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You look at the plots showing where gaze is directed.
I've use eye-trackers. The instructions tend to be very simple. Usually fixate on a central point--a red dot, black cross, whatever. Sometimes there's an instruction to look at an object when it appears; more often the fixation point vanishes shortly before the subject's gaze is to be allowed to wander and track the subject's attention. Usually the software does a check for gaze: If the subject's gaze isn't on the fixation point when the recording starts then the trial's discarded. All subject's begin at the same point. What they do after that varies a lot. But it should vary randomly until the stimulus is presented, then the variation should rapidly diminish. Typically test runs are done to make sure that the screens prior to the presentation of the stimulus don't induce some sort of bias.
Then you average by condition, not over all trials.
There are four possibilities: The gaze cue points towards the object, subject follows gaze (predicted fast average time to acquisition of target); the gaze cue points from the object, subject follows gaze (predicted delayed average time to acquisition of target); no gaze cue, subject finds target independently (default/neutral average time to acquisition of target); gaze cue ignored, subject finds target independently (default/neutral average time to acquisition of target). Because the screen is neutral, those who randomly let their gaze wander after the fixation point vanishes are as likely to have it wander away from as towards where the target will appear.
You'd probably expect the last two to show similar acquisition times, but you'd keep them separate just in case: Perhaps those who ignore gaze cues are just slower to respond to the stimuli. The researchers seem to be saying there is no difference. The first two should show highly divergent acquisition times, clearly separated from the third and fourth. The eyetrackers we used, IIRC, resolved to 3.33 msec.
I don't know that they're addressing cause/effect. Just correlation, with speculation. The thing is, this kind of study is risky. Conservative/liberal isn't a clear-cut, binary opposition unless they did a survey and weeded out all but those at the extremes; if they did that, they may have introduced other factors into the subject pool.
How you read the results depends on what you want to read. Take responsiveness to gaze cues. Upstream a poster pointed out that it likely reflect empathy. (Perhaps, perhaps not. Empathy and response to gaze cues are clearly related but aren't the same thing.) Solidarity, an assumption of like interests, lots of other things could be assumed. "Empathy" is a current buzz-word. On the other hand, gaze cues can easily be used to redirect a person's attention, and if it happens outside of the person's control the word is "gullible." Alternatively, over-valuing others' opinions, dependence, wanting to be part of the herd, and many other negative traits could also be adduced. Similarly, a set of positive and negative traits for lack of responsiveness are possible. (Odds are that it's mixed; purely negative traits would most likely be weeded out of the gene pool, so both have been found useful for the species' survival and success.)
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