A tutorial on tidy cross-validation with R
Analyzing NetHack data, part 1: What kills the players
Analyzing NetHack data, part 2: What players kill the most
Building a shiny app to explore historical newspapers: a step-by-step guide
Classification of historical newspapers content: a tutorial combining R, bash and Vowpal Wabbit, part 1
Classification of historical newspapers content: a tutorial combining R, bash and Vowpal Wabbit, part 2
Curly-Curly, the successor of Bang-Bang
Dealing with heteroskedasticity; regression with robust standard errors using R
Easy time-series prediction with R: a tutorial with air traffic data from Lux Airport
Exporting editable plots from R to Powerpoint: making ggplot2 purrr with officer
Fast food, causality and R packages, part 1
Fast food, causality and R packages, part 2
For posterity: install {xml2} on GNU/Linux distros
Forecasting my weight with R
From webscraping data to releasing it as an R package to share with the world: a full tutorial with data from NetHack
Get text from pdfs or images using OCR: a tutorial with {tesseract} and {magick}
Getting data from pdfs using the pdftools package
Getting the data from the Luxembourguish elections out of Excel
Going from a human readable Excel file to a machine-readable csv with {tidyxl}
Historical newspaper scraping with {tesseract} and R
How Luxembourguish residents spend their time: a small {flexdashboard} demo using the Time use survey data
Imputing missing values in parallel using {furrr}
Intermittent demand, Croston and Die Hard
Looking into 19th century ads from a Luxembourguish newspaper with R
Making sense of the METS and ALTO XML standards
Manipulate dates easily with {lubridate}
Manipulating strings with the {stringr} package
Maps with pie charts on top of each administrative division: an example with Luxembourg's elections data
Missing data imputation and instrumental variables regression: the tidy approach
Modern R with the tidyverse is available on Leanpub
Objects types and some useful R functions for beginners
Pivoting data frames just got easier thanks to `pivot_wide()` and `pivot_long()`
R or Python? Why not both? Using Anaconda Python within R with {reticulate}
Searching for the optimal hyper-parameters of an ARIMA model in parallel: the tidy gridsearch approach
Some fun with {gganimate}
Statistical matching, or when one single data source is not enough
The best way to visit Luxembourguish castles is doing data science + combinatorial optimization
The never-ending editor war (?)
The year of the GNU+Linux desktop is upon us: using user ratings of Steam Play compatibility to play around with regex and the tidyverse
Using Data Science to read 10 years of Luxembourguish newspapers from the 19th century
Using a genetic algorithm for the hyperparameter optimization of a SARIMA model
Using cosine similarity to find matching documents: a tutorial using Seneca's letters to his friend Lucilius
Using linear models with binary dependent variables, a simulation study
Using the tidyverse for more than data manipulation: estimating pi with Monte Carlo methods
What hyper-parameters are, and what to do with them; an illustration with ridge regression
{pmice}, an experimental package for missing data imputation in parallel using {mice} and {furrr}
Building formulae
Functional peace of mind
Get basic summary statistics for all the variables in a data frame
Getting {sparklyr}, {h2o}, {rsparkling} to work together and some fun with bash
Importing 30GB of data into R with sparklyr
Introducing brotools
It's lists all the way down
It's lists all the way down, part 2: We need to go deeper
Keep trying that api call with purrr::possibly()
Lesser known dplyr 0.7* tricks
Lesser known dplyr tricks
Lesser known purrr tricks
Make ggplot2 purrr
Mapping a list of functions to a list of datasets with a list of columns as arguments
Predicting job search by training a random forest on an unbalanced dataset
Teaching the tidyverse to beginners
Why I find tidyeval useful
tidyr::spread() and dplyr::rename_at() in action
Easy peasy STATA-like marginal effects with R
Functional programming and unit testing for data munging with R available on Leanpub
How to use jailbreakr
My free book has a cover!
Work on lists of datasets instead of individual datasets by using functional programming
Method of Simulated Moments with R
New website!
Nonlinear Gmm with R - Example with a logistic regression
Simulated Maximum Likelihood with R
Bootstrapping standard errors for difference-in-differences estimation with R
Careful with tryCatch
Data frame columns as arguments to dplyr functions
Export R output to a file
I've started writing a 'book': Functional programming and unit testing for data munging with R
Introduction to programming econometrics with R
Merge a list of datasets together
Object Oriented Programming with R: An example with a Cournot duopoly
R, R with Atlas, R with OpenBLAS and Revolution R Open: which is fastest?
Read a lot of datasets at once with R
Unit testing with R
Update to Introduction to programming econometrics with R
Using R as a Computer Algebra System with Ryacas

This blog post is an excerpt of my ebook Modern R with the tidyverse that you can read for free here. This is taken from Chapter 8, in which I discuss advanced functional programming methods for modeling.

As written just above (note: as written above *in the book*), `map()`

simply applies a function
to a list of inputs, and in the previous
section we mapped `ggplot()`

to generate many plots at once. This approach can also be used to
map any modeling functions, for instance `lm()`

to a list of datasets.

For instance, suppose that you wish to perform a Monte Carlo simulation. Suppose that you are dealing with a binary choice problem; usually, you would use a logistic regression for this.

However, in certain disciplines, especially in the social sciences, the so-called Linear Probability Model is often used as well. The LPM is a simple linear regression, but unlike the standard setting of a linear regression, the dependent variable, or target, is a binary variable, and not a continuous variable. Before you yell “Wait, that’s illegal”, you should know that in practice LPMs do a good job of estimating marginal effects, which is what social scientists and econometricians are often interested in. Marginal effects are another way of interpreting models, giving how the outcome (or the target) changes given a change in a independent variable (or a feature). For instance, a marginal effect of 0.10 for age would mean that probability of success would increase by 10% for each added year of age.

There has been a lot of discussion on logistic regression vs LPMs, and there are pros and cons of using LPMs. Micro-econometricians are still fond of LPMs, even though the pros of LPMs are not really convincing. However, quoting Angrist and Pischke:

“While a nonlinear model may fit the CEF (population conditional expectation function) for LDVs
(limited dependent variables) more closely than a linear model, when it comes to marginal effects,
this probably matters little” (source: *Mostly Harmless Econometrics*)

so LPMs are still used for estimating marginal effects.

Let us check this assessment with one example. First, we simulate some data, then run a logistic regression and compute the marginal effects, and then compare with a LPM:

```
set.seed(1234)
x1 <- rnorm(100)
x2 <- rnorm(100)
z <- .5 + 2*x1 + 4*x2
p <- 1/(1 + exp(-z))
y <- rbinom(100, 1, p)
df <- tibble(y = y, x1 = x1, x2 = x2)
```

This data generating process generates data from a binary choice model. Fitting the model using a logistic regression allows us to recover the structural parameters:

`logistic_regression <- glm(y ~ ., data = df, family = binomial(link = "logit"))`

Let’s see a summary of the model fit:

`summary(logistic_regression)`

```
##
## Call:
## glm(formula = y ~ ., family = binomial(link = "logit"), data = df)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -2.91941 -0.44872 0.00038 0.42843 2.55426
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 0.0960 0.3293 0.292 0.770630
## x1 1.6625 0.4628 3.592 0.000328 ***
## x2 3.6582 0.8059 4.539 5.64e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 138.629 on 99 degrees of freedom
## Residual deviance: 60.576 on 97 degrees of freedom
## AIC: 66.576
##
## Number of Fisher Scoring iterations: 7
```

We do recover the parameters that generated the data, but what about the marginal effects? We can
get the marginal effects easily using the `{margins}`

package:

```
library(margins)
margins(logistic_regression)
```

`## Average marginal effects`

`## glm(formula = y ~ ., family = binomial(link = "logit"), data = df)`

```
## x1 x2
## 0.1598 0.3516
```

Or, even better, we can compute the *true* marginal effects, since we know the data
generating process:

```
meffects <- function(dataset, coefs){
X <- dataset %>%
select(-y) %>%
as.matrix()
dydx_x1 <- mean(dlogis(X%*%c(coefs[2], coefs[3]))*coefs[2])
dydx_x2 <- mean(dlogis(X%*%c(coefs[2], coefs[3]))*coefs[3])
tribble(~term, ~true_effect,
"x1", dydx_x1,
"x2", dydx_x2)
}
(true_meffects <- meffects(df, c(0.5, 2, 4)))
```

```
## # A tibble: 2 x 2
## term true_effect
## <chr> <dbl>
## 1 x1 0.175
## 2 x2 0.350
```

Ok, so now what about using this infamous Linear Probability Model to estimate the marginal effects?

```
lpm <- lm(y ~ ., data = df)
summary(lpm)
```

```
##
## Call:
## lm(formula = y ~ ., data = df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.83953 -0.31588 -0.02885 0.28774 0.77407
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.51340 0.03587 14.314 < 2e-16 ***
## x1 0.16771 0.03545 4.732 7.58e-06 ***
## x2 0.31250 0.03449 9.060 1.43e-14 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3541 on 97 degrees of freedom
## Multiple R-squared: 0.5135, Adjusted R-squared: 0.5034
## F-statistic: 51.18 on 2 and 97 DF, p-value: 6.693e-16
```

It’s not too bad, but maybe it could have been better in other circumstances. Perhaps if we had more
observations, or perhaps for a different set of structural parameters the results of the LPM
would have been closer. The LPM estimates the marginal effect of `x1`

to be
0.1677134 vs 0.1597956
for the logistic regression and for `x2`

, the LPM estimation is 0.3124966
vs 0.351607. The *true* marginal effects are
0.1750963 and 0.3501926 for `x1`

and `x2`

respectively.

Just as to assess the accuracy of a model data scientists perform cross-validation, a Monte Carlo study can be performed to asses how close the estimation of the marginal effects using a LPM is to the marginal effects derived from a logistic regression. It will allow us to test with datasets of different sizes, and generated using different structural parameters.

First, let’s write a function that generates data. The function below generates 10 datasets of size 100 (the code is inspired by this StackExchange answer):

```
generate_datasets <- function(coefs = c(.5, 2, 4), sample_size = 100, repeats = 10){
generate_one_dataset <- function(coefs, sample_size){
x1 <- rnorm(sample_size)
x2 <- rnorm(sample_size)
z <- coefs[1] + coefs[2]*x1 + coefs[3]*x2
p <- 1/(1 + exp(-z))
y <- rbinom(sample_size, 1, p)
df <- tibble(y = y, x1 = x1, x2 = x2)
}
simulations <- rerun(.n = repeats, generate_one_dataset(coefs, sample_size))
tibble("coefs" = list(coefs), "sample_size" = sample_size, "repeats" = repeats, "simulations" = list(simulations))
}
```

Let’s first generate one dataset:

`one_dataset <- generate_datasets(repeats = 1)`

Let’s take a look at `one_dataset`

:

`one_dataset`

```
## # A tibble: 1 x 4
## coefs sample_size repeats simulations
## <list> <dbl> <dbl> <list>
## 1 <dbl [3]> 100 1 <list [1]>
```

As you can see, the tibble with the simulated data is inside a list-column called `simulations`

.
Let’s take a closer look:

`str(one_dataset$simulations)`

```
## List of 1
## $ :List of 1
## ..$ :Classes 'tbl_df', 'tbl' and 'data.frame': 100 obs. of 3 variables:
## .. ..$ y : int [1:100] 0 1 1 1 0 1 1 0 0 1 ...
## .. ..$ x1: num [1:100] 0.437 1.06 0.452 0.663 -1.136 ...
## .. ..$ x2: num [1:100] -2.316 0.562 -0.784 -0.226 -1.587 ...
```

The structure is quite complex, and it’s important to understand this, because it will have an
impact on the next lines of code; it is a list, containing a list, containing a dataset! No worries
though, we can still map over the datasets directly, by using `modify_depth()`

instead of `map()`

.

Now, let’s fit a LPM and compare the estimation of the marginal effects with the *true* marginal
effects. In order to have some confidence in our results,
we will not simply run a linear regression on that single dataset, but will instead simulate hundreds,
then thousands and ten of thousands of data sets, get the marginal effects and compare
them to the true ones (but here I won’t simulate more than 500 datasets).

Let’s first generate 10 datasets:

`many_datasets <- generate_datasets()`

Now comes the tricky part. I have this object, `many_datasets`

looking like this:

`many_datasets`

```
## # A tibble: 1 x 4
## coefs sample_size repeats simulations
## <list> <dbl> <dbl> <list>
## 1 <dbl [3]> 100 10 <list [10]>
```

I would like to fit LPMs to the 10 datasets. For this, I will need to use all the power of functional
programming and the `{tidyverse}`

. I will be adding columns to this data frame using `mutate()`

and mapping over the `simulations`

list-column using `modify_depth()`

. The list of data frames is
at the second level (remember, it’s a list containing a list containing data frames).

I’ll start by fitting the LPMs, then using `broom::tidy()`

I will get a nice data frame of the
estimated parameters. I will then only select what I need, and then bind the rows of all the
data frames. I will do the same for the *true* marginal effects.

I highly suggest that you run the following lines, one after another. It is complicated to understand what’s going on if you are not used to such workflows. However, I hope to convince you that once it will click, it’ll be much more intuitive than doing all this inside a loop. Here’s the code:

```
results <- many_datasets %>%
mutate(lpm = modify_depth(simulations, 2, ~lm(y ~ ., data = .x))) %>%
mutate(lpm = modify_depth(lpm, 2, broom::tidy)) %>%
mutate(lpm = modify_depth(lpm, 2, ~select(., term, estimate))) %>%
mutate(lpm = modify_depth(lpm, 2, ~filter(., term != "(Intercept)"))) %>%
mutate(lpm = map(lpm, bind_rows)) %>%
mutate(true_effect = modify_depth(simulations, 2, ~meffects(., coefs = coefs[[1]]))) %>%
mutate(true_effect = map(true_effect, bind_rows))
```

This is how results looks like:

`results`

```
## # A tibble: 1 x 6
## coefs sample_size repeats simulations lpm true_effect
## <list> <dbl> <dbl> <list> <list> <list>
## 1 <dbl [3]> 100 10 <list [10]> <tibble [20 × … <tibble [20 × …
```

Let’s take a closer look to the `lpm`

and `true_effect`

columns:

`results$lpm`

```
## [[1]]
## # A tibble: 20 x 2
## term estimate
## <chr> <dbl>
## 1 x1 0.228
## 2 x2 0.353
## 3 x1 0.180
## 4 x2 0.361
## 5 x1 0.165
## 6 x2 0.374
## 7 x1 0.182
## 8 x2 0.358
## 9 x1 0.125
## 10 x2 0.345
## 11 x1 0.171
## 12 x2 0.331
## 13 x1 0.122
## 14 x2 0.309
## 15 x1 0.129
## 16 x2 0.332
## 17 x1 0.102
## 18 x2 0.374
## 19 x1 0.176
## 20 x2 0.410
```

`results$true_effect`

```
## [[1]]
## # A tibble: 20 x 2
## term true_effect
## <chr> <dbl>
## 1 x1 0.183
## 2 x2 0.366
## 3 x1 0.166
## 4 x2 0.331
## 5 x1 0.174
## 6 x2 0.348
## 7 x1 0.169
## 8 x2 0.339
## 9 x1 0.167
## 10 x2 0.335
## 11 x1 0.173
## 12 x2 0.345
## 13 x1 0.157
## 14 x2 0.314
## 15 x1 0.170
## 16 x2 0.340
## 17 x1 0.182
## 18 x2 0.365
## 19 x1 0.161
## 20 x2 0.321
```

Let’s bind the columns, and compute the difference between the *true* and estimated marginal
effects:

```
simulation_results <- results %>%
mutate(difference = map2(.x = lpm, .y = true_effect, bind_cols)) %>%
mutate(difference = map(difference, ~mutate(., difference = true_effect - estimate))) %>%
mutate(difference = map(difference, ~select(., term, difference))) %>%
pull(difference) %>%
.[[1]]
```

Let’s take a look at the simulation results:

```
simulation_results %>%
group_by(term) %>%
summarise(mean = mean(difference),
sd = sd(difference))
```

```
## # A tibble: 2 x 3
## term mean sd
## <chr> <dbl> <dbl>
## 1 x1 0.0122 0.0370
## 2 x2 -0.0141 0.0306
```

Already with only 10 simulated datasets, the difference in means is not significant. Let’s rerun the analysis, but for difference sizes. In order to make things easier, we can put all the code into a nifty function:

```
monte_carlo <- function(coefs, sample_size, repeats){
many_datasets <- generate_datasets(coefs, sample_size, repeats)
results <- many_datasets %>%
mutate(lpm = modify_depth(simulations, 2, ~lm(y ~ ., data = .x))) %>%
mutate(lpm = modify_depth(lpm, 2, broom::tidy)) %>%
mutate(lpm = modify_depth(lpm, 2, ~select(., term, estimate))) %>%
mutate(lpm = modify_depth(lpm, 2, ~filter(., term != "(Intercept)"))) %>%
mutate(lpm = map(lpm, bind_rows)) %>%
mutate(true_effect = modify_depth(simulations, 2, ~meffects(., coefs = coefs[[1]]))) %>%
mutate(true_effect = map(true_effect, bind_rows))
simulation_results <- results %>%
mutate(difference = map2(.x = lpm, .y = true_effect, bind_cols)) %>%
mutate(difference = map(difference, ~mutate(., difference = true_effect - estimate))) %>%
mutate(difference = map(difference, ~select(., term, difference))) %>%
pull(difference) %>%
.[[1]]
simulation_results %>%
group_by(term) %>%
summarise(mean = mean(difference),
sd = sd(difference))
}
```

And now, let’s run the simulation for different parameters and sizes:

`monte_carlo(c(.5, 2, 4), 100, 10)`

```
## # A tibble: 2 x 3
## term mean sd
## <chr> <dbl> <dbl>
## 1 x1 -0.00826 0.0291
## 2 x2 -0.00732 0.0412
```

`monte_carlo(c(.5, 2, 4), 100, 100)`

```
## # A tibble: 2 x 3
## term mean sd
## <chr> <dbl> <dbl>
## 1 x1 0.00360 0.0392
## 2 x2 0.00517 0.0446
```

`monte_carlo(c(.5, 2, 4), 100, 500)`

```
## # A tibble: 2 x 3
## term mean sd
## <chr> <dbl> <dbl>
## 1 x1 -0.00152 0.0371
## 2 x2 -0.000701 0.0423
```

`monte_carlo(c(pi, 6, 9), 100, 10)`

```
## # A tibble: 2 x 3
## term mean sd
## <chr> <dbl> <dbl>
## 1 x1 -0.00829 0.0546
## 2 x2 0.00178 0.0370
```

`monte_carlo(c(pi, 6, 9), 100, 100)`

```
## # A tibble: 2 x 3
## term mean sd
## <chr> <dbl> <dbl>
## 1 x1 0.0107 0.0608
## 2 x2 0.00831 0.0804
```

`monte_carlo(c(pi, 6, 9), 100, 500)`

```
## # A tibble: 2 x 3
## term mean sd
## <chr> <dbl> <dbl>
## 1 x1 0.00879 0.0522
## 2 x2 0.0113 0.0668
```

We see that, at least for this set of parameters, the LPM does a good job of estimating marginal effects.

Now, this study might in itself not be very interesting to you, but I believe the general approach is quite useful and flexible enough to be adapted to all kinds of use-cases.

Hope you enjoyed! If you found this blog post useful, you might want to follow me on twitter for blog post updates and buy me an espresso or paypal.me.