# 23 Model basics

## 23.1 Introduction

The goal of a model is to provide a simple low-dimensional summary of a dataset. In the context of this book we’re going to use models to partition data into patterns and residuals. Strong patterns will hide subtler trends, so we’ll use models to help peel back layers of structure as we explore a dataset.

However, before we can start using models on interesting, real, datasets, you need to understand the basics of how models work. For that reason, this chapter of the book is unique because it uses only simulated datasets. These datasets are very simple, and not at all interesting, but they will help you understand the essence of modelling before you apply the same techniques to real data in the next chapter.

There are two parts to a model:

1. First, you define a family of models that express a precise, but generic, pattern that you want to capture. For example, the pattern might be a straight line, or a quadratic curve. You will express the model family as an equation like y = a_1 * x + a_2 or y = a_1 * x ^ a_2. Here, x and y are known variables from your data, and a_1 and a_2 are parameters that can vary to capture different patterns.

2. Next, you generate a fitted model by finding the model from the family that is the closest to your data. This takes the generic model family and makes it specific, like y = 3 * x + 7 or y = 9 * x ^ 2.

It’s important to understand that a fitted model is just the closest model from a family of models. That implies that you have the “best” model (according to some criteria); it doesn’t imply that you have a good model and it certainly doesn’t imply that the model is “true”. George Box puts this well in his famous aphorism:

All models are wrong, but some are useful.

It’s worth reading the fuller context of the quote:

Now it would be very remarkable if any system existing in the real world could be exactly represented by any simple model. However, cunningly chosen parsimonious models often do provide remarkably useful approximations. For example, the law PV = RT relating pressure P, volume V and temperature T of an “ideal” gas via a constant R is not exactly true for any real gas, but it frequently provides a useful approximation and furthermore its structure is informative since it springs from a physical view of the behavior of gas molecules.

For such a model there is no need to ask the question “Is the model true?”. If “truth” is to be the “whole truth” the answer must be “No”. The only question of interest is “Is the model illuminating and useful?”.

The goal of a model is not to uncover truth, but to discover a simple approximation that is still useful.

### 23.1.1 Prerequisites

In this chapter we’ll use the modelr package which wraps around base R’s modelling functions to make them work naturally in a pipe.

library(tidyverse)

library(modelr)
options(na.action = na.warn)

## 23.2 A simple model

Lets take a look at the simulated dataset sim1, included with the modelr package. It contains two continuous variables, x and y. Let’s plot them to see how they’re related:

ggplot(sim1, aes(x, y)) +
geom_point()

You can see a strong pattern in the data. Let’s use a model to capture that pattern and make it explicit. It’s our job to supply the basic form of the model. In this case, the relationship looks linear, i.e. y = a_0 + a_1 * x. Let’s start by getting a feel for what models from that family look like by randomly generating a few and overlaying them on the data. For this simple case, we can use geom_abline() which takes a slope and intercept as parameters. Later on we’ll learn more general techniques that work with any model.

models <- tibble(
a1 = runif(250, -20, 40),
a2 = runif(250, -5, 5)
)

ggplot(sim1, aes(x, y)) +
geom_abline(aes(intercept = a1, slope = a2), data = models, alpha = 1/4) +
geom_point() 

There are 250 models on this plot, but a lot are really bad! We need to find the good models by making precise our intuition that a good model is “close” to the data. We need a way to quantify the distance between the data and a model. Then we can fit the model by finding the value of a_0 and a_1 that generate the model with the smallest distance from this data.

One easy place to start is to find the vertical distance between each point and the model, as in the following diagram. (Note that I’ve shifted the x values slightly so you can see the individual distances.)

This distance is just the difference between the y value given by the model (the prediction), and the actual y value in the data (the response).

To compute this distance, we first turn our model family into an R function. This takes the model parameters and the data as inputs, and gives values predicted by the model as output:

model1 <- function(a, data) {
a[1] + data$x * a[2] } model1(c(7, 1.5), sim1) #> [1] 8.5 8.5 8.5 10.0 10.0 10.0 11.5 11.5 11.5 13.0 13.0 13.0 14.5 14.5 14.5 #> [16] 16.0 16.0 16.0 17.5 17.5 17.5 19.0 19.0 19.0 20.5 20.5 20.5 22.0 22.0 22.0 Next, we need some way to compute an overall distance between the predicted and actual values. In other words, the plot above shows 30 distances: how do we collapse that into a single number? One common way to do this in statistics to use the “root-mean-squared deviation”. We compute the difference between actual and predicted, square them, average them, and then take the square root. This distance has lots of appealing mathematical properties, which we’re not going to talk about here. You’ll just have to take my word for it! measure_distance <- function(mod, data) { diff <- data$y - model1(mod, data)
sqrt(mean(diff ^ 2))
}
measure_distance(c(7, 1.5), sim1)
#> [1] 2.665212

Now we can use purrr to compute the distance for all the models defined above. We need a helper function because our distance function expects the model as a numeric vector of length 2.

sim1_dist <- function(a1, a2) {
measure_distance(c(a1, a2), sim1)
}

models <- models %>%
mutate(dist = purrr::map2_dbl(a1, a2, sim1_dist))
models
#> # A tibble: 250 x 3
#>       a1      a2  dist
#>    <dbl>   <dbl> <dbl>
#> 1 -15.2   0.0889  30.8
#> 2  30.1  -0.827   13.2
#> 3  16.0   2.27    13.2
#> 4 -10.6   1.38    18.7
#> 5 -19.6  -1.04    41.8
#> 6   7.98  4.59    19.3
#> # … with 244 more rows

Next, let’s overlay the 10 best models on to the data. I’ve coloured the models by -dist: this is an easy way to make sure that the best models (i.e. the ones with the smallest distance) get the brighest colours.

ggplot(sim1, aes(x, y)) +
geom_point(size = 2, colour = "grey30") +
geom_abline(
aes(intercept = a1, slope = a2, colour = -dist),
data = filter(models, rank(dist) <= 10)
)

We can also think about these models as observations, and visualising with a scatterplot of a1 vs a2, again coloured by -dist. We can no longer directly see how the model compares to the data, but we can see many models at once. Again, I’ve highlighted the 10 best models, this time by drawing red circles underneath them.

ggplot(models, aes(a1, a2)) +
geom_point(data = filter(models, rank(dist) <= 10), size = 4, colour = "red") +
geom_point(aes(colour = -dist))

Instead of trying lots of random models, we could be more systematic and generate an evenly spaced grid of points (this is called a grid search). I picked the parameters of the grid roughly by looking at where the best models were in the plot above.

grid <- expand.grid(
a1 = seq(-5, 20, length = 25),
a2 = seq(1, 3, length = 25)
) %>%
mutate(dist = purrr::map2_dbl(a1, a2, sim1_dist))

grid %>%
ggplot(aes(a1, a2)) +
geom_point(data = filter(grid, rank(dist) <= 10), size = 4, colour = "red") +
geom_point(aes(colour = -dist)) 

When you overlay the best 10 models back on the original data, they all look pretty good:

ggplot(sim1, aes(x, y)) +
geom_point(size = 2, colour = "grey30") +
geom_abline(
aes(intercept = a1, slope = a2, colour = -dist),
data = filter(grid, rank(dist) <= 10)
)

You could imagine iteratively making the grid finer and finer until you narrowed in on the best model. But there’s a better way to tackle that problem: a numerical minimisation tool called Newton-Raphson search. The intuition of Newton-Raphson is pretty simple: you pick a starting point and look around for the steepest slope. You then ski down that slope a little way, and then repeat again and again, until you can’t go any lower. In R, we can do that with optim():

best <- optim(c(0, 0), measure_distance, data = sim1)
best$par #> [1] 4.222248 2.051204 ggplot(sim1, aes(x, y)) + geom_point(size = 2, colour = "grey30") + geom_abline(intercept = best$par[1], slope = best$par[2]) Don’t worry too much about the details of how optim() works. It’s the intuition that’s important here. If you have a function that defines the distance between a model and a dataset, an algorithm that can minimise that distance by modifying the parameters of the model can find the best model. The neat thing about this approach is that it will work for any family of models that you can write an equation for. There’s one more approach that we can use for this model, because it’s a special case of a broader family: linear models. A linear model has the general form y = a_1 + a_2 * x_1 + a_3 * x_2 + ... + a_n * x_(n - 1). So this simple model is equivalent to a general linear model where n is 2 and x_1 is x. R has a tool specifically designed for fitting linear models called lm(). lm() has a special way to specify the model family: formulas. Formulas look like y ~ x, which lm() will translate to a function like y = a_1 + a_2 * x. We can fit the model and look at the output: sim1_mod <- lm(y ~ x, data = sim1) coef(sim1_mod) #> (Intercept) x #> 4.220822 2.051533 These are exactly the same values we got with optim()! Behind the scenes lm() doesn’t use optim() but instead takes advantage of the mathematical structure of linear models. Using some connections between geometry, calculus, and linear algebra, lm() actually finds the closest model in a single step, using a sophisticated algorithm. This approach is both faster, and guarantees that there is a global minimum. ### 23.2.1 Exercises 1. One downside of the linear model is that it is sensitive to unusual values because the distance incorporates a squared term. Fit a linear model to the simulated data below, and visualise the results. Rerun a few times to generate different simulated datasets. What do you notice about the model? sim1a <- tibble( x = rep(1:10, each = 3), y = x * 1.5 + 6 + rt(length(x), df = 2) ) 2. One way to make linear models more robust is to use a different distance measure. For example, instead of root-mean-squared distance, you could use mean-absolute distance: measure_distance <- function(mod, data) { diff <- data$y - model1(mod, data)
mean(abs(diff))
}

Use optim() to fit this model to the simulated data above and compare it to the linear model.

3. One challenge with performing numerical optimisation is that it’s only guaranteed to find one local optimum. What’s the problem with optimising a three parameter model like this?

model1 <- function(a, data) {

### 23.4.2 Interactions (continuous and categorical)

What happens when you combine a continuous and a categorical variable? sim3 contains a categorical predictor and a continuous predictor. We can visualise it with a simple plot:

ggplot(sim3, aes(x1, y)) +
geom_point(aes(colour = x2))

There are two possible models you could fit to this data:

mod1 <- lm(y ~ x1 + x2, data = sim3)
mod2 <- lm(y ~ x1 * x2, data = sim3)

When you add variables with +, the model will estimate each effect independent of all the others. It’s possible to fit the so-called interaction by using *. For example, y ~ x1 * x2 is translated to y = a_0 + a_1 * x1 + a_2 * x2 + a_12 * x1 * x2. Note that whenever you use *, both the interaction and the individual components are included in the model.

To visualise these models we need two new tricks:

1. We have two predictors, so we need to give data_grid() both variables. It finds all the unique values of x1 and x2 and then generates all combinations.

2. To generate predictions from both models simultaneously, we can use gather_predictions() which adds each prediction as a row. The complement of gather_predictions() is spread_predictions() which adds each prediction to a new column.

Together this gives us:

grid <- sim3 %>%
data_grid(x1, x2) %>%
gather_predictions(mod1, mod2)
grid
#> # A tibble: 80 x 4
#>   model    x1 x2     pred
#>   <chr> <int> <fct> <dbl>
#> 1 mod1      1 a      1.67
#> 2 mod1      1 b      4.56
#> 3 mod1      1 c      6.48
#> 4 mod1      1 d      4.03
#> 5 mod1      2 a      1.48
#> 6 mod1      2 b      4.37
#> # … with 74 more rows

We can visualise the results for both models on one plot using facetting:

ggplot(sim3, aes(x1, y, colour = x2)) +
geom_point() +
geom_line(data = grid, aes(y = pred)) +
facet_wrap(~ model)

Note that the model that uses + has the same slope for each line, but different intercepts. The model that uses * has a different slope and intercept for each line.

Which model is better for this data? We can take look at the residuals. Here I’ve facetted by both model and x2 because it makes it easier to see the pattern within each group.

sim3 <- sim3 %>%
gather_residuals(mod1, mod2)

ggplot(sim3, aes(x1, resid, colour = x2)) +
geom_point() +
facet_grid(model ~ x2)

There is little obvious pattern in the residuals for mod2. The residuals for mod1 show that the model has clearly missed some pattern in b, and less so, but still present is pattern in c, and d. You might wonder if there’s a precise way to tell which of mod1 or mod2 is better. There is, but it requires a lot of mathematical background, and we don’t really care. Here, we’re interested in a qualitative assessment of whether or not the model has captured the pattern that we’re interested in.

### 23.4.3 Interactions (two continuous)

Let’s take a look at the equivalent model for two continuous variables. Initially things proceed almost identically to the previous example:

mod1 <- lm(y ~ x1 + x2, data = sim4)
mod2 <- lm(y ~ x1 * x2, data = sim4)

grid <- sim4 %>%
data_grid(
x1 = seq_range(x1, 5),
x2 = seq_range(x2, 5)
) %>%
gather_predictions(mod1, mod2)
grid
#> # A tibble: 50 x 4
#>   model    x1    x2   pred
#>   <chr> <dbl> <dbl>  <dbl>
#> 1 mod1   -1    -1    0.996
#> 2 mod1   -1    -0.5 -0.395
#> 3 mod1   -1     0   -1.79
#> 4 mod1   -1     0.5 -3.18
#> 5 mod1   -1     1   -4.57
#> 6 mod1   -0.5  -1    1.91
#> # … with 44 more rows

Note my use of seq_range() inside data_grid(). Instead of using every unique value of x, I’m going to use a regularly spaced grid of five values between the minimum and maximum numbers. It’s probably not super important here, but it’s a useful technique in general. There are two other useful arguments to seq_range():

• pretty = TRUE will generate a “pretty” sequence, i.e. something that looks nice to the human eye. This is useful if you want to produce tables of output:

seq_range(c(0.0123, 0.923423), n = 5)
#> [1] 0.0123000 0.2400808 0.4678615 0.6956423 0.9234230
seq_range(c(0.0123, 0.923423), n = 5, pretty = TRUE)
#> [1] 0.0 0.2 0.4 0.6 0.8 1.0
• trim = 0.1 will trim off 10% of the tail values. This is useful if the variables have a long tailed distribution and you want to focus on generating values near the center:

x1 <- rcauchy(100)
seq_range(x1, n = 5)
#> [1] -115.86934  -83.52130  -51.17325  -18.82520   13.52284
seq_range(x1, n = 5, trim = 0.10)
#> [1] -13.841101  -8.709812  -3.578522   1.552767   6.684057
seq_range(x1, n = 5, trim = 0.25)
#> [1] -2.17345439 -1.05938856  0.05467728  1.16874312  2.28280896
seq_range(x1, n = 5, trim = 0.50)
#> [1] -0.7249565 -0.2677888  0.1893788  0.6465465  1.1037141
• expand = 0.1 is in some sense the opposite of trim() it expands the range by 10%.

x2 <- c(0, 1)
seq_range(x2, n = 5)
#> [1] 0.00 0.25 0.50 0.75 1.00
seq_range(x2, n = 5, expand = 0.10)
#> [1] -0.050  0.225  0.500  0.775  1.050
seq_range(x2, n = 5, expand = 0.25)
#> [1] -0.1250  0.1875  0.5000  0.8125  1.1250
seq_range(x2, n = 5, expand = 0.50)
#> [1] -0.250  0.125  0.500  0.875  1.250

Next let’s try and visualise that model. We have two continuous predictors, so you can imagine the model like a 3d surface. We could display that using geom_tile():

ggplot(grid, aes(x1, x2)) +
geom_tile(aes(fill = pred)) +
facet_wrap(~ model)

That doesn’t suggest that the models are very different! But that’s partly an illusion: our eyes and brains are not very good at accurately comparing shades of colour. Instead of looking at the surface from the top, we could look at it from either side, showing multiple slices:

ggplot(grid, aes(x1, pred, colour = x2, group = x2)) +
geom_line() +
facet_wrap(~ model)
ggplot(grid, aes(x2, pred, colour = x1, group = x1)) +
geom_line() +
facet_wrap(~ model)

This shows you that interaction between two continuous variables works basically the same way as for a categorical and continuous variable. An interaction says that there’s not a fixed offset: you need to consider both values of x1 and x2 simultaneously in order to predict y.

You can see that even with just two continuous variables, coming up with good visualisations are hard. But that’s reasonable: you shouldn’t expect it will be easy to understand how three or more variables simultaneously interact! But again, we’re saved a little because we’re using models for exploration, and you can gradually build up your model over time. The model doesn’t have to be perfect, it just has to help you reveal a little more about your data.

I spent some time looking at the residuals to see if I could figure if mod2 did better than mod1. I think it does, but it’s pretty subtle. You’ll have a chance to work on it in the exercises.

### 23.4.4 Transformations

You can also perform transformations inside the model formula. For example, log(y) ~ sqrt(x1) + x2 is transformed to log(y) = a_1 + a_2 * sqrt(x1) + a_3 * x2. If your transformation involves +, *, ^, or -, you’ll need to wrap it in I() so R doesn’t treat it like part of the model specification. For example, y ~ x + I(x ^ 2) is translated to y = a_1 + a_2 * x + a_3 * x^2. If you forget the I() and specify y ~ x ^ 2 + x, R will compute y ~ x * x + x. x * x means the interaction of x with itself, which is the same as x. R automatically drops redundant variables so x + x become x, meaning that y ~ x ^ 2 + x specifies the function y = a_1 + a_2 * x. That’s probably not what you intended!

Again, if you get confused about what your model is doing, you can always use model_matrix() to see exactly what equation lm() is fitting:

df <- tribble(
~y, ~x,
1,  1,
2,  2,
3,  3
)
model_matrix(df, y ~ x^2 + x)
#> # A tibble: 3 x 2
#>   (Intercept)     x
#>           <dbl> <dbl>
#> 1             1     1
#> 2             1     2
#> 3             1     3
model_matrix(df, y ~ I(x^2) + x)
#> # A tibble: 3 x 3
#>   (Intercept) I(x^2)     x
#>           <dbl>    <dbl> <dbl>
#> 1             1        1     1
#> 2             1        4     2
#> 3             1        9     3

Transformations are useful because you can use them to approximate non-linear functions. If you’ve taken a calculus class, you may have heard of Taylor’s theorem which says you can approximate any smooth function with an infinite sum of polynomials. That means you can use a polynomial function to get arbitrarily close to a smooth function by fitting an equation like y = a_1 + a_2 * x + a_3 * x^2 + a_4 * x ^ 3. Typing that sequence by hand is tedious, so R provides a helper function: poly():

model_matrix(df, y ~ poly(x, 2))
#> # A tibble: 3 x 3
#>   (Intercept) poly(x, 2)1 poly(x, 2)2
#>           <dbl>         <dbl>         <dbl>
#> 1             1     -7.07e- 1         0.408
#> 2             1     -7.85e-17        -0.816
#> 3             1      7.07e- 1         0.408

However there’s one major problem with using poly(): outside the range of the data, polynomials rapidly shoot off to positive or negative infinity. One safer alternative is to use the natural spline, splines::ns().

library(splines)
model_matrix(df, y ~ ns(x, 2))
#> # A tibble: 3 x 3
#>   (Intercept) ns(x, 2)1 ns(x, 2)2
#>           <dbl>       <dbl>       <dbl>
#> 1             1       0           0
#> 2             1       0.566      -0.211
#> 3             1       0.344       0.771

Let’s see what that looks like when we try and approximate a non-linear function:

sim5 <- tibble(
x = seq(0, 3.5 * pi, length = 50),
y = 4 * sin(x) + rnorm(length(x))
)

ggplot(sim5, aes(x, y)) +
geom_point()

I’m going to fit five models to this data.

mod1 <- lm(y ~ ns(x, 1), data = sim5)
mod2 <- lm(y ~ ns(x, 2), data = sim5)
mod3 <- lm(y ~ ns(x, 3), data = sim5)
mod4 <- lm(y ~ ns(x, 4), data = sim5)
mod5 <- lm(y ~ ns(x, 5), data = sim5)

grid <- sim5 %>%
data_grid(x = seq_range(x, n = 50, expand = 0.1)) %>%
gather_predictions(mod1, mod2, mod3, mod4, mod5, .pred = "y")

ggplot(sim5, aes(x, y)) +
geom_point() +
geom_line(data = grid, colour = "red") +
facet_wrap(~ model)

Notice that the extrapolation outside the range of the data is clearly bad. This is the downside to approximating a function with a polynomial. But this is a very real problem with every model: the model can never tell you if the behaviour is true when you start extrapolating outside the range of the data that you have seen. You must rely on theory and science.

### 23.4.5 Exercises

1. What happens if you repeat the analysis of sim2 using a model without an intercept. What happens to the model equation? What happens to the predictions?

2. Use model_matrix() to explore the equations generated for the models I fit to sim3 and sim4. Why is * a good shorthand for interaction?

3. Using the basic principles, convert the formulas in the following two models into functions. (Hint: start by converting the categorical variable into 0-1 variables.)

mod1 <- lm(y ~ x1 + x2, data = sim3)
mod2 <- lm(y ~ x1 * x2, data = sim3)
4. For sim4, which of mod1 and mod2 is better? I think mod2 does a slightly better job at removing patterns, but it’s pretty subtle. Can you come up with a plot to support my claim?

## 23.5 Missing values

Missing values obviously can not convey any information about the relationship between the variables, so modelling functions will drop any rows that contain missing values. R’s default behaviour is to silently drop them, but options(na.action = na.warn) (run in the prerequisites), makes sure you get a warning.

df <- tribble(
~x, ~y,
1, 2.2,
2, NA,
3, 3.5,
4, 8.3,
NA, 10
)

mod <- lm(y ~ x, data = df)
#> Warning: Dropping 2 rows with missing values

To suppress the warning, set na.action = na.exclude:

mod <- lm(y ~ x, data = df, na.action = na.exclude)

You can always see exactly how many observations were used with nobs():

nobs(mod)
#> [1] 3

## 23.6 Other model families

This chapter has focussed exclusively on the class of linear models, which assume a relationship of the form y = a_1 * x1 + a_2 * x2 + ... + a_n * xn. Linear models additionally assume that the residuals have a normal distribution, which we haven’t talked about. There are a large set of model classes that extend the linear model in various interesting ways. Some of them are:

• Generalised linear models, e.g. stats::glm(). Linear models assume that the response is continuous and the error has a normal distribution. Generalised linear models extend linear models to include non-continuous responses (e.g. binary data or counts). They work by defining a distance metric based on the statistical idea of likelihood.

• Generalised additive models, e.g. mgcv::gam(), extend generalised linear models to incorporate arbitrary smooth functions. That means you can write a formula like y ~ s(x) which becomes an equation like y = f(x) and let gam() estimate what that function is (subject to some smoothness constraints to make the problem tractable).

• Penalised linear models, e.g. glmnet::glmnet(), add a penalty term to the distance that penalises complex models (as defined by the distance between the parameter vector and the origin). This tends to make models that generalise better to new datasets from the same population.

• Robust linear models, e.g. MASS::rlm(), tweak the distance to downweight points that are very far away. This makes them less sensitive to the presence of outliers, at the cost of being not quite as good when there are no outliers.

• Trees, e.g. rpart::rpart(), attack the problem in a completely different way than linear models. They fit a piece-wise constant model, splitting the data into progressively smaller and smaller pieces. Trees aren’t terribly effective by themselves, but they are very powerful when used in aggregate by models like random forests (e.g. randomForest::randomForest()) or gradient boosting machines (e.g. xgboost::xgboost.)

These models all work similarly from a programming perspective. Once you’ve mastered linear models, you should find it easy to master the mechanics of these other model classes. Being a skilled modeller is a mixture of some good general principles and having a big toolbox of techniques. Now that you’ve learned some general tools and one useful class of models, you can go on and learn more classes from other sources.