probability definition
Hello, I have a question regarding the definition of probability. If I understand correctly, probability may be defined using just axioms. However, my textbook also uses a relative frequency definition, in which a probability is defined as being the proportion of times an outcome occurs in repeated trials of an experiment. This makes sense when one flip of the coin is one trial, and in repeated trials, the proportion of heads is 1/2. But what about a situation (an ex. in my textbook) where the probability of rain tomorrow is 0.70. How do you define this experiment? Perhaps you measure rainfall, temperature, pressure, etc. for each day over a long time period. Then the probability of rain tomorrow is the proportion of times that rain occurred on days with similar values for temp., humidity, etc.? This seems a bit awkard to me. Also, how many trials must one perform an experiment, before you know that the proportion converges to a particular fraction? Any help on interpretation of relative frequency probabilities would be greatly appreciated. In many cases, it seems difficult, at least for textbook examples, to define what the actual experiment is. ___ Send a cool gift with your E-Card http://www.bluemountain.com/giftcenter/ = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: Cronbach's alpha and sample size
The effect of N on alpha is minimal unless the assumptions for alpha are not met. If you have a multidimensional construct then the alpha will tend to go down as the sample size decreases. At leaset I have observed this in monte carlo analyses. At 12:08 PM 2/28/01 +0100, you wrote: >How is Cronbach's alpha affected by the sample size apart from questions >related to generalizability issues? > >Ifind it hard to trace down the mathmatics related to this question >clearly, and wether there migt be a trade off between N of Items and N >of sujects (i.e. compensating for lack of subjects by high number of >items). > >Any help is appreciated, > >Thanks, Nico >-- > > >= >Instructions for joining and leaving this list and remarks about >the problem of INAPPROPRIATE MESSAGES are available at > http://jse.stat.ncsu.edu/ >= > Paul R. Swank, PhD. Professor Advanced Quantitative Methodologist UT-Houston School of Nursing Center for Nursing Research Phone (713)500-2031 Fax (713) 500-2033 = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: probability definition
Hi Alex, Can you provide the definition of probability under each way? In other words, can you explain a little more on each way of defining probability? As it is, some of them are clear (e.g., Frequentist theory) and others are not clear to me. Thanks. Siddeek Alex Yu wrote: Probability can be defined in at least five different ways: 1. Classical Laplacean theory of probability: The prob.is derived from the fairness assumption e.g. a fair coin. It is also called equiproability. 2. Frequentist theory: It is developed by von Mises and Reichenbach. Prob. is the relative frequency in the long run by limiting observations. 3. Propensity: It is based upon the physical or the objective property of the events. 4. Logical: developed by Carnap. Prob. is defined like Y logically entails X. 5. Subjective or Bayesian: degree of belief There is no easy answer to your question. It depends on which point of view you chose. Chong-ho (Alex) Yu, Ph.D., MCSE, CNE Academic Research Professional/Manager Educational Data Communication, Assessment, Research and Evaluation Farmer 418 Arizona State University Tempe AZ 85287-0611 Email: [EMAIL PROTECTED] URL:http://seamonkey.ed.asu.edu/~alex/ = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ = = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: Satterthwaite-newbie question
On Tue, 27 Feb 2001, Allyson Rosen wrote:
I need to compare two means with unequal n's. Hayes (1994) suggests using a
formula by Satterthwaite, 1946. I'm about to write up the paper and I can't
find the full reference ANYWHERE in the book or in any databases or in my
books. Is this an obscure test and should I be using another?
Perhaps it refers to:
F. E. Sattherwaite, 1946: An approximate distribution of estimates of
variance components. Biometrics Bulletin, 2, 110-114.
According to Casella Berger (1990, pp. 287-9), "this approximation
is quite good, and is still widely used today." However, it still may
not be valid for your specific analysis: I suggest reading the
discussion in Casella Berger ("Statistical Inference", Duxbury Press,
1990). There are more commonly used methods for comparing means with
unequal n available, and you should make sure that they can't be used
in your problem before resorting to Sattherwaite.
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Re: basic stats question
Re probability/independence, I've found that the most effective way to communicate this concept to my students (College of Education, not heavily math-oriented) is the following: Consider the student population of your university. Perhaps there is a fairly equal split of males and females in the student body. Now, put a condition upon the student body -- only those majoring in, say, psychology. Do you find the same proportion of students who are male within only psych majors, compared with the proportion of students in the entire student body who are male? If gender and psych major are independent, then the probability of a randomly chosen person at the university being male should equal the probability of a randomly chosen psych major being male. That is, p(male) = p(male|psych major) ==(p. of male, given you're looking at psych majors) Then you can move to an example of racial profiling. Out of all the people in your city who drive, what proportion are African-American? [p(African-American).] Now, GIVEN that you look only at drivers who are pulled over, what proportion of these people are African American? [p(African-American|pulled over).] If being black and being pulled over are independent events, then the probabilities should be equal. You can illustrate this graphically by drawing a large box to represent all the drivers, then mark the proportion representing African-American drivers. Then draw a smaller box representing the people being pulled over, with a proportion of the box marked to represent the African-American drivers who are pulled over. If the proportions of each box are equal, then the events are independent. So now, I would welcome comments from the more mathematically/statistically rigorous list members among us! ~~~ Lise DeShea, Ph.D. Assistant Professor Educational and Counseling Psychology Department University of Kentucky 245 Dickey Hall Lexington KY 40506 Email: [EMAIL PROTECTED] Phone: (859) 257-9884
Re: Regression with repeated measures
I don't have an answer, but I'm very glad this question was asked because I'm having a similar problem. I have 14 grids, values from which are to be used as the dependent variable in a regression. Each 6x6 grid consists of 36 observation points. Their are some fairly strong spatial correlations among the values at each grid, so I certainly can't treat them as if they were independent, yet reducing each grid to a single mean value (the other extreme) seems like a foolish waste of power. I'm trying to figure out how to use all of the observations, but also use the estimated spatial autocorrelations to weight them in the regression. (The design was originally created to answer a very different question, which is how I got into this mess.) I hope that there's a single answer to both of our questions. Rich Strauss At 10:54 AM 2/28/01 -0600, Michael M. Granaas wrote: I have a student coming in later to talk about a regression problem. Based on what he's told me so far he is going to be using predicting inter-response intervals to predict inter-stimulus intervals (or vice versa). What bothers me is that he will be collecting data from multiple trials for each subject and then treating the trials as independent replicates. That is, assuming 10 tials/S and 10 S he will act as if he has 100 independent data points for calculating a bivariate regression. Obviously these are not independent data points. Is the non-independence likely to be severe enough to warrant concern? If yes, is there some method that will allow him to get the prediction equation he wants? Thanks Michael Dr Richard E Strauss Biological Sciences Texas Tech University Lubbock TX 79409-3131 Email: [EMAIL PROTECTED] (formerly [EMAIL PROTECTED]) Phone: 806-742-2719 Fax: 806-742-2963 = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: Regression with repeated measures
Linear mixed models (aka multilelvel models, random coefficient models, etc) as implemented by many software products: SAS PROC MIXED, MIXREG, MLwiN, HLM, etc. You might want to look at some links on my website http://sites.netscape.net/segregorich/index.html Steve Gregorich Obviously these are not independent data points. Is the non-independence likely to be severe enough to warrant concern? If yes, is there some method that will allow him to get the prediction equation he wants? Thanks Michael M. Granaas = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: Regression with repeated measures
Hi Professor Granaas! The observations that your student is collecting is indeed a problem. Because they are correlated (being collected over time), the standard errors for the regression approach that he is planning to use are probably too low creating type I error problems. You should look into a repeated measures type of design. Given what he is trying to do, it is difficult to suggest what steps he might take exactly. Hope this helps. Gary Winkel City University of New York = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: Cronbach's alpha and sample size
On Wed, 28 Feb 2001 12:08:55 +0100, Nicolas Sander [EMAIL PROTECTED] wrote: How is Cronbach's alpha affected by the sample size apart from questions related to generalizability issues? - apart from generalizability, "not at all." Ifind it hard to trace down the mathmatics related to this question clearly, and wether there migt be a trade off between N of Items and N of sujects (i.e. compensating for lack of subjects by high number of items). I don't know what you mean by 'trade-off.' I have trouble trying to imagine just what it is, that you are trying to trace down. But, NO. Once you assume some variances are equal, Alpha can be seen as a fairly simple function of the number of items and the average correlation -- more items, higher alpha. The average correlation has a tiny bias by N, but that's typically, safely ignored. -- Rich Ulrich, [EMAIL PROTECTED] http://www.pitt.edu/~wpilib/index.html = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
FW: Regression with repeated measures
These both sound to me as if multi-level models would be appropriate to handle the type of data to which you are referring. Look at this site for some basic info on multi-level models (MLM): http://www.ioe.ac.uk/multilevel/ Interested in learning more... then dowload this classic text on MLM for free: http://www.arnoldpublishers.com/support/goldstein.htm Finally, If you decide this method is what you are looking for, then have a look at the following text that describes Linear MLM or as they call it Hierarchichal Linear Models (HLM)--the multilevel equivalent of linear regression: Bryk,A.S., and Raudenbush,S.W. (1992). Hierarchical Linear Models. Newbury Park, Sage. -Original Message- From: Rich Strauss [mailto:[EMAIL PROTECTED]] Sent: Wednesday, February 28, 2001 2:40 PM To: [EMAIL PROTECTED] Subject: Re: Regression with repeated measures I don't have an answer, but I'm very glad this question was asked because I'm having a similar problem. I have 14 grids, values from which are to be used as the dependent variable in a regression. Each 6x6 grid consists of 36 observation points. Their are some fairly strong spatial correlations among the values at each grid, so I certainly can't treat them as if they were independent, yet reducing each grid to a single mean value (the other extreme) seems like a foolish waste of power. I'm trying to figure out how to use all of the observations, but also use the estimated spatial autocorrelations to weight them in the regression. (The design was originally created to answer a very different question, which is how I got into this mess.) I hope that there's a single answer to both of our questions. Rich Strauss At 10:54 AM 2/28/01 -0600, Michael M. Granaas wrote: I have a student coming in later to talk about a regression problem. Based on what he's told me so far he is going to be using predicting inter-response intervals to predict inter-stimulus intervals (or vice versa). What bothers me is that he will be collecting data from multiple trials for each subject and then treating the trials as independent replicates. That is, assuming 10 tials/S and 10 S he will act as if he has 100 independent data points for calculating a bivariate regression. Obviously these are not independent data points. Is the non-independence likely to be severe enough to warrant concern? If yes, is there some method that will allow him to get the prediction equation he wants? Thanks Michael Dr Richard E Strauss Biological Sciences Texas Tech University Lubbock TX 79409-3131 Email: [EMAIL PROTECTED] (formerly [EMAIL PROTECTED]) Phone: 806-742-2719 Fax: 806-742-2963 = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ = = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: basic stats question
In article [EMAIL PROTECTED], Richard A. Beldin [EMAIL PROTECTED] wrote: This is a multi-part message in MIME format. --20D27C74B83065021A622DE0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit I have long thought that the usual textbook discussion of independence is misleading. In the first place, the most common situation where we encounter independent random variables is with a cartesian product of two indpendent sample spaces. Example: I toss a die and a coin. I have reasonable assumptions about the distributions of events in either case and I wish to discuss joint events. I have tried in vain to find natural examples of independent random variables in a smple space not constructed as a cartesian product. I think that introducing the word "independent" as a descriptor of sample spaces and then carrying it on to the events in the product space is much less likely to generate the confusion due to the common informal description "Independent events don't have anything to do with each other" and "Mutually exclusive events can't happen together." Comments? The usual definition of "independence" is a computational convenience, but an atrocious definition. A far better way to do it, which conveys the essence, is to use conditional probability. Random variables, or more generally partitions, are independent if, given any information about some of them, the conditional probability of any event formed from the others is the same as the unconditional probability. This is the way it is used. As for a "natural" example not coming from a Cartesian product, consider drawing a hand from an ordinary deck of cards. On another newsgroup, someone asked for a proof that the number of aces and the number of spades was uncorrelated; they are not independent. The proof I posted used that for the i-th and j-th cards dealt, the rank of the i-th card and the suit of the j-th are independent. For i=j, this can be looked upon as a product space, but not for i and j different. There are other examples. The independence of the sample mean and sample variance in a sample from a normal distribution is certainly an important example. The independence of the various sample variances in an ANOVA model is another. The independence for each t of X(t) and X'(t) in a stationary differentiable Gaussian process is another. This is thrown together off the cuff. There are lots of others. -- 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/ =
Re: probability definition
In article [EMAIL PROTECTED], James Ankeny [EMAIL PROTECTED] wrote: Hello, I have a question regarding the definition of probability. If I understand correctly, probability may be defined using just axioms. However, my textbook also uses a relative frequency definition, in which a probability is defined as being the proportion of times an outcome occurs in repeated trials of an experiment. This makes sense when one flip of the coin is one trial, and in repeated trials, the proportion of heads is 1/2. But what about a situation (an ex. in my textbook) where the probability of rain tomorrow is 0.70. How do you define this experiment? Perhaps you measure rainfall, temperature, pressure, etc. for each day over a long time period. Then the probability of rain tomorrow is the proportion of times that rain occurred on days with similar values for temp., humidity, etc.? This seems a bit awkard to me. Also, how many trials must one perform an experiment, before you know that the proportion converges to a particular fraction? Any help on interpretation of relative frequency probabilities would be greatly appreciated. In many cases, it seems difficult, at least for textbook examples, to define what the actual experiment is. I think it is dangerous, and even useless, to ATTEMPT to define probability. In physics, one no longer even tries to define length or mass, just specify their properties. It is the same with probability. A quantum mechanical model has a joint probability distribution for observations, but is worse between them. Just as we use the postulated properties for length and mass, we should use those for probabilities. We do have the nasty problem that there is no way we can accurately calculate probabilities, unless very strong additional assumptions are made. -- 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/ =
The currency market may be your answer. 30528
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