I conducted a sound propagation experiment in which recorded maned wolves calls were broadcasted at different sites(x3), hours(x6: 17h,18h,23h,05h,06h,11h), and with different speaker position (x2: straight forward and inclined upward 45o). The intensity of the calls were measured at 1.25m, 20m, 40m, 80m, 160m, 320m, and 640m from the speaker.

My goal is to analyze the effect of site, hour and speaker position on the sound propagation quality (a smaller drop of intensity by distance means a better propagation). To that end I wish to use a linear mixed model, with "distance", "site", "hour", and "position" as the fixed effects and the individual call being the random effect.

The problem is that as the call propagates it gets fainter and fainter and eventually undetectable (when its intensity drops below or near the background noise level, which is around 20-30 dB). The maximum detectable distance varies depending on site, hour and speaker position. Considering all conditions, at 160m around 95% of roar-barks are detectable, but at 320m only around 45%, and at 640m only around 20%.

How do I deal with this missing data on my response/outcome variable (intensity)? It is obviously not missing at random and the missing pattern reveals information on the conditions that sound better propagates. Should I use the data only up to 160m, were almost all roar-barks have been detected? Should (or can) I use multiple imputations to complete the outcomes, for instance using Hmisc package in R? Should I simply run the model with the outcomes I have (with unequal number of observations in each distance)?

PS: no predictor variable value is missing.

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    $\begingroup$ I'm a little confused as to what data is missing here. It sounds like you have intensity measurements for the distant positions, they're just zero. Missing data would occur if you had sensors randomly shorting out and not reporting any intensity when there actually is some. It seems to me like all your measurements are reflective of what's actually happening. A measurement of zero is very different from a missing measurement. $\endgroup$ – Nuclear Wang May 4 '18 at 20:36
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    $\begingroup$ This sounds like a sort of censoring problem, with the catch that that you don't know the exact threshold. Maybe using a 30 dB threshold is good enough. You can then use Tobit regression. $\endgroup$ – Kodiologist May 4 '18 at 20:41
  • $\begingroup$ I considered marking the intensity of undetected calls as 0, but I think this would make the model very inaccurate. The calls become undetectable when they reach an intensity near or below the background noise, which varies (20-30dB) but is not 0. I will definitively research this Tobit regression. Thank you both! $\endgroup$ – Luane May 5 '18 at 19:15

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