Determining the Ultimate Dog Treat via Statistical Analysis

By Adam Wespiser | Published June 19, 2026

Meet Bebop: a 33-inch tall, 83lb Greyhound with three primary passions: sprinting, shadowing my every move around the house, and consuming treats. Whether it is a dedicated chew, a piece of pizza scavenged from an unsuspecting child at a party, or a stray dish of cat food, Bebop possesses both the olfactory precision and the athletic prowess to secure his prize.

Despite years of observation, I realized I lacked a definitive answer to a crucial question: Which treat is actually his favorite? Since Bebop cannot communicate his preferences verbally, I turned to mathematics.

The Theoretical Framework: The Bradley-Terry Model

To quantify "treat strength," I utilized the Bradley-Terry model, which derives a global ranking from a series of pairwise comparisons.

How it Works

The model assigns a positive strength value, p_i, to each competitor (in this case, each treat). When two treats, $i$ and $j$, are pitted against each other, the probability that $i$ is chosen over $j$ is expressed as:

$Pr(i > j) = \frac{p_i}{p_i + p_j}$

Alternatively, if we represent strength as an exponential score where $p_i = e^{\beta_i}$, the formula becomes:

$Pr(i > j) = \frac{e^{\beta_i}}{e^{\beta_i} + e^{\beta_j}}$

Core Insight: The difference between the latent strengths of two items determines the log-odds of one winning over the other.

Bradley-Terry vs. Elo

You may be familiar with Elo ratings (common in chess), which use a similar logic:

$Pr(i > j) = \frac{10^{R_i/400}}{10^{R_i/400} + 10^{R_j/400}}$

However, Elo is an online system designed for continuous updates. After a match, a rating is adjusted based on the prediction error:

R_A' = R_A + K(S_A - E_A)

• S_A: Actual outcome (1 for win, 0.5 for draw, 0 for loss).
• E_A: Expected outcome.
• K: A constant determining the magnitude of the rating shift.

While Elo is perfect for games that never end, the Bradley-Terry model is superior for small, static datasets where we can fit the model all at once.

Real-World Applications

This logic isn't just for dogs; it's used in:

1. FaceSmash: The fictional (and real) early experiment by Mark Zuckerberg depicted in The Social Network.
2. Chatbot Arena: A system used to rank LLM performance via head-to-head battles.

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The Experimental Design

The goal was simple: present two treats and see which one Bebop chose.

The Protocol

• Timing: Every night at approximately 11:00 PM.
• The Cue: I would say the word "choice".
• The Presentation: One treat in each hand.
• The Rule: Bebop could only take one; the other returned to the bag.

The Variables

I selected a mix of long-time favorites (like Greenies) and various options sourced from Amazon.

Note: I ignored differences in treat size for simplicity. To mitigate "hunger bias," trials occurred two hours after dinner, ensuring Bebop was full and the results mirrored real-world treat-giving.

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Data Collection & Refinement

I maintained a strict schedule of head-to-head matchups. For example, a single day's data might look like this:

Trial 1: Treat C vs Treat B -> Winner: B
Trial 2: Treat E vs Treat B -> Winner: E

As the experiment progressed, I noticed a trend: Treat C (Pork Chomps) and Treat B (Greenies) were consistently losing. To increase the statistical power of the remaining contenders, I skipped planned trials involving B and C and replaced them with more matchups between A, D, and E.

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Results and Analysis

The "Right-Side" Bias

Interestingly, in trials where Bebop was presented with two identical treats, he almost always chose the one in my left hand (his right side).

• Observation: This indicates a right-side preference.
• Caveat: This doesn't necessarily mean he is "right-pawed."
• Hypothesis: My kitchen layout is asymmetrical; a window fan on the left may have influenced his choice.

The Winner

Treat E is the current champion. However, the victory isn't a landslide. Treat A remains a strong contender, as the head-to-head record between E and A is a close 3–2.

The model suggests the probability of E beating A is only 57.5%.

Bootstrap Validation

To determine how stable these rankings were, I performed a bootstrap analysis (resampling the data multiple times and re-fitting the model).

Conclusion: While Treat E is the statistical leader, further testing should focus almost exclusively on the E vs. A matchup to definitively crown the winner.