Re: EdStat: Triangular coordinates
On Tue, 10 Jul 2001, Alex Yu wrote: I am trying to understand Triangular coordinates -- a kind of graph which combines four dimensions into 2D You meant, condenses four dimensions into 3D, didn't you? Your subsequent description indicates three dimensions all together, two of them used to represent 3 variables: by joining three axes to form a triangle while the Y axis stands up. The Y axis can be hidden if the plot is depicted as a contour plot or a mosaic plot rather than a surface plot. I have a hard time to follow how a point is determined with the three axes as a triangle. There must be constraints on the values of the three variables. Commonly used for situations like a chemical mixture of 3 components. Each component can have a relative concentration between 0% and 100%, but if component A is at 100%, components B and C must both be at 0%, and the point (100%, 0%, 0%) falls at one apex of the triangle. The formal restriction, of course, is that the sum of all three concentrations equals 100%, so that there are really only two dimensions' worth of information available: (A, B, (100%-A-B)), (A, (100%-A-C), C), or ((100%-B-C), B, C). Since there is usually no reason to treat any component as more (or less) important than any other, triangular coordinates are often displayed on an equilateral triangle, and special graph paper can be purchased that has such a grid. In the absence of such paper, one can plot, say, A and B at right angles to each other and let the 45-degree line from (100,0) to (0,100) represent the C axis (and the upper boundary of the space of possible points). When there is not some such constraint on the values of the three variables, triangular coordinates don't make a whole lot of sense and may be extremely misleading. -- DFB. Donald F. Burrill [EMAIL PROTECTED] 184 Nashua Road, Bedford, NH 03110 603-471-7128 = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Bayesian analyses in education
As a teacher of research methodology in (music) education I am interested in the relation between traditional statistics and the bayesian approach. Bayesians claim that their approach is superior compared with the traditional, for instance because it does not assume normal distributions, is intuitively understandable, works with small samples, predicts better in the long run etc. If this is so, why is it so rare in educational research? Are there some hidden flaws in the approach or are the researchers just ignorant? Comments? Kai Karma Professor of music education Sibelius Academy Helsinki, Finland = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: EdStat: Triangular coordinates
From: [EMAIL PROTECTED] (Donald Burrill) There must be constraints on the values of the three variables. Commonly used for situations like a chemical mixture of 3 components. Each component can have a relative concentration between 0% and 100%, but if component A is at 100%, components B and C must both be at 0%, and the point (100%, 0%, 0%) falls at one apex of the triangle. The formal restriction, of course, is that the sum of all three concentrations equals 100%, so that there are really only two dimensions' worth of information available: (A, B, (100%-A-B)), (A, (100%-A-C), C), or ((100%-B-C), B, C). Since there is usually no reason to treat any component as more (or less) important than any other, triangular coordinates are often displayed on an equilateral triangle, and special graph paper can be purchased that has such a grid. In the absence of such paper, one can plot, say, A and B at right angles to each other and let the 45-degree line from (100,0) to (0,100) represent the C axis (and the upper boundary of the space of possible points). Snip -- DFB. These ternary plots are common in petrology, where the vertices are % sand, % clay, % silt and in population genetics, where the 3 vertices are AA aa and Aa (individuals from a population in Hardy-Weinberg equilibrium fall along a curve on this plot; departures from H-W equilibrium are readily evident). Middleton's (2000, p. 181-185); Data Analysis in the earth sciences using Matlab) provides Matlab code for plotting these ternary diagrams. = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: Bayesian analyses in education
KKARMA wrote: As a teacher of research methodology in (music) education I am interested in the relation between traditional statistics and the bayesian approach. Bayesians claim that their approach is superior compared with the traditional, for instance because it does not assume normal distributions, Sometimes it does; and sometimes classical stats doesn't. is intuitively understandable, Some bits are. Perhaps it's more accurate to say that while the premises of Bayesian statistics are more intuitive at first glance, and the conclusions come closer to what people intuitively want to know (what is the probability that this is correct?) Bayesian stats has its own thorny philosophical problems. Frequentist stats says that only statements about the outcome of random variables have probability; it is legitimate to say that the probability that this die shows a 6 on this roll is 1/6 but not (unless the die itself was drawn at random from a well-defined collection) there is an 80% probability that this die rolls 6 more often than 1. Bayesians do allow the broader use of probability to describe levels of belief about something that was not generated by a well-defined sampling operation, but the cost of this is that the pump must be primed with a prior probability representing your level of belief before the observations, and this is necessarily subjective. The cost is not as great as it appears, because as the data accumulates the impact of the prior becomes less and less; that is, rational observers with initially different beliefs come to more or less agree after observing the data. The justification for this is that a Bayesian interpretation of an opinion poll can actually be The probability that the Garden Party would get more than 40% of the votes in this election is x% [if it were held today and voting patterns matched polling response patterns] whereas - despite the fact that this is intuitively the answer to the question people *want* to ask- frequentist stats cannot. The frequentist can only assign probabilities to samples from well-defined populations. So the frequentist analysis of the same poll might be IF the Garden Party would have exactly 40% popular support [if the election were held today and polling response patterns matched voting patterns] the probability of getting this result or one less favorable to the Garden Party in an opinion poll done in this way would be y%. works with small samples, To some extent; and so does frequentist stats, to some extent. Both tend to be inconclusive with small samples, and to get some of whatever power they have from assumptions that the data cannot justify. That's the way the universe works: you want answers, first get enough data. If this is so, why is it so rare in educational research? Are there some hidden flaws in the approach or are the researchers just ignorant? (1) Propagation delay. What statisticians are writing about in theoretical journals today will be used by statisticians in their practical work in a few years. It'll be in upper-level stats textbooks for stats majors in a decade. Maybe in two decades significant numbers of social science PhD's will start to hear about it; maybe another ten years later somebody will be bold enough to put it in an applied stats textbook; at that point it becomes well enough known to be used widely. Maybe. (2) Encapsulation. The philosophical complications of the frequentist method are well-hidden for most users inside phrases such as confidence interval and significance level. You can construct a 95% confidence interval correctly while believing that this guarantees a 95% probability that this particular interval contains the true value (which is not so); and you can even state this in your paper and many referees and editors will let it pass. Similarly, you can do a hypothesis test at the 5% significance level while believing that 5% is the probability that your data are wrong (it isn't.). If people were required to truly understand hypothesis tests and confidence intervals before using them there might be more impetus for change. Note: This is NOT entirely a valid argument for change, any more than saying that if people were required to understand the workings of their vehicles there would be a lot more bikes on the road and a lot fewer cars is. On the other hand, if there were no mechanics, it might be. (3) Standardization: There are genuine advantages to everybody singing from the same hymnbook, which tends to lead to change being slow. -Robert Dawson = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/
Re: Bayesian analyses in education
In article [EMAIL PROTECTED], KKARMA [EMAIL PROTECTED] wrote: As a teacher of research methodology in (music) education I am interested in the relation between traditional statistics and the bayesian approach. Bayesians claim that their approach is superior compared with the traditional, for instance because it does not assume normal distributions, is intuitively understandable, works with small samples, predicts better in the long run etc. If this is so, why is it so rare in educational research? Are there some hidden flaws in the approach or are the researchers just ignorant? Comments? Bayesian analysis is not that simple, nor is it claimed to be. It IS intuitively understandable, but there is the question of its justification. Any non-Bayesian procedure can be improved in any real sense by the limit of Bayesian procedures, which limit need not be quite a Bayesian procedure. There are no restrictions on sample size. In this form, it might be very difficult to use. It is rare in most places, as it disagrees with what are mistakenly given as the criteria for a statistical procedure. Testing does not result in fixing a significance level, for example, although it often corresponds to some standard test at SOME level. The level varies with sample size, and varies considerably. The straightforward Bayes approach is to take a prior distribution on states of nature. Then one can use Bayes' Theorem to obtain a posterior distribution. This is a simple probability result, but is rarely taught in the standard statistical methods courses, which often avoid probability. But this does not justify it. One can get a justification by assuming self consistent actions; this makes the quantity to be minimized the expected value of the loss with respect to some prior measure over the states of nature, which is Bayesian action. This can often be approximated even if strict Bayesian computation is too difficult to carry out. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399 [EMAIL PROTECTED] Phone: (765)494-6054 FAX: (765)494-0558 = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
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