And that’s my problem. It’s statistically unlikely because of what? What makes it statistically unlikely? If the X factor that makes it statistically unlikely is present, then the statistic needs to reflect this.
I get the impression that there’s this feeling that black skin is some sort of invulerabity against being ganged raped by white men.
This is probably due to my stupid typo in my first post, but I was really just trying to clarify things. Sensitivity and specificity are most useful in population-based studies (e.g. epidemiology). However, if you’re trying to translate population-based studies down to an individual level, the positive and negative predictive values are more relevant if you’re using Bayesian analysis.
Which means that Malacandra’s example is essentially correct. Let’s look at the stats problem he presented:
From this presentation, we are given that:
The witness’s “sensitivity to maroon” is 95% (19/20 maroon cars identified as such).
The witness’s “specificity for maroon” is 95% (19/20 not-maroon [i.e. red] cars identified as such).
The prevalence of “maroon” is 2%.
Let’s now use a population sample of 1000 cars.
Of this population, 20 cars are maroon, and 980 are red.
Of the 20 cars which are truly maroon, the witness will identify 19 of them as maroon, and 1 as not-maroon (red) with a sensitivity of 95%.
Of the 980 cars which are truly not-maroon (red), the witness will identify 49 of them as maroon, and 931 as not-maroon (red) with a specificity of 95%.
Therefore, the witness has identified a total of 19+49 = 68 cars as maroon, but only 19 of them are truly maroon. Thus, the positive predictive value of a “maroon” result is 19/68 = 27.9%.
This is why general population screening for rare diseases is often found to be not worth the trouble, because the majority of positive results are false positives.
I apologize in advance if I screwed up the numbers; it’s not really my field and I’m doing this off the cuff.
My formula terms Your formula terms
True negatives reds identified as red
True negatives reds IDed as red
False positives reds IDed as maroons
True positives maroons identified as maroon
True positives maroons IDed as maroon
False negatives maroons IDed as red
These seem to be to be identical. What am I missing?
Given the “givens” in the problem statement, it seems to me that all of the terms are known values. This is based on the assumption, which I may have incorrectly made, that the 95% accuracy applied to distinguishing red from maroon and maroon from red. On preview, I realize I have no idea how to make columns in a post.
Let’s go with this for a minute. Let’s say, for the sake of argument, that only 5% of all rapes in the US are of White men raping Black women. How are you going to use that to determine if this case is credible unless you know something about the number of times that type of rape is falsely reported? What if only about 5% of all reported rapes are of Black women by White men? That would indicate that when a Black woman reports such a rape, it generally holds true-- even if the event itself is relatively rare.
So, if you wanted to make some general statement about how believable your random White on Black rape is, you need to know something about how often false claims of rape are made in those circumstances-- which is something we don’t know. It’s not the rarity of the event that matters, because even rare events do happen (remember the 5% assumption).
But let me be clear that even that type of statstical analysis is a gross oversimplification for all the reasons we’ve been talking about. Wwe realy don’t know what motivates your “average” rapist or whether the guys in question could be considered your “average” rape suspect. What is an “average rape suspect” anyway?
Superb analysis there, monstro.
Doesn’t it mean a lot in the hypothetical though? The witness has stated that the car in question is maroon. Based on the numbers provided, it’s actually much more likely that it’s red (regardless of what the witness said) and thus the police should continue to focus the majority of their investigation on red cars. Right?
I’m also curious about your comment here:
You and I agree that we will pick 1 American citizen out of a voter registration roll. If that citizen is obese, you will pay me $5. If not, I will pay you $1. Who makes money and why? Does this not equate to a 33% that any given American citizen is obese?
Like your entry into this thread? Like your insistences that stats are relevant for determining the likelihood of the Duke accusation? Why should she admit to making mistakes she hasn’t made, when you can’t even concede that you are on the wrong side of an argument?
But she DIDN’T bungle that. Pazu and even Dragon Ash both noticed the same flaw that YWTF caught. The only difference is that YWTF insisted it be acknowledged by Malacandra. And guess what? He DID.
Why can’t you?
The only one in denial is you. You know your argument stinks to heaven, but you’re too weak to admit it.
What a minute. Earlier, didn’t you say that stats are useful only in the absence of evidence. And now you say they are but “one data pointed to be factored in”. In with what? Evidence?
If I have evidence that I got hit by a Ford Pinto, will stats showing that only 0.005% of the cars on the road are Ford Pintos need to be factored in to assess my allegation?
I thought Huerta’s constantly overlooked point was that stats are helpful in the ABSENCE OF EVIDENCE. You dig your hole deeper whenever you post, Weirddave.
emphasis mine
(Note that I’m quoting you directly so that I can’t be said to be “miscontruing” your posts)
Can I ask you to clarify what the bolded part means? Because–and I might be wrong–it sounds like what you mean is that statistics should only be used when it corraborates, not contradicts, evidence. And if this is indeed what you’re saying, then you once again point to your own shoddy reasoning. Isn’t doing that cherry picking the data?
Me being raped by a white man right now does not mean that a black woman in California is suddenly more at risk of being attacked by a white guy.
Let me get the most elementary example I can think of:
I have a bag full of marbles. YWTF, living thousands of miles away from me, has another bag of marbles. Both of our bags have nineteen white marbles, one red marble. The chance that either of us will draw out a red marble is 1/20.
Mind you, this is set by the rules of probability, not a historical data set (aka statistics).
I draw a red marble on my first attempt. What’s YWTF’s chances of drawing a red marble?
1/20. The same probability as it was before.
This year, we might see a sudden increase in the incidents of white-on-black rapes, leading to a concomittant jump in the crime stats. This does not mean that black women are suddenly more at risk of being raped. Nor does it mean that an allegation of white-on-black rape is more plausible, more probable, or more likely. White rapists in Atlanta, GA are independent of white rapists in Suffolk County, NY, just like my red marble exists independently of YWTF’s red marble.
And this hasn’t been asked yet, but what do you consider “likely”? Let’s say the crime stats say white teenagers represent 40% of those charged with shoplifting. Your teenage daughter is accused of shoplifting. Do you assume, based on the stats, that she’s a likely culprit? Does that particular statistic tell you anything about how many white teenagers are NOT shoplifters? Wouldn’t this be more important a statistic than the other one?
Seriously, if you can’t see where your arguments fall short after this post, I will have to give up.
I hope you can see that this hypothetical contains more meaningful info that the other that you presented.
Using my formulas this is what I get:
specificity = reds identified as red/( reds IDed as red+ reds IDed as maroons)
= 19/(19+1) = .95
sensitivity = maroons identified as maroon/(maroons IDed as maroon + maroons IDed as red)
= 19/(19+1) = .95
Which means that 95% of the time the witness should be able to correctly identify a maroon car.
The prevalence of maroon cars is 2%. The formula for positive predictive value is:
PPV = (maroon prevalence–MP)(sensitivity)/(MP)(sens) + (1-MP)(1-sens) =
=(.02)(.95)/(.02)(.95) + (.98)(.05)
= .28 or 28%
This is a relatively low number, so the witnesses testimony should be treated accordingly. Other evidence will be needed to pull up the slack, but they still would be good to put on the witness stand. Someone who can spot a maroon car correctly 95% is a reliable witness.
As I hope is clear by now, Malacandra’s orginal hypothetical did not provide the information necessary to reach this final conclusion.
I think you misunderstand her. If we have a stat showing that 33 % of Americans are obese, then yes–it will be safe to assume that a random draw of Americans will show that 33% of them are obese.
But an individual does not have a 33% chance of being obese. A figure like this is based on an assessment of risk.
One out of every four women has been sexually assaulted or abused. However, that doesn’t mean that women have a 25% chance of being raped.
Actually you’re right in that they are identical. Mea culpa. The problem comes in when you assume facts that are not in evidence such as the “number of maroons IDed as maroon”.
Okay, I’ve got the obese thing. You’re right I did misunderstand the point.
Not necessarily, if only for the fact that rare events do happen.
This brings us back to the point that Ellis Dee was trying to argue about a murdered child. Just because a parent is statistically more likely to be the murderer than a stranger, it does not mean that any accusation that a stranger did it should be viewed skeptically. Particularly if there is a witness who can testify to that affect.
The car thing is no different. A witness’s testimony is one data point, but it’s not enough by itself to bring a case down. A witness with 95% accuracy in IDing maroon cars is hardly something to dismiss, though. That doesn’t change just because maroon cares are rare.
Your set up is equivalent to picking someone at random, which I already said should give you 33% odds.
The problem would be if I was to say that every single American, regardless of underlying factors, has a 33% of being obese. This is not true at all. A young, nonsmoking, vegetarian male with an active lifestyle is not as likely to be obese as a middle-aged, smoking, omnivorous female with a sedentary lifestyle.
No, the “flaw” was that neither poster was evaluating the relevant statistic, which is the positive predictive value. The difference was that I thought it was pretty clear that Malacandra was getting at it, in a roundabout way.
I could not even evaluate the PPV without sensitivity, and that was not assumable from the 95% statistic initial given. Is that clear to you?
You already went on the record saying that this isn’t your area, so please watch how strongly you make your assertions.
We’re getting clearer now! Cool. I made that assumption about the 95% applying to IDing both colors. I think that assumption was intended in the hypothetical. Maybe not, or maybe it was, but not clearly stated. I still think you would have been better off solving the simple math problem and then laying out the reasons it didn’t apply, rather than arguing against the structure of the problem first. That’s what I was reacting to (plus maybe a little bit of condescension of your part). I’m sorry if my choice of words offended you. That’s how I saw it at that moment, and we’re in the pit. Through the many pages of this thread, I’ve agreed (silently) with your position on the relevance of stats in the Duke case.
Hey, what do you mean, ‘even’? This might be the pit but let’s keep it civil.
Hoo-rah. Finally we get down to the meat and potatoes: that even a reliable witness’s report of a statistically unlikely event should be viewed with scepticism. And it’s not a bit of good putting the witness on the stand if counsel for the defence knows his math, for he can easily demonstrate that despite her 95% reliability in case of a balanced sample, the real-world figures are such that she is much more likely to be wrong than not. Of course, you can hope that the jury’s grasp of math is on a par with your own.
The original hypothetical did provide the information necessary. It’s just that having got the answer wrong by being wedded to your notion that a 95% reliable witness must still be 95% likely to be right no matter what, you had to cast about you desperately to come up with a justification for your view. You even resorted to blurting out “Well, the witness could just say ‘red’ all the time and be right 980 times out of a thousand!”… without explaining how the witness thereby came to identify the car as maroon in the first place. :dubious: Well, finally we’ve managed to fight ignorance, and better late than never.
Really, I wasn’t out to spring a “gotcha” on you with the face so much as to demonstrate a perfectly simple point: reports of unlikely events are probably mendacious or mistaken. Even if my “taste exact for faultless fact/Amounts to a disease”, the smart line to take if I start talking about the conversation I had this morning with a brontosaurus-riding Martian is to assume I’m not telling the truth, no?
Talking of “bring it on”, either produce substantive evidence that I am a racist or shove it up your stinkhole, cretin. The first four words after your last comma was where that sentence should have ended. :wally
You’re correct; I had simply assumed that his initial statement meant that the 95% applied both to sensitivity and specificity, but it’s true that there’s no reason to assume so.
It isn’t my area, true; I’m not a statistician. I’m a pathologist, so I’m mostly familiar with clinical testing characteristics and how to translate them to clinicians who call and ask me questions like “The test was positive; how likely is it that my patient has disease X?”
So can you explain to my simple mind why positive predictive value isn’t what we should be looking at?
Maybe it’s just me, but I don’t see how this is any shape or form relevant to the Duke rape allegation and the usefulness of stats in evaluating the merits of this case. We’re just naval gazing now.
We were talking about population-level statistics like the DOJ crime stats. How did we get so far off track? We’re just naval-gazing now.
Just trying to steer the conversation back to the original topic.
“Skepticism” is not the right word, though. Being skeptical suggests that something doesn’t sound right, e.g doesn’t sound reasonable or plausible, and therefore should not be believed.
The appropriate word to use in your hypothetical is “weight”. Meaning, how many weight do you give the witness’s testimony?. Is it damning evidence or is it relatively light-weight stuff?
This is a mistake. What lawyer in his right mind would turn away a witness who with 95% accuracy can identify a vehicle matching the defendant’s description? The PPV is relatively low, but the witness is still credible (reliable) and therefore, still adds evidence to the prosecution’s side. That’s all that is necessary.
Your hypothetical doesn’t support this conclusion. To an extent, all allegations are unusual and unique. That doesn’t necessarily make them false.
**There is nothing, on its face, improbable about a maroon car being involved in a hit-and-run. This doesn’t change if the maroon prevalence is 2% or 80%. **
I don’t know what else to say if you still disagree with me on this point.