In your case, for one particular meal where the subjects had probably fasted beforehand, the lag is just under 2⁄3 of the half-time. If you accept Umani Rochi’s half-time of 360-420 minutes, then the lag could be 2⁄3 of that, or 240+ minutes. Of course, for all I know Umani Rochi could have been referring to the lag time, or the final gastric emptying time, rather than the half-time. Lags could easily be much smaller, or larger, than 2⁄3 of the half-time in this case.
As best I can determine, Ronchi was talking about total emptying time, not half-time (let alone lag time). This is unquestionably what would make the most sense, given not only the term used (“gastric emptying”), but also the averages presented, for example, here:
50% of stomach contents emptied: 2.5 to 3 hours
Total emptying of the stomach: 4 to 5 hours
Given this, a total emptying time of 6-7 hours under some cirumstances doesn’t seem outside the bounds of possibility. Extrapolating in such a way as to preserve ratios, we could then imagine a half-time of up to 4.5 hours, say. But 2⁄3 of that would give us 3 hours, or 180 minutes—not the 4-5 hours we need for the prosecution theory.
It sounds like you might disagree with not just with Umani Rochi (a court-appointed expert), and Raffaele’s consultant Vinci, but also with another of Raffaele’s consultants, Introna, who placed the start of attack between 21:30 and 22:30.
Not according to p. 180 of Massei-Cristiani, where Introna is described as placing it between 21:00 and 21:30. Raffaele’s appeal document argues for 21:30 − 22:00; this is apparently obtained by averaging the 2-3 hour (from last meal) figure of Lalli and Introna, and the 3-4 hour figure of Ronchi (whose testimony was incorrectly interpreted by Massei and Cristiani, according to the appeal: Raffaele’s lawyers cite passages where he appeared to agree that 4 hours is the normal limit). My “disagreement” with Raffaele’s lawyers in this context is of little import, for several reasons: (1) I am in perfect agreement with Introna, as reported by Massei; (2) We’re talking about confidence intervals anyway; I think 21:00-21:30 is most likely, but 21:30-22:00 is not ruled out nearly as strongly in my model as anything after 22:00 is; (3) Ronchi, whose estimate figured into their calculation, was talking about total emptying time anyway, not lag time (or even half time).
As for Vinci, he was looking at different criteria for the time of death (not specifically gastric contents), and simply gave a wider range, not an incompatible one. No disagreement here that I am aware of.
Note that stress (such as being attacked) can increase lag time, so we might be talking about the time the attack started rather than the time of death.
Sure; just remember that the computer evidence provides an alibi up to nearly 21:30.
In addition to the starchiness of the meal, I would claim [various ways lag time could be increased beyond study results]
No doubt you’ve identified some of the ways Meredith’s digestive process could have been slowed (although there is no evidence of significant alcohol or drug consumption), in the event that there actually was a lag time of 4-5 hours. The question, however, is how likely such an extraordinary retardation is. According to standard data (see below), it should be a highly unusual event. So how does this information (the fact that a time of death—or, if you like, attack time—after 23:00 requires a lag time of over 4 hours) affect your probability of guilt? It seems to me that it should go down noticeably, unless your model was already incorporating both the studies on digestion and the fact that Meredith’s duodenum was empty (i.e. you weren’t surprised by either datum). (For what it’s worth, I think the Massei court erred in this regard by ignoring lag time, and also by using uncertainty as an excuse to smuggle in probability for their preferred conclusion.)
Anyway, what’s your model here: What do you personally estimate the lag to be based solely on digestion (assuming no slippage)? Maybe you can give a mean and a standard deviation, and we can start by modeling it as a normal distribution?
Although the paper I cited explicitly stated that the results did not fit to a normal distribution, the percentiles given are fairly well approximated by assuming a mean of 81.5 and a standard deviation of 30. Under these assumptions, the lag time required for the Massei guilt scenario would be at least a five- or six-sigma event.
Now I know you doubt that the conditions of the study hold here, but don’t you find this at least a little bit confusing? To make the Massei narrative reasonable, you would basically have to assume that (1) the 6-7 hours allowed by Ronchi for total emptying of farinaceous meals is typical rather than exceptional; (2) this extrapolates to a typical lag time of 3 hours or more, as in my calculation above; and (3) the variance is large enough to make a 4-5-hour lag time a reasonable exception (in which case we would probably be talking about a total emptying time of 8-9 hours or more). Each of these seems highly doubtful, to say nothing of their conjunction. Regarding (2) in particular, note that the study data suggests that the ratio of lag time to total emptying time is closer to 1⁄3 than to 1⁄2 (implying a more concave relationship between elapsed time and percentage of contents emptied; suggesting perhaps that lag time may be short even when total emptying time is long).
How much does application of the ligatures reduce the probability of slippage? If ligatures were not applied, how likely do you think complete slippage would be? If they are applied, what are the odds that (1) the slippage occurs before the ligatures are applied, or (2) the slippage occurs anyway after the ligatures are applied, perhaps due to improper application?
I imagine that preventing slippage is at least part of the purpose of the ligatures, and so I assume they reduce the probability significantly. But even in the worst-case scenario here, the amount of slippage can’t have been very large, because the stomach contents could easily have constituted the entire meal on their own. In the unlikely event that the ligatures were improperly applied, we can infer that Meredith may have just passed her lag time, and that a few pieces of food had just started to pass into her duodenum. This is of minimal help to the prosecution, because on their timeline, we should have been long into that stage, and the stomach should not have been nearly as full as it was—indeed, we should have expected with significant probability that the stomach would be completely empty.
To illustrate further, if as much as half of Meredith’s meal had passed into the duodenum, and we assume a normally-distributed half-time with median 127 minutes and standard deviation 40 (the median taken from the study), the finding would still have put her well within the slowest 1% under the prosecution theory (while only in the slowest 10% to 50% under the defense theory).
As best I can determine, Ronchi was talking about total emptying time, not half-time (let alone lag time). This is unquestionably what would make the most sense, given not only the term used (“gastric emptying”), but also the averages presented, for example, here:
50% of stomach contents emptied: 2.5 to 3 hours
Total emptying of the stomach: 4 to 5 hours
Given this, a total emptying time of 6-7 hours under some cirumstances doesn’t seem outside the bounds of possibility. Extrapolating in such a way as to preserve ratios, we could then imagine a half-time of up to 4.5 hours, say. But 2⁄3 of that would give us 3 hours, or 180 minutes—not the 4-5 hours we need for the prosecution theory.
Not according to p. 180 of Massei-Cristiani, where Introna is described as placing it between 21:00 and 21:30. Raffaele’s appeal document argues for 21:30 − 22:00; this is apparently obtained by averaging the 2-3 hour (from last meal) figure of Lalli and Introna, and the 3-4 hour figure of Ronchi (whose testimony was incorrectly interpreted by Massei and Cristiani, according to the appeal: Raffaele’s lawyers cite passages where he appeared to agree that 4 hours is the normal limit). My “disagreement” with Raffaele’s lawyers in this context is of little import, for several reasons: (1) I am in perfect agreement with Introna, as reported by Massei; (2) We’re talking about confidence intervals anyway; I think 21:00-21:30 is most likely, but 21:30-22:00 is not ruled out nearly as strongly in my model as anything after 22:00 is; (3) Ronchi, whose estimate figured into their calculation, was talking about total emptying time anyway, not lag time (or even half time).
As for Vinci, he was looking at different criteria for the time of death (not specifically gastric contents), and simply gave a wider range, not an incompatible one. No disagreement here that I am aware of.
Sure; just remember that the computer evidence provides an alibi up to nearly 21:30.
No doubt you’ve identified some of the ways Meredith’s digestive process could have been slowed (although there is no evidence of significant alcohol or drug consumption), in the event that there actually was a lag time of 4-5 hours. The question, however, is how likely such an extraordinary retardation is. According to standard data (see below), it should be a highly unusual event. So how does this information (the fact that a time of death—or, if you like, attack time—after 23:00 requires a lag time of over 4 hours) affect your probability of guilt? It seems to me that it should go down noticeably, unless your model was already incorporating both the studies on digestion and the fact that Meredith’s duodenum was empty (i.e. you weren’t surprised by either datum). (For what it’s worth, I think the Massei court erred in this regard by ignoring lag time, and also by using uncertainty as an excuse to smuggle in probability for their preferred conclusion.)
Although the paper I cited explicitly stated that the results did not fit to a normal distribution, the percentiles given are fairly well approximated by assuming a mean of 81.5 and a standard deviation of 30. Under these assumptions, the lag time required for the Massei guilt scenario would be at least a five- or six-sigma event.
Now I know you doubt that the conditions of the study hold here, but don’t you find this at least a little bit confusing? To make the Massei narrative reasonable, you would basically have to assume that (1) the 6-7 hours allowed by Ronchi for total emptying of farinaceous meals is typical rather than exceptional; (2) this extrapolates to a typical lag time of 3 hours or more, as in my calculation above; and (3) the variance is large enough to make a 4-5-hour lag time a reasonable exception (in which case we would probably be talking about a total emptying time of 8-9 hours or more). Each of these seems highly doubtful, to say nothing of their conjunction. Regarding (2) in particular, note that the study data suggests that the ratio of lag time to total emptying time is closer to 1⁄3 than to 1⁄2 (implying a more concave relationship between elapsed time and percentage of contents emptied; suggesting perhaps that lag time may be short even when total emptying time is long).
I imagine that preventing slippage is at least part of the purpose of the ligatures, and so I assume they reduce the probability significantly. But even in the worst-case scenario here, the amount of slippage can’t have been very large, because the stomach contents could easily have constituted the entire meal on their own. In the unlikely event that the ligatures were improperly applied, we can infer that Meredith may have just passed her lag time, and that a few pieces of food had just started to pass into her duodenum. This is of minimal help to the prosecution, because on their timeline, we should have been long into that stage, and the stomach should not have been nearly as full as it was—indeed, we should have expected with significant probability that the stomach would be completely empty.
To illustrate further, if as much as half of Meredith’s meal had passed into the duodenum, and we assume a normally-distributed half-time with median 127 minutes and standard deviation 40 (the median taken from the study), the finding would still have put her well within the slowest 1% under the prosecution theory (while only in the slowest 10% to 50% under the defense theory).
Just replied here: http://lesswrong.com/r/discussion/lw/6k7/experiment_knox_case_debate_with_rolf_nelson/4liq