Haven’t read Part One of this series? Click here.

The Context

You are the sole proprietor of Pearson’s Pizza, a local pizza shop. Out of nepotism and despite his weak math skills, you’ve hired your nephew Lloyd to run the joint. And because you want your business to succeed, you decide this is a good time to strengthen your stats knowledge while you teach Lloyd – after all,

“In learning you will teach, and in teaching you will learn.”

– Latin Proverb and Phil Collins

Your pizzas are priced as follows:

Cheese pizza (no toppings): $15

Additional toppings: $1/each for regular, $1.50 for premium

When we left off, you and Lloyd were exploring the relationship between the number of toppings to the pizza price using a sample of possible scenarios.

The Purpose(s) of a “Regression” Line

When investigating data sets of two continuous, numerical variables, a scatterplot is the typical go-to graph of choice. (See Daniel Zvinca’s article for more on this, and other options.)

So. When do we throw in a “line of best fit”? The answer to that question may surprise you:

A “line of best” fit, a regression line, is used to: (1) assess the relationship between two continuous variables that may respond or interact with each other (2) predict the value of y based on the value of x.

In other words, a regression line may not add value to Lloyd’s visualization if it won’t help him predict pizza prices from the number of toppings ordered. 

The equation: pizza price = 1.25*Toppings +15

Recall the slope of the line above says that for every additional topping ordered the price of the pizza will increase by $1.25.

In the last post you discussed some higher-order concepts with Lloyd, like the correlation coefficient (R) and R-Squared. Using the data above, you said, “89.3% of the variability (differences) in pizza prices can be explained by the number of toppings.” Which also means 10.7% of the variability can be explained by other variables, in this case the two types of toppings.

Since there is a high R-Squared value, does Pearson’s Pizza have a solid model for prediction purposes? Before you answer, consider the logic behind “least-squares regression.”

Least-Squares Regression

You and Lloyd now understand that “trend lines”, “lines-of-best-fit”, and “regression lines” are all different ways of saying, “prediction lines.”

The least-squares regression line, the most common type of prediction line, uses regression to minimize the sum of the squared vertical distances from each observation (each point) to the regression line. These vertical distances, called residuals, are found simply by subtracting the predicted pizza price from the actual pizza price for each observed pizza purchase.

The magnitude of each residual indicates how much you’ve over- or under- predicted the pizza price, or prediction error.

Note the green lines in the plot below:

Lloyd learns about residuals

Recall, the least-squares regression equation:

pizza price = 1.25(toppings) + 15

Lloyd says he can predict the price of a pizza with 12 toppings:

pizza price =1.25*12 + 15

pizza price = $30

Sure, it’s easy to take the model and run with it. But what if the customer ordered 12 PREMIUM toppings? Logic says that’s (1.50)*12 + 15 = $33.

You explain to Lloyd that the residual here is 33 – 30, or $3. When a customer orders a pizza with 12 premium toppings, the model UNDER predicts the price of the pizza by $3. 

How valuable is THIS model for prediction purposes? Answer: It depends how much error is acceptable to your business and to your customer. 

Why the Residuals Matter

To determine if a linear model is appropriate, protocol* says to create a residual plot and check the graph of residuals. That is, graph all x-values (# of toppings) against the residuals and look for any obvious patterns. Create a residual plot with your own data here.

Ideally, the graph will show a cloud of points with no pattern. Patterns in residual plots suggest a linear model may NOT be suitable for prediction purposes.

You notice from the residual plot above, as the number of toppings increase, the residuals increase. You realize the prediction error increases as we predict for more toppings. For Pearson’s Pizza, the least-squares regression line may not be very helpful for predicting price from toppings as the number of toppings increases.

Is a residual plot necessary? Not always. The residual plot merely “zooms in” on the pattern surrounding the prediction line. Becoming more aware of residuals and the part they play in determining model fit helps you look for these patterns in the original plots. In larger data sets with more variability, however, patterns may be difficult to find.

Lloyd says, “But the p-value is significant. It’s < 0.0001. Why look at the visualization of the residual plot when the p-value is so low?”

Is Lloyd correct?! Find out in Part 3 of this series.

Summary

Today Lloyd learned a regression line has adds little to no value to his visualization if it won’t help him predict pizza prices from the number of toppings ordered.

As the owner of a prestigious pizza joint, you realize the importance of visualizing both the scatterplot and the residual plot instead of flying blind with correlation, R-Squared, and p-values alone.

Understanding residuals is one key to determining the success of your regression model. When you decide to use a regression line, keep your ultimate business goals in mind – apply the model, check the residual plot, calculate specific residuals to judge prediction error. Use context to decide how much faith to place in the magical maths.

*Full list of assumptions to be checked for the use of linear regression and how to check them here.

Want to have your own least-squares fun? This Rossman-Chance applet provides hours of entertainment for your bivariate needs. 

—Anna Foard is a Business Development Consultant at Velocity Group

The Basics

Let’s say you own Pearson’s Pizza, a local pizza joint. You hire your nephew Lloyd to run the place, but you don’t exactly trust Lloyd’s math skills. So, to make it easier on the both of you, you price pizza at $15 and each topping at $1.

On a scatterplot you see a positive, linear pattern:

Interpreting Software Output

Trend lines are used for prediction purposes (more on that later). In this example, you wouldn’t need a trend line to determine the cost of a pizza with, say, 10 toppings. But let’s say Lloyd needs some math help and you dabble in the black art of statistics.

Most software calculates this line of best fit using a method to minimize the squared vertical distances from the points to that line (called least-squares regression). In the pizza parlor example, little is needed to find the line of best fit since the points line up perfectly.

The Equation of the Trend Line…

…may take you back to 9th grade Algebra

y = mx +b

Price = 1*NumberofToppings + 15

The price of the pizza (y) depends on the the number of toppings ordered (x). The independent (x) variable is always multiplied by the slope of the line. Here, the slope is $1. For every additional topping, the price of the pizza is predicted to increase by $1.

The price of the pizza without any toppings is $15. In the equation above, 15 is the y-intercept –The price of a cheese pizza, to be more specific to the example.

We’ll also refer to this equation as the “linear model.”

R and R-Squared (or, The Coefficients of Confusion)

The second value listed is called R-squared. But before you interpret R-squared (R^2) for Lloyd, you need to give him an idea of R since R-Squared is based on R.

R has many names: Pearson’s Coefficient, Pearson’s R, Pearson’s Product Moment, Correlation Coefficient

Why R? Pearson begins with a P…

No, Pearson wasn’t a Pirate. The Greek letter Ρ is called “Rho,” and translates to English as an “R”.

Pearson’s R measures correlation – the strength and direction of a linear relationship. Emphasis on LINEAR.

-From LinkedIn Learning 

Since R-Squared = 1, you’ve probably figured out R = √1, or ±1. Positive 1 here, since there is a positive association between number of toppings and price. There is a perfect positive correlation between the number of toppings ordered and the price of the pizza.

Since the price of pizza goes up as the number of toppings increases, the slope is positive and therefore the correlation coefficient is positive (there is a mathematical relationship between the two – not going to bore you with the calculations). It is interesting to note the calculation for correlation does not distinguish between independent and dependent variables — that means, mathematically, correlation does not imply causation*. 

The p-value of this output tells you the significance of the association between the two variables – specifically, the slope. Did the slope of 1 happen by chance? No, not at all. It’s significant because the two variables are associated in a perfectly linear pattern. This particular software gives “N/A” in this situation, but other software will give p < 0.000000. (P-values deserve their own blog post – no room here.)

R-Squared has another name: The coefficient of determination

Often you’ll hear R-Squared reported as a %. In this case, R-Squared = 100%. So why is R-Squared 100% here? Look at the graph – no points stray from the line! There is absolutely no variability (differences) whatsoever between the actual points and the linear model! Which makes it easy to understand the interpretation of R-Squared here:

100% of the variability (differences) in pizza prices can be explained by the different number of toppings.

Hearing this, you tell Lloyd that R-Squared tells us how useful this linear equation is for predicting pizza prices from number of toppings.

But in real life…R-Squared is NOT 100%.

Business Complications

Problem: Your customers start asking for “gourmet” toppings. And to profit, you’ll have to charge $1.50 for these gourmet toppings. You’ll still offer the $1 “regular” toppings as well.

Now, the relationship between a pizza’s price and number of toppings could vary substantially:

Lloyd is gonna freak.

As the number of toppings increase, there is more and more dispersion of points along the line. That’s because the combination of regular and gourmet toppings differs more with as number of toppings increase.

Lloyd says a customer wants 4 toppings. He forgot to write down exactly which toppings. Four regular toppings will come to $19. But 4 gourmet toppings is a little pricier at $21. The prediction line says it’s $20. We’re only within a couple dollars, but that’s a good bit of variability. Over time, Pearson’s Pizza may lose money or piss off customers (losing more money) if Lloyd chooses the prediction line over getting the order right.

R-Squared (Again):

It’s all about VARIABILITY – the differences between the actual points and the line. And this is why predicting with a trend line is to be done with caution:

89.29% of the variability (differences) in pizza prices can be explained by the different number of toppings. Other reasons (like the type of topping chosen) cause the price differences, not just the number of toppings.

What R-Squared isn’t:

And that doesn’t mean the model will get it right 89.29% of the time (it’s not a probability). R-Squared also doesn’t tell us the percent of the points the line goes through (a common misunderstanding).

Non-linear Models – 3 Warnings

How does gas mileage change as your car speed increases?

Even though we can see the points are not linear, let’s slap a trend line on there to make certain, for LOLs:

Hint: Horizontal trend lines tell you NOTHING. If slope = 0, R = 0.

And now you also understand why the R-squared value is equal to 0:

0% of the variability in gas mileage can be explained by the change in speed of the vehicle.

Wait a second…

CLEARLY there is a relationship! AKA, Why we visualize our data and don’t trust the the naked stats.

Mathematically, the trick is to “transform” the curve into a line to find the appropriate model. It typically involves logarithms, square roots, or the reciprocal of a predictor variable.

I won’t do that here.

As you can see, technology is amazing and created this model from a 3rd degree polynomial…

Warning #1

Which is TOTALLY FINE if you’re going to interpolate – predict for mileage only between the speeds of 20 and 60 mph. In case you are wondering why you wouldn’t extrapolate – predict for speeds outside the 20 – 60 mph range, I brought in a special guest.

Third degree polynomials have 2 turns:

The R-Squared value here is 0.9920 – this value is based on the transformed data (when the software temporarily made it linear behind your back). Remember the part about R (and therefore R-Squared) describing only LINEAR models? The R-Squared is still helpful in determining a model fit, but context changes a bit to reflect the mathematical operations used to make the fit. So use R-Squared as a guide, but the interpretation isn’t going to make sense in the context of the original variables anymore. Though no need to worry about all that if you stick to interpolation!

Warnings #2 and #3

What if my software uses nonlinear regression?

This can get confusing so I’ll keep it brief. Full disclosure: I thought nonlinear regression and curve-fitting with linear regression yielded the same interpretation until Ben Jones pointed out my mistake!

R-Squared does NOT make sense for nonlinear regression. R-Squared is mathematically inaccurate for nonlinear models and can lead to erroneous conclusions. Many statistical software packages won’t include R-Squared for nonlinear models – please ignore it if your software kicks it out to you.

Consequently, ignore the p-value for nonlinear regression models as well – the p-value is based on an association using Pearson’s R, which is robust for linear relationships only.

The explanation of warnings 2 and 3 are beyond the scope of this post – but if you’d like to learn more about the “why,” let me know!

Thanks for sticking around until the end. Send me a message if you have a suggestion for the next topic!

*Even though number of toppings does cause the price to increase in this use case, we cannot apply that logic to correlation universally. Since correlation does not differentiate between the independent and dependent variables, the correlation value itself could erroneously suggest pizza prices cause the number of toppings to increase.

Bonus Resource: An excellent video I found explaining R-squared using a similar pizza example! If what I said still leaves you confused, Mr. Nystrom will certainly give you pizza mind! 

—Anna Foard is a Business Development Consultant at Velocity Group