Quadratic Regression

Quadratic regression is a statistical method used to model a relationship between variables with a parabolic best-fit curve, rather than a straight line. It's ideal when the data relationship appears curvilinear. The goal is to fit a quadratic equation $y = a x^{2} + b x + c$ to the observed data, providing a nuanced model of the relationship. Contrary to historical or biological connotations, "regression" in this mathematical context refers to advancing our understanding of complex relationships among variables, particularly when data follows a curvilinear pattern.

What is quadratic regression?

We have already covered linear regression, which we use to predict the value of a variable based on the value of another variable. This gives us the "line of best fit" and allows us to make accurate predictions about our data.

The same general concept applies to quadratic regression. The only difference is that instead of trying to find the line of best fit, we're trying to find the parabola of best fit.

So how exactly does this work?

Instead of getting a simple linear equation, quadratic regression leaves us with this:

If you already know about the "least squares method," you can use the exact same process with quadratic regression.

We need to find the values of $a$ , $b$ , and $c$ so that the squared vertical distance between each point $(x_{i}, y_{i})$ and the quadratic curve $y = a x^{2} + b x + c$ is minimal.

We can use a matrix equation to represent our quadratic curve:

We can also use these formulas to find the values of $a$ , $b$ , and $c$ :

In these equations:

$x$ and $y$ are our variables
$a$ , $b$ , and $c$ are our coefficients for our quadratic equation
$n =$ the number of elements
$\sum x =$ the sum of $x$ values
$\sum y =$ = the sum of $y$ values
$\sum x^{2} =$ the sum of the squares of $x$ values
$\sum x^{3} =$ the sum of the cubes of $x$ values
$\sum x^{4} =$ the sum of the fourth powers of the $x$ values

These calculations can be represented by a system of equations in a matrix form, and the coefficients a, b, and c can be determined using these equations. It's important to note that the equations and methods used to calculate these coefficients can be complex, involving several sums and squares of various combinations of the data points. Thus, these computations are typically done with the assistance of software.

Note that the relative predictive power of a quadratic model is denoted by R². Our predictive power tells us how accurate our predictions truly are. It is the power of a scientific theory to generate testable predictions. In the case of quadratic regression, this applies to the predictions related to our parabola.

The formula for our predictive power is as follows:

R^{2} = 1 - \frac{SSE}{SST}

In this formula:

$SSE = \sum_{i = 1}^{n} {(y_{i} - a - b x_{i} - c {x_{i}}^{2})}^{2}$

$SST = \sum_{i = 1}^{n} {(y_{i} - \bar{y})}^{2}$

Our relative predictive power $R^{2}$ lies anywhere between 0 and 1. The closer it is to 1, the more accurate our model is.

We should also note that the "bar" above the y value in the SST equation represents the average of all y values.

Working with quadratic regression

As we might have guessed, these calculations can become quite complex and tedious. We have just gone over a few very detailed formulas, but the truth is that we can handle these calculations with a graphing calculator. This saves us from having to go through so many steps -- but we still must understand the core concepts at play.

Let's try a practice problem that includes quadratic regression. Consider the following set of data:

\{(-3, 7.5), (-2, 3), (-1, 0.5), (0, 1), (1, 3), (2, 6), (3, 14)\}

Can we determine the quadratic regression for this set?

Our first step is to enter our x-coordinates and y-coordinates into our graphing calculator. We can then carry out our operation for a quadratic equation. This will give us the equation of the parabola that best approximates the points:

y = 1.1071 x^{2} + x + 0.5714

Great! Now all we need to do is plot our graph. We should be left with something like this:

We also know that our relative predictive power ( $R^{2}$ ) is 0.9942. That's pretty accurate -- and it tells us that our calculations for quadratic regression worked!

Topics related to the Quadratic Regression

Developing a Probability Distribution from Empirical Data

Line of best fit(Eyeball Method)

Fitting Equations to Data

Flashcards covering the Quadratic Regression

Statistics Flashcards

Common Core: High School - Statistics and Probability Flashcards

Practice tests covering the Quadratic Regression

Probability Theory Practice Tests

Common Core: High School - Statistics and Probability Diagnostic Tests

Pair your student with a tutor who understands quadratic regression

Quadratic regression is similar to linear regression, but there are a few distinctions that can catch students off guard. The best way to address these potential points of confusion is alongside a qualified, experienced math tutor in a 1-on-1 learning environment. Tutors can slow things down to a more manageable pace for your student, allowing them to walk through the concepts step-by-step. They can also break down more complicated steps into manageable chunks. In addition, a tutor can give your student plenty of opportunities to ask questions -- something that isn't always possible in a full classroom. While tutoring can make things more manageable, it can also present interesting challenges -- and your student can take the time to practice their skills. Speak with our Educational Directors to learn more, and rest assured: Varsity Tutors will pair your student with a suitable tutor.