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I am comparing sperm motility between 2 sets of samples (2 groups). Motility just means weather the sperm are alive or not, in this test I am just looking at that. If its moving, then its alive (I have the motility values as percentages, for example an ejaculate from 1 person might have 20% motile sperm and 80 immotile). It is important to mention that sperm motility decreases with time after ejaculation. I have 2 different groups of people. The control group has had sperm motility (percentage of moving sperm cells in ejaculate) measured in various time periods (1–8 hours after ejaculation). The second group is the treatment group. Here, an extender liquid has been added to the ejaculate which should prolong the motility time (the sperm should stay alive for longer). The sperm motility of this group has been measured later than the control group`s (6–24 hours after ejaculation). I want to test weather the extender liquid has a significant effect on motility over time. How should I approach this?

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One experimental design problem is that you don't have much overlap in the observation times between the two groups. Another problem (maybe less important if you have a lot of specimens), is that each specimen either did or did not have the "extender liquid" added. It seems that it should have been possible to treat half of each specimen with the "extender liquid" for a more direct test of the treatment.

I recommend comparing the two groups at some shared evaluation time between 6 and 8 hours. One could consider a joint model of survival over time that incorporates the treatment as a predictor variable, but the treated group provides no information before 6 hours and the untreated group provides no information after 8 hours. Of course, if the untreated group has exactly 0 survival beyond 8 hours you pretty much have shown the superiority of the "extender liquid" anyway.

An alive/dead outcome ideally should be evaluated by a binomial model based on the observed counts. You have much more confidence in a value of 90% survival, for example, if you have 100 observations instead of 10 observations; the analysis should take that into account. For a single time point (e.g., 7 hours) that would just be a standard binomial test. If you want to model over time, a logistic regression would be appropriate. For modeling over time, if you made multiple motility measurements on the same specimen you would need to take that within-specimen correlation of outcomes into account, e.g. with a mixed model treating specimens as random effects.

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