Clustering a single binary variable? I am trying to design a study which will involve asking people to watch videos and then stop them at points that they think relevant (looking for specific behaviours). The idea is to see if people stop the video at the same time - I’m struggling to think how I can add some statistical analysis to this? I thought a cluster analysis but as the data will be largely binary (ie I will have each second of the video with a stop / not stop identifier) I’m not sure what I can do?
 A: I would simply do KDE on the time of stoppage.
You'll see something like YouTube videos show these days:

The grey bell shaped bumps on the image are the representation of frequency at which people jump into a point in the video.
A: I am not sure what you mean with clustering using one binary variable in your case.
You could do a time-to-event analysis (aka survival analysis) to investigate the rate at which people stop the video. If you have some information about this people, such as sex or other characteristics, you can use a Cox Proportional Hazard model (that allows the inclusion of covariates) to study the effects of these characteristics on rate of these stops. You can also use Kaplan-Meier curves to investigate how a categorical variable influences the rate of stopping.
Even more simple, you could disregard the time and run a logistic regression using the covariates.
You can also look from another perspective. Imagine that you can divide the video in different parts based on some characteristics, for example the first part is introduction, the second part shows a lot of nature and so on (I am just imagining now). Then you can compare the characteristics of groups of subjects based on the section of video they stopped at.
This is not easy though, you might use some help from someone with experience in data analysis, if possible!
