T'ai Chi : Some math/stat questions for you
BillHoyt said:
No I didn't, I said "other".
In any case, what is your answer,
1.
2.
3.
or other?
Why are you so un-specific on this point? 1, 2, 3, or other.
Oh, by the way, I don't think having infinity in a closed interval makes sense...
Re: Re: Re: Re: T'ai Chi : Some math/stat questions for you
Bill, from a little ways back:
Do you feel that my answer (about the use of the t-test) "especially in comparing the means of two groups" is different from your answer of "The most usual use is to compare two samples for sameness." ?
About your question "What does the standard deviation have to do with efficiency?": Do you still disagree with my answer of: "Ratios of the standard deviations (their squares, the variances anyway) are used to determine the relative efficiency of experiments." ?
Bill, from a little ways back:
Do you feel that my answer (about the use of the t-test) "especially in comparing the means of two groups" is different from your answer of "The most usual use is to compare two samples for sameness." ?
About your question "What does the standard deviation have to do with efficiency?": Do you still disagree with my answer of: "Ratios of the standard deviations (their squares, the variances anyway) are used to determine the relative efficiency of experiments." ?
davefoc
Philosopher
BillHoyt,
Count me as one who has missed the point also.
As I see it you challenged T'ai Chi's claim of statistical expertise by asking him some questions relating to his statistical expertise and thereby implying some statistical knowledge on your part to judge his answers.
T'ai Chi answered your questions and now challenges your ability to judge his statistical expertise with some questions of his own.
Following your rules you can at this point, as you have done, refuse to answer, but claiming that T'ai Chi has missed the point seems pretty lame (although not as lame as your little quip about integers or something).
What point is it that he has missed? I thought your challenge was interesting and creative, but I didn't realize that there was a special Bill Hoyt rule that meant it could only be one way.
Have you talked this over with your fellow strip tease club bouncers? What have they said? Perhaps some of them have offered to help you with some of the questions.
Count me as one who has missed the point also.
As I see it you challenged T'ai Chi's claim of statistical expertise by asking him some questions relating to his statistical expertise and thereby implying some statistical knowledge on your part to judge his answers.
T'ai Chi answered your questions and now challenges your ability to judge his statistical expertise with some questions of his own.
Following your rules you can at this point, as you have done, refuse to answer, but claiming that T'ai Chi has missed the point seems pretty lame (although not as lame as your little quip about integers or something).
What point is it that he has missed? I thought your challenge was interesting and creative, but I didn't realize that there was a special Bill Hoyt rule that meant it could only be one way.
Have you talked this over with your fellow strip tease club bouncers? What have they said? Perhaps some of them have offered to help you with some of the questions.
davefoc said:
I'm sorry you've missed the point here, davefoc. Woodini had the right to refuse to play. Those are not my rules; they are Randi's. Woodini claimed degrees and had the audacity to demand others reveal their own. That is an arrogant appeal to authority. The new call that I prove an ability to judge statistical expertise is yet another appeal to authority.
What is your point in supporting this call? Nobody should post here until they prove competence in a particular subject? That appears to be Woodini's idiotic stance.
I have made no claims regarding degrees. I will make none now. I will not play authority games, and neither should anybody here at JREF. I will always support my claims and support my objections to woo claims with references to papers from peer-reviewed journals, textbooks or other trustworthy sources. Not with the laughable woodini suggestion of "scanned in" degrees.
People here will simply have to deal with me as a bouncer at a local strip club, and examine statements I make about particular topics on their own merits.
Cheers,
Bill, are you stating that you refuse to "play"? Do you refuse to answer my questions? If so, then state it, and I'll post my answers, because others are interested in discussing the topics.
Bill, you had the audacity and arrogance to state that my degrees are worthless, if I even had any, and critique my statistical knowledge (at times very incorrectly) when you, yourself, apparently know a fraction of what I do in that area. You've also had the audacity and arrogance to not admit that I got more questions correct then you are giving me credit for (ie, the t-test one, and the standard deviation and efficiency one, for starters).
It is no appeal to authority Bill, it is an appeal to determine how much statistics you know. If you are critiquing others, I'd hope you'd know enough, otherwise you are just critiquing without any real knowledge from which to critique.
Bill, we've been off of the topic of degrees for a long time now, Bill.
Could you please answer the questions I've asked, or clearly state your refusal?
Or ignore all the questions like you've been doin'.
Bill, you had the audacity and arrogance to state that my degrees are worthless, if I even had any, and critique my statistical knowledge (at times very incorrectly) when you, yourself, apparently know a fraction of what I do in that area. You've also had the audacity and arrogance to not admit that I got more questions correct then you are giving me credit for (ie, the t-test one, and the standard deviation and efficiency one, for starters).
It is no appeal to authority Bill, it is an appeal to determine how much statistics you know. If you are critiquing others, I'd hope you'd know enough, otherwise you are just critiquing without any real knowledge from which to critique.
Bill, we've been off of the topic of degrees for a long time now, Bill.
Could you please answer the questions I've asked, or clearly state your refusal?
Or ignore all the questions like you've been doin'.
Re: Re: T'ai Chi : Some math/stat questions for you
I was re-reading the thread, and saw this.
t^2 = F, only with an F-distribution with 1 degree of freedom for the numerator. (and any degree of freedom for the denomenator). Most people doing statistics encounter this first in simple linear regression computer output.
bpesta22 said:
I was re-reading the thread, and saw this.
t^2 = F, only with an F-distribution with 1 degree of freedom for the numerator. (and any degree of freedom for the denomenator). Most people doing statistics encounter this first in simple linear regression computer output.
davefoc
Philosopher
T'ai Chi,
I think that Bill Hoyt has made his intentions clear enough. He doesn't intend to answer any of the questions.
I'm going on a short vacation for a few days so if you don't see me posting it's not because I'm not interested.
Perhaps, if Bill has some free time between his gigs as a strip club bouncer he'll see fit to comment a bit.
Meanwhile, how about an answer to my Sherlock Holmes question?
Perhaps one of the other strip club bouncers that participate in this forum will start a thread with questions to challenge Bill's claim that he really is a strip club bouncer.
Dave
I think that Bill Hoyt has made his intentions clear enough. He doesn't intend to answer any of the questions.
I'm going on a short vacation for a few days so if you don't see me posting it's not because I'm not interested.
Perhaps, if Bill has some free time between his gigs as a strip club bouncer he'll see fit to comment a bit.
Meanwhile, how about an answer to my Sherlock Holmes question?
Perhaps one of the other strip club bouncers that participate in this forum will start a thread with questions to challenge Bill's claim that he really is a strip club bouncer.
Dave
I believe Bill when he says that he is one. I picture a muscle man, yet brainy, that could pound me into the asphalt pretty easily with one hand, and adjust his sunglasses with the other.
With my minimal martial arts skills, statistical knowledge, small bladder, and mystical leanings, and with Bill's bouncer skills, street smarts, skeptical know-how, and iron will (hey, you'd have to have that for behaving in a strip club), Bill and I could become an unstoppable duo in fighting crime!
Do I sense a sitcom??
davefoc
Philosopher
Hmm T'ai Chi,
I see something a little different here. A sort of sleazy fellow, a little too old for the job, his skin has a deep ruddy complection caused by years of severe alcoholism. His youthful days as an elite mathmetician only a distant memory now. A hacking cough brought on by his relentless smoking. A raunchy tatoo on his right foream acquired after a night of binge drinking.
His only real happiness anymore comes from the discussion of esoteric statistics problems with his fellow bouncers and beating the crap out of Scientologists that try to recruit him.
Incidentally, are you dodging the SH question?
I see something a little different here. A sort of sleazy fellow, a little too old for the job, his skin has a deep ruddy complection caused by years of severe alcoholism. His youthful days as an elite mathmetician only a distant memory now. A hacking cough brought on by his relentless smoking. A raunchy tatoo on his right foream acquired after a night of binge drinking.
His only real happiness anymore comes from the discussion of esoteric statistics problems with his fellow bouncers and beating the crap out of Scientologists that try to recruit him.
Incidentally, are you dodging the SH question?
T'ai Chi said:
Could you come up with a list of questions for Bill and I then?
No thanks. For starters, my own knowledge of statistics is much worse than it should be and the only redeeming feature is that statistics is not very important on my own field (computational logic).
But anyway, I think that the whole concept of comparing respective levels of knowledge via lists of questions is, well, completely unproductive. Especially given that there is no way of knowing who actually solved them.
Here are my thoughts on these questions since Bill is absent.
1. For a mound-shaped distribution, what is a decent way to estimate the standard deviation from the range?
Assuming approximate normality, take the range/4 for small data sets, or range/5 or range/6 for larger datasets.
2. If you were to do a test to see if Passing and Failing an examination were independent of Before and After applying a drug, what is the name of a test you would use?
It is like a chi-square test for independence, but it is looking at Before and After, so it is actually a McNemar's test.
3. If you wanted to see if a digital blood pressure cuff's results can be used in place of a traditional blood pressure cuff's results (ie. are "the same"), what statistic/test would you use?
It is an error to use the correlation coefficient (because one cuff could always register 5 points higher than the other. There would be perfect correlation, but the cuffs wouldn't be reading the same at all). It is also an error to use a t-test by itself or in a regression. I did my oral exam on this topic, it is called a Overall Concordance Correlation Coefficient, and looks at the expected value of the squared difference between measurements. Lin (1989 in Biometrics and later) did a lot of work on this subject.
4. Is it possible to have a two-sided alternative hypothesis for a test using the chi-square distribution?
Yes. When testing variances, for example.
5. Is it possible for degrees of freedom to be a non-integer?
Yes. The Satterwaithe approximation.
6. P(x) = e^(a+bx)/[1+e^(a+bx)] is a linear model, true or false.
True. The linear part refers to how the parameters (the a and b) enter in the model.
7. If two individual 2x2 tables in a chi-square test for independence showed non-significance, but the combined 2x2 table (ie, literally table 1 + table 2) showed significance, how would you interpret that?
You usually go with the individual tables, because the combined tables ignores effects. For example, if we were looking at a combined 2x2 table of gender and education, we'd be ignore the gender effects of male and female.
8. Why is testing if the population correlation is 0 equivalent to testing if the population slope coefficient is 0?
In simple linear regression, b = r*(Sy/Sx), so testing if b=0 is the same as testing if r=0.
9. If we only observe the outcomes from coin flipping (Heads = 1, Tails = 0): 1000100010101, what is the most sensible estimate of the probability of Heads? What general mathematical technique would you use here?
5/13 would be a sensible estimate. Call each Head, a sucess, p, and each tail, a failure, as 1-p. From our string of 1's and 0's, our likelihood function (multiplying them all together) is p^5*(1-p)^8. Differentiating this and setting it equal to 0, yields p^=5/13. (of course, you still have to show it is a maximum)
The question arose of 'What do we do if we get a string like 111111, or 0000? Does the above Maximum Likelihood estimation method break down? There is something called a Wilson estimate that overcomes this. It says that instead of estimating p as X/n, where X is the number of successes, it estimates p as p=(X+2)/(n+4).
10. If I am studying bugs in a statistics class, what statistics am I studying?
BUGS is a computer program for Bayesian inference Under Gibbs Sampling.
11. What was the name of the person who gave an example of the correlation coefficients be the same for multiple sets of data, but their plots looking completely different?
Anscombe. A very well-known example.
12. If X's are distributed normally, what do you do to them to get to a log-normal distribution?
e^X.
13. If X and Y are individually distributed as chi-squares, what do you do to them to get to a F distribution?
(X/v1) / (Y/v2), where v1 and v2 are their respective degrees of freedom.
14. If X and Y are individually distributed normally, what do you do to them to get to Cauchy?
X/Y
15. How do you transform between a similarity measure and a distance measure?
It is called the Standard Transformation. If you have objects A and B, with distance measure d(A,b) and similarity measure s(A,B), then:
d(A,B) = sqrt(s(A,A)-2s(A,B)+s(B,B)).
16. If X's are distributed as an Exponential, what do you do to them to get to a Double Exponential?
Take their difference, like X1-X2.
17. Are partial derivatives important in asymptotic normality? If so, how?
Yes. They are used in calculating the variance.
18. What are the conditions for a distribution belonging to the Regular Exponential Family?
It is from an exponential family if we can write its probability function as:
f(x;theta) = a(theta)*h(x)*exp{SUM b_j(theta)*R_j(x) (j=1 to k) }
It is regular if:
a) k=p (where p is the number of parameters)
b) theta contains a p-dimensional rectangle
c) b_j(theta) are differentiable
19. How do you determine if a statistic, T(x), is sufficient for estimating, say, the parameter, theta?
Take the ratio f_x(x,theta)/f_T(T(x),theta), and see if theta cancels out.
20. Why is ancillarity important in theoretical statistics?
Basu's Theorem for one (determining independence knowing some other conditions)
21. Canberra and Bhattacharyya formulas are used for...?
Distance measures. There are many others too, like Euclidean, city block distance, and various types of squared distances.
22. What is a sensible plotting technique in a repeated measure analysis?
Profile plots.
23. Why is the F distribution called an F distribution?
Fischer.
24. Let's say forty subjects are randomly assigned to four treatment groups, ten to each group. Three responses are measured on each subject. What specific distribution would you use to draw inferences about the differences between means in the different treatment groups?
An F-distribution, with 3 and 36 degrees of freedom. This is from multivariate statistics, using Hotelling's T^2. In general, use F with q and n1+n2+n3+n4-q-1 degrees of freedoms, where q is the number of responses, and the n's are the number of people in each group.
25. Who worked with the quincunx?
Galton.
26. Does this make sense: "The probability of the population mean being in the interval [26.4, 28.8]mg's is 95%"?
Yes, but only in a Bayesian interpretation.
27. What statistician had a ladder?
Tukey. (used in transformations)
28. What are the differences between a confidence interval, prediciton interval, and a statistical tolerance interval?
Ugg, too long to answer this one completely. But confidence intervals are used to say that we are so and so confident that the parameter is between so and so endpoints. A prediction interval is used to say that we are so and so confident that a future value of the response is between so and so endpoints. A tolerance interval is used to see if a process is out of control or not, like a machine producing weights that are outside so and so endpoints hints that the process needs to be examined.
29. Is a MVUE accurate, or precise?
Both. It is minimum variance and unbiased.
30. For what distributions is range/SD >= sqrt(2) ?
All of them. I found this in my notes from a theory class (without proof). I'm trying to prove it, but am having a hard time.
31. How do you interpret: "The Pearson correlation coefficient of age and gender is .96."
You interpret it as an error. The Pearson correlation coefficient can only be done on two continuous variables. Gender is not a continuous variable.
32. What is the average of a dataset with n elements if you repeatedly take a 1st level Winsorized mean?
I think the maximum of the dataset.
33. Why do we worry about S (the sample standard deviation)? Why not just focus on S^2?
Because it gets us back to the original units of the data.
34. If X_i = K*Y_i+C, then what is the mean and the standard deviation of X?
("_i" is a subscript)
E[X_i] = E[K*Y_i+C] = K*E[Y_i]+C
V[X_i] = V[K*Y_i+C] = K^2*V[Y_i], so the standard deviation is sqrt(K^2*V[Y_i]) = K*SD[Y_i]
35. What is the general name for the mean of the means?
The grand mean.
36. Show mathematically that for a standard normal random variable Z, that the variance of Z is 1.
Properly, it is showed by an integration. But, I'll do it just using a standardized variable:
Z = (x-mu)/s, so
V[Z] = V[(x-mu)/s] =
1/s^2*V[x-mu] =
1/s^2*V[x] = s^2/s^2 = 1.
1. For a mound-shaped distribution, what is a decent way to estimate the standard deviation from the range?
Assuming approximate normality, take the range/4 for small data sets, or range/5 or range/6 for larger datasets.
2. If you were to do a test to see if Passing and Failing an examination were independent of Before and After applying a drug, what is the name of a test you would use?
It is like a chi-square test for independence, but it is looking at Before and After, so it is actually a McNemar's test.
3. If you wanted to see if a digital blood pressure cuff's results can be used in place of a traditional blood pressure cuff's results (ie. are "the same"), what statistic/test would you use?
It is an error to use the correlation coefficient (because one cuff could always register 5 points higher than the other. There would be perfect correlation, but the cuffs wouldn't be reading the same at all). It is also an error to use a t-test by itself or in a regression. I did my oral exam on this topic, it is called a Overall Concordance Correlation Coefficient, and looks at the expected value of the squared difference between measurements. Lin (1989 in Biometrics and later) did a lot of work on this subject.
4. Is it possible to have a two-sided alternative hypothesis for a test using the chi-square distribution?
Yes. When testing variances, for example.
5. Is it possible for degrees of freedom to be a non-integer?
Yes. The Satterwaithe approximation.
6. P(x) = e^(a+bx)/[1+e^(a+bx)] is a linear model, true or false.
True. The linear part refers to how the parameters (the a and b) enter in the model.
7. If two individual 2x2 tables in a chi-square test for independence showed non-significance, but the combined 2x2 table (ie, literally table 1 + table 2) showed significance, how would you interpret that?
You usually go with the individual tables, because the combined tables ignores effects. For example, if we were looking at a combined 2x2 table of gender and education, we'd be ignore the gender effects of male and female.
8. Why is testing if the population correlation is 0 equivalent to testing if the population slope coefficient is 0?
In simple linear regression, b = r*(Sy/Sx), so testing if b=0 is the same as testing if r=0.
9. If we only observe the outcomes from coin flipping (Heads = 1, Tails = 0): 1000100010101, what is the most sensible estimate of the probability of Heads? What general mathematical technique would you use here?
5/13 would be a sensible estimate. Call each Head, a sucess, p, and each tail, a failure, as 1-p. From our string of 1's and 0's, our likelihood function (multiplying them all together) is p^5*(1-p)^8. Differentiating this and setting it equal to 0, yields p^=5/13. (of course, you still have to show it is a maximum)
The question arose of 'What do we do if we get a string like 111111, or 0000? Does the above Maximum Likelihood estimation method break down? There is something called a Wilson estimate that overcomes this. It says that instead of estimating p as X/n, where X is the number of successes, it estimates p as p=(X+2)/(n+4).
10. If I am studying bugs in a statistics class, what statistics am I studying?
BUGS is a computer program for Bayesian inference Under Gibbs Sampling.
11. What was the name of the person who gave an example of the correlation coefficients be the same for multiple sets of data, but their plots looking completely different?
Anscombe. A very well-known example.
12. If X's are distributed normally, what do you do to them to get to a log-normal distribution?
e^X.
13. If X and Y are individually distributed as chi-squares, what do you do to them to get to a F distribution?
(X/v1) / (Y/v2), where v1 and v2 are their respective degrees of freedom.
14. If X and Y are individually distributed normally, what do you do to them to get to Cauchy?
X/Y
15. How do you transform between a similarity measure and a distance measure?
It is called the Standard Transformation. If you have objects A and B, with distance measure d(A,b) and similarity measure s(A,B), then:
d(A,B) = sqrt(s(A,A)-2s(A,B)+s(B,B)).
16. If X's are distributed as an Exponential, what do you do to them to get to a Double Exponential?
Take their difference, like X1-X2.
17. Are partial derivatives important in asymptotic normality? If so, how?
Yes. They are used in calculating the variance.
18. What are the conditions for a distribution belonging to the Regular Exponential Family?
It is from an exponential family if we can write its probability function as:
f(x;theta) = a(theta)*h(x)*exp{SUM b_j(theta)*R_j(x) (j=1 to k) }
It is regular if:
a) k=p (where p is the number of parameters)
b) theta contains a p-dimensional rectangle
c) b_j(theta) are differentiable
19. How do you determine if a statistic, T(x), is sufficient for estimating, say, the parameter, theta?
Take the ratio f_x(x,theta)/f_T(T(x),theta), and see if theta cancels out.
20. Why is ancillarity important in theoretical statistics?
Basu's Theorem for one (determining independence knowing some other conditions)
21. Canberra and Bhattacharyya formulas are used for...?
Distance measures. There are many others too, like Euclidean, city block distance, and various types of squared distances.
22. What is a sensible plotting technique in a repeated measure analysis?
Profile plots.
23. Why is the F distribution called an F distribution?
Fischer.
24. Let's say forty subjects are randomly assigned to four treatment groups, ten to each group. Three responses are measured on each subject. What specific distribution would you use to draw inferences about the differences between means in the different treatment groups?
An F-distribution, with 3 and 36 degrees of freedom. This is from multivariate statistics, using Hotelling's T^2. In general, use F with q and n1+n2+n3+n4-q-1 degrees of freedoms, where q is the number of responses, and the n's are the number of people in each group.
25. Who worked with the quincunx?
Galton.
26. Does this make sense: "The probability of the population mean being in the interval [26.4, 28.8]mg's is 95%"?
Yes, but only in a Bayesian interpretation.
27. What statistician had a ladder?
Tukey. (used in transformations)
28. What are the differences between a confidence interval, prediciton interval, and a statistical tolerance interval?
Ugg, too long to answer this one completely.
But confidence intervals are used to say that we are so and so confident that the parameter is between so and so endpoints. A prediction interval is used to say that we are so and so confident that a future value of the response is between so and so endpoints. A tolerance interval is used to see if a process is out of control or not, like a machine producing weights that are outside so and so endpoints hints that the process needs to be examined.29. Is a MVUE accurate, or precise?
Both. It is minimum variance and unbiased.
30. For what distributions is range/SD >= sqrt(2) ?
All of them. I found this in my notes from a theory class (without proof). I'm trying to prove it, but am having a hard time.
31. How do you interpret: "The Pearson correlation coefficient of age and gender is .96."
You interpret it as an error.
The Pearson correlation coefficient can only be done on two continuous variables. Gender is not a continuous variable.32. What is the average of a dataset with n elements if you repeatedly take a 1st level Winsorized mean?
I think the maximum of the dataset.
33. Why do we worry about S (the sample standard deviation)? Why not just focus on S^2?
Because it gets us back to the original units of the data.
34. If X_i = K*Y_i+C, then what is the mean and the standard deviation of X?
("_i" is a subscript)
E[X_i] = E[K*Y_i+C] = K*E[Y_i]+C
V[X_i] = V[K*Y_i+C] = K^2*V[Y_i], so the standard deviation is sqrt(K^2*V[Y_i]) = K*SD[Y_i]
35. What is the general name for the mean of the means?
The grand mean.
36. Show mathematically that for a standard normal random variable Z, that the variance of Z is 1.
Properly, it is showed by an integration. But, I'll do it just using a standardized variable:
Z = (x-mu)/s, so
V[Z] = V[(x-mu)/s] =
1/s^2*V[x-mu] =
1/s^2*V[x] = s^2/s^2 = 1.
Number Six
JREF Kid
- Joined
- Sep 5, 2001
- Messages
- 5,016
13 and 14 also need independence to be true.
16 needs independence and identical Exponential distributions.
Re. 26 there was a thread on the JREF board that sorta went like this. Shuffle the cards and place them face down on the table. What is the probability that the top card is the 3 of Clubs. Most said "1 in 52" but I thought that a better answer was "Either 0 or 1" because after the cards are sitting on the table the top card is either the 3 of Clubs or is not the 3 of Clubs. Similarly I think saying there is a 95% probability that the mean is in the interval isn't accurate. I know that Bayesian statistics has a different framework and all that but I think it's basically just a re-definition of the word "probability" to say that a true but unknown parameter has a 95% chance of lying within a specified interval. It either does or it doesn't. I realize that that can be taken to be semantics to some degree.
16 needs independence and identical Exponential distributions.
Re. 26 there was a thread on the JREF board that sorta went like this. Shuffle the cards and place them face down on the table. What is the probability that the top card is the 3 of Clubs. Most said "1 in 52" but I thought that a better answer was "Either 0 or 1" because after the cards are sitting on the table the top card is either the 3 of Clubs or is not the 3 of Clubs. Similarly I think saying there is a 95% probability that the mean is in the interval isn't accurate. I know that Bayesian statistics has a different framework and all that but I think it's basically just a re-definition of the word "probability" to say that a true but unknown parameter has a 95% chance of lying within a specified interval. It either does or it doesn't. I realize that that can be taken to be semantics to some degree.