ Check Your Understanding — Question 5

A power simulation returns 72% power at n = 40 participants. A colleague suggests simply running more simulations (e.g. 2000 instead of 500) to get the power estimate above 80%. Is this correct?

1. Yes — more simulations always increase power
2. No — more simulations reduce the uncertainty in the power estimate but do not change the underlying power level. To increase power, you need to increase the real sample size (n participants or n items)
3. Yes — the simulation estimate is biased downward with only 500 replications
4. No — you should instead increase α from .05 to .10

Answer

b) Running more simulations reduces the Monte Carlo standard error of the power estimate (the uncertainty around the 72% figure) but does not change the underlying statistical power. Power is a property of the study design — the true effect size, sample size, random effects structure, and α level. To achieve 80% power, you need a larger real sample or a larger assumed effect. Option (d) would work mathematically but at the cost of a higher false positive rate, which is not generally desirable.

---

Generating Synthetic Text Data

Section Overview

What you will learn: How to generate synthetic sentences using string templates; how to simulate a synthetic corpus with Zipfian word frequency properties; and how to build a bigram language model to generate more linguistically plausible text sequences

Template-Based Sentence Generation

The simplest approach to synthetic text is direct string construction using paste() and sprintf():

set.seed(2026)

subjects <- c("The speaker", "Each participant", "Every respondent", "The learner")
verbs    <- c("produced", "uttered", "used", "avoided")
objects  <- c("the target form", "an error", "a hesitation", "the amplifier")

n_sentences <- 20
sentences <- paste(
  sample(subjects, n_sentences, replace = TRUE),
  sample(verbs,    n_sentences, replace = TRUE),
  sample(objects,  n_sentences, replace = TRUE),
  "."
)
head(sentences, 5)

[1] "The speaker produced the amplifier ."      
[2] "The speaker avoided an error ."            
[3] "The speaker uttered an error ."            
[4] "Each participant uttered the target form ."
[5] "The speaker uttered the amplifier ."       

# Numbered sentences with sprintf
sprintf("Sentence %d, Speaker %s: %s",
        1:6,
        sample(LETTERS[1:4], 6, replace = TRUE),
        sample(sentences, 6))

[1] "Sentence 1, Speaker A: Every respondent used a hesitation ."  
[2] "Sentence 2, Speaker A: The learner produced the target form ."
[3] "Sentence 3, Speaker C: The learner produced the amplifier ."  
[4] "Sentence 4, Speaker A: The speaker uttered an error ."        
[5] "Sentence 5, Speaker D: The learner avoided an error ."        
[6] "Sentence 6, Speaker D: The learner used a hesitation ."       

Simulating a Corpus with Controlled Frequency Properties

set.seed(2026)

# Vocabulary with Zipfian probabilities
vocab <- data.frame(
  word = c("the","language","corpus","analysis","text",
           "speaker","frequency","very","quite","shows",
           "contains","reveals","significant","common","rare"),
  prob = c(0.15,0.12,0.10,0.09,0.08,
           0.07,0.07,0.06,0.05,0.05,
           0.04,0.04,0.03,0.03,0.02)
)
cat("Probability sum:", sum(vocab$prob), "\n")

Probability sum: 1 

# Generate one text of n_words
generate_text <- function(n_words, vocab_df) {
  paste(sample(vocab_df$word, n_words, replace = TRUE, prob = vocab_df$prob),
        collapse = " ")
}

# Corpus of 10 texts with varying lengths
text_lengths <- round(runif(10, 50, 200))
synth_corpus <- data.frame(
  text_id  = paste0("SYNTH_", stringr::str_pad(1:10, 2, pad = "0")),
  n_tokens = text_lengths,
  text     = sapply(text_lengths, generate_text, vocab_df = vocab),
  stringsAsFactors = FALSE
)

head(synth_corpus[, c("text_id", "n_tokens")], 5)

   text_id n_tokens
1 SYNTH_01      155
2 SYNTH_02      133
3 SYNTH_03       71
4 SYNTH_04       93
5 SYNTH_05      133

cat("\nSample text:\n", synth_corpus$text[1], "\n")

Sample text:
 the very language reveals language corpus frequency the corpus the corpus corpus language the analysis language analysis the the frequency contains language very the frequency analysis language text shows analysis analysis reveals analysis analysis the the common corpus text language significant speaker significant contains text very shows the significant language corpus contains frequency analysis analysis the speaker text frequency speaker reveals analysis very reveals language speaker very frequency language contains the significant shows significant the significant speaker common language language common very shows corpus corpus corpus shows corpus speaker analysis the quite analysis the language language speaker frequency language frequency analysis reveals the corpus language the frequency the language shows analysis the frequency shows analysis the analysis text the analysis analysis speaker the the reveals the language text frequency corpus very corpus text speaker contains the language speaker speaker the language the rare text shows very corpus very shows the the text common the the 

Bigram Language Model

A bigram model makes word selection dependent on the previous word, producing smoother and more natural-looking (though semantically arbitrary) text:

set.seed(2026)

word_types  <- c("<START>","the","corpus","analysis","shows",
                 "very","significant","results","<END>")

# Transition probability matrix: P(next word | current word)
transitions <- matrix(
  c(
    0.00,0.80,0.10,0.05,0.00,0.00,0.00,0.00,0.05,  # <START>
    0.00,0.00,0.40,0.35,0.00,0.00,0.00,0.00,0.25,  # the
    0.00,0.10,0.00,0.00,0.60,0.00,0.00,0.00,0.30,  # corpus
    0.00,0.00,0.00,0.00,0.70,0.00,0.00,0.00,0.30,  # analysis
    0.00,0.00,0.00,0.00,0.00,0.50,0.20,0.20,0.10,  # shows
    0.00,0.00,0.00,0.00,0.00,0.00,0.90,0.00,0.10,  # very
    0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.90,0.10,  # significant
    0.00,0.10,0.10,0.10,0.00,0.00,0.00,0.00,0.70,  # results
    0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.00,1.00   # <END>
  ),
  nrow = length(word_types), byrow = TRUE,
  dimnames = list(word_types, word_types)
)

generate_bigram_sentence <- function(trans, max_len = 20) {
  words   <- "<START>"
  current <- "<START>"
  repeat {
    probs  <- trans[current, ]
    next_w <- sample(colnames(trans), 1, prob = probs)
    if (next_w == "<END>" || length(words) > max_len) break
    words   <- c(words, next_w)
    current <- next_w
  }
  paste(words[-1], collapse = " ")
}

bigram_sents <- replicate(10, generate_bigram_sentence(transitions))
for (i in seq_along(bigram_sents)) cat(sprintf("  %2d. %s\n", i, bigram_sents[i]))

   1. the analysis shows very significant results
   2. corpus shows results
   3. the corpus
   4. the corpus shows very significant results
   5. the corpus shows very significant results
   6. the analysis
   7. the analysis shows results
   8. the analysis
   9. the corpus
  10. the analysis shows very

---

Citation and Session Info

Schweinberger, Martin. 2026. Simulating Data with R. Brisbane: The Language Technology and Data Analysis Laboratory (LADAL). url: https://ladal.edu.au/tutorials/simulate/simulate.html (Version 2026.05.01).

@manual{schweinberger2026simdat,
  author       = {Schweinberger, Martin},
  title        = {Simulating Data with R},
  note         = {https://ladal.edu.au/tutorials/simulate/simulate.html},
  year         = {2026},
  organization = {The Language Technology and Data Analysis Laboratory (LADAL)},
  address      = {Brisbane},
  edition      = {2026.05.01}
}

sessionInfo()

R version 4.4.2 (2024-10-31 ucrt)
Platform: x86_64-w64-mingw32/x64
Running under: Windows 11 x64 (build 26200)

Matrix products: default


locale:
[1] LC_COLLATE=English_United States.utf8 
[2] LC_CTYPE=English_United States.utf8   
[3] LC_MONETARY=English_United States.utf8
[4] LC_NUMERIC=C                          
[5] LC_TIME=English_United States.utf8    

time zone: Australia/Brisbane
tzcode source: internal

attached base packages:
[1] stats     graphics  grDevices datasets  utils     methods   base     

other attached packages:
[1] lme4_1.1-36   Matrix_1.7-2  MASS_7.3-61   ggplot2_4.0.2 purrr_1.0.4  
[6] stringr_1.5.1 tidyr_1.3.2   dplyr_1.2.0  

loaded via a namespace (and not attached):
 [1] utf8_1.2.4         generics_0.1.3     renv_1.1.7         stringi_1.8.4     
 [5] lattice_0.22-6     digest_0.6.39      magrittr_2.0.3     evaluate_1.0.3    
 [9] grid_4.4.2         RColorBrewer_1.1-3 fastmap_1.2.0      jsonlite_1.9.0    
[13] mgcv_1.9-1         scales_1.4.0       codetools_0.2-20   reformulas_0.4.0  
[17] Rdpack_2.6.2       cli_3.6.4          rlang_1.1.7        rbibutils_2.3     
[21] splines_4.4.2      withr_3.0.2        yaml_2.3.10        tools_4.4.2       
[25] nloptr_2.1.1       minqa_1.2.8        boot_1.3-31        vctrs_0.7.1       
[29] R6_2.6.1           lifecycle_1.0.5    htmlwidgets_1.6.4  pkgconfig_2.0.3   
[33] pillar_1.10.1      gtable_0.3.6       glue_1.8.0         Rcpp_1.1.1        
[37] xfun_0.56          tibble_3.2.1       tidyselect_1.2.1   rstudioapi_0.17.1 
[41] knitr_1.51         farver_2.1.2       htmltools_0.5.9    nlme_3.1-166      
[45] rmarkdown_2.30     labeling_0.4.3     compiler_4.4.2     S7_0.2.1          

AI Transparency Statement

This tutorial was written with the assistance of Claude (claude.ai), a large language model created by Anthropic. Claude was used to split an existing combined loading/saving/simulation tutorial into two separate files and to substantially expand the simulation content. All content was reviewed and approved by the named author (Martin Schweinberger), who takes full responsibility for its accuracy.

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References

• Cedergren, H. J. & Sankoff, D. (1974). Variable rules: Performance as a statistical reflection of competence. Language, 50(2), 333–355.
• Sankoff, D. & Labov, W. (1979). On the uses of variable rules. Language in Society, 8(2), 189–222.
• DeBruine, L. & Barr, D. J. (2021). Understanding mixed-effects models through data simulation. Advances in Methods and Practices in Psychological Science, 4(1). https://doi.org/10.1177/2515245920965119
• Green, P. & MacLeod, C. J. (2016). SIMR: an R package for power analysis of generalised linear mixed models by simulation. Methods in Ecology and Evolution, 7(4), 493–498.