Shawn Lee

What does it take to be a good fighter?

Perhaps I, typing away on my desk sipping some hot cocoa, may not quite know.

What I do know is a bit of computational economics, and through a model known as logistic regression, we are going to dig into the numbers to see if we can find out something new that perhaps even the most veteran coach or skilled fighter may not know yet.

We have a dataset of 5,512 fights taken from Kaggle user mdabbert and cleaned slightly by user Andrew Ritchie here. Each row describes a bout between two fighters including their physical attributes, fight record, betting odds, and the eventual outcome of the fight.

Our analysis begins by treating this as a binary classification problem. We want to look at several features mentioned above and determine if it has a significant impact on one fighter's chances of winning. Enter:

The Logistic regression¶

A logistic regression is a statistical tool that helps us predict binary classes. Unlike linear regression which predicts a continuous variable, a logistic regression can be used to predict classes which make it for predicting a yes/no or a win/loss.

This works by putting a linear regression into a sigmoid function. The sigmoid function takes our input $y$ and returns $\sigma$ which is always a value between 0 to 1. The closer the output is to 0, the stronger the prediction is that the outcome is a loss for one fighter and vice-versa.

Linear regression:

$ \large y = β_0 + β_1X_1 + β_2X_2 + ... + β_nX_n $

Sigmoid function:

$ \large \sigma = \frac{1}{1 + \large e^{-y}} $

Logistic regression:

$ \large \sigma = \frac{1}{1 + \large e^{-(β_0 + β_1X_1 + β_2X_2 + ... + β_nX_n)}} $

Let's try recreating that in python:

You might notice one part of the code we haven't discussed yet which is estimating the beta coefficients. To do this, we will use something called:

Maximum Likelihood Estimation (MLE)¶

MLE is an estimation method that takes our dataset's parameters, such as $\hat{\mu}$ and $\hat{\sigma}$, as our given conditions and tests different sets of coefficients $\beta$ to see which one is most likely to produce our actual dataset.

In statistical terms, if the probability density function is given as:

$$ Pr (data | distribution) $$

Then likelihood is:

$$ L (distribution | data) $$

We're not out of the woods yet because we still need a way of approaching the best likelihood value as we test through multiple guesses or iterations of $\beta$. I'll spare you the calculus and present to you the loglikelihood function which will do just that:

Okay that was a lot to take in (especially if you're writing this in the morning like I am) so let's take a step back and look at why we're writing all these functions in the first place. We want to find out:

What does it take to be the
statistically advantageous Fighter?

Well, we can start by asking the question "what do we normally think of as being advantaegous in a fight?"

Traditionally in combat sports, we have what is called "the tale of the tape" which highlights a few key stats about each fighter that may influence the outcome of the fight. In the UFC, one of the biggest MMA promotions today, these stats are: age, height, weight, and reach.

Conventional wisdom dictates that a weight disparity between two combatants would intuitively have the most significant impact over who wins a fight. Because of this, the pool of professional fighters are always divided into weightclasses which means this variable is pretty much held constant for us for every observation.

(Side note: it is common these days for atheletes to "cut weight" for fights. They would shed a ridiculous amount of body weight when weighing in for their division, then immediately rehydrate and bulk back up until the actual fight a few days later. It would be interesting to see how the regular "walkaround" weight disparity between fighters may play a role in determining the outcome of a fight but we'll leave that scraping task for another day.)

What does our data look like?¶

Our table contains 5,512 observations of UFC fights beginning March 21, 2010 (billed as UFC Live: Vera vs. Jones) until December 10, 2022 (UFC 282). There are some inconsistencies and inaccuracies with the data which I won't get into here, but we either will be cleaning those features as we go or not using the feature altogether.

Let's start by seeing the spread of observations across our four features: height, reach, age, and betting odds. Because we have two fighters for every row of observation (Blue fighter, Red fighter), we will create a df combined_feats which appends the rows vertically for the four features so that our df has shape (11024, 4).

We'll clean the data a bit with the dropna() method, then we'll create four histograms side-by-side. We'll also add in boxplots at the bottom so we can tease out any outliers.

Since fighters tend to compete within one weightclass for their whole career, their matchups with other fighters tend to be limited to whoever else is in their division. This leads to certain trends that may form within each division and can be an interesting way of clustering the fights together. Let's see how our observations are spread across weightclasses.

Logistic Regression¶

We'll start by making some adjustments to our data. First, we create an outcome column that gives us '1' if the Red fighter wins and '0' if the Blue fighter wins.

We'll also create odds_dif to present the differential between Red's betting odds and Blue's betting odds (lower odds represent the favorite to win).

Now we'll standardize all our ordinal features by their z-scores.

Thankfully, the data we have already comes with a column of 1's constant_1 but we could have easily done that ourselves with df['intercept'] = 1.

To make things neater, we will create df_logit to contain only our intercept, independent, and dependent variables.

Finally, we will use statsmodels.api to perform our logistic regression for us.

We simply define X as our independent variable columns, and y as our dependent variable column.

We pass the two into sm.Logit().fit() and use the summary() method. Like how we discussed MLE, this method will test multiple iterations of coefficients for our X variabes to see which iteration has the highest likelihood of producing our current dataset.

Real quickly, let's check if the functions we defined earlier would generate the same coefficients.

Awesome! We got the same results as our statsmodel package.

Interpretting the logistic regression results¶

Before we draw conclusions with our coefficients, we must first understand that we are working with a logistic regression and not a linear regression. This means that y does not change by $β$ per unit change in x.

A unit change x $\cdot \beta$ is passed into the sigmoid function which will have some positive/negative effect on the output. At most we can say that the positivity/negativity of the effect will be in the same direction as $\beta$'s sign, but that's about all we can say at this point.

Lucky for us, we have the get_margeff() method which translates our coefficients to more linear terms in dy/dx:

Of course we will have another caveat in that our data was standardized by their z-scores, so every unit of change is measured in terms of standard deviations $\sigma$.

For example, the standard deviation of height differences observed is 9.30cm. With a reach_dif_std dy/dx of -0.0192, that means that red's probability of winning improves for every -0.0192 * 9.30cm = -0.17856 cm reach advantage they get or 0.0192cm reach they give up to their opponent. Controversially, this suggests that having shorter arms than your opponent may actual have an advantage in professional fighting.

Indeed, our analysis seems consistent for all other features:

• age_dif_std with negative dy/dx suggests that being younger is advantageous
• height_dif_std with positive dy/dx suggests that being taller is advantageous
• odds_dif_std with negative dy/dx suggests that having a more negative betting odd (which is considered the favorite) is advantageous

Before we close the book on this analysis, however, we cannot just take our results as the absolute truth. With any statistical test, it is important to also evaluate our p-value to see if we have enough evidence in the data to support our claim. In this case, height_dif_std has a surprisingly high p-value.

A p-value of 0.480 is way above the conventional threshold of 0.05. This means, we do not have enough evidence to say that height has an impact on the outcome of a fight!

Clustering data for additional analysis¶

Earlier we mentioned the importance of weight class and how each weight class may form different trends. Out of curiosity, let's see if we can code up a way to show how each weight class' logistic regression results vary.

Note that at this stage, we are working with significantly smaller datasets so we'll have to take any conclusions made with a grain of salt.

Interesting yet unsurprisingly, parsing down our 5000+ observations down to just 200-300 per cluster significantly affects our p-values and our ability to reject the null-hypothesis.

While this might seem anticlimactic, we actually do learn new tidbits of information. For example:

• Heavyweights are the only fighters who gain a statistically significant advantage from being taller.
• On the other hand, Heavyweights and Light Heavyweights are the only ones to gain an advantage in reach differential, and the advantage is seen for those with reachers shorter than their opponents!
• With p-values virtually equal to 0, betting odds are a great predictor for fight outcome.
• Interestingly, catch weight fights (which often take place on short-notice or because a fighter failed their weightcut) have a less non-zero p-value of 0.0001. This suggests that the uncertain nature of catch weight fights are also difficult to read for bettors!

Final thoughts¶

There's still so much we can dig through and learn from studying fight data, but at the end of the day, it's great to learn that physical attributes and conventional wisdom only play a minor role in determining the outcome of a fight. Training, experience, matchups, and a lot of other factors likely play a much larger role in helping you win a fight. But until we get more granular data with bigger datasets, we will leave those features shrouded in the noise of our model.