---
title: "Introduction to `ordbetareg`"
author: "Robert Kubinec"
date: "`r Sys.Date()`"
output: 
  rmarkdown::html_vignette:
  toc: true
vignette: >
  %\VignetteIndexEntry{package_introduction}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---
  
```{r setup, include = FALSE}

library(ordbetareg)
library(dplyr)
library(ggplot2)
library(haven)
library(brms)
library(tidyr)
library(stringr)
library(Hmisc)
library(modelsummary)
library(marginaleffects)


knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  warn=FALSE
)

knitr::opts_chunk$set(
  fig.align="center",
  fig.width=7,
  fig.height=3
)
options(rmarkdown.html_vignette.check_title = FALSE)

options(modelsummary_factory_default = "gt")

set.seed(628871)

```


# Overview

The ordered beta regression model is designed explicitly for data with upper and lower bounds, such as survey slider scales, dose/response relationships, and anything that can be considered a proportion or a percentage. This type of data cannot be fit with the standard beta regression model because the beta distribution does not allow for any observations at the bounds, such as a percentage/proportion of 0% or 100%. The ordered beta regression model solves this problem by combining the beta distribution with an ordinal distribution over continuous and discrete, or degenerate, observations at the bounds. It is an efficient model that produces intelligible estimates that also respect the bounds of the dependent variable. For more information, I refer you to my [paper on the model](https://www.cambridge.org/core/journals/political-analysis/article/ordered-beta-regression-a-parsimonious-wellfitting-model-for-continuous-data-with-lower-and-upper-bounds/89F4141DA16D4FC217809B5EB45EEE83) for more details (also there is an ungated version available [here](https://osf.io/preprints/socarxiv/2sx6y/)).

This notebook contains instructions for running the ordered beta regression model in the R package `ordbetareg`. `ordbetareg` is a front-end to `brms`, a very powerful regression modeling package based on the [Stan Hamiltonian Markov Chain Monte Carlo](https://mc-stan.org/) sampler. I only show in this vignette a small part of the features which are available via `brms`, and I refer the user to the [copious documentation describing the features of the brms package](https://paul-buerkner.github.io/brms/). Suffice it to say that most kinds of regression models can be fit with the software, including hierarchical, dynamic, nonlinear and multivariate models (or all of the above in combination). The `ordbetareg` package allows for all of these features to be used with the ordered regression model distribution by adding this distribution to `brms`.

The reproducible Rmarkdown file this vignette is compiled from can be accessed from [the package Github site](https://github.com/saudiwin/ordbetareg_pack/blob/master/vignettes/package_introduction.Rmd).

If you use the model in a paper, please cite it as:
  
  Kubinec, Robert. "Ordered Beta Regression: A Parsimonious, Well-Fitting Model for Continuous Data with Lower and Upper Bounds." *Political Analysis.* 2022. Forthcoming.

```{r runmod}

# whether to run models from scratch
run_model <- F

```

# Define Custom Family

If you prefer to run `brms` directly rather than use this package, you can run the underlying code via the R script `define_ord_betareg.R` in the [paper Github repo](https://github.com/saudiwin/ordbetareg). Please not that I cannot offer support for this alternative as it is not a complete integration with `brms`.

# Data Preparation

In this vignette I use an empirical application from data from a Pew Forum survey built into the package to show how the package works. It is also possible to generate simulated ordered beta data using the `rordbeta` function, as I do later in this guide. The following plot shows a histogram of respondents' views towards college professors based on a bounded 0 to 100 scale:

```{r load_data}

data("pew")

pew %>% 
  ggplot(aes(x=as.numeric(therm))) +
  geom_histogram(bins=100) +
  theme_minimal() + 
  theme(panel.grid=element_blank()) +
  scale_x_continuous(breaks=c(0,25,50,75,100),
                     labels=c("0","Colder","50","Warmer","100")) +
  ylab("") +
  xlab("") +
  labs(caption=paste0("Figure shows the distribution of ",sum(!is.na(pew$therm))," non-missing survey responses."))

```

The distributions of feelings towards college professors contains both degenerate (0 and 100) and continuous responses between 0 and 100. To model it, the outcome needs to be re-scaled to lie strictly between 0 and 1. However, it is not necessary to do that ahead of time as the `ordbetareg` package will do that re-normalization internally. I also do some other data processing tasks:

```{r munge_data}

model_data <- select(pew,therm,age="F_AGECAT_FINAL",
                        sex="F_SEX_FINAL",
                        income="F_INCOME_FINAL",
                        ideology="F_IDEO_FINAL",
                        race="F_RACETHN_RECRUITMENT",
                        education="F_EDUCCAT2_FINAL",
                     region="F_CREGION_FINAL",
                        approval="POL1DT_W28",
                       born_again="F_BORN_FINAL",
                       relig="F_RELIG_FINAL",
                        news="NEWS_PLATFORMA_W28") %>% 
    mutate_at(c("race","ideology","income","approval","sex","education","born_again","relig"), function(c) {
      factor(c, exclude=levels(c)[length(levels(c))])
    }) %>% 
    # need to make these ordered factors for BRMS
    mutate(education=ordered(education),
           income=ordered(income))

```

The completed dataset has `r nrow(model_data)` observations. 

# Run In `ordbetareg`

The `ordbetareg` function will take care of normalizing the outcome and adding additional information necessary to estimate the distribution. Any additional arguments can be passed to the underlying `brm` function to use specific `brm` features. For example, in the code below I use the `backend="cmdstanr"` argument to `brm()`, which allows me to use the R package `cmdstanr` for estimating models. `cmdstanr` tends to have the most up to date version of Stan, though you must [install it yourself](https://mc-stan.org/cmdstanr/). 

What you need to pass to the `ordbetareg` function are the standard components of any R model: a formula and a dataset that has all the variables mentioned in the formula. There are additional parameters that allow you to modify the priors, such as for the dispersion parameter `phi`. While in most cases these priors are sensible defaults, if you have data with an unusual scale (such as very large or very small values), you may want to change the priors to ensure they are not having an outsize effect on your estimates. If you want to assign a prior to specific regression coefficient or another part of the model, you can use the `extra_prior` option and pass a `brms`-compliant prior object.

If you want to use some of `brms` more powerful techniques, such as multivariate modeling, you can also pass the result of a `bf` function call to the `formula` argument. I refer you to the `brmsformula()` [function help](https://paul-buerkner.github.io/brms/reference/brmsformula.html) for more details.

To demonstrate some of the power of using `brms` as a regression engine, I will model education and income as ordinal predictors by using the `mo()` function in the formula definition. By doing so, we can get a single effect for education and income instead of having to use dummies for separate education/income categories. As a result, I can include an interaction between the two variables to see if wealthier more educated people have better views towards college professors than poorer better educated people. Finally, I include varying (random) census region intercepts.

```{r run_ordbetareg}

if(run_model) {
  
  ord_fit_mean <- ordbetareg(formula=therm ~ mo(education)*mo(income) +
                               (1|region), 
                       data=model_data,
                       control=list(adapt_delta=0.95),
                cores=1,chains=1,iter=500,
                refresh=0)
                # NOTE: to do parallel processing within chains
                # add the options below
                #threads=5,
                #backend="cmdstanr"
                #where threads is the number of cores per chain
                # you must have cmdstanr set up to do so
                # see https://mc-stan.org/cmdstanr/
  
} else {
  
  data("ord_fit_mean")
  
}
```

Because this model is somewhat complicated with multiple ordinal factors and multilevel intercepts, I use the option `adapt_delta=0.95` to improve sampling and remove divergent transitions. 

## Changing priors

The default priors in `ordbetareg` are weakly informative for data that is scaled roughly between $[-10,10]$. Outside of those scales, you may want to increase the SD of the Normal prior on regression coefficients via the `coef_prior_sd` and/or `phi_coef_prior_sd` options to `ordbetareg` (depending on whether the prior is for the coefficients in the main outcome model or the auxiliary dispersion `phi` regression). If you need to set a specific prior on the intercept, pass values to the `intercept_prior_mean`/`phi_intercept_prior_mean` and `intercept_prior_sd`/`phi_intercept_prior_sd` parameters (you need to have values for both for the prior to be set correctly) to set a Normally-distributed prior on the intercept. You can also zero out the intercept by setting a tight prior around 0, such as `intercept_prior_mean=0` and `intercept_prior_sd=0.001`.

If you need to do something more involved with priors, you can use the `extra_prior` option. This argument can take any `brms`-specified prior, including any of the many `brms` options for priors. These options are beyond the scope of the vignette, but I refer the reader to the [`brms` documentation for more info](https://www.jstatsoft.org/article/view/v080i01).

# Post-Estimation

## Visualization

We can visualize the model cutpoints by showing them relative to the empirical distribution. To do so, we have to transform the cutpoints using the inverse logit function in R (`plogis`) to get back values in the scale of the response, and I have to exponentiate and add the first cutpoint to get the correct value for the second cutpoint. The plot shows essentially how spread-out the cutpoints are relative to the data, though it should be noted that the analogy is inexact--it is not the case that observations above or below the cutpoints are considered to be discrete or continuous. Rather, as distance from the cutpoints increases, the probability of a discrete or continuous response increases. Cutpoints that are far apart indicate a data distribution that shows substantial differences between continuous and discrete observations.

```{r plot_cut}

all_draws <- prepare_predictions(ord_fit_mean)

cutzero <- plogis(all_draws$dpars$cutzero)
cutone <- plogis(all_draws$dpars$cutzero + exp(all_draws$dpars$cutone))

pew %>% 
  ggplot(aes(x=therm)) +
  geom_histogram(bins=100) +
  theme_minimal() + 
  theme(panel.grid=element_blank()) +
  scale_x_continuous(breaks=c(0,25,50,75,100),
                     labels=c("0","Colder","50","Warmer","100")) +
  geom_vline(xintercept = mean(cutzero)*100,linetype=2) +
  geom_vline(xintercept = mean(cutone)*100,linetype=2) +
  ylab("") +
  xlab("") +
  labs(caption=paste0("Figure shows the distribution of ",sum(!is.na(pew$therm))," non-missing survey responses."))


```

The plot shows that the model sees significant heterogeneity between the discrete responses at the bounds and the continuous responses in the plot. 

The best way to visualize model fit is to plot the full predictive distribution relative to the original outcome. Because ordered beta regression is a mixed discrete/continuous model, a separate plotting function, `pp_check_ordbeta`, is included that accurately handles the unique features of this distribution. This function returns a list with two plots, `discrete` and `continuous`, which can either be printed and plotted or further modified as `ggplot2` objects. 

The discrete plot, which is a bar graph, shows that the posterior distribution accurately captures the number of different types of responses (discrete or continuous) in the data. For the continuous plot, shown as a density plot with one line per posterior draw, the model can't capture all of the modality in the distribution -- there are effectively four separate modes -- but it is reasonably accurate over the middle responses and the responses near the bounds.

```{r post_predict,message=F}

# new theme option will add in new ggplot2 themes or themes
# from other packages

plots <- pp_check_ordbeta(ord_fit_mean,
                          ndraws=100,
                          outcome_label="Thermometer Rating",
                          new_theme=ggthemes::theme_economist())

plots$discrete
plots$continuous

```
The `pp_check_ordbeta` function can also show each posterior draw one at a time in an animated plot if the `animate` option is set to TRUE. This option requires the installation of the additional packages `gganimate` and `transformr`. 

## Coefficients and Marginal Effects Reporting

We can see the coefficients from the model in table form using the `modelsummary` package, which has support for `brms` models. We'll specify only confidence intervals as other frequentist statistics have no Bayesian analogue (i.e. p-values). We'll also specify only the main effects of the ordinal predictors, and give them more informative names. 

```{r coef_plot}

library(modelsummary)

modelsummary(ord_fit_mean,statistic = "conf.int",
             metrics = "RMSE",
             coef_map=c("b_Intercept"="Intercept",
                        "bsp_moeducation"="Education",
                        "bsp_moincome"="Income",
                        "bsp_moeducation:moincome"="EducationXIncome"))

```

`modelsummary` tables have many more options, including output to both html and latex formats. I refer the reader to the [package documentation](https://vincentarelbundock.github.io/modelsummary/articles/modelsummary.html) for more info.

### Marginal Effects

There is a related package, `marginaleffects`, that allows us to convert these coefficients into more meaningful marginal effect estimates, i.e., the effect of the predictors expresses as the actual change in the outcome on the 0 - 1 scale. Because we have two variables in our model that are both ordinal in nature, `marginaleffects` will produce an estimate of the marginal effect of each value of each ordinal predictor. I use the `avg_slopes` function to convert the `ordbetareg` model to a data frame of marginal/conditional effects in the scale of the outcome that can be easily printed:
  
```{r marg_effect}

avg_slopes(ord_fit_mean, variables="education") %>%
  select(Variable="term",
         Level="contrast",
         `5% Quantile`="conf.low",
         `Posterior Mean`="estimate",
         `95% Quantile`="conf.high") %>% 
  knitr::kable(caption = "Marginal Effect of Education on Professor Thermometer",
               format.args=list(digits=2),
               align=c('llccc'))
```

We can see that we have separate marginal effects for each level of education due to modeling it as an ordinal predictor. At present the `avg_slopes()` function cannot calculate a single effect, though that is possible manually. As we can see with the raw coefficient in the table above, the marginal effects are all positive, though the magnitude varies across different levels of education.

Note as well that the `slopes` function can calculate unit-level (row-wise) estimates of the predictors on the outcome (see `marginaleffects` documentation for more info).

# Multiple Imputation

`ordbetareg` supports `brms` functions for using multiple imputed datasets and for multivariate (multiple response) modeling, at least with two response variables. `brms` is able to apply the model to a list of multiple-imputed datasets (created with other packages such as `mice`) and then combine the resulting models into a single posterior distribution, which makes it very easy to do inference. I first demonstrate briefly how to do this and also multivariate modeling, but I refer the user to the `brms` documentation for more details.

In the code below I use the `rordbeta` function to simulate some ordered beta responses given a covariate, then create missing values by replacing some values at random with `NA`. I use the `mice` package to impute this missing data, creating two imputed datasets which I then feed into `ordbetareg` and set the `use_brm_multiple` option to `TRUE`:
  
```{r mult_impute}

# simplify things by using one covariate within the [0,1] interval

X <- runif(n = 100,0,1)
outcome <- rordbeta(n=100,mu = 0.3 * X, phi =3, cutpoints=c(-2,2))

# set 10% of values of X randomly to NA

X[runif(n=100)<0.1] <- NA

# create a list of two imputed datasets with package mice

mult_impute <- mice::mice(data=tibble(outcome=outcome,
                                      X=X),m=2,printFlag = FALSE) %>% 
  mice::complete(action="all")

# pass list to the data argument and set use_brm_multiple to TRUE

if(run_model) {
  
  fit_imputed <- ordbetareg(formula = outcome ~ X,
                            data=mult_impute,
                            use_brm_multiple = T,
                            cores=1,chains=1, iter=500)
  
} else {
  
  data('fit_imputed')
  
}

# all functions now work as though the model had only one dataset
# imputation uncertainty included in all results/analyses
# marginal effects, though, only incorporate one imputed dataset

knitr::kable(avg_slopes(fit_imputed))

modelsummary(fit_imputed,statistic = 'conf.int',metrics="all")

```

# Multivariate Modeling and Mediation Analysis

In the following code, I show how to model a Gaussian/Normally-distributed variable and an ordered beta regression model together. This is useful when examining the role of a mediator, which we will simulate here as a Normally-distributed variable `Z`. In order to specify a different distribution than ordered beta, the `family` parameter must be specified in the `bf` formula function. Otherwise the function assumes that the distribution is ordered beta.

```{r mult_variate}

# generate a new Gaussian/Normal outcome with same predictor X and mediator
# Z

X <- runif(n = 100,0,1)

Z <- rnorm(100, mean=3*X)

# use logit function to map unbounded continuous data to [0,1] interval
# X is mediated by Z

outcome <- rordbeta(n=100, mu = plogis(.4 * X + 1.5 * Z))


# use the bf function from brms to specify two formulas/responses
# set_rescor must be FALSE as one distribution is not Gaussian (ordered beta)

# OLS for mediator
mod1 <- bf(Z ~ X,family = gaussian)
# ordered beta
mod2 <- bf(outcome ~ X + Z)

if(run_model) {
  
  fit_multivariate <- ordbetareg(formula=mod1 + mod2 + set_rescor(FALSE),
                                 data=tibble(outcome=outcome,
                                             X=X,Z=Z),
                                 cores=1,chains=1, iter=500)
  
}

# need to calculate each sub-model's marginal effects separately

knitr::kable(avg_slopes(fit_multivariate,resp="outcome"))
knitr::kable(avg_slopes(fit_multivariate, resp="Z"))

suppressWarnings(modelsummary(fit_multivariate,statistic = "conf.int",
             metrics="none"))

```

To estimate the indirect effect of `X` on `outcome` that is mediated by `Z`, we can use the package `bayestestR`:
  
```{r mediation}

bayestestR::mediation(fit_multivariate)

```
Because the effect of `Z` on the outcome is positive, and `X` has a positive direct (unmediated effect) on the outcome, the total effect of `X` on the outcome is larger than the direct effect of `X` without considering mediation.

# Understanding Clustering/Polarization of Respondents

As I explain in the paper, one of the main advantages of using a Beta regression model is its ability to model the dispersion among respondents not just in terms of variance (i.e. heteroskedasticity) but also the shape of dispersion, whether it is U or inverted-U shaped. Conceptually, a U shape would imply that respondents are bipolar, moving towards the extremes. An inverted-U shape would imply that respondents tend to cluster around a central value. We can predict these responses conditionally in the sample by adding predictors for `phi`, the scale/dispersion parameter in the Beta distribution. Higher values of `phi` imply a uni-modal distribution clustered around a central value, with increasing `phi` implying more clustering. Lower values of `phi` imply a bi-modal distribution with values at the extremes. Notably, these effects are calculated independently of the expected value, or mean, of the distribution, so values of `phi` will produce different shapes depending on the average value. 

The one change we need to make to fit this model is to add a formula predicting `phi` in the code below. Because we now have two formulas--one for the mean and one for dispersion--I use the `bf` function to indicate these two sub-models. I also need to specify `phi_reg` to be TRUE because some of the priors will change. 

Because there is no need to model the mean, I leave the first formula as `therm ~ 1` with the 1 representing only an intercept, not a covariate. I then specify a separate model for `phi` with an interaction between `age` and `sex` to see if these covariates are associated with dispersion. I set `phi_reg` to `"only"` because I am only including covariates for predicting `phi` instead of also `therm` (in which case I would want to set the option to `"both"`).


```{r run_brms_phi}

if(run_model) {
  
  ord_fit_phi <- ordbetareg(bf(therm ~ 1, 
                               phi ~ age + sex),
                            phi_reg = "only",
                            data=model_data,
                            cores=2,chains=2,iter=500,
                            refresh=0)
  # NOTE: to do parallel processing within chains
  # add the options below
  #threads=threading(5),
  #backend="cmdstanr"
  #where threads is the number of cores per chain
  # you must have cmdstanr set up to do so
  # see https://mc-stan.org/cmdstanr/
  
} else {
  
  data("ord_fit_phi")
  
}


```

We can quickly examine the raw coefficients:
  
```{r phicoef}

summary(ord_fit_phi)

```

However, these are difficult to interpret as they relate to the Beta distribution, which is highly nonlinear. Generally speaking, higher values of `phi` mean the distribution is more concentrated around a single point. Lower values imply the distribution is more dispersed to the point that it actually becomes bi-modal, meaning that responses could be close to either 0 or 1 but are unlikely in the middle.

Because `phi` is a dispersion parameter, by definition the covariates have no effect on the average value. As a result, we'll need to use the `posterior_predict` function in `brms` if we want to get an idea what the covariates do. We don't have any fancy packages to do this for us, so we'll have to pass in two data frames, one with sex equal to female and one with sex equal to male. We'll want each data frame to have each unique value of age in the data. 

```{r plot_phi_sim}

# we can use some dplyr functions to make this really easy

female_data <- distinct(model_data, age) %>% 
  mutate(sex="Female")

male_data <- distinct(model_data, age) %>% 
  mutate(sex="Male")

to_predict <- bind_rows(female_data,
                        male_data) %>% 
  filter(!is.na(age))

pred_post <- posterior_predict(ord_fit_phi,
                               newdata=to_predict)

# better with iterations as rows

pred_post <- t(pred_post)
colnames(pred_post) <- 1:ncol(pred_post)

# need to convert to a data frame

data_pred <- as_tibble(pred_post) %>% 
  mutate(sex=to_predict$sex,
         age=to_predict$age) %>% 
  gather(key="iter",value='estimate',-sex,-age)

data_pred %>% 
  ggplot(aes(x=estimate)) +
  geom_density(aes(fill=sex),alpha=0.5,colour=NA) +
  scale_fill_viridis_d() +
  theme(panel.background = element_blank(),
        panel.grid=element_blank())

```

We can see that the female distribution is more clustered around a central value -- 0.75 -- than are men, who are somewhat more likely to be near the extremes of the data. However, the movement is modest, as the value of the coefficient suggests. Regression on `phi` is useful when examining polarizing versus clustering dynamics in the data.

# Power Analysis Via Simulation

Finally, we can also simulate data from the ordered beta regression model with the `sim_ordbeta` function. This is useful either for examining how different parameters interact with each other, or more generally for power calculation by iterating over different possible sample sizes. I demonstrate the function here, though note that the vignette loads saved simulation results unless `run_model` is set to TRUE. Because each simulation draw has to estimate a model, it can take some time to do calculations. Using multiple cores a la the `cores` option is strongly encouraged to reduce processing time. 

To access the data simulated for each run, the `return_data=TRUE` option can be set. To get a single simulated dataset, simply use this option combined with a single iteration and set of parameter values. The data are saved as a list in the column `data` in the returned data frame. The chunk below examines the first 10 rows of a single simulated dataset (note that the rows are repeated `k` times for each iteration, while there is one unique simulated dataset per iteration). Each predictor is listed as a `Var` column from 1 to `k`, while the simulated outcome is in the `outcome` column.

```{r check_data,eval=FALSE}

# NOT RUN IN THE VIGNETTE

single_data <- sim_ordbeta(N=100,iter=1,
                           return_data=T)

# examine the first dataset

knitr::kable(head(single_data$data[[1]]))

```

By default, the function simulates continuous predictors. To simulate binary variables, such as in a standard experimental design, use the `beta_type` function to specify `"binary"` predictors and `treat_assign` to determine the proportion assigned to treatment for each predictor. For a standard design with only one treatment variable, we'll also specify that `k=1` for a single covariate. Finally, to estimate a reasonable treatment effect, we will specify that `beta_coef=0.5`, which equals an increase of .5 on the logit scale. While it can be tricky to know a priori what the marginal effect will be (i.e., the actual change in the outcome), the function will calculate marginal effects and report them, so you can play with other options to see what gets you marginal effects/treatment effects of interest.

```{r sim_data_full}

if(run_model) {
  
  sim_data <- sim_ordbeta(N=c(250,500,750),
                          k=1,
                          beta_coef = .5,
                          iter=100,cores=10,
                          beta_type="binary",
                          treat_assign=0.3)
  
} else {
  
  data("sim_data")
  
}


```

For example, in the simulation above, the returned data frame stores the true marginal effect in the `marg_eff` column, and lists it as `r round(sim_data$marg_eff[1], 3)`, which is quite a large effect for a $[0,1]$ outcome. The following plot shows some of the summary statistics derived by aggregating over the iterations of the simulation. Some of the notable statistics that are included are power (for all covariates $k$), S errors (wrong sign of the estimated effect) and M errors (magnitude of bias). As can be seen, issues of bias decline markedly for this treatment effect size and a sample of 500 or greater has more than enough power.

```{r sim_plot}

sim_data %>% 
    select(`Proportion S Errors`="s_err",N,Power="power",
         `M Errors`="m_err",Variance="var_marg") %>% 
  gather(key = "type",value="estimate",-N) %>%
  ggplot(aes(y=estimate,x=N)) +
  #geom_point(aes(colour=model),alpha=0.1) +
  stat_summary(fun.data="mean_cl_boot") + 
  ylab("") +
  xlab("N") +
  scale_x_continuous(breaks=c(250,500,750)) +
  scale_color_viridis_d() +
  facet_wrap(~type,scales="free_y",ncol = 2) +
  labs(caption=stringr::str_wrap("Summary statistics calculated as mean with bootstrapped confidence interval from simulation draws. M Errors  and S errors are magnitude of bias (+1 equals no bias) and incorrect sign of the estimated marginal effect respectively. Variance refers to estimated posterior variance (uncertainty) of the marginal effect(s).",width=50)) +
  guides(color=guide_legend(title=""),
         linetype=guide_legend(title="")) +
  theme_minimal() +
  theme(plot.caption = element_text(size=7),
        axis.text.x=element_text(size=8))

```

While the `sim_ordbeta` function has the ability to iterate over `N`, it is of course possible to do more complex experiments by wrapping the function in a loop and passing different values of other parameters, such as `treat_assign`.

