Sean Trott

High-level goals

This began as a subsection in another tutorial on mixed models in R. While writing that tutorial, I realized that a lengthier discussion of model comparison approaches probably deserves its own post.

So keeping that in mind, the main goals of this tutorial are to address the following questions:

What is model comparison?
What's the conceptual intuition behind a model comparison?
What are some of the different kinds of model comparison approaches?

Additionally, this tutorial will likely change over time as I learn more about the domain (mostly in the direction of including more details and content). So if you see anything missing, please let me know and I will try to add it.

If you want to see the slides from a presentation of this tutorial, see here.

Logistics

First, logistics. We'll be using four libraries: library(tidyverse), library(lme4), library(lmtest), and library(ggridges). If you don't have them installed already, use install.packages("tidyverse"), install.packages("lme4"), install.packages("lmtest"), and install.packages("ggridges") to install them.

Also, if you want to clone a copy of this tutorial locally on your computer, you can do so with git:

git clone https://github.com/seantrott/model_comparison.git

Introduction

Science involves building models of the world. In fact, some would say that's the goal of science.

These models range from verbal descriptions of a phenomenon to statistical models that make quantitative predictions. I'll be focusing more on the latter kind of model today---though it's worth keeping in mind that the term "model" is often used loosely.

Typically, statistical models include terms that are meant to stand in for some theoretical construct. We then use our algorithm of choice to fit a model and find the set of parameters (i.e., coefficients) \(\theta\) that minimize the prediction error between our actual \(Y\) values and the values predicted by the model, \(\hat{Y}\).

A good statistical model should satisfy the following criteria¹:

Accuracy: A model should describe data accurately and generate accurate predictions.
Parsimony: All else being equal, we should prefer a simpler model with fewer assumptions (i.e., fewer parameters).

Thus, from this point of view, the goal of a model selection procedure is to settle on the simplest model that best explains your data².

Model comparison: Background

The core idea behind the model comparison approach is to compare the explanatory power of two or more models, with the goal of identifying the model that best explains variance in \(Y\). Of course, a model with more parameters will almost always explain slightly more variance than a model with less. That is, if \(N\) is the number of observations and \(k\) is the number of parameters, the model will necessarily improve its predictions as \(k\) approaches \(N\). This can result in overfitting: a model whose parameters are too tuned to the variance specific to a particular dataset, and thus exhibits poor generalizability across datasets. So reducing variance in \(Y\) isn't our only goal: we also want a parsimonious model.

Theoretically, you can compare any two models. Here, we'll be comparing nested models fit to the same data, in which only a single parameter differs, e.g., \(M_{k}\) and \(M_{k-1}\). This allows us to infer the explanatory power of the parameter added in the full model \(M_{k}\): if such a parameter substantially improves the model by some measure of model fit, we can infer that this parameter explains variance in \(Y\).

There are a number of different model comparison methods, each designed with the twin goals of:

Comparing the amount of variance explained by \(M_{k}\) and \(M_{k-1}\).
Enforcing model parsimony; all else being equal, we prefer a simpler model (e.g., \(M_{k-1}\)) over a model with more parameters (\(M_{k}\)).

These goals correspond to the high-level goals of accuracy and parsimony mentioned earlier.

Below, I introduce and describe a few different appraoches to model comparison. The section on likelihood ratio tests is particularly long, but that's because I also introduce/describe what likelihood is. Fortunately, those insights should transfer to understanding AIC and BIC as well.

Likelihood ratio tests

The likelihood of a model, \(\mathcal{L}(M_{k})\), is a way of quantifying how well a model fits the data. It's akin to asking: what is the likelihood that a given model (or hypothesis) generated a given dataset?

A likelihood ratio test, then, compares the likelihoods of different models for the same data. The basic intuition is that we should prefer the model under which the data are more likely; if the data are equally likely under both models, then there's no reason to prefer one model over another (except for, say, one model having fewer parameters).

Edwards (1992), quoted in Etz (2018), puts this better than I could:

The Law of Likelihood states that "within the framework of a statistical model, a particular set of data supports one statistical hypothesis better than another if the likelihood of the first hypothesis, on the data, exceeds the likelihood of the second hypothesis" (Edwards, 1992, p. 30).

Calculating likelihood

Model likelihood is calculated using a likelihood function: for a predictor matrix \(X\), parameters \(\theta\), and observations \(y\), compute the likelihood \(\mathcal{L}\)of \(y\) given \(X\) and \(\theta\), i.e., \(\mathcal{L}(y | \theta, X)\). In maximum likelihood estimation (MLE), the goal is to (as the name implies) maximize this likelihood³, i.e., identify the set of parameter values for predictors \(X\) that will best explain the data \(y\).

Comparison to OLS regression

If you're familiar with Ordinary Least Squares (OLS) regression, this might sound familiar. With OLS, the goal is to find the set of coefficient values that minimize the sum of squared residuals (i.e., the difference between predictions \(\hat{y}\) and \(y\)), where that sum is defined as follows:

\(\sum{(\hat{y} - y)^2}\)

In essence, the sum of squared residuals measures the "distance" between the predictions from your model and the actual values in your data. OLS thus makes intuitive sense: with linear regression, we want to identify the best-fitting line, where "best fit" is the line with the smallest deviations in the real data.

Let's illustrate this quickly with an example. I'll first generate a set of data according to a linear equation, with some amount of normally distributed noise added:

\(Y = 10 + 2X + \epsilon\)

set.seed(1)
X = c(1:100)
Y = 10 + 2 * X + rnorm(n = length(X), sd = 20)

df_xy = data.frame(X, Y)

df_xy %>%
  ggplot(aes(x = X,
             y = Y)) +
  geom_point(alpha = .6) +
  theme_minimal()

Now let's draw a couple different lines on top of this data, i.e., using different intercepts and different slopes. Keep in mind that the ground truth line, i.e., the function that's responsible for generating the data, is defined by \(\beta_0 = 10, \beta_1 = 2\); this line is shown in blue.

df_xy %>%
  ggplot(aes(x = X,
             y = Y)) +
  geom_point(alpha = .6) +
  geom_abline(intercept = 10, slope = 2, color = "blue", alpha = .6) +
  geom_abline(intercept = 20, slope = 1.5, color = "red", alpha = .6, linetype = "dotted") +
  geom_abline(intercept = 5, slope = 3, color = "red", alpha = .6, linetype = "dotted") +
  geom_abline(intercept = 40, slope = .5, color = "red", alpha = .6, linetype = "dotted") +
  theme_minimal()

It's pretty obvious that some of these lines are better fits than others. We can actually plug in the slope/intercept values for each line into a linear equation, and just measure the residuals from the predictions for each one.

y_real = 10 + 2*X
y1 = 20 + 1.5 * X
y2 = 5 + 3 * X
y3 = 40 + .5 * X

df_rss = data.frame(
  real = sum((Y - y_real)**2),
  m1 = sum((Y - y1)**2),
  m2 = sum((Y - y2)**2),
  m3 = sum((Y - y3)**2)) %>%
  pivot_longer(c(real, m1, m2, m3), names_to = "model", values_to = "rss")
  

df_rss %>%
  ggplot(aes(x = reorder(model, rss),
             y = rss)) +
  geom_bar(stat = "identity") +
  labs(x = "Model",
       y = "RSS") +
  theme_minimal()

As you'd expect, the residual sum of squares is lowest for the predictions produced by the real equation. The RSS is still non-zero, simply because the actual data have some degree of noise. But it's still a much better fit than the other models; hopefully this helps build your intuition for what it means for a model to have worse or better fit, and what exactly something like RSS is measuring. You can also check out my tutorial on the bias-variance trade-off for more details on this.

OLS vs. MLE

OLS is actually a derivation of a more general approach to parameter estimation: maximum likelihood estimation (MLE). With MLE---as in OLS---we want to identify a set of parameters that maximizes the likelihood of our data.

This requires calculating the likelihood of a model, \(\mathcal{L}\):

\(\mathcal{L}(\theta | x_1, x_2, ..., x_n) = f(x_1, x_2, ..., x_n | \theta)\)

In particular, we want to calculate the joint probability of each data point, under the model's estimated parameters. We can write that as follows:

\(\mathcal{L}(\theta | X) = \prod\limits_{i=1}^N p(X_i | \theta)\)

One issue with this formulation is that the probability densities for each data point can quickly become either very small or very large, depending on the sample size and the variance of the underlying distribution. Computers can only store up to so many digits before "rounding" a number to either 0 or Inf. But this is clearly non-ideal, as the product of any sequence of numbers including 0 will be 0 (and Inf for a sequence of numbers including Inf). Furthermore, as described here, this formulation requires us to find parameter estimates that maximize the likelihood of a model, which is a harder optimization problem than minimizing a value (in this case, minimizing the negative log likelihood).

Thus, typically we rewrite this as the sum of log probabilities.

\(\mathcal{L}(\theta | X) = \sum\limits_{i=1}^N log(p(X_i | \theta))\)

But how exactly is \(p(X_i | \theta)\) calculated? The details will depend on aspects of the data (e.g., whether it's discrete or continuous) and the class of model being applied (e.g., whether we're using linear or logistic regression). To build intuition for how this works, it helps to back up just a bit: instead of thinking about calculating the likelihood of data under a complicated statistical model, let's think about how to calculate the likelihood of data under various probability distributions.

Back to basics: likelihood and distributions

We can think of a probability distribution as a generative process: for any possible value \(X_i\), a probability function assigns some probability mass (or density) to that value. This can be used to generate data, or evaluate the likelihood of a given data point under that distribution.

Continuous distributions

Let's start with a normal distribution. A normal distribution is a real-valued, continuous probability distribution parameterized by a mean, \(\mu\), and standard deviation, \(\sigma\); the mean is equal to the median and the mode. We can simulate a set of normally distributed data using rnorm:

set.seed(1)
X = rnorm(n = 1000, mean = 0, sd = 10)
hist(X)

This distribution has the following parameters: \(\mu = 0, \sigma = 10\).

Just looking at this distribution, it's very clear that some values are more likely than others. For example, we're much more likely to obtain a value close to 0 (the mean of the distribution) than a value close to 20.

We can use dnorm to calculate the probability density for each of these values of \(X\). The density function for some value \(x\) is defined as the following:

\(f(x) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}(\frac{x - \mu}{\sigma})^2}\)

But fortunately, we can also just use the dnorm function in R. As you'd expect with the parameters of this normal distribution, values closer to 0 are more likely than those close to 20:

plot(X, dnorm(X, mean = 0, sd = 10))

Each of those values of \(X\) is assigned a likelihood. To find the likelihood of all our data, i.e., \(\mathcal{L}(X|\theta)\), we would have to multiply each of these probability values---or sum their log probabilities. We can do this by first applying log to the output of the dnorm function, then using sum to add up all those logged probabilities:

sum(log(dnorm(X, mean = 0, sd = 10)))

## [1] -3756.581

This number may not make a ton of sense out of context. But it's worth thinking about exactly what we just did:

First, we generated a bunch of data by drawing samples from a normal distribution with parameters \(\mu = 0, \sigma = 10\).
Then, for each of those data points, we calculated the probability density of that value under a normal distribution with parameters \(\mu = 0, \sigma = 10\).
Finally, we summed all of those log probabilities.

That final step gives us the log probability, \(\mathcal{L}\), of our parameters, \(\theta\)!

Now imagine that we didn't know the parameters used to generate our dataset \(X\), and we wanted to estimate them. How would you go about this?

One intuitive way to do this is to search the space of possible parameters, i.e., values for \(\mu\) and \(\sigma\), and for each parameter set \(\theta\), calculate the log likelihood of the data under those parameters. We can then select the parameters that yield the best likelihood. In the simplest iteration, we can do something called a grid search⁴.

For simplicity, let's assume we already know the standard deviation, \(\sigma\), and we want to just estimate the mean, \(\mu\). Let's consider all the integers between -50 and 50⁵:

df_likelihood = data.frame(
  mu = seq(from = -50, to = 50, length.out = 101)
)

Now, for each hypothesized value \(\mu_i\), let's calculate the log likelihood of the observed data \(X\).

LL = function(mean, sd) {
  # Calculate log likelihood for dataset X
  sum(dnorm(x = X, mean = mean, sd = sd, log = TRUE))
}

# Apply log likelihood function for each hypothesized value of mu (and fix SD at 10).
df_likelihood$ll <- mapply(LL, mean=df_likelihood$mu, sd = 10)

# Plot resulting distribution
df_likelihood %>%
  ggplot(aes(x = mu,
             y = ll)) +
  geom_point() +
  theme_minimal()

And of course, if were trying to run this through an optimization algorithm, we'd want the negative log likelihood instead:

df_likelihood %>%
  ggplot(aes(x = mu,
             y = -ll)) +
  geom_point() +
  theme_minimal()

But either way, what's clear is that the best log likelihood value is obtained when \(\mu = 0\) (or at least is close to 0).

This is good news! Remember that our original distribution was generated with \(\mu = 0\). Thus, it makes sense that the data generated from that distribution would be most likely under a model with similar parameter values.

We can thus use this approach to find the best set of parameters \(\theta\) for a given dataset \(X\).

Discrete distributions

Now, we could also apply this same logic to discrete distributions, such as the binomial distribution.

Imagine we have a coin, and we want to identify the probability \(p\) of tossing a heads. Before, we wanted to identify the parameters that comprise the normal distribution (\(\mu\) and \(\sigma\)); now, we want to identify \(p\).

Now we observe 20 independent tosses of the coin. Of those 20 tosses, 14 come up heads. What is the best estimate for \(p(H)\)?

Again, we can calculate the likelihood of observing that outcome for various values of \(p\). But this is dealing with binomial data, not normally distributed data; so instead of using dnorm, we'll use dbinom.

DATA = 14 ## 14 observed heads
NUM_EVENTS = 20 ## 20 tosses
df_probs = data.frame(
  p = seq(0, 1, .01)) ## Various hypotheses for p(H)
df_probs = df_probs %>%
  mutate(likelihood = dbinom(DATA , size=NUM_EVENTS, prob=p)) 

df_probs %>%
  ggplot(aes(x = p,
             y = likelihood)) +
  geom_point() +
  theme_minimal()

We see that the highest likelihood is assigned to a value of \(p\) somewhere around 0.7 (note that this is of course what you'd get if you simply divided the number of heads, 14, by the number of tosses, 20). If this were a Bayesian analysis, we might multiply each of these likelihoods by a prior probability assigned to that hypothesized value of \(p\)---i.e., maybe we're extremely confident a priori that this is an unbiased coin. But based on the data alone, we should assign the highest likelihood to \(p = 0.7\).

Likelihoods for statistical models

Hopefully, the above sections were a useful refresher on calculating the probability of some set of observations, given some assumptions about the distribution that generated those observations.

Think of a statistical model as a generalization of the simpler distributions we were just working with. Each statistical model is itself generative: the parameters of that model, when applied to some dataset \(X\), entail certain predictions, \(\hat{Y}\). Or, put another way: a model's parameters entail certain probabilities for the possible values that \(Y\) could take on!

For example, let's say we want to predict the value of some continuous-valued outcome variable, e.g., height. One of the simplest ways to make predictions about that outcome would be to assume it's normally distributed, and then estimate the mean of that distribution. This is the approach we took above. Of course, such an approach won't give us very much leverage in terms of predictions, but if we have limited information, it can be a good first step. Below, I first simulate a normal distribution with parameters \(\mu = 60, \sigma = 10\), then visualize that distribution on top of a single prediction: the mean of that distribution.

set.seed(1)
df_height = data.frame(
  height = rnorm(n = 100, mean = 60, sd = 10)
)

df_height$pred_mean = 60
df_height %>%
  ggplot(aes(x = pred_mean,
             y = height)) +
  geom_point() +
  theme_minimal()

This approach won't get us too far. That's because our predictions are conditioned only a single variable: the mean, \(\mu\), of the target distribution in question.

But let's assume we know something else about our data---i.e., we don't just know the heights of each individual, we also know their mother's height. If those two variables are related, it will give us more leverage in predicting height. Let's assume that the two variables are indeed related:

df_height$m_height = rnorm(n=100, df_height$height, sd = 10) 

df_height %>%
  ggplot(aes(x = m_height,
             y = height)) +
  geom_point() +
  labs(x = "Mother's height",
       y = "Child's height") +
  theme_minimal()

Now, we can estimate additional parameters. Our predictions \(\hat{Y}\) won't just be set to the mean of \(Y\), i.e., \(\mu\).

Rather, our value for \(\mu\) will be conditioned on the values of \(X\)---specifically, each prediction \(\hat{Y_i}\) will be determined from the value \(X_i\) and a slope coefficient relating \(X\) to \(Y\). And similarly, our value for \(\sigma\) will be based on our average prediction error (aka the residual standard error).

Let's fit a linear model to this data:

m = lm(data = df_height,
       height ~ m_height)
m$coefficients

## (Intercept)    m_height 
##  32.6854721   0.4678476

We can use this model to generate predictions, \(\hat{Y}\).

df_height$y_pred = predict(m)

df_height %>%
  ggplot(aes(x = m_height,
             y = height)) +
  geom_point() +
  geom_smooth(method = "lm") +
  labs(x = "Mother's height",
       y = "Actual height") +
  theme_minimal()

## `geom_smooth()` using formula 'y ~ x'

rss = sum((df_height$y_pred - df_height$height)**2)
rse = sqrt(rss / (nrow(df_height) - 2))
rse

## [1] 6.588668

Our predictions aren't perfect, of course. On average, we can expect a prediction error of about 6.59.

How would we evaluate the likelihood of this model? Well, as noted above, we can think of each predicted value \(\hat{Y}_i\) as representing the mean of a normal distribution, conditioned on a particular value \(X_i\); we assume that \(\sigma\) is constant across values of \(X\) ⁶.

Thus, for each actual value \(Y_i\), we can calculate the probability of that value occurring under a normal distribution with parameters \(\mu_i, \sigma\), where \(\mu_i = \beta_1 * X_i + \beta_0\). Put together, this gives us the likelihood function:

\(\mathcal{L}(Y|\theta) = \sum\limits_{i=1}^N log(p(Y_i | \theta))\)

Where \(\theta\), in this case, is \(\beta_0, \beta_1\). Thus, \(p(Y_i | \theta)\) is akin to asking, for some value of \(X_i\):

\(p(Y_i) | \beta_0, \beta_1) = p(Y_i | Normal(\mu=\beta_0 + X_i\beta_1, \sigma=RSE))\)

I.e., what's the probability of getting some value \(Y_i\) from a normal distribution parameterized by the predicted value \(\hat{Y}_i\) and residual standard error (RSE)? That's exactly what the code below calculates:

ll = sum(dnorm(x = df_height$height, 
               mean = df_height$y_pred, 
               sd = rse, 
               log = TRUE))
ll

## [1] -329.429

We can double-check this using the logLik function in R:

logLik(m)

## 'log Lik.' -329.4188 (df=3)

Zooming in

If the above explanation doesn't make sense, we can also try to illustrate what's going on graphically.

Below, I'll visualize some of the points from the above graph again, but only select a random subset of them (for ease of visualization). Along with the actual observations (shown in blue), I show the predicted values \(\hat{Y}\) (in red), which are arranged along the line of best fit. I also show the residuals (the dotted red lines), i.e., the difference between each actual observation and the predicted value.

And most crucially, I'll also draw a normal distribution around the predicted value \(\hat{Y}_i\), and hopefully that gives intuition for this likelihood calculation: the further each actual value (the blue dots) is from the center of the corresponding normal distribution, the less likely that data point is under the model. All credit for the normal distribution visualizations goes to this answer on StackOverflow.

Note that the width of each normal distribution is determined by the the residual standard error ⁷. The likelihood calculation then asks: what's the probability of the blue point (i.e., the actual data) occurring under that normal distribution?

set.seed(1)
df_sub_sample = df_height %>%
  sample_n(10) %>%
  filter(m_height < 60)


# For every row in `df`, compute a rotated normal density centered at `y` and shifted by `x`
curves <- lapply(seq_len(nrow(df_sub_sample)), function(i) {
  mu <- df_sub_sample$y_pred[i]
  range <- mu + c(-rse, rse)
  seq <- seq(range[1], range[2], length.out = 100)
  data.frame(
    x = -1 * dnorm(seq, mean = mu) + df_sub_sample$m_height[i],
    y = seq,
    grp = i
  )
})

# Combine above densities in one data.frame
curves <- do.call(rbind, curves)


df_sub_sample$x = df_sub_sample$m_height
df_sub_sample$y = df_sub_sample$height

ggplot(df_sub_sample, aes(x, y)) +
  geom_point(size = 3, alpha = .6, color = "blue") +
  # geom_point(aes(x = m_height, y = height), ) +
  geom_point(aes(x = m_height, y = y_pred), size = 3, alpha = .4, color = "red") +
  # The path draws the curve
  geom_path(data = curves, aes(group = grp))  +
  geom_polygon(data = curves, aes(y = scales::oob_squish(y, c(0, Inf)),group = grp)) +
  geom_abline(slope = m$coefficients["m_height"],
            intercept =m$coefficients["(Intercept)"],
            color = "red",
            alpha = .6) +
  geom_segment(aes(xend = m_height, yend = y_pred), color = "red", linetype = "dotted") +
  labs(x = "Mother's height",
       y = "Child's height") +
  theme_minimal()

Conducting a likelihood ratio test

Now that we know what log likelihood is, and (more or less) how to calculate it, we can finally apply this knowledge to our first model comparison technique: the likelihood ratio test, or LRT.

The basic intuition of a likelihood ratio test here is as follows: given a choice between two models, \(M_1\) and \(M_2\), we should prefer the model that best accounts for some set of data \(D\). In particular, a LRT is typically used to select between two nested models, i.e., where one model (\(M_{k-1}\)) is considered a "special case" of a larger model (\(M_k\)); here, \(k\) refers to the number of parameters. I'll be focusing on the comparison of nested models specifically, though see Lewis (2011) for a discussion of how to compare two non-nested models with a likelihood ratio test.

Formal description and walkthrough

Intuitively, the likelihood ratio is a fraction expressing how much more likely the data are under one model vs. another model. Let's first define this ratio term:

\(\lambda = \frac{\mathcal{L}(M_{k-1})}{\mathcal{L}(M_{k})}\)

Because \(M_{k-1}\) is nested within \(M_{k}\) (i.e., \(M_{K}\) has all the parameters that \(M_{k-1}\) has, plus at least one), its likelihood should never be any larger than \(M_{k}\). That means that the ratio \(\lambda\) should be bound between \((0, 1)\).

What would various values of \(\lambda\) tell us about the relative likelihoods about the models? Conceptually, larger values of \(\lambda\) (closer to \(1\)) tells us that \(\mathcal{L}(M_{k-1})\) isn't much less likely than \(\mathcal{L}(M_{k})\); in other words, the "full" model isn't much better than the reduced, "special case" model⁸. If \(\lambda\) is very closer to \(0\), it suggests that the full model is a big improvement over the reduced model (i.e., the data are many times many "likely" under this full model).

To calculate the full likelihood ratio (let's call this \(\Delta\)), there's just a few more steps:

\(\Delta = -2 *log(\frac{\mathcal{L}(M_{k-1})}{\mathcal{L}(M_{k})})\)

Let's unpack this formula a bit. Above, we've already seen that a smaller ratio between the likelihoods indicates a greater improvement of the full model over the reduced, "special case" model. Correspondingly, the log of this ratio should be larger (i.e., more negative) when the full model represents a larger improvement.

We can visualize that below. First, let's set up a vector of possible values of \(\lambda\), bounded between \((0, 1)\).

lambdas = seq(.01, 1, by=.001)

We then take the log of these \(\lambda\) values:

log_lambdas = log(lambdas)

And finally, we calculate the values of \(\Delta\):

ll_ratio = -2 * log_lambdas
df_loglik_example = data.frame(lambdas = lambdas,
                               log_lambdas = log_lambdas,
                               lr = ll_ratio)

As you'd expect, larger lambdas (closer to \(1\)) yield values of \(log(\lambda)\) closer to \(0\) (a smaller absolute value).

df_loglik_example %>%
  ggplot(aes(x = lambdas,
             y = log_lambdas)) +
  geom_point() +
  labs(x = "Lambda",
       y = "Log(lambda)") +
  theme_minimal()

Ultimately, larger lambdas also correspond to smaller likelihood ratios \(\Delta\).

df_loglik_example %>%
  ggplot(aes(x = lambdas,
             y = lr)) +
  geom_point() +
  labs(x = "Lambda",
       y = "-2 * Log(lambda)") +
  theme_minimal()

So there we have it: The better a full model is than a smaller, "special case" model, the larger the likelihood ratio between those models will be.

Quick note: recall that during Maximum Likelihood Estimation (MLE), we typically calculate the log likelihood, rather than the raw likelihood. This means we can express the above formula in the following way:

\(\Delta = -2 * (log(\mathcal{L}(M_{k-1})) - log(\mathcal{L}(M_{k})))\)

It's the same formula, but the log likelihood values are the values we typically optimize over (or technically, the negative log likelihood).

Hypothesis testing

So what do we do with this LRT? Many scientists use the paradigm of Null Hypothesis Significance Testing (NHST). With more traditional tests, this involves comparing some critical test statistic (e.g., an F-ratio) to a sampling distribution representing the null hypothesis. If that test statistic is sufficiently unlikely to occur under that sampling distribution (typically \(p < .05\)), we reject the null hypothesis. Put another way: if the data are not very likely under the null hypothesis, we reject the null as the best explanation of the data.

There are a number of critiques of NHST and, of course, its reliance on p-values (Wagenmakers, 2007, McShane et al, 2019), but this is not a tutorial about the relative merits of NHST vs. Bayesian methods. Given that you're reading this section, you probably want to know: how do I use a likelihood ratio test? That is, how do I tell whether the full model is significantly better than the reduced model?

It turns out to be pretty similar to evaluating the significance of an F-ratio or t-statistic. In fact, Wilks' theorem states: As the sample increases, the distribution of the test statistic \(\delta\) approaches the chi-squared distribution parameterized by the difference in degrees of freedom between the models.

The chi-squared (\(\chi^2\)) distribution is well-understood, and is of course used to calculate the p-value for chi-squared tests. That means that all we have to do is look up our log-likelihood ratio (\(\delta\)) in the appropriate chi-squared distribution. Here, "appropriate" means the distribution with the correct degrees of freedom---and that corresponds, essentially, to the difference in number of parameters between the full model and the reduced model.

To help illustrate this, let's consider several different chi-squared distributions with different degrees of freedom:

df_chi = data.frame(
  "df 1" = rchisq(1000, 1),
  "df 5" = rchisq(1000, 5),
  "df 8" = rchisq(1000, 8)
) %>%
  pivot_longer(cols = c("df.1", "df.5", "df.8"),
               names_to = "df",
               values_to = "chi")

df_chi %>%
  ggplot(aes(x = chi,
             fill = df)) +
  geom_density(alpha = .4) +
  geom_vline(xintercept = 4, linetype = "dotted") +
  labs(title = "Null chi-squared distributions for varying DF",
       x = "Chi-squared statistic") +
  theme_minimal()

Hopefully, this plot helps illustrate an important point: the degrees of freedom (df) is a really important parameter for determining the shape of the null distribution! Distributions with fewer degrees of freedom (like \(df = 1\)) tend to have relatively smaller chi-squared values than distributions with more degrees of freedom (like \(df = 8\)). Since NHST works by looking up a given test statistic in a null distribution, this means that the same test statistic will be differentially likely (i.e., have a different p-value) under different distributions.

For example, the dotted line is located at \(\chi^2 = 4\). This is relatively more likely when \(df = 1\) (p = 0.046) than when \(df = 8\) (p = 0.857).

Just to illustrate this further, let's calculate the p-value for each of the log-likelihood ratios we simulated earlier, assuming a \(df = 1\):

# Obtain p-value by looking up actual value in a chi-squared distribution, where DF = 1
df_loglik_example$p1 = 1 - pchisq(df_loglik_example$lr, 1)

df_loglik_example %>%
  ggplot(aes(x = lr,
             y = p1)) + 
  geom_point() +
  labs(title = "P-values of different log-likelihood ratios (DF = 1)",
       x = "Log likelihood ratio (LR)",
       y = "p-value (assuming DF=1)") +
  geom_hline(yintercept = .05, linetype = "dotted") +
  theme_minimal()

We see that as the log-likelihood ratio increases, the p-value decreases. In other words, the probability of obtaining a log likelihood ratio that large is less and less likely under the null distribution.

The same would be true with a larger value for \(df\) (e.g., \(df = 3\)), but our critical threshold would be higher:

# Obtain p-value by looking up actual value in a chi-squared distribution, where DF = 3
df_loglik_example$p3 = 1 - pchisq(df_loglik_example$lr, 3)

df_loglik_example %>%
  ggplot(aes(x = lr,
             y = p3)) + 
  geom_point() +
  labs(title = "P-values of different log-likelihood ratios (DF = 3)",
       x = "Log likelihood ratio (LR)",
       y = "p-value (assuming DF = 3)") +
  geom_hline(yintercept = .05, linetype = "dotted") +
  theme_minimal()

Example 1: Simple linear regression

Remember that distribution of heights we simulated earlier?

df_height %>%
  ggplot(aes(x = m_height,
             y = height)) +
  geom_point() +
  labs(x = "Mother's height",
       y = "Child's height") +
  theme_minimal()

Recall that we built a model predicting child's height from mother's height (let's just rebuild this here to refresh our memory):

m = lm(data = df_height,
       height ~ m_height)

We also calculated the log-likelihood of this model:

logLik(m)[1]

## [1] -329.4188

Now let's say we want to compare this model to some reduced model. For example, what if we just included a freely varying intercept and no other parameters? Let's build that model below, and calculate its log-likelihood.

m2 = lm(data = df_height,
        height ~ 1)

logLik(m2)[1]

## [1] -360.9135

As expected, the log likelihood of the full model is lower than that of the reduced model. But how much lower? Let's calculate the log likelihood ratio:

\(\Delta = -2 * (log(\mathcal{L}(M_{k-1})) - log(\mathcal{L}(M_{k})))\)

delta = -2 * (logLik(m2)[1] - logLik(m)[1])
delta

## [1] 62.98936

Recall that our models also differ in one degree of freedom. The full model contains a freely varying intercept, and a freely varying slope coefficient for the mother's height variable; the reduced model only contains only a freely varying intercept.

So we can calculate the p-value for this \(\delta\) value by looking it up in a chi-squared distribution with the appropriate degrees of freedom (\(df = 1\)):

pchisq(delta, 1, lower.tail = FALSE)

## [1] 2.078258e-15

That's a very low p-value! This means that we're very likely to have achieved a value this large under the null distribution. We can double-check our work using the lrtest function:

lrtest(m, m2)

## Likelihood ratio test
## 
## Model 1: height ~ m_height
## Model 2: height ~ 1
##   #Df  LogLik Df  Chisq Pr(>Chisq)    
## 1   3 -329.42                         
## 2   2 -360.91 -1 62.989  2.078e-15 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Example 2: Linear mixed effects models

Log likelihood ratio tests are, as far as I know, the standard approach to evaluating mixed (i.e., multilevel) models. Let's run through a really quick example; for more details, see my mixed models in R tutorial for more details, which has more information.

The sleepstudy dataset has information about the average reaction time for participants given different degrees of sleep deprivation. There are multiple responses per subject, which means we probably want to add at least a random intercept for each subject---and possibly a random slope for the effect of Days of sleep deprivation on RT.

Let's build our two models below:

model_full = lmer(data = sleepstudy,
                  Reaction ~ Days + (1 + Days | Subject),
                  REML = FALSE)

model_reduced = lmer(data = sleepstudy,
                  Reaction ~ (1 + Days | Subject),
                  REML = FALSE)

Note that the full model contains a fixed effect of Days, as well as by-subject random slopes for the effect of Days, and random intercepts for Subject. The reduced model omits only the fixed effect of Days, meaning the two models differ only in one parameter.

We can use the handy anova function to conduct a log likelihood ratio test:

anova(model_full, model_reduced)

## Data: sleepstudy
## Models:
## model_reduced: Reaction ~ (1 + Days | Subject)
## model_full: Reaction ~ Days + (1 + Days | Subject)
##               npar    AIC    BIC  logLik deviance  Chisq Df Pr(>Chisq)    
## model_reduced    5 1785.5 1801.4 -887.74   1775.5                         
## model_full       6 1763.9 1783.1 -875.97   1751.9 23.537  1  1.226e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

If you want to report such a result, I'd go with something like: "A fixed effect of Days significantly improved model fit [\(\chi^2(1) = 23.53, p < .001\)] above and beyond a model omitting this effect."

Akaike Information Criterion (AIC)

Another common model evaluation (and comparison) approach is the Akaike Information Criterion, or AIC (Akaike, 1974). The formula is as follows:

\(AIC = 2k - 2ln(L)\)

Where \(k\) is the number of parameters, and \(\mathcal{L}\) is the likelihood of the model. Don't worry if this doesn't seem intuitive yet---the sections below will explore the philosophy and implications of this formula.

Philosophy of AIC

Disclaimer: I'm by no means an expert in information theory and certainly not these information-theoretic methods (or their relationship to likelihood ratio tests). Anything you see below was adapted from papers I read to understand the topic: Burnham & Anderson (2002), Burnham & Anderson (2004), and Lewis et al (2011).

Furthermore, this background may not be strictly necessary for using AIC. But if you only want to read one thing, I recommend this 2002 paper, which is a high-level overview of common pitfalls people make when calculating AIC:

Anderson, D. R., & Burnham, K. P. (2002). Avoiding pitfalls when using information-theoretic methods. The Journal of Wildlife Management, 912-918.

NHST vs. strength of evidence

One important difference between the log-likelihood ratio test (at least in terms of its NHST manifestation) and AIC lies in their underlying philosophies. As Burnham and Anderson (2002) note, AIC and other information-theoretic approaches are not "tests"; although one can rank models according to their AIC, there is no assumption that the relative differences or ratios in their AIC values is associated with some known distribution.

Thus, unlike a log-likelihood ratio test, there is no focus on rejecting a "null hypothesis"---rather, the idea is to compare among multiple, plausible models of the data, and select either the model (or models) on the basis of strength of evidence. In this sense, AIC (and BIC) are not inherently tied to some decision rule, whereas NHST is explicitly about decision thresholds---though standard statistical tests, such as t-tests, can in fact be replicated using AIC values.

KL-Divergence

That said, there is a deep relationship between the log-likelihood ratio and AIC.

AIC (Akaike, 1974) is rooted in the notion of Kullback-Leibler Divergence, or KL-D. KL-D (or KL-information) is a measure of the degree to which one probability distribution differs from one another---it is formally operationalized as the difference in number of "bits" required for encoding some probability distribution \(f\) using some model/theory \(g\) as opposed to using \(f\). Put another way, KL information (or \(I(f, g)\)) is the information lost when some model \(g\) is used to approximate \(f\), or "true reality" (Burnham & Anderson, 2004).

One might intuit why such a measure would be useful for evaluating models: ideally, we want the model that loses the least amount of information about "reality" relative to other models under consideration. But we obviously don't have access to "true reality"; so as Burnham et al (2011) point out, rather than calculating the divergence between each model in a set of candidate models, we can calculate the divergence between each model and the best model in that set.

KL-information is expressed as follows:

\(I(f, g) =\int f(x)log(f(x))dx - \int f(x)log(g(x|\theta))dx\)

As Burnham & Anderson (2004) write, this can also be expressed as:

\(I(f, g) = E_{f}[log(f(x))] - E_{f}[log(g(x|\theta))]\)

Where the expectation \(E\) is taken with respect to \(f(x)\), "the truth". Critically, the value \(E_{f}[log(f(x))]\) is constant across all the different models we might be comparing, since it depends solely on the truth---i.e., it doesn't change, and can be represented as some constant \(C\). This means that to compare models, we really only need to calculate \(E_{f}[log(g(x|\theta))]\).

Akaike's (1974) insight was that this term \(E_{f}[log(g(x|\theta))]\) could be estimated---albeit with some bias---using the maximum likelihood value. And it turns out that this bias is approximately equal to the number of parameters (\(K\)) in the model.

This last bit is explored further below, but the point of this section is to drive home the connection between the underlying philosophy of AIC and model likelihood. Akaike's goal was to find some "metric" that would capture the degree of information lost across different models about the data. His starting point was KL-Divergence, but he ended up converging on model likelihood as a biased approximation of this relative information loss.

Walking through the formula

Where does this formula come from? To start, recall that Akaike demonstrated the following equivalence:

\(\hat{E}(KL) = log(\mathcal{L}) - K\)

This states that the relative information loss is equal to the log likelihood of a model, minus the number of parameters (\(K\)). According to Burnham & Anderson (2004), Akaike then multiplied this result by \(-2\) "for historical reasons", giving us the final value:

\(AIC = -2*log(L) + 2k = 2k - 2*log(\mathcal{L})\)

Using this definition, a lower AIC is "better". Recall that smaller negative log-likelihood values are better, and the formula is essentially calculating the negative log-likelihood (\(-2 * log(L)\)). To get a better sense of how exactly AIC, likelihood, and number of parameters relate, let's simulate some "models" with varying likelihoods and number of parameters.

## Possible likelihood values
l = seq(.1, .99, length.out = 50)
## Possible number of parameters
k = seq(1, 100, length.out = 25)

## Expand into a "grid", with each row corresponding to a particular model.
df_models = expand.grid(likelihood = l, 
                        parameters = k)

## Formula to calculate AIC
aic_og = function(l, k) {
  aic = 2*k - 2*log(l)
  aic
}

## Calculate AIC
df_models = df_models%>%
  mutate(aic = aic_og(likelihood, parameters))

Okay, now let's plot \(AIC\) by \(Likelihood\), and color each dot according to the number of \(parameters\).

df_models %>%
  ggplot(aes(x = likelihood,
             y = aic,
             color = parameters)) +
  geom_point(alpha = .8) +
  theme_minimal()

What's the takeaway from this graph?

First, models with larger likelihood values have lower AICs in general. In other words, the more likely the data are under a particular model, the lower that model's AIC value. This drives home the point from above: a lower AIC corresponds to a "better" model.

Second, models with more parameters (the lighter dots) have larger AIC values than models with fewer parameters (the darker dots), on average. This is because of the bias correction term, \(2K\). Intuitively, adding more parameters to a model will increase the size of this term, yielding a larger AIC. So two models with the same likelihood but different numbers of parameters will have different AIC values.

This latter point can also be illustrated more directly by just plotting \(AIC\) by \(parameters\):

df_models %>%
  ggplot(aes(x = parameters,
             y = aic)) +
  geom_point(alpha = .5) +
  theme_minimal()

Correction for small samples

Note that if the number of parameters (\(K\)) is quite large relative to the sample size (\(n\)), there is a "small-sample" version of AIC called \(AIC_c\). This looks essentially like the formula above, but adds another term:

\(AIC_c = AIC + \frac{2K(K + 1)}{n - K - 1}\)

What exactly does this additional "correction" term do? The basic idea is that it adds a correction that's scaled to the relative difference btween the number of parameters and the number of observations.

If \(n\) is much larger than \(K\), then the additional term will result in a smaller ratio---thus adding less to the ultimate AIC calculation; as \(n \to \infty\), this term approaches \(0\). Conversely, as \(K \to n\), this correction term gets larger and larger.

Just to illustrate exactly how this works, I'll plot out various values of \(n\), \(K\), and the resulting correction.

n = seq(100, 1000, length.out = 10)
k = seq(1, 10, length.out = 10)

df_example = expand.grid(n=n, k=k)
df_example = df_example %>%
  mutate(correction = (2 * k**2 + 2*k) / (n - k - 1))

df_example %>%
  ggplot(aes(x = n,
             y = k,
             size = correction)) +
  geom_point() +
  theme_minimal()

This plot illustrates a couple key insights:

Larger values of \(n\) correspond to smaller "corrections".
Larger values of \(K\) correspond to larger corrections.

Thus, the largest corrections are in the upper left (small \(n\), large \(K\)), and the smallest corrections are in the bottom right (big \(n\), small \(K\)).

Important Note: If the model is too "saturated" (i.e., if \(K = n - 1\), or even \(K = n\)), this correction will yield nonsensical results. For example, in the case where \(K = n - 1\), the correction term involves division by \(0\); in the case where \(K = n\), the denominator ends up negative, which would appear to favor models with the same number of parameters as data points. This is obviously wrong, as the case where \(K = n\) is a prime example of overfitting; any model can achieve perfect fit if there are as many parameters as data points.

Okay, now let's see how this additional penalty impacts AIC. We've already seen that models with larger likelihoods have smaller AIC values (holding \(K\) constant), and also that models with more parameters have larger AIC values (holding likelihood constant). How does accounting for our sample size (\(n\)) impact our value for \(AIC_c\)? And how does this compare to the more "vanilla" AIC calculation?

First, let's simulate some model parameters.

## Possible likelihood values
l = seq(.1, .99, length.out = 20)
## Sample sizes
n = seq(20, 400, length.out = 20)
## Number of parameters
k = seq(1, 10, length.out = 20)

## Extended AIC
aic_c = function(l, k, n) {
  aic = 2*k - 2*log(l) + (2*k * (k + 1)) / (n - k - 1)
  aic
}

## Create grid of models
df_models = expand.grid(n=n,
                        k=k,
                        likelihood = l)
## Calculate AIC
df_models = df_models %>%
  mutate(aic1 = aic_og(likelihood, k),
         aic2 = aic_c(likelihood, k, n))

Now let's compare the "vanilla" \(AIC\) to \(AIC_c\):

df_models %>%
  ggplot(aes(x = aic1,
             y = aic2,
             color = k)) +
  geom_abline(aes(slope = 1, intercept = 0), 
              linetype = "dotted") +
  geom_point(alpha = .5) +
  labs(x = "AIC",
       y = "AIC (corrected)") +
  theme_bw() +
  scale_x_continuous(limits = c(0, max(df_models$aic2))) +
  scale_y_continuous(limits = c(0, max(df_models$aic2))) +
  facet_wrap(~n)

The dotted line represents \(Y = X\). So points that lie along that dotted line have roughly the same value for \(AIC_c\) as \(AIC\). In contrast, points that lie above the line have larger values for \(AIC_c\)---this indicates that they received relatively harsher penalties in the correction term.

Looking closely at the graph, we see that the largest deviations from the line of parity are for the smallest sample sizes (e.g., \(n = 20\)). This makes sense, given what we described earlier: as \(K \to n\), the value \(\frac{2K(K + 1)}{n - K - 1}\) gets larger and larger, and thus \(AIC_c\) gets a larger penalty.

When should you use corrected AIC?

Burnham and Anderson (2004) suggest that you should use the corrected term \(AIC_c\) unless \(\frac{n}{K} > 40\), i.e., you have roughly 40x as many data points as parameters. So if you have 60 observations and 20 model parameters, you'll probably want to go with \(AIC_c\). Whereas if you have more than, say, 1000 observations---and still just have 20 parameters at the most---then you might be okay using \(AIC\).

But as we've seen above, the penalty term ends up approaching \(0\) as \(n \to \infty\), which from my perspective, suggests that using \(AIC_c\) is usually a safe bet!

Comparing AIC values

Ultimately, we want to evaluate the strength of evidence for each model in our set of candidates \(M = {m_1, m_2, ..., m_n}\), allowing us to select the model that best explains some dataset \(D\). How do we do this?

We've already established that a lower AIC reflects better model fit. And unlike with log-likelihood ratio tests, the calculation for AIC explicitly incorporates the number of parameters, so we don't have to worry about adding a further penalty for each models' respective degrees of freedom. From one perspective, this makes model comparison quite straightforward: we can just rank each model from best (lowest AIC) to worst (highest AIC). The "best" model is the one that minimizes K-L information loss (Burnham et al, 2011).

AIC Delta

Note, however, that the absolute value of AIC doesn't really matter, as it'll be a function of the model likelihood, which of course depends on properties of the data and the model itself. What really matters if the relative value of AIC across our models. Thus, to better evaluate models, people sometimes rescale \(AIC\) according to the minimum AIC value, \(AIC_{min}\):

\(\Delta_i = AIC_i - AIC_{min}\)

This means that the best model will have its \(\Delta = 0\), while the other models will all have positive values. Each successive value of \(\Delta_i\) will reflect how much "worse" (in units of \(AIC\)) a given model is from the very best model. As noted earlier, the philosophical motivation for this is rooted in the notion of Kullback-Leibler information. Ideally, we want to minimize the information loss between reality and our model. Since we don't have access to reality, we instead compare each model to the best model; see Figure 1 in Burnham et al (2011) for a really helpful illustration of this.

So for example, if our AIC values were as follows:

models = seq(1000, 1010, by = 2)
models

## [1] 1000 1002 1004 1006 1008 1010

We would rescale them in the following way:

models_rescaled = models - min(models)
models_rescaled

## [1]  0  2  4  6  8 10

Relative likelihood

This means that we can quantify the relative likelihood of each model in relation to the best model (Burnham et al, 2011). Note, here, that "likelihood" is now being used to refer to a different quantity than what we called likelihood earlier. That is, we now want to calculate \(\ell = \mathcal{L}(m_i | D)\), i.e., the likelihood of a model, given the data; earlier, we calculated \(\mathcal{L}(D | m_i)\), the probability of the data, given the model. We can calculate the relative likelihood by simply exponentiating the \(\Delta_i\) value:

\(\ell_i = \mathcal{L}(m_i | D) = exp(-\frac{1}{2}\Delta_i)\)

By comparing two values \(\ell_i, \ell_j\), we can quantify the relative likelihood of model \(m_i\) and model \(m_j\) (i.e., "Model 3 is 20 times more likely than model 4").

Let's calculcate \(\ell_i\) for each of our rescaled models:

L = exp(-(1/2) * models_rescaled)
L

## [1] 1.000000000 0.367879441 0.135335283 0.049787068 0.018315639 0.006737947

This also allows us to calculate the relative probability of each model, given the data and our set of candidate models \(M\), by normalizing each likelihood value:

\(p(m_i | D) = \frac{\ell_i}{\sum\limits_{j=1}^M\ell_j}\)

Where \(M\) is our total set of candidate models. Below, I calculate \(p(m_i | D)\) for each of the likelihood values simulated above.

p = L / sum(L)
p

## [1] 0.633691323 0.233122010 0.085760795 0.031549633 0.011606461 0.004269779

These probabilities are sometimes called Akaike weights, for reasons described in the "Multimodel Inference" section below.

Finally, using either the relative likelihoods or the relative probabilities of each model, we can calculate something called the evidence ratio:

L[2] / L[3]

## [1] 2.718282

p[2] / p[3]

## [1] 2.718282

This is sometimes interpreted as meaning the "empirical support" for one model is \(X\) times stronger than another model, i.e., the empirical support for model \(m_2\) is 2.72 stronger than model \(m_3\).

In addition to comparing each model to each other, one can compare each model to the best model. The resulting evidence ratio reveals the degree to which the evidence is stronger for the best model relative to the model in question:

ev = L[1] / L
ev

## [1]   1.000000   2.718282   7.389056  20.085537  54.598150 148.413159

Final notes on AIC

Hopefully this brief discussion of AIC helps illuminate the theoretical motivation, as well as what the different parameters (i.e., \(\mathcal{L}\), \(n\), and \(K\)) mean in terms of the ultimate value. As I noted earlier, I'm not an expert on information-theoretic methods for model comparison---much of what I know, I had to learn for this tutorial! The following papers were invaluable:

Anderson, D. R., & Burnham, K. P. (2002). Avoiding pitfalls when using information-theoretic methods. The Journal of Wildlife Management, 912-918.
Burnham, K. P., & Anderson, D. R. (2004). Multimodel inference: understanding AIC and BIC in model selection. Sociological methods & research, 33(2), 261-304.
Burnham, K. P., Anderson, D. R., & Huyvaert, K. P. (2011). AIC model selection and multimodel inference in behavioral ecology: some background, observations, and comparisons. Behavioral ecology and sociobiology, 65(1), 23-35.

Common pitfalls

Burnham and Anderson (2002) list several common pitfalls people encounter when using information-theoretic methods, including AIC. I'll summarize just a few below:

Information-theoretic models are not a "test". I've described this above and below, but neither \(AIC\) nor any of the calculations derived with it (e.g., the evidence ratio) have a theoretical justification for anything like a "significance test".

Use of AIC instead of corrected AIC. If the sample size is small relative to the number of parameters, one should use \(AIC_c\) rather than \(AIC\)! See the simulation above for an illustration of how incorporating \(n\) into the calculation penalizes \(AIC_c\) (relative to "vanilla" \(AIC\)).

Incorrect estimation of number of parameters. A common mistake with approaches like linear regression is to forget that the residuals are considered a parameter. So a linear model with an intercept \(\beta_0\), a slope \(\beta_1\), and some residual variance \(\epsilon\) has \(3\) parameters.

Comparing AIC across datasets. This might go without saysing, but you can't compare information criteria across different datasets! Burnham and Anderson (2002) write (pg. 917):

The data must be considered fixed, and then models in the set can be compared and ranked.

This is especially important if some of your predictors contain missing data, such that different models will actually be fit to different subsets of the dataset. Not all software packages alert you when certain rows of data are ignored or omitted from a model, so this is something to be aware of.

Check out the paper for more advice!

Relationship to likelihood ratio tests

The most obvious connection between AIC and likelihood ratio tests (LRT) is that AIC incorporates likelihood into its formula; thus, AIC clearly depends on likelihood.

Further, both LRTs and AIC account in some way for the number of parameters in a model. AIC does this explicitly, using the number of parameters \(K\) for the final calculation. But LRTs also do this, in a way: as we saw earlier, the significance test for an LRT involves looking a given \(\chi^2\) value in the chi-squared distribution with the appropriate degrees of freedom (\(df\)), where \(df\) is defined as the difference in number of parameters between the full model and the reduced model. Thus, the same likelihood ratio will "mean" something different depending on the difference in number of parameters across the models.

As Lewis et al (2011) point out, the connection is deeper than just a conceptual analogy. The difference in AIC between models \(m_i\) and \(m_j\) is equivalent to the likelihood ratio between those models, plus a term correcting for the difference in number of parameters:

\(AIC_i - AIC_j = Q + 2(K_i - K_j)\)

Where \(Q\) is the likelihood ratio between \(m_i\) and \(m_j\), and \(K_i\) and \(K_j\) are the number of the parameters in each model, respectively⁹.

Of course, unlike LRTs, AIC is (as far as I know) not typically used in an NHST framework. For people trained under an NHST paradigm (like me), this sometimes feels frustrating---we crave binary decision rules ("Is it significant or not? Do I reject the null hypothesis or not?")---but I do think there's value in trying to become more comfortable with continuity. The world is fuzzy and uncertain, so perhaps it's good to acquaint ourselves with thinking about the relative strength of evidence for different models, rather than approaching theory construction from the perspective of which models are "correct" vs. "incorrect".

Bayesian Information Criterion (AIC)

Another well-known information-theoretic criterion for model selection is Bayesian information criterion, or BIC. I'm not going to go into great detail about BIC here, but it's defined as follows:

\(BIC = -2*log(\mathcal{L}) + K*log(n)\)

Where \(\mathcal{L}\) is the likelihood of the data given the model, \(K\) is the number of parameters in the model, and \(n\) is the sample size.

BIC vs. AIC

This formula looks pretty similar to the formula for \(AIC\)! Both formulas incorporate the term \(-2 * log(\mathcal{L})\), and both also incorporate the number of parameters \(K\). Like \(AIC\), smaller values of \(BIC\) reflect better model fit. The key difference is that unlike "vanilla" \(AIC\), \(BIC\) incorporates information about the sample size (\(n\)). Of course, we saw that corrected AIC, i.e., \(AIC_c\), does account for sample size.

Is there a theoretical reason to prefer one over the other? Burnham and Anderson (2004) suggest that many Bayesian-oriented people prefer BIC because it is a Bayesian procedure; as they point out, however, model selection rooted in AIC can also be Bayesian (pg. 283-284):

The difference is in the prior distribution placed on the model set. Hence, for a Bayesian procedure, the argument about BIC versus AIC must reduce to one about priors on the models.

Personally, I have a hard time getting intuition about different formulas until I actually see what values they produce. Before, we compared vanilla \(AIC\) with \(AIC_c\). Here, let's compare \(BIC\) to both.

## Possible likelihood values
l = seq(.1, .99, length.out = 20)
## Sample sizes
n = seq(20, 400, length.out = 20)
## Number of parameters
k = seq(1, 10, length.out = 20)

## Extended AIC
bic = function(l, k, n) {
  bic = -2*log(l) + k *log(n)
  bic
}

## Create grid of models
df_models = expand.grid(n=n,
                        k=k,
                        likelihood = l)
## Calculate AIC
df_models = df_models %>%
  mutate(aic1 = aic_og(likelihood, k),
         aic2 = aic_c(likelihood, k, n),
         bic = bic(l, k, n))

Let's see how these values compare:

df_models %>%
  ggplot(aes(x = aic1,
             y = bic,
             color = k)) +
  geom_abline(aes(slope = 1, intercept = 0), 
              linetype = "dotted") +
  geom_point(alpha = .5) +
  labs(x = "AIC (vanilla)",
       y = "BIC") +
  theme_bw() +
  scale_x_continuous(limits = c(0, max(df_models$bic))) +
  scale_y_continuous(limits = c(0, max(df_models$bic))) +
  facet_wrap(~n)

df_models %>%
  ggplot(aes(x = aic2,
             y = bic,
             color = k)) +
  geom_abline(aes(slope = 1, intercept = 0), 
              linetype = "dotted") +
  geom_point(alpha = .5) +
  labs(x = "AIC (corrected)",
       y = "BIC") +
  theme_bw() +
  scale_x_continuous(limits = c(0, max(df_models$bic))) +
  scale_y_continuous(limits = c(0, max(df_models$bic))) +
  facet_wrap(~n)

We see that relative to vanilla \(AIC\), \(BIC\) is larger for both larger values for \(n\) and \(K\); this makes sense, because vanilla \(AIC\) doesn't penalize small sample sizes at all. And the \(K*log(n)\) multiplier means that models with more parameters (larger values of \(K\)) will be penalized more in \(BIC\) than \(AIC\) (unless \(log(n) < 2\)).

\(BIC\) is also larger than \(AIC_c\) for most sample sizes, except for very small samples (like \(n = 20\)).

Multimodel inference

So far, we've discussed various approaches for selecting the best model from a set of candidate models \(M\), with the ultimate goal of using \(m_i\) to perform inference. Burnham & Anderson (2004) describe the "four steps" of model selection as follows:

We want to perform model-based inference from data.
There are \(M\) possible models, but we don't know which one should be used.
We select among these "competing" models using the data at hand.
Using the "best" model, we draw inferences from it "as if it were a priori the only model fit to the data" (pg. 298).

Burnham & Anderson (2004) criticize the logic of steps (3) and (4). They argue that there's no philosophical justification for performing inference from a single model, as if it was the only model under consideration. Of course, there is (by definition) always one model that is "best", in terms of having the lowest AIC or BIC value---or, in the case of an NHST framework, a model with significantly better fit to the data than some reduced model. But that doesn't mean we should ignore all the other models under consideration, especially if the second-best or third-best models come close to the very best model. Often, these second-best or third-best models even capture variance that's not explained by the best model (at least in the case of non-nested models). Why should we throw that explanatory power away?

The alternative to model selection is called multimodel inference, and the basic idea is that inference should be based on all the models in the a priori set, not just the best model. What exactly does this mean?

Averaging model predictions

One approach to multimodel inference is called model averaging, and is best-suited to evaluating the predictions of our models¹⁰. Specifically, we can derive a set of predictions \(\hat{y}_i\) for each model \(m_i\) in our candidate set \(M\). We then average those predictions together, weighting the predictions across the models according to the probability of the model (see the section on \(AIC\) for details on calculating model probability):

\(\bar{Y} = \sum\limits_{i=1}^Mp_i\hat{y_i}\)

This makes intuitive sense. We want to account for the predictions of all our models, but we also want to account for the fact that some models are more likely (given the data) than others. Altogether, predictions derived from weighted averages across our models (as opposed to a single model) yield a "more honest measure of precision and reduced bias" (Burnham and Anderson, 2004, pg. 274).

Let's illustrate this with a quick example. Imagine we have three models, \(M = [m_1, m_2, m_3]\), which have the following (rescaled) \(AIC\) values:

aic_rescaled = c(0, 2, 4)

We can calculate the likelihood \(\ell_i\) of each model given the data, and then calculate thir normalized probabilities \(p_i\) (also known as Akaike weights):

L = exp(-(1/2) * aic_rescaled)
L

## [1] 1.0000000 0.3678794 0.1353353

p = L / sum(L)
p

## [1] 0.66524096 0.24472847 0.09003057

Now let's suppose that each model has a set of predictions \(\hat{Y}_i\). Here, we won't worry about what the "real" data look like, as the point is simply to illustrate how to generate weighted predictions. To make this simpler, let's suppose that each model is just producing a normal distribution:

y1 = rnorm(mean = 70, sd = 3, n = 50)
y2 = rnorm(mean = 68, sd = 5, n = 50)
y3 = rnorm(mean = 73, sd = 8, n = 50)

While we could just average these predictions in an unweighted way, we know that these models have pretty different fits. Thus, we should be more confident about predictions from \(m_1\), which has a normalized probability of 0.67, than \(m_3\), which has a normalized probability of 0.09.

Below, I'll compute both the unweighted and weighted average predictions:

y_unweighted = (y1 + y2 + y3)/3
y_weighted = y1 * p[1] + y2 * p[2] + y3 * p[3]

Now let's visuaslize all these predictions next to each other. We see that the predictions from our best model (\(m_1\)) are closest in shape to the weighted average (and also to the unweighted average), which makes sense because the predictions from that model were weighted most heavily. Feel free to play around with the parameters of each model above, as well as the Akaike weights.

df_all_preds = data.frame(
  y1 = y1,
  y2 = y2, 
  y3 = y3,
  y_weighted = y_weighted,
  y_unweighted = y_unweighted
) %>%
  pivot_longer(cols = c(y1, y2, y3, y_weighted, y_unweighted),
               names_to = "model", values_to = "prediction")

df_all_preds %>%
  ggplot(aes(x = prediction,
             y = model,
             fill = model)) +
  geom_density_ridges2(aes(height = ..density..), 
                       color=gray(0.25), alpha = 0.5, 
                       scale=0.85, size=0.75, stat="density") +
  theme_minimal()

Averaging model parameters

If some parameter \(\theta\) is shared across all models, we can take the weighted average of estimates for that parameter value across those models (just as we did for predictions):

\(\bar{\theta} = \sum\limits_{i=1}^Mp_i\hat{\theta_i}\)

The logic here is the same as for averaging model predictions: we can obtain a better estimate of the true population parameter if we average across many estimates of that parameter.

Obviously, if you were forced to choose only one parameter estimate, you'd probably want to choose the one from the better model (i.e., the one that fits the data better). But if you've already constructed multiple candidate models, and if each model is theoretically justified in its own way, and especially if the spread of AIC values is not that wide (i.e., the best model isn't that much worse than the worst model), you'll often have better luck using the average parameter estimate.

As I understand this (and I could be wrong!), it's similar to the Central Limit Theorem: taking the mean of many sample means gets you closer to the population mean than relying on any given sample mean alone. If you had many different datasets, you might similarly imagine fitting the same model \(m_i\) to each dataset and taking the average estimate of some parameter \(\theta\). The situation here is the reverse---we fit many different models to the same dataset, and take the average estimate of \(\theta\) across those models. And again, because it's a weighted average, it will reflect how well each model fits the data.

Many models vs. one "big" model

One question that arises is what the "point" of multimodel inference is, at least when it comes to estimating a given parameter. Rather than build many small models with various predictors, why not simply construct one "big", global model with all the factors you could possibly be interested in?

Burnham et al (2011, pg. 30) argue that datasets are limited in information. Each time another new parameter is added and estimated, the information left in the dataset is reduced---and the probability of finding spurious effects is increased. You can certainly estimate new parameters, but with increasing uncertainty. Thus, they recommend building a number of theoretically motivated models to explain the dependent variable, then calculating AIC for each model and averaging the predictors (or parameters) across these models using the Akaike weights.

To be honest, I'm not sure what current "best practices" are here, especially with the rise of techniques like multilevel modeling. Additionally, the rationale given by Burnham et al (2011) basically comes down to datasets containing limited information---and I assume this argument holds less water when you're working with massive datasets. That said, the point that predictions and parameter estimates have less bias when you're averaging across multiple models may still well be true, and worth considering.

Contact

This tutorial is by no means exhaustive. Right now, there's no discussion of Bayes Factors, an increasingly popular approach to assessing ength of evidence for two models. There's also no in-depth discussion of Bayesian information criterion. Those are both things I hope to add.

If you'd like to see anything else in this tutorial, or have various questions, comments, or criticisms, don't hesitate to contact me by email: the username is "sttrott"," at ucsd.edu. I'll likely augment this tutorial with more details (or pointers to other resources) over time.

References

Akaike, H. (1974). A new look at the statistical model identification. IEEE transactions on automatic control, 19(6), 716-723.

Anderson, D. R., & Burnham, K. P. (2002). Avoiding pitfalls when using information-theoretic methods. The Journal of Wildlife Management, 912-918.

Burnham, K. P., & Anderson, D. R. (2004). Multimodel inference: understanding AIC and BIC in model selection. Sociological methods & research, 33(2), 261-304.

Burnham, K. P., Anderson, D. R., & Huyvaert, K. P. (2011). AIC model selection and multimodel inference in behavioral ecology: some background, observations, and comparisons. Behavioral ecology and sociobiology, 65(1), 23-35.

Edwards, A. W. (1992). F. 1972. Likelihood. London: The Johns HopkinsUniversity PressEdwardsLikelihood1972.

Etz, A. (2018). Introduction to the concept of likelihood and its applications. Advances in Methods and Practices in Psychological Science, 1(1), 60-69.

Lewis, F., Butler, A., & Gilbert, L. (2011). A unified approach to model selection using the likelihood ratio test. Methods in Ecology and Evolution, 2(2), 155-162.

McShane, B. B., Gal, D., Gelman, A., Robert, C., & Tackett, J. L. (2019). Abandon statistical significance. The American Statistician, 73(sup1), 235-245.

Wagenmakers, E. J. (2007). A practical solution to the pervasive problems of p values. Psychonomic bulletin & review, 14(5), 779-804.

Citation

For citation, please cite as:

Trott, S. (2021). Model comparison.

Footnotes

I'm not going to devote too much space to justifying why these critera are important. Accuracy is fairly straightforward: from a theory-neutral perspective, we'd prefer a better description of the data to a worse description. Defenses of parsimony usually cite Occam's razor; there's a long history of various philosophers defending parsimony, with the arguments given ranging from aesthetic preferences to pragmatic considerations. I'll just say it's not a completely settled matter, but I guess I lean towards the pragmatic side: I worry that the inclusion of too many parameters leads to models that overfit the data. But of course, the real world is indeed very complex, and sometimes the more complicated model is preferable or even necessary. Ultimately it's not an issue that can be resolved purely algorithmically, independent of context. See also the bias-variance trade-off.↩
One useful way to think about a statistical model is as a description of a phenomenon. There are many ways to describe the same phenomenon, and these distinct descriptions (or explanations) involve different factors.↩
Or, in practice, minimize the negative log-likelihood (NLL). This is because it is apparently easier to build optimization algorithms that minimize a function.↩
Note that as our parameter space increases, a grid search becomes computationally intractable↩
Put in more Bayesian terms, this is our set of hypotheses. A Bayesian might also assign different priors to each possible value of \(\mu\), based on what we know about that distribution or the data-generating process.↩
Interestingly, this is why homoscedasticity is such a critical assumption of linear regression↩
Again, error is assumed to be constant across the regression line. This assumption is called homoscedasticity.↩
Note that one could also define the ratio with the full model in the numerator, and the reduced model in the denominator. The main thing this will change down the line is whether we multiply the ratio by -2 or just 2.↩
I haven't actually worked through the algebra on this one, but I assume it's correct!↩
I'm not positive about this, but I assume this should be done on a held-out set. If anyone reading this knows whether or not this is true, please let me know!↩

Introduction to model comparisons

Sean Trott

High-level goals

Logistics

Introduction

Model comparison: Background

Likelihood ratio tests

Calculating likelihood

Comparison to OLS regression

OLS vs. MLE

Back to basics: likelihood and distributions

Continuous distributions

Discrete distributions

Likelihoods for statistical models

Zooming in

Conducting a likelihood ratio test

Formal description and walkthrough

Hypothesis testing

Example 1: Simple linear regression

Example 2: Linear mixed effects models

Akaike Information Criterion (AIC)

Philosophy of AIC

NHST vs. strength of evidence

KL-Divergence

Walking through the formula

Correction for small samples

When should you use corrected AIC?

Comparing AIC values

AIC Delta

Relative likelihood

Final notes on AIC

Common pitfalls

Relationship to likelihood ratio tests

Bayesian Information Criterion (AIC)

BIC vs. AIC

Multimodel inference

Averaging model predictions

Averaging model parameters

Many models vs. one "big" model

Contact

References

Citation

Footnotes