*This is a guest post by Matthew Winn:*

One of the more useful skills I’ve learned in the past couple years is growth curve analysis (GCA), which helps me analyze eye-tracking data and other kinds of data that take a functional form. Like some other advanced statistical techniques, it is a procedure that can be done without complete understanding, and is likely to demand more than one explanation before you really “get it”. In this post, I will illustrate the way that I think about it, in hopes that it can “click” for some more people. The objective is to break down a complex curve into individual components.

You probably already know how to break down components of a function. For example, suppose we have the following function: (y=2x + 3)

This can be broken down into the linear slope component (2 * x) and the intercept component (+3)

When you add the left panel and the center panel, you get the right panel. You didn’t need to read this post to figure it out. But the point is that GCA can be broken down just as simply as this!

## A Clean Hypothetical Example

Here’s a complex curve of some hypothetical data of pupil dilation measured over time. Time moves along from -2 seconds to a maximum of 1 second. These time values were set relative to a specific landmark. We are interested in modeling the growth of pupil dilation over time, which looks something like this...

using components like these...

The pupil diameter curve has a linear (slope) and a quadratic (curved) component, as well as an intercept (overall level). Our job is to measure the contributions of each.

The pupil diameter curve has a linear (slope) and a quadratic (curved) component, as well as an intercept (overall level). Our job is to measure the contributions of each.

Now we can see that each of those time polynomials show up in the complex plot. Sure, the quadratic component looks upside-down, but that just means the coefficient is negative.

Recall the simple y = ax + b plot; if you add all of these components reading left to right, you will end up with the complex plot on the right.

Here are a few other clean hypothetical examples with different coefficients for the intercept, linear and quadratic components. Think of each as simple addition reading left to right.

Now we can see that orthogonal polynomials are just different components being added together.

## Putting It To Use: Comparing Across Conditions

Now that we see the components visualized separately, it could be useful to compare the components across your different experimental conditions. After all, we report statistical differences between the individual components.

Study background: Participants heard sentences that were processed to have varying degrees of signal quality that we call spectral resolution. Basically, the sound spectrum is broken down into a discrete number of frequency channels, where a greater number of channels gives clearer sound quality than fewer channels. Think of it as the number of pixels in an image; more is better. The sentences were 2 – 4 seconds long, and after each sentence, there was a 1.5-second delay, after which participants repeated back the sentence. The outcome measure (in addition to the accuracy level) was change in pupil diameter during the sentence. Greater pupil dilation is known to correspond to (among other things) greater cognitive load during various tasks. We hypothesized that sentences processed with fewer channels would elicit greater pupil dilation because they would require more effort to understand.

The following plot suggests that this hypothesis was correct. The aggregated data are points with lines for standard error, and the barred white lines represent the statistical model to be described below.

Growth curve analysis was done to describe the portion of this plot from -2000 to +500 ms relative to the sentence offset. This corresponded to the growth of pupil dilation from baseline to roughly the max level.

As you might expect, we reported differences in the intercept, linear and quadratic components across all these conditions. There were a large number of significant effects that emerged, but that’s not my point here (you can read the paper for that!). The point is that a helpful tool in understanding these differences is visualizing them individually rather than in aggregate. After all, a systematic increase in the intercept across the conditions is not easy to see when there are sloped and curved components that distract the eye.

In the following grid of plots, we have flipped the scheme of the previous plot by 90 degrees. Now the different polynomial components (intercept, linear, quadratic) don’t move left to right, but instead move top to bottom.

The objective of this plot is to let the reader directly compare the components of each condition’s curve side by side. The model coefficients for each component are indicated by the number in the box in each panel. One can imagine the addition of cubic and quartic components for other data.

With this plot, it is much easier to see that:

1. The intercept level grows systematically across the conditions

2. Each condition’s linear component gets steeper, moving from left to right.

3. Each condition gets progressively more curved, as the quadratic component gets increasingly more negative.

I don’t expect this to be a standard way to visualize GCA, if only because journal space is in short supply. But as a blog post, or as supplemental material, perhaps this approach can help de-mystify GCA.

With this plot, it is much easier to see that:

1. The intercept level grows systematically across the conditions

2. Each condition’s linear component gets steeper, moving from left to right.

3. Each condition gets progressively more curved, as the quadratic component gets increasingly more negative.

I don’t expect this to be a standard way to visualize GCA, if only because journal space is in short supply. But as a blog post, or as supplemental material, perhaps this approach can help de-mystify GCA.

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