Re: 95% confidence interval
Radford Neal wrote: In article [EMAIL PROTECTED], James Ankeny [EMAIL PROTECTED] wrote: ... if the distribution is heavily skewed to the right, say like income, why do we want an interval for the population mean, when we are taught that the median is a better measure of central tendency for skewed distributions? As another poster has said, one reason is technical convenience. A more fundamental reason, though, is that the median is probably not the best thing to look at, whatever you might have been taught. But that's not what he said - he said it was _better_ than the mean, as a criticism of using the mean. -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: The meaning of the p value
Will Hopkins wrote: I accept that there are unusual cases where the null hypothesis has a finite probability of being be true, but I still can't see the point in hypothesizing a null, not in biomedical disciplines, anyway. If only we could replace the p value with a probability that the true effect is negative (or has the opposite sign to the observed effect). The easiest way would be to insist on one-tailed tests for everything. Then the p value would mean exactly that. An example of two wrongs making a right. No, a one-tailed test doesn't work; it is still computed using the null value. To find what you want you need Bayesian techniques... but then (if your prior distribution is valid) you can answer the question you *really* wanted to answer - "what is the probability that the effect exists?" Or even "what is the distribution of the parameter value?" -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: 95% confidence interval
Radford Neal wrote: ... the median is probably not the best thing to look at, whatever you might have been taught. What to look at depends not just on the shape of the distribution, but also on what your purpose is. Ask yourself whether there are very many purposes for which it would make no difference if the upper 49% of the incomes were doubled, leaving the median unchanged. Alan Hutson [EMAIL PROTECTED] wrote: Yes, the median may not change, but the confidence interval and/or the variance estimate for the median will reflect the fact that something is different in the upper tail. Also, in your hypothetical above you are assuming one would estimate the median with "middle" order statistic. My point is not anything to do with problems of estimating the median, but rather with whether you should be interested in the median at all. Since the population median would be the same even if the upper 49% of the values were doubled, it seems unlikely that it's what you really are interested in. In contrast, the mean corresponds to the sort of aggregate that is of interest in many contexts. Mean income, for instance, will tell you something about how much money is available to spend on goods. Of course, for many purposes, no single number will be an adequate description of the situation. As a technical point, a confidence interval for the median is not necessarily going to be influenced by the upper tail. If you are interested in the shape of the tails, you should look at them. Radford Neal Radford M. Neal [EMAIL PROTECTED] Dept. of Statistics and Dept. of Computer Science [EMAIL PROTECTED] University of Toronto http://www.cs.utoronto.ca/~radford = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Matrixer econometric software
Please, take notice of new version of Matrixer econometric software. System requirements: Windows 95/98/2000/NT Current version: 4.0 beta Size: 527 Kb (zip) Author: Alexander Tsyplakov Matrixer sites: matrixer.narod.ru and mtxr.hypermart.net E-mail: [EMAIL PROTECTED] What's Matrixer supposed to do? ~~~ Matrixer is a piece of software especially suited for teaching econometrics and doing medium-scale applied research. Position it somewhere between EViews and EasyReg (both in price and functionality). The program provides many classical as well as more recent econometric techniques and models # Linear regression # Binomial logit and probit # Poisson regression # Regression with multiplicative heteroskedasticity # GARCH regression # Regression with ARMA errors # Box-Jenkins model (ARIMA) # (Generalized) instrumental variables estimator # Nonparametric regression # Nonparametric density estimation # Quantile regression # Simultaneous equations, 2SLS, FIML and 3SLS # Nonlinear regression # Method of maximum likelihood The program includes many other helpful features to handle and analyze data # Descriptive statistics (mean, variance, etc.) # Various data plots # Visual data description (histogram, spectral density estimate, autocorrelation function) # Table of correlations between variables # Probability calculator (normal, Student t, chi-square, Fisher F) # Dickey-Fuller test (additional files are needed) # Vector and matrix operations (that is why it was called Matrixer). # Small build-in programming language for writing macros # Table editor # Fast drawing of function plots according to formulas # Import data from text files # History of commands About the current beta version ~~ The program originally had Russian interface. Recently it was translated into English. Translation of the help file is now in progress. As soon as it will be finished the first complete English version will be released. The program is planned to be shareware. Price will be approximately $30-$40 per copy. The current beta version has two limitations. First, it has very short help file. However, any person with average computer skills would be able to use it, because Matrixer is menu-driven. Second, the number of observations was artificially limited to a maximum of 400. This was made in order to encourage paying for the future complete version of Matrixer. The purpose of releasing this beta is to test whether the English translation of the program interface is adequate. The author would be grateful for comments and bug reports. Several parts of Matrixer help system are available at Matrixer sites as HTML. Please, spend a minute to look through them to find possible language errors. = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Multivariate Multiple Regression w/ Dummy Variables
I have a very interesting question...I am currently trying to run a Multivariate Multiple Regression model (i.e. each trial has more than one dependent variable) and the type of independent variables that I am using in this model are dummy variables. Now my question is why does the square of the design matrix (i.e. the Z'Z matrix) become non- invertable if an entire column of the design matrix (i.e. Z matrix)is either all ones or all zero's. That is, why does Z'Z become singular if your entrie sample population has the same answer for any given dummy variable. Thanks Sent via Deja.com http://www.deja.com/ = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: Simulating T tests for Likert scales
Will, I gotta reply to this one! I've done this type of thing a number of times. Will Hopkins wrote: I have an important (for me) question, but first a preamble and hopefully some useful info for people using Likert scales. A week or so ago I initiated a discussion about how non-normal the residuals have to be before you stop trusting analyses based on normality. Someone quite rightly pointed out that it depends on the sample size, because the sampling distribution of almost every statistic derived from a variable with almost any distribution is near enough to normal for a large enough sample, thanks to the central limit theorem. Therefore you get believable confidence limits from t statistics. The distribution of the average of 12 observations, taken from a 'saw tooth' population, is about 1 significant line width away from a normal population. n, the sample size, doesn't have to be very big. But how non-normal, and how big a sample? I have been doing simulations to find out. I've limited the simulations to t tests for Likert scales with only a few levels, because these crop up often in research, and Likert-scale variables with responses stacked up at one end are not what you call normally distributed. Yes, I know you can and maybe should analyze these with logistic regression, but it's hard work for statistically challenged research students, and the outcomes (odds ratios) are hard for all but statisticians to understand. Scoring the levels with integers and working out averages is so much easier to do and interpret. My simulations have produced some seemingly amazingly good results. For example, with a 3-point scale with values of 1, 2 and 3, samples of as few as 10 in each of two groups give accurate confidence intervals for the difference in the means of the groups when both means are ~2.0 (i.e. in the middle) and SDs are ~0.7 (i.e. the majority of observations on 2, with a reasonable number on 1 and 3). They are still accurate even when one of the groups is stacked up at one end with a mean of 2.6 (and SD ~0.5). If both means are stacked up at one or either end, sample sizes of 20 or more are needed, depending on how extreme the stacking is. Likert scales with more than 3 levels work perfectly for anything except responses stacked up in the same extreme way at either end. Aren't these getting over toward some kind of binary distribution? Now, my question. Suppose in real life I have a sample of 10 observations of, say, a 5-point scale scored as 1 to 5. Suppose I get 1 response on 3, 5 responses on 4 and 4 responses on 5. You have assumed that a response must be integer - i.e., ordinal scale. The best 'resolution' of your scale is, roughly, 20% - one unit in 5. If I knew enough math, I might be able to show what is the least difference in two means that you could use, to demonstrate a difference in those means. For a given sample size. The mean is therefore 4.3. Suppose the other group is no problem (e.g., 10 or more responses spread around the middle somewhere). Now, according to my simulations, it's OK for me to do a t test to get the confidence limits for the difference, isn't it? Now suppose the first group was stacked more extremely, with 2 on 4 and 8 on 5. The mean for this group is now 4.8. According to my simulations, that's too extreme to apply the t test, with a sample of 10, anyway. Suppose I have 5 coins, weighted so p(heads) = .96. Count a head as 1, a tail as 0. Toss 5 and add up the coins. Multile times. Average: 4.8 Could I use the binary caluclations to determine the sample size requried before the Student 't' and normal dist. could apply? You bet! Is this the correct way to apply the results of my simulations? I can see how it could fall over: you could in principle get a sample of 1x3, 5x4 and 4x5 when the true distribution has a mean of 4.8, but the chance of that happening is small. To put the question in a more general context of simulation: if the observed sample has a value of the outcome statistic that simulations show has an accurate confidence interval for the given sample size when that value is the population value of the statistic, is the resulting confidence interval accurate? Will I'm not clear why you 'give' away information by making your Likert scale into an ordinal value, instead of accepting fractional units, such as 0.5 (2.5, 4.5, etc.). Whenever a survey respondent puts the 'x' mark part way between the box for 'neutral' and 'somewhat agree,' they are trying to tell you that they use a continuous scale. This additional information the researcher throws away when they shift the 'x' to 'neutral' or worse, throw it out altogether. If you say that this additional information is not 'real,' because the respondent cannot be that 'fine' in their accuracy of response, then I would urge that additional
Re: The meaning of the p value
I've been involved in off-list discussion with Duncan Murdoch. At one stage there I was about to retire in disgrace. But sighs of relief... his objection is Bayesian. OK. The p value is a device to put in a publication to communicate something about precision of an estimate of an effect, under the assumption of no prior knowledge of the magnitude of the true value of the effect. If we assume no prior knowledge of the true value, then my claim stands: the p value for a one-tailed test is the probability of an opposite true effect--any true effect opposite in sign or impact to that observed. I can't see how a Bayesian perspective dilutes or invalidates this interpretation. The same Bayesian perspective would make you re-evaluate the p value under its conventional interpretation. In other words, if you have some other reason for believing that the true value has the same sign as the observed value, reduce the p value in your mind. Or if you believe it has opposite sign, increase it. If we are stuck with p values, then I believe we should start showing one-tailed p values, along with 95% confidence limits for the effect. Both these are far far easier to understand than hypothesis testing and statistical significance. Put a note in the Methods saying something like: "The p values, which were all derived from one-tailed tests, represent the probability that the true value of the effect is opposite in sign (correlations; differences or changes in means) or impact (relative risks, odds ratios) to that observed." Will = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
