"I had a discussion with another illustrator over composition recently. Specifically about the usage of directional lines and shapes to lead the eye to the focal point of the painting or cinematic frame in a movie.
     "I know you refer to it as spokewheeling and shapewelding. I learned all this back in college and it was drilled into us based on the old Loomis books.
     "Anyway, my friend says that all of that is nonsense due to eye tracking and that it doesn't matter where the lines are going because the brain is going to look for a face or random points of interest every time.
     "So do you think eye tracking negates spokewheeling etc or is it all a combination of elements to lead the viewer through a composition?
     "I'm firmly in the camp that it doesn't. I've been studying the work of some famous cinematographers lately and they compose whole frames through use of strong light, shadow, color and directional shapes to lead the viewer.
     "I don't know....So I'm writing you. Hope you can settle this for me or at least offer some insight."

Hi, Eric,
That's a fascinating question, and I'm glad you asked it. Here's the short answer: I believe that scientific insights from eyetracking challenges a lot of the art-school dogma about how we look at pictures. But don't throw out the compositional toolkit just yet. Many of those compositional devices are probably still valid.

Eye tracking heatmap in a bar. Viewers apparently want to know what brands of beer are on tap

You and your friend are both right. Your friend is right that faces (or other psychologically important objects) will attract the most attention wherever you place them in the design. Eye tracking proves that. It also shows that the way each viewer explores the picture is highly individual. No two viewers will experience the picture in the same way.

Venice by Turner. I'd love to see an eyetracking heatmap of this painting. I believe  that I'm most attracted to the light buildings on the light background, not to the areas of highest contrast. But maybe I'm misreporting my experience, and maybe I look at this painting differently than others do.

The scanpath (the track of eye movements over time) of a given viewer depends to a great extent on what psychological or narrative expectations he or she brings to the interpretation of the image. Contrary to many dogmatic assertions that we learned in art school, the eye's path through the picture does not really follow passively along the directional lines. Instead it jumps around in unpredictable jagged leaps all over the picture. While we customarily speak about "leading the eye" or "forcing the viewer" or "directing the attention" by means of leading lines, we have to remember that the eyes are not driven in a deterministic way, like a train on a track.

Eyes are active extensions of a hungry brain.

Does this mean that those traditional compositional devices have no effect on our experience of the picture?

No, and here is where I think you are also right. I believe that most of the classical design devices (including  spokewheeling, chromatic accents, edge control, value organization, etc.) can influence the way we perceive a composition. When used intelligently, they can help the average viewer decode what's important in a picture, and they accentuate the viewer's satisfaction in having their attention anchored to the centers of interest as they further explore subordinate areas.

Yarbus's data originally published in "Eye movements and vision" (read more) .

But it's difficult to know exactly how we're influenced by such devices. I suspect that we perceive them by means of our peripheral vision, even if we don't perceive them directly with our center of vision.

For example, let's look at the two paintings in this post. In "The Last Supper," Leonardo's placement of the vanishing point behind Christ's head seems to reinforce our focus on that important center of intererst.

But in the case of "They Did Not Expect Him," Repin doesn't place the vanishing point behind any of the major heads, but that doesn't seem to compromise the ability of viewers to find what is important in his painting.

Yarbus showed that people looked at the the Repin painting many different ways (right) depending on what question they were prompted with first.

Viewers are perhaps more influenced by leading questions than leading lines.

Science is beginning to reveal that visual processing of any image—but especially a realistic, narrative image—involves many areas of the brain. How we look at a picture appears to be affected by several interrelated factors, such as lines, tones, lighting, color, psychology, title, caption, and other factors. The leading lines and the shapes are just two of those elements.

My advice
Science can help us bayonet sacred cows, but it can't guide us very much in designing pictures. How we look at artwork is a topic that is still mostly unexplored by cognitive scientists using modern technology. Until more studies are carried out, we can't fully understand the logic behind pictorial design. My advice is to be skeptical when you hear any dogmatic assertions about composition. Instead, follow your instincts. Don't concern yourself with following compositional "rules," and don't bother with making your pictures pleasing or harmonious. Instead just work to make your picture interesting. Figure out what you want to say and say it emphatically.

If a graduate student in neurobiology is reading this and wants to devise some experiments, please contact me! I'll volunteer some of my paintings as guinea pigs.
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More info
Previous posts:
Spokewheeling 
Shapewelding
Eyetracking and Composition (series)
Books: Vision and Art (Updated and Expanded Edition)
Imaginative Realism: How to Paint What Doesn't Exist