When Metrics Collide: A Clash of Regression Titans in the Spotlight

In the quiet, data-driven town of Statia, three brothers lived in a grand, glass-walled house at the heart of the city. These brothers were no ordinary residents—they were the renowned evaluators of regression models: MSE (mean squared error), MAE (mean absolute error), and R² (R-squared). They were the Regression Titans! Each had their own unique perspective on how to judge the performance of a predictive model, and their discussions often turned into heated debates.

The Gathering

In the morning of a sunny day, the brothers gathered in their spacious living room, a holographic model projected in the center of the room. It represented a regression model that predicted house prices based on features like area, location, and number of bedrooms.

MSE, the eldest, adjusted his glasses and looked at the model with a critical eye. “Well, my bros, let’s get to work. This model needs evaluation. I’ll go first,” he said, his voice steady and authoritative.

“Of course you want to go first,” MAE teased, crossing her arms. “You always want to make everything about your squares.” There were some rumors that MAE was jealous of MSE from time to time. But after all, they were siblings, that was all.

R² smirked but stayed quiet, observing the scene like the reserved philosopher he was. Despite his youth, he was the most mature of them all.

MSE’s Take

MSE cleared his throat and started to talk. “Listen, my approach is simple yet effective. I measure how far the predictions are from the actual values, and then I also square the differences. This way, larger errors get penalized heavily. It’s the best way to highlight extreme errors.”

He pointed to a hologram showing the model’s predictions. “See here? The mean squared error of this model is 1200. That’s quite high. It means the model is struggling with large errors.”

MAE rolled her eyes. “Ah, my brother, you blow things out of proportion again, typical of you. Squaring the errors can make small mistakes look insignificant compared to a few large ones. That’s not fair.”

“It should be!” said MSE.

“Don’t you think you are making the model’s performance look unnecessarily bad just because you took the square of just a couple of big errors; maybe a result of bad luck?” said MAE. He was proud of his argument.

“Ah, little brother, your habit of associating rare big mistakes with bad luck will cause you much pain one day. And you can not know; they may not always be as rare as you think!” said MSE.

“If the number of big errors in a model is too high, then my method would also evaluate them fairly, but whatever,” MAE said in an annoyed tone.

“Tell me your method, then,” said MSE.

MAE’s Perspective

MAE stepped forward, his approach as straightforward as his personality. “Unlike my big brother, MSE, I don’t complicate things. I just take the absolute differences between the predictions and actual values. No squaring, no exaggerating. For this model, the mean absolute error is 20. It’s much easier to interpret. It means, on average, the model is off by 20 units.”

MSE snorted. “Sure, but your approach treats all errors equally. You’re not punishing the big mistakes enough! Imagine if a model predicts a house price of $100,000 when the actual value is $500,000. You’d let that slide way too easily.”

MAE stood her ground. “And you’d make that one mistake the center of attention, ignoring the rest. That’s not balanced, either.”

R² Enters the Chat

R² was watching this discussion with boredom. Finally, he decided to join in the discussion.

“Why wouldn’t you work together?” he asked.

“What do you mean?” MSE answered him with another question.

“Your approaches are actually very similar, and they complete each other in some way. If you work together, you know both the average of the absolute errors and the average of the squared errors. If the difference between them is too high, then you would know there are several big errors, and then you can start to explore those to understand why they exist.”

“What if the difference is not too high?” asked MAE.

“Then move on,” said R².

“Actually, it makes sense,” said MSE, and looking at his brother with compassion, he extended his hand to him. MAE also took his big brother’s hand warmly.

R² Reveals His Unique Approach

“However,” said R², “you two are always so focused on errors. Have you ever considered how much better—or worse—the model is compared to a simple baseline?”

MSE and MAE looked at him, intrigued but slightly annoyed.

“What baseline?” asked MAE.

“The mean. Simply, the mean of all the house prices in the training dataset!”

“I don’t understand,” said MSE.

R² was excited. “Let’s imagine that all predictions of the model are the same and that this prediction is the average of known house prices. Do you think this is a good model?”

“Probably not,” answered MAE, “it does not reveal any insights.”

“But it is also something,” said MSE, “not efficient and informative, but, at the end, a simple prediction approach no matter good or bad.”

“Yes! You are getting closer!” said R², and continued, “what I’m trying to do is figure out how much better the predictions the model makes are than just predicting the mean all the time.”

MAE and MSE stared in astonishment.

“I measure how well the model explains the variability in the data. If my value is 1, the model is perfect. If it’s 0, it’s no better than guessing the mean. For this model, the R² is 0.85, meaning it explains 85% of the variance in house prices. That’s a strong performance.”

“What if your value is less than 0? Is it possible?” asked MAE.

“Ohh, yes, it is sometimes possible! Then the model is completely bullshit. Predicting the mean all the time is even better!” answered R², and the three brothers laughed heartily for minutes.

After a while, MSE frowned. “But you’re so abstract. People don’t always understand what 0.85 means.”

MAE nodded. “And you rely on comparing to the mean baseline. What if the baseline itself is misleading?”

R² smiled serenely. “No metric is perfect. But my approach provides context. I show how much better—or worse—a model is than doing nothing. That’s invaluable.”

The Agreement

The brothers debated for hours, each defending their method with passion. Eventually, they reached a conclusion.

“As your big brother, I can admit that, we all have strengths and weaknesses,” MSE said. “I am great at highlighting large errors, but I sometimes can be too sensitive to outliers.”

MAE nodded. “And I am simple and intuitive, but I might not punish extreme mistakes enough.”

R² added, “And I provide a sense of improvement over the baseline, but I am not as direct as either of you.”

The brothers agreed that no single metric could capture the full story of a model’s performance. Instead, they decided to work together, offering their perspectives as a team.

And so, in Statia, MSE, MAE, and R² continued their work, ensuring that every regression model was evaluated fairly and comprehensively, their unique voices forming a harmonious symphony of metrics.