Every minute counts

[Fisherman]

Hi Fisherman,

It does not necessarily follow that a killer a minute before Cross/Lechmere would be unlikelier, and while I agree the difference between them would be small, it is entirely possible for Cross/Lechmere to be the less likely.

Again, let's say we have 0.1% of the cases stop bleeding in under 1 minute. And 0.4% stop bleeding between minute 1 and 2, and 1 % between minute 3 and 4, etc. So working backwards, it would be the later killer that is more probable than the earlier one at these short intervals.

If I extended my above example series, it would reach a point where the later killer does become less likely, but we would need to know at what point that happens, not just pick a number. This is what I'm trying to explain to you, without knowing the density function of the probabilities, we can make no inferences about which would be more likely.

I like your idea of trying to use this information. It is a good one in principle, though it may be we have too little information to really do it properly, the general gist of it is sound. However, the nature of the probabilities you're employing are the wrong tool, and it does your good idea no benefit. Probabilities, and the inferences we can make from them, are not the sort of thing where intuition is very helpful, and very often what appears to be "common sense tells me this is correct" turns out to be very very wrong.

Anyway, I'm trying to help you develop your idea not tear it down or suggest the data you're looking at couldn't provide some interesting ways to look at things. Because I do think it might be possible to develop something. Because of that, I think it would be better if it was done using the proper tools (which is what the probabilities are, tools to inform us). Sadly, I rather suspect there will not be a study that will provide us with those probability distributions, making it hard to turn the idea, which is good in principle, into an actual analysis. But, I could be wrong, and there could be information that one could use, or adapt, to give a suitable estimate of the distributions. I would, as I say, be highly surprised if, after all of that, the differences between the times associated with Cross/Lechmere and JtR as Other end up being at all different from each other. But that's only my prediction, it remains to be tested, and I would be interested in seeing how it turns out should you pursue this.

- Jeff

I look at it like this, and you are welcome to tell me when science is of another meaning.

1. For a minute of bleeding to exist after the opening minute of bleeding (bleeding minute number one, as it were, the first 60 seconds after the cut is opened up), it takes that the preceding minute must also have been a bleeding minute. If she bled in minute four, then she must also have bled in minute three and all the preceding minutes. If she bled in minute ten, then she also bled in minute nine and all the preceding minutes.

I hope we agree on that.

2. Since we can work from the factual knowledge that the bleeding will stop at some stage, we can also say that IF we are to detract time from the bleeding process, then it must be detracted from the last minute/s of the process. Meaning that if somebody suggests a bleeding time of ten minutes and we think that is too much, instead suggesting eight minutes as the probable maximum, then it is minutes nine and ten we want to take away from the process. Not minute one and two.

I hope we agree on that too.

3. This means that the minutes of bleeding that may be discussed or be under contention as having existed, are the last minutes of the process.

I once again hope we agree?

4. The outcome of these facts (which I hope you agree ARE facts) is that when we suggest adding a minute to a bleeding process, then the later in that process that minute is, the lesser the likelihood is that it really existed. Example: If you say that Nichols could have bled for twelve minutes, and I say thirteen, and Gary Barnett says fourteen, and Simon Wood says fifteen and so on, then generally speaking, Simon is the person least likely to be correct. His suggestion stretches things the most and is therefore less likely than the other suggestions, at elast as long as we accept that the standard curve, so to speak, is in line with what the pathologists said in the Nichols case: that the likeliest end of the bleeding process would occur at around 3-5 minutes.

Let me know if you agree with this too.

Here comes the last point, Jeff:

5. If we insert an alternative killer into the Nichols case, then we add to the total tally of minutes of bleeding, at least if we make the presumption that this alternative killer cut Nicholsīs throat before Lechmere arrived.
We can then of course say that the alternative killer was the one who provided the number one minute of bleeding and that there is nothing stranga about Nichols bleeding at that stage at all.
But what we must note with such a killer is that he will reinterpret the last minutes of the bleeding. What was minute number nine with Lechmere as the cutter at 3.45, suddenly becomes minute number ten - if Mr Alternative did HIS cutting a minute before Lechmere only. If, as most people out here say, the throat came BEFORE the abdominal cuts and if he covered the cuts before he left the site, then we should perhaps add two minutes, making what was previously number nine of bleeding numer eleven instead. And if the pathologists were correct, then every minute that is added beyond minute number five of bleeding must be regarded as an unexpected one. Consequentially, to suggest another killer adds to the tally of improbable bleeding minutes.
Of course, Lechmeres suggested cutting is also placed at an improbable remove in time - but LESS improbable anyway.

The outcome of all of this is that we cannot say that there was probably another killer, based on the bleeding process only. The fact is that what we must say is that it was probably NOT another killer - although such a person is not impossible per se.

The existing man of flesh and blood, who we KNOW was there must always be a red hot bid if the pathologists were correct. The alteernative killer, who is a theoretical construction only, could have been the killer, but he cannot match Lechmere in terms of overall credibility, based on the bleeding process.

Itīs a good thing there is not a lot of empirical material to compare from, though ...

[JeffHamm]

Hi Fisherman,

What would be important is knowing the distribution of times it takes for the bleeding to stop. What percentage of cases does the bleeding stop in minute 1? That will be very low. What percentage stop in minute 2? that will still be low, but slightly greater than stopping in minute 1. You are still working with cumulative distributions, and they do not provide the probabilities needed here. You're talking about doing an inferential statistical analysis, and how your doing it is not how it's done. You've got a good idea, you're just implementing it wrong. I know what you're saying, and I know it feels intuitively correct, but sadly, it is not. I've outlined before how it would be done, so there's no need to repeat myself. Honest, I do this for a living, and I teach it (well, not bleeding times, but inferential statistics is a big part of my job). I do like your basic idea of looking at bleeding times. But in the end, it's a "time of death estimation", or in this case "time of wound estimation" analysis, and those have wide margins of error, which is why I'm positive that in the end what we would find is that the analysis would end up telling us that Cross/Lechmere and JtR as other cannot be distinguished from each other. Anyway, if you want to continue as you are, feel free. It is wrong, and you'll convince yourself, but it won't be correct. If you want to do it correctly, as a proper statistical test, you would need to find a distribution of "time until bleeding stops" and use the density function not the cumulative probability function, as you are currently doing.

- Jeff

[Fisherman]

Hi Fisherman,

What would be important is knowing the distribution of times it takes for the bleeding to stop. What percentage of cases does the bleeding stop in minute 1? That will be very low. What percentage stop in minute 2? that will still be low, but slightly greater than stopping in minute 1. You are still working with cumulative distributions, and they do not provide the probabilities needed here. You're talking about doing an inferential statistical analysis, and how your doing it is not how it's done. You've got a good idea, you're just implementing it wrong. I know what you're saying, and I know it feels intuitively correct, but sadly, it is not. I've outlined before how it would be done, so there's no need to repeat myself. Honest, I do this for a living, and I teach it (well, not bleeding times, but inferential statistics is a big part of my job). I do like your basic idea of looking at bleeding times. But in the end, it's a "time of death estimation", or in this case "time of wound estimation" analysis, and those have wide margins of error, which is why I'm positive that in the end what we would find is that the analysis would end up telling us that Cross/Lechmere and JtR as other cannot be distinguished from each other. Anyway, if you want to continue as you are, feel free. It is wrong, and you'll convince yourself, but it won't be correct. If you want to do it correctly, as a proper statistical test, you would need to find a distribution of "time until bleeding stops" and use the density function not the cumulative probability function, as you are currently doing.

- Jeff

I am not interested in doing something wrong, Jeff, for obvious reasons. As far as I understand, though, what you want to include in the picture is the possibility that the bleeding does not follow a straight line, in terms of volume. Would that be correct? Of course, I am working to the general idea that the bleeding will taper off over time, not necessarily evenly so, but generally speaking. I am also working to the idea that Nichols did not materially differ from the average in terms of bleeding. Can I take it that your model introduces the possibility that she did? I also work from the conception that the pathologists are correct in saying that the bleeding is more likely to go on in minutes three and five than in minute seven. Does your model work from the same assumption?

I understand that weighing in more factors may change the picture, but to begin with, would not such a thing be just as likely to increase the likelihood of guilt on Lechmereīs behalf as it would be to diminish it? Or does it only work in one way? And just how would a scenario look, practically speaking, that would point to Lechmere being LESS likely to be the killer than someone who cut Nichols at an earlier stage?

You say that a "time of wound" estimation has a wide margin of error. Does that mean that you involve the possibility of an alternative killer making that wound AFTER Lechmere and Paul examined the body? Or are you, like me, accepting that an alternative killer must have been in place before that stage?

PS. You failed to say whether or not you agree with the points I made in my last post. Why is that?

[JeffHamm]

I am not interested in doing something wrong, Jeff, for obvious reasons.

I didn't think you were. I am familiar with how intuition about probabilities, though, can convince people they are doing it right when they're not. There's a lot of underlying statistical theory with regards to how analyses are done so that they get to the inferences we're trying to make. The maths themselves are usually not too complicated at the analysis stage itself.

As far as I understand, though, what you want to include in the picture is the possibility that the bleeding does not follow a straight line, in terms of volume.
Would that be correct?

Well, not volume per se, but duration since we're interested in time. I suspect the amount of blood that is lost would also be greater near the point of cutting for all sorts of reasons, but it's the duration of the bleed that we're interested in. How long in time, from start to stop, should we expect someone to bleed under these circumstances. Some cases will bleed a short time, some a long time. So yes, in the cumulative sense, earlier minutes have more bleeders, but we'll be adding fewer and fewer as we go earlier and earlier because as we approach the point where the cut was made, almost nobody will stop bleeding in the first minute. And it's that aspect that one needs to look at to make an inference about when things happened; which is the density function.

Of course, I am working to the general idea that the bleeding will taper off over time, not necessarily evenly so, but generally speaking. I am also working to the idea that Nichols did not materially differ from the average in terms of bleeding. Can I take it that your model introduces the possibility that she did?

No, we would have to work with the idea that Nichols was a "typical bleeder". We would need a lot of information about her conditions and such to suggest she's otherwise, and we don't have that. When trying to model something, if you don't know the specific value, one is always safest to use a population average of some sort (well, sometimes one chooses the median if the population is known to be skewed, and there can be reasons to pick the mode, but those are picky details for our purposes, no matter which measure of central tendency one picks you're still just trying to get the most representive value for the typical population, so we would be assuming Nichols was typical).

I also work from the conception that the pathologists are correct in saying that the bleeding is more likely to go on in minutes three and five than in minute seven. Does your model work from the same assumption?

That has to be the case because everyone bleeding in minute 7 was also bleeding in 3 and 5, but not everyone in 3 and 5 will be in 7. But again, that's thinking in terms of cumulative data (running totals, because we're counting the same people over and over again;minute 3 & 5 recount those in minute 7). What we need is how many stop bleeding (not have stopped, but stop so during minute 3 how many switch from being a bleeder to non-bleeder). Now, each person can only fall into one time bin, so we won't be recounting the same people over and over and over. Once we do that, there may very well be more people in minute 7 than in minute 3, because very few people actually stop bleeding in minute 3.

But, yes, we could easily recover your value from these data, because if I just take those percentages of "% who switch" in each bin and add them up, that will give your value. But that's not the value used to make inferences like the one we're talking about.

I understand that weighing in more factors may change the picture, but to begin with, would not such a thing be just as likely to increase the likelihood of guilt on Lechmereīs behalf as it would be to diminish it? Or does it only work in one way? And just how would a scenario look, practically speaking, that would point to Lechmere being LESS likely to be the killer than someone who cut Nichols at an earlier stage?

As I say, depending upon the distribution, the outcome could increase or decrease the probability of Cross/Lechmere relative to JtR as Other. It does not guarantee an outcome. The cumulative probabilities, because they are not the right ones to use, always appear to favour an earlier time, but that's an arefact.

You say that a "time of wound" estimation has a wide margin of error. Does that mean that you involve the possibility of an alternative killer making that wound AFTER Lechmere and Paul examined the body? Or are you, like me, accepting that an alternative killer must have been in place before that stage?

I would assume it does simply because the distribution of pretty much every biological function has huge variability. It's all very complicated processes, and so crude measures like duration of bleed will reflect that. Doesn't mean it can't be tried though, if we had the data.

A probability distribution could very well end up suggesting that a time after Cross/Lechmere is more probable. But then, that's exactly what using a cummulative distribution would suggest too. The cummulative approach would be "she's bleeding now; everyone bleeds at the time they are cut, so she has the highest probability of having been curt right before our eyes!", which is clearly nonsense, and perhaps illustrates why the cumulative values are not the ones to go with. But yes, the most probable time might be afterwards, which could only occur if Nichols wasn't cut and mutilated until after Cross/Lechmere leave but before she's found by the police.

I think I've seen that suggested by some, but not sure if they are serious or just presenting an argument for consideration (i.e. how do we know this didn't happen ...). But we're interested in comparing the probability of two specific time points. And real information (she was found at a certain time) overrides a probability estimated time.

PS. You failed to say whether or not you agree with the points I made in my last post. Why is that?

That's because I focused on the more important issue I think we need to sort out. You're presentation was based upon the cumulative distribution, and they have a seductive intuitive feel to them, but again, even if every one of the statements you present are correct, the inferences about which hypothesis is more likely would still be incorrect because, as I say, you're using the wrong tool to make those inferences. It has to do with the repeated counting of the same data, the earlier bins are not just the probability she's bleeding, but is the total of all people who will bleed longer than that time. So if she's bleeding in minutes 7, then that is counting all of the people who will bleed in minute 9 as well, and they will make up a greater proportion of those "still bleeding at minute 7" than those "still bleeding at minute 3" (because we will have removed everyone who stopped bleeding in between, but none of those who will continue to bleed longer). That means, if she's bleeding at minute 7 there is a greater chance that she will be bleeding in minute 9 compared to if she's still bleeding at minute 3. So, the probability of being a "long bleeder" goes up the longer she's bleeding.

Again, to compare the two options, we would need a distribution of the "duration of the bleed", and if we had that, we could compare the two time points. And the outcome of that test could be in either direction, although I'm as certain as one can be without actually having the data, that the result would end up being "no difference in probability", meaning they would be effectively equal.

- Jeff

[jerryd]

Thank you Jeff and nice post.

So, in the case of Alice McKenzie, a witness that states when he saw the body the blood from her neck was "spurting", "gushing" or "running vey fast". In this case the cutter would have been at the body very recently? Correct? By the way, this witness didn't arrive at the body until a few minutes after it was discovered by PC Andrews.

[Fisherman]

I didn't think you were. I am familiar with how intuition about probabilities, though, can convince people they are doing it right when they're not. There's a lot of underlying statistical theory with regards to how analyses are done so that they get to the inferences we're trying to make. The maths themselves are usually not too complicated at the analysis stage itself.

Well, not volume per se, but duration since we're interested in time. I suspect the amount of blood that is lost would also be greater near the point of cutting for all sorts of reasons, but it's the duration of the bleed that we're interested in. How long in time, from start to stop, should we expect someone to bleed under these circumstances. Some cases will bleed a short time, some a long time. So yes, in the cumulative sense, earlier minutes have more bleeders, but we'll be adding fewer and fewer as we go earlier and earlier because as we approach the point where the cut was made, almost nobody will stop bleeding in the first minute. And it's that aspect that one needs to look at to make an inference about when things happened; which is the density function.

No, we would have to work with the idea that Nichols was a "typical bleeder". We would need a lot of information about her conditions and such to suggest she's otherwise, and we don't have that. When trying to model something, if you don't know the specific value, one is always safest to use a population average of some sort (well, sometimes one chooses the median if the population is known to be skewed, and there can be reasons to pick the mode, but those are picky details for our purposes, no matter which measure of central tendency one picks you're still just trying to get the most representive value for the typical population, so we would be assuming Nichols was typical).

That has to be the case because everyone bleeding in minute 7 was also bleeding in 3 and 5, but not everyone in 3 and 5 will be in 7. But again, that's thinking in terms of cumulative data (running totals, because we're counting the same people over and over again;minute 3 & 5 recount those in minute 7). What we need is how many stop bleeding (not have stopped, but stop so during minute 3 how many switch from being a bleeder to non-bleeder). Now, each person can only fall into one time bin, so we won't be recounting the same people over and over and over. Once we do that, there may very well be more people in minute 7 than in minute 3, because very few people actually stop bleeding in minute 3.

But, yes, we could easily recover your value from these data, because if I just take those percentages of "% who switch" in each bin and add them up, that will give your value. But that's not the value used to make inferences like the one we're talking about.

As I say, depending upon the distribution, the outcome could increase or decrease the probability of Cross/Lechmere relative to JtR as Other. It does not guarantee an outcome. The cumulative probabilities, because they are not the right ones to use, always appear to favour an earlier time, but that's an arefact.

I would assume it does simply because the distribution of pretty much every biological function has huge variability. It's all very complicated processes, and so crude measures like duration of bleed will reflect that. Doesn't mean it can't be tried though, if we had the data.

A probability distribution could very well end up suggesting that a time after Cross/Lechmere is more probable. But then, that's exactly what using a cummulative distribution would suggest too. The cummulative approach would be "she's bleeding now; everyone bleeds at the time they are cut, so she has the highest probability of having been curt right before our eyes!", which is clearly nonsense, and perhaps illustrates why the cumulative values are not the ones to go with. But yes, the most probable time might be afterwards, which could only occur if Nichols wasn't cut and mutilated until after Cross/Lechmere leave but before she's found by the police.

I think I've seen that suggested by some, but not sure if they are serious or just presenting an argument for consideration (i.e. how do we know this didn't happen ...). But we're interested in comparing the probability of two specific time points. And real information (she was found at a certain time) overrides a probability estimated time.

That's because I focused on the more important issue I think we need to sort out. You're presentation was based upon the cumulative distribution, and they have a seductive intuitive feel to them, but again, even if every one of the statements you present are correct, the inferences about which hypothesis is more likely would still be incorrect because, as I say, you're using the wrong tool to make those inferences. It has to do with the repeated counting of the same data, the earlier bins are not just the probability she's bleeding, but is the total of all people who will bleed longer than that time. So if she's bleeding in minutes 7, then that is counting all of the people who will bleed in minute 9 as well, and they will make up a greater proportion of those "still bleeding at minute 7" than those "still bleeding at minute 3" (because we will have removed everyone who stopped bleeding in between, but none of those who will continue to bleed longer). That means, if she's bleeding at minute 7 there is a greater chance that she will be bleeding in minute 9 compared to if she's still bleeding at minute 3. So, the probability of being a "long bleeder" goes up the longer she's bleeding.

Again, to compare the two options, we would need a distribution of the "duration of the bleed", and if we had that, we could compare the two time points. And the outcome of that test could be in either direction, although I'm as certain as one can be without actually having the data, that the result would end up being "no difference in probability", meaning they would be effectively equal.

- Jeff

Okay, I think things are (hopefully) beginning to clear up for me now. And in my world, Lechmere is still the likelier killer than someone who preceded him. Which is sort of where I go wrong, statistically speaking.
But let me explain.

I should have used the density method instead of the cumulative methos, I am told, and this may be so. But what I must also do to make things work statistically is to abandon any preconceptions I have. And one such preconception is that the alternative killer, if there was one, will have cut Nichols before Lechmere arrived. You now tell me that the statistical method I need to use weighs in the possibility that she was cut by the alternative killer after Lechmere was in place but before the police arrived.

And thatīs where the crux lies. I have always looked upon it like this:

If we reason that Lechmere cut Nichols at 3.45, and if the pathologists were correct in saying that the likely time of bleeding would be 3-5 minutes, then Nichols bled longer than the pathologists expected.

To introduce a killer who cut Nichols even earlier than Lechmere did, but who would nevertheless be as likely or likelier a cutter than Lechmere, would accordingly be impossible since it would require Nichols to have bled even longer, and so such a bleeding would be even less expected.

It is an easy enough equation.

But if we allow for Nichols to have been cut AFTER Lechmere left the site, then of course there will be a possibility for this killer to cut her at a time that is more consistent with the suggestions of the pathologists.

So this will be where such an alternative killer can be said to be on par with or possibly even better as a bid for the killers role than Lechmere.

If we instead shut that door and predispose that an alternative killer must have been active in the time leading up to Lechmereīs arrival, then - if I am correct - regardless of which method we use, Lechmere will be the likelier killer.

Before somebody tells me not to shut any doors, I must point out that I want to know whether this factor has the bearing I think it does, in order to better understand the finer points of the statistics and methods involved. Nothing else.

This aside, there are a few matters that I fail to understand in your post:

"...in the cumulative sense, earlier minutes have more bleeders, but we'll be adding fewer and fewer as we go earlier and earlier because as we approach the point where the cut was made, almost nobody will stop bleeding in the first minute."

How can the earlier minutes NOT have more bleeders, Jeff? It is a process where the numbers of bleeders taper off as more time is added. It cannot be a process where the numbers of bleeders grow as the minutes pass by, can it? Or are you simply saying that we are likely to find more people who finish their bleedings if we travel further away in time, whereas such people will be fewer the closer we get to the cutting stage?

"No, we would have to work with the idea that Nichols was a "typical bleeder". We would need a lot of information about her conditions and such to suggest she's otherwise, and we don't have that. When trying to model something, if you don't know the specific value, one is always safest to use a population average of some sort (well, sometimes one chooses the median if the population is known to be skewed, and there can be reasons to pick the mode, but those are picky details for our purposes, no matter which measure of central tendency one picks you're still just trying to get the most representive value for the typical population, so we would be assuming Nichols was typical).

That has to be the case because everyone bleeding in minute 7 was also bleeding in 3 and 5, but not everyone in 3 and 5 will be in 7. But again, that's thinking in terms of cumulative data (running totals, because we're counting the same people over and over again;minute 3 & 5 recount those in minute 7). What we need is how many stop bleeding (not have stopped, but stop so during minute 3 how many switch from being a bleeder to non-bleeder). Now, each person can only fall into one time bin, so we won't be recounting the same people over and over and over. Once we do that, there may very well be more people in minute 7 than in minute 3, because very few people actually stop bleeding in minute 3."

What the pathologists suggested was that minutes 3 and 5 were the ones most likely to see the bleeding stop, and so why you say that very few people actually stop bleeding in minute 3 is a bit beyond me. They also said that it was likelier for people to stop bleeding in minute 3 than in minute 7, so it seems wrong when you suggest that more people could have stopped bleeding in that minute than in minute 3. Maybe you just chose the wrong numbers? If you had spoken of minute 2 and minute 5, it would make more sense to me.

[JeffHamm]

Okay, I think things are (hopefully) beginning to clear up for me now. And in my world, Lechmere is still the likelier killer than someone who preceded him. Which is sort of where I go wrong, statistically speaking.
But let me explain.

I should have used the density method instead of the cumulative methos, I am told, and this may be so. But what I must also do to make things work statistically is to abandon any preconceptions I have. And one such preconception is that the alternative killer, if there was one, will have cut Nichols before Lechmere arrived. You now tell me that the statistical method I need to use weighs in the possibility that she was cut by the alternative killer after Lechmere was in place but before the police arrived.

And thatīs where the crux lies. I have always looked upon it like this:

If we reason that Lechmere cut Nichols at 3.45, and if the pathologists were correct in saying that the likely time of bleeding would be 3-5 minutes, then Nichols bled longer than the pathologists expected.

To introduce a killer who cut Nichols even earlier than Lechmere did, but who would nevertheless be as likely or likelier a cutter than Lechmere, would accordingly be impossible since it would require Nichols to have bled even longer, and so such a bleeding would be even less expected.

It is an easy enough equation.

But if we allow for Nichols to have been cut AFTER Lechmere left the site, then of course there will be a possibility for this killer to cut her at a time that is more consistent with the suggestions of the pathologists.

So this will be where such an alternative killer can be said to be on par with or possibly even better as a bid for the killers role than Lechmere.

If we instead shut that door and predispose that an alternative killer must have been active in the time leading up to Lechmereīs arrival, then - if I am correct - regardless of which method we use, Lechmere will be the likelier killer.

Again, not quite. When one uses the density function whether or not Cross/Lechmere or JtR as other, becomes the more probable is possible. It would depend upon what that density function looks like, something neither of us knows because that would require an empirical study from which to obtain that information. I'm not presupposing it would come out either way, although I do predict that it will be associated with such large variation that the small amount of time we're talking about will result in near identical probability values, leading us to be unable to assert one as more probable than the other. I could be wrong, but I would be highly surprised if I was on that point (the variability of the data).

Before somebody tells me not to shut any doors, I must point out that I want to know whether this factor has the bearing I think it does, in order to better understand the finer points of the statistics and methods involved. Nothing else.

This aside, there are a few matters that I fail to understand in your post:

"...in the cumulative sense, earlier minutes have more bleeders, but we'll be adding fewer and fewer as we go earlier and earlier because as we approach the point where the cut was made, almost nobody will stop bleeding in the first minute."

How can the earlier minutes NOT have more bleeders, Jeff? It is a process where the numbers of bleeders taper off as more time is added. It cannot be a process where the numbers of bleeders grow as the minutes pass by, can it? Or are you simply saying that we are likely to find more people who finish their bleedings if we travel further away in time, whereas such people will be fewer the closer we get to the cutting stage?

The earlier minutes will have more bleeders because you are counting the same people over and over again (so really, you don't have "more bleeders" rather you just have bigger numbers due to recounting the "same bleeders" over and over again). Someone bleeding in minutes 7 is getting counted in minutes 1-6 as well. So the counts will be higher becomes some will stop bleeding at some point before minute 7. But you are re-counting the same people, so really, there are not "more bleeders" there is just the fact that if you recount the same people over and over you get a bigger number.

This is why the cumulative distribution leads to the wrong answer with regards to making an inference. it recounts the same people over and over and over. That has a massive impact on how you can treat the values converted to percentages.

"No, we would have to work with the idea that Nichols was a "typical bleeder". We would need a lot of information about her conditions and such to suggest she's otherwise, and we don't have that. When trying to model something, if you don't know the specific value, one is always safest to use a population average of some sort (well, sometimes one chooses the median if the population is known to be skewed, and there can be reasons to pick the mode, but those are picky details for our purposes, no matter which measure of central tendency one picks you're still just trying to get the most representive value for the typical population, so we would be assuming Nichols was typical).

That has to be the case because everyone bleeding in minute 7 was also bleeding in 3 and 5, but not everyone in 3 and 5 will be in 7. But again, that's thinking in terms of cumulative data (running totals, because we're counting the same people over and over again;minute 3 & 5 recount those in minute 7). What we need is how many stop bleeding (not have stopped, but stop so during minute 3 how many switch from being a bleeder to non-bleeder). Now, each person can only fall into one time bin, so we won't be recounting the same people over and over and over. Once we do that, there may very well be more people in minute 7 than in minute 3, because very few people actually stop bleeding in minute 3."

What the pathologists suggested was that minutes 3 and 5 were the ones most likely to see the bleeding stop, and so why you say that very few people actually stop bleeding in minute 3 is a bit beyond me. They also said that it was likelier for people to stop bleeding in minute 3 than in minute 7, so it seems wrong when you suggest that more people could have stopped bleeding in that minute than in minute 3. Maybe you just chose the wrong numbers? If you had spoken of minute 2 and minute 5, it would make more sense to me.

I was just using numbers as an example, if 2 and 5 get the point I was making across feel free to change them to 2 and 5. All I'm saying is that you cannot use the cumulative distribution to make the inferences you are. Do you know what study the pathologists are working from? If you could find that study we might be able to use the data reported in it to construct the appropriate distribution. Pending upon what is reported in that study, I might be able to easily extract those values and we could have a look. That would be interesting, and kind of cool.

- Jeff

[Fisherman]

This is getting more and more absorbing by the minute.

So hereīs a question for you, Jeff:

You say that the two suggested killers, Lechmere and Mr Alternative, can probably not be told apart in tertms of viability as the killer and this owes to how we cannot establish the expanse of the bleeding time and use the density method.

Assuming that the alternative killer was in place before Lechmere was and accepting that the pathologists were correct when they said that the bleeding was most likely to end around minutes 3-5, plus accepting that we cannot say when the bleeding stopped, therefore leaving that particular parameter totally open, what happens if we move that alternative killer along the time axis? Does he at any stage become less viable than Lechmere as the killer? If we say that he was in place 5, 15, 25 or 100 minutes before Lechmere, can we at any of those stages say, working with the density method, that Mr Alternative is not as likely as killer as Lechmere? If we can conclude such a thing, just how large must the time gap be? Or can it be of any size, and the two are nevertheless equally viable as long as we leave the bleeding time open?

[Abby Normal]

lest anyone get lost in the minutia, its clear that the fact that nichols was still bleeding puts lech clearly in tje frame for her murder because we all agree at some point bleeding stops. she clearly wasnt murdered a half hour before lech arrived. hes seen standing alone near a recently killed victim before trying to raise any alarm. paul dosnt see or hear him walking in front of him. by lechs own admission on when he left his house he has time to commit the murder. there is no other evidence of anyone else around.
lech is clearly in the frame for her murder.

[Paddy Goose]

This is called guilt by association. He was walking to work and discovered the victim. He is not guilty.

Cross stopped because he noticed her laying there. Something was horribly wrong. Soon Paul came along and he called him over to have a look see, too. Together they went and found the first policeman and notified him.

It's simple. Also, a good deep study of that time and place where multiple people would be walking to work at that hour wouldn't hurt.

Paddy

[Fisherman]

This is called guilt by association. He was walking to work and discovered the victim. He is not guilty.

Cross stopped because he noticed her laying there. Something was horribly wrong. Soon Paul came along and he called him over to have a look see, too. Together they went and found the first policeman and notified him.

It's simple. Also, a good deep study of that time and place where multiple people would be walking to work at that hour wouldn't hurt.

Paddy

It would be outrageous if somebody said that Lechmere proved himself the killer - or even a suspect - by way of finding Nicholsīs body on his way to work. I am a hundred per cent with you on that score.

My main problem, though, is that people do not look at the whole picture when assessing the carman.

Yes, anybody can have the poor luck of finding a dead body if it is laying around in a public space. The fact is, somebody WILL find it.

But letīs take a look on the matters that line the road of Lechmereīs finding Nicholsīs body.

To begin with, he found it at a stage where she was still bleeding and would go on to do so for many minutes. That does not per se make Lechmere any guiltier. If you can find a long cold body, then you can find a warm and bleeding body too. But if you DO find a warm and bleeding body, then you will become a person of interest if that dead body belongs to a person who has been murdered, if the killer has not been identified and if there are no other circumstances that make it clear that you are not the killer.

This isolates a group of people who have found or claim to have found, or who have themselves been found with a murdered person at a remove in time that is consistent with being the killer. Some of these people will be innocent, some will be guilty, no specific figures stated. Charles Lechmere belongs to this group of people. We cannot take his word as gospel when it comes to how he said that Robert Paul arrived immediately after himself, thereby supplying an alibi. The reason is that if Lechmere was guilty, then he would likely lie about such a thing in order to evade responsibility.

These are the basics of what kind of status you get when finding a dead body, adding the varying circumstances into the picture.

As most people will gather, if the police cannot find a killer and if they have a person of interest who found or claimed to have found or was found by a freshly dead body, they will look into the specifics of this person of interest and try to see if they can eliminate him or her from their investigation. If they cannot, they wil instead see if there is evidence pointing in his or her direction as being the killer. And in Lechmereīs case, we have a whole array of such matters. We have the correlation between the area he traversed and three further murders, we have the correlation between his mothers/daughters place and the Stride murder and the fact that the Eddwes case happened more or less along his old working trek from James Street to Broad Street. There is also the fact that the four murders that happened along his logical working treks also happened on what was normally early working day mornings, whereas the murders that happened on a Saturday - which was nornally a day off - did not happen along his work treks. So that all fits the picture.

Then we have the fact that he alerted Paul to the body - but would not assist in propping Nichols up.

Then we have the fact that he disagreed with Mizen about what was said on the murder morning. And interestingly, what he said according to Mizen was in perfect line with trying to pass the police by unquestioned.

Then we have the fact that he did not use the name he was registered by when speaking to the police, whereas he seems to have used the name Lechmere in all other contacts with the authorities.

Then we have the fact that the clothes were pulled down over the wounds on Nicholsīs body, something that would be essential if he wanted to con Robert Paul.

Then we have the fact that Nichols seemingly bled for so long a period that it becomes hard to believe in an earlier killer.

Then we have the fact that once a torso victim was dumped on Ripper territory, it was dumped on the exact street where Lechmere and his immediate family had a very large presence over the years.

Then we have the fact that the rag from Eddowesīs apron was dumped in a place that indicated that the killer was walking northeast - towards the area where Lechmere lived.

Then we have the fact that another rag was found the day after the Pinchin Street torso was found - and lo and behold, that rag was dumped in a direct line from the railway arch in Pinchin Street up to 22 Doveton Street.

Now, by all means, we can pick any of these matters and supply it with an innocent explanation. Take the last one, for example:

"Nah, we donīt even know that this rag was connected to the Pinchin Street murder, and it could have been anyone who threw it away there. If you want to throw a rag away, you have to do it somewhere, and regardless of where you do it, it WILL point in SOME direction and there WILL be someone living along the line it points out. Does that make them all killers?"

This, though, is not how a proper investigation is made when looking at a suspect. A proper investigation weighs these matters together and look at whether the collected weight is enough to make somebody a suspect. it does not pick the matters off, one by one, supplying them all with - easily enough made up - innocent explanations.

At the end of the day, either that rag is a further absolutely massive coincidence in an ocean of other coincidences all wrongfully pointing to Lechmere, or it is circumstantial evidence pointing a killer out. And at the end of that self same day, that is precisely why we should not say that somebody had to find the body and speak of guilt by association. There is an overwhelming collection of indicators pointing in the exact same direction, and it lies upon us to do a sound weighing up of them instead of isolating them and picking them off one by one. It is the easisest thing in the world to do, but it is also a surefire way to exonerate any potential killer, regardless of the amount and quality of the evidence.

[JeffHamm]

This is getting more and more absorbing by the minute.

So hereīs a question for you, Jeff:

You say that the two suggested killers, Lechmere and Mr Alternative, can probably not be told apart in tertms of viability as the killer and this owes to how we cannot establish the expanse of the bleeding time and use the density method.

Assuming that the alternative killer was in place before Lechmere was and accepting that the pathologists were correct when they said that the bleeding was most likely to end around minutes 3-5, plus accepting that we cannot say when the bleeding stopped, therefore leaving that particular parameter totally open, what happens if we move that alternative killer along the time axis? Does he at any stage become less viable than Lechmere as the killer? If we say that he was in place 5, 15, 25 or 100 minutes before Lechmere, can we at any of those stages say, working with the density method, that Mr Alternative is not as likely as killer as Lechmere? If we can conclude such a thing, just how large must the time gap be? Or can it be of any size, and the two are nevertheless equally viable as long as we leave the bleeding time open?

Hi Fisherman,

I can't say how far along one would have to move him because that requires actually knowing the density function for this kind of data, and to know that we don't just have to know the most common time range (you've mentioned 3-5 minutes) but the variability of the times as well. So, I can't answer that question because we don't have the information available to us to work out what the answer is. We would, however, have to know when the bleeding stopped and work back in time from there. While the bleeding is ongoing, we can't know where in the process we are (is this minute 3 of bleeding or is it minute 5? If we knew that, we don't need a test, do we?)

However, in general terms, as I've mentioned above, yes, it is entirely possible for either Cross/Lechmere or JtR as other to end up being the more likely (or the less likely if you prefer). That's one of the signals that this is an actual test of two ideas, where you were headed with using the cumulative probabilities is along a line that can only imply the more recent is the more probable, even in situations where that's not true. That's not a very good test.

Again, even though I don't know what the density function probabilities actually are, having looked at a few types of data based upon biological processes of this sort, I do know they are associated with very high variation, and a small time difference of 30-60 seconds is just going to be too small to be detected (meaning, we wouldn't be able to tell with any certainty which was actually the more probable).

- Jeff

[Fisherman]

Hi Fisherman,

I can't say how far along one would have to move him because that requires actually knowing the density function for this kind of data, and to know that we don't just have to know the most common time range (you've mentioned 3-5 minutes) but the variability of the times as well. So, I can't answer that question because we don't have the information available to us to work out what the answer is. We would, however, have to know when the bleeding stopped and work back in time from there. While the bleeding is ongoing, we can't know where in the process we are (is this minute 3 of bleeding or is it minute 5? If we knew that, we don't need a test, do we?)

However, in general terms, as I've mentioned above, yes, it is entirely possible for either Cross/Lechmere or JtR as other to end up being the more likely (or the less likely if you prefer). That's one of the signals that this is an actual test of two ideas, where you were headed with using the cumulative probabilities is along a line that can only imply the more recent is the more probable, even in situations where that's not true. That's not a very good test.

Again, even though I don't know what the density function probabilities actually are, having looked at a few types of data based upon biological processes of this sort, I do know they are associated with very high variation, and a small time difference of 30-60 seconds is just going to be too small to be detected (meaning, we wouldn't be able to tell with any certainty which was actually the more probable).

- Jeff

Okay. You donīt know how far you would have to move the alternative killer before he becomes a less likely killer than Lechmere.

Does that mean that he WILL become a less likely killer sooner or later as we move him along the scale? Or do we have to have the bleeding times fixed before we know that?

[JeffHamm]

Okay. You donīt know how far you would have to move the alternative killer before he becomes a less likely killer than Lechmere.

Does that mean that he WILL become a less likely killer sooner or later as we move him along the scale? Or do we have to have the bleeding times fixed before we know that?

Hi Fisherman,

Presumably, yes, if you move JtR as other two days further away for example, and leave Cross/Lechmere in place, I'm pretty sure the data would indicate JtR as Other would be far less likely. But that's hardly the alternative theory though, is it? Proving Cross/Lechmere is better than an impossible situation doesn't mean much, but if you're really just wondering if there would be be a gap large enough that the test could detect, then yes, there would be time differences that could be reliably separated. However, as I've been saying, I'm positive those time differences would be much larger than the ones being considered here and that the 30-60 second gap is below the sensitivity of a test to differentiate. Biological processes like bleeding are just not that precise of a measurement.

Basically, and again, without us actually having data on this, including at a bare minimum, the average time for bleeding to stop and the variation of those times (we must have both), there's no way to know how it would come out. And even if we had both of those, we would have to assume that bleeding duration is normally distributed, which we know it can't be (because it can't be negative time, or bleeding started before the injury). It will probably be a skewed distribution of some sort, so we would need to know the mean, variance, and the skew in order to construct our density function model. If we had that, I could answer such questions, but without those, it's anybody's guess.

- Jeff

[Fisherman]

Hi Fisherman,

Presumably, yes, if you move JtR as other two days further away for example, and leave Cross/Lechmere in place, I'm pretty sure the data would indicate JtR as Other would be far less likely. But that's hardly the alternative theory though, is it? Proving Cross/Lechmere is better than an impossible situation doesn't mean much, but if you're really just wondering if there would be be a gap large enough that the test could detect, then yes, there would be time differences that could be reliably separated. However, as I've been saying, I'm positive those time differences would be much larger than the ones being considered here and that the 30-60 second gap is below the sensitivity of a test to differentiate. Biological processes like bleeding are just not that precise of a measurement.

Basically, and again, without us actually having data on this, including at a bare minimum, the average time for bleeding to stop and the variation of those times (we must have both), there's no way to know how it would come out. And even if we had both of those, we would have to assume that bleeding duration is normally distributed, which we know it can't be (because it can't be negative time, or bleeding started before the injury). It will probably be a skewed distribution of some sort, so we would need to know the mean, variance, and the skew in order to construct our density function model. If we had that, I could answer such questions, but without those, it's anybody's guess.

- Jeff

Okay, thanks for the lesson. Statistics is a funny discipline at times. Much can be ”proven” by it. In the case at hand, I think I will simply stay with the oldfashioned take on things, telling me that if the forensic specialists tell me in a random case that I probably have a cutting time of around 00.05, then if a man X has been at the cutting site at a time of 00.00 only to then disappear, and another man Y has been in place at the cutting site at 00.01 and then left it, then Y is the one who is in place at the cutting site at a time that is closer to the estimation made by the forensics, and that will make me favor Y over X as the likelier cutter.
Similarly, since the alternative Bucks Row killer will add time to the bleeding, time not expected by the pathologists, I will favor Lechmere over him because it saves an unspecified amount of bleeding time. (Of course, I also tend to prioritize people whos existence is proven over people made up out of thin air, but that is another matter).

I make for a poor statistician, I’ m afraid.