Student's t vs. z tests
(This note is largely in support of points made by Rich Ulrich and Paul Swank.) I disagree with the claim (expressed in several recent postings) that z-tests are in general superseded by t-tests. The t-test (in simple one-sample problems) is developed under the assumption that independent observations are drawn from a normal distribution (and hence the mean and sample SD are independent and have specific distributional forms). It is widely applicable because it is fairly robust against violations of this assumptions. However, there are also situations in which the t-test is clearly inferior to a z-test. Consider first a set of measurements taken with a measuring instrument whose sampling errors have a known standard deviation (and approximately normal distribution). In this case, with a few observations (let's say 1 or 2, if you want to make it very clear), the z-based procedure that uses the known SD will give much more useful tests or intervals than a t-based procedure (which estimates the SD from the data at hand). Now consider estimation of a proportion. Using the information that the data consist only of 0's and 1's, and an approximate value of the proportion, we can calculate an approximate standard error more accurately (for p near 1/2) than we could without this information. The interval based on the usual variance formula p(1-p) and the z distribution is therefore better than the one based on the t distribution. This is why (as Paul pointed out) everybody uses z tests in comparing proportions, not t tests. The same applies to generalizations of tests of proportions as in logistic regression. On the pedagogical issue, if you want to motivate the z-test all you need is the formula for the variance of the mean and the fact (accepted without proof in an elementary course) that a mean of normals is normal. To get to the t-distribution you need all of this and also have to talk about the sampling distribution of the SE estimate in the denominator and how they combine to give yet another distribution which is free of the mean and the nuisance parameter (a fact that depends on subtle properties of the normal). One could take the cynical view that most intro students will get neither of these, but short of that, the Z seems easier to motivate. When I taught out of Moore and McCabe, I usually tried to give some motivation along these lines for the Z test/interval, and then when I got to the t I waved my hands and said "when we estimate the variance instead of knowing it in advance, the intervals have to be spread out a bit more as shown in this table". Alan Zaslavsky Harvard Med School = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: Student's t vs. z tests
Alan: Could you please give us an example of such a situation? "Consider first a set of measurements taken with a measuring instrument whose sampling errors have a known standard deviation (and approximately normal distribution)." Jon At 01:10 PM 4/20/01 -0400, you wrote: (This note is largely in support of points made by Rich Ulrich and Paul Swank.) I disagree with the claim (expressed in several recent postings) that z-tests are in general superseded by t-tests. The t-test (in simple one-sample problems) is developed under the assumption that independent observations are drawn from a normal distribution (and hence the mean and sample SD are independent and have specific distributional forms). It is widely applicable because it is fairly robust against violations of this assumptions. However, there are also situations in which the t-test is clearly inferior to a z-test. Consider first a set of measurements taken with a measuring instrument whose sampling errors have a known standard deviation (and approximately normal distribution). In this case, with a few observations (let's say 1 or 2, if you want to make it very clear), the z-based procedure that uses the known SD will give much more useful tests or intervals than a t-based procedure (which estimates the SD from the data at hand). snip Alan Zaslavsky Harvard Med School = 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: Student's t vs. z tests
Alan: I don't understand your comments about the estimation of a proportion. It sounds to me as if you are using the estimated standard error. (Surely you are not assuming a known standard error.) You are presumably, also using the normal approximation to the binomial (or perhaps the hypergeometric.) To do so requires a "large" sample size in which case it doesn't matter whether you use the normal or t distribution. Both would be acceptable approximations. (and both would be approximations.) So what is your point? Once more I think you need to separate the issues of what statistic to use and what distribution to use. Jon At 01:10 PM 4/20/01 -0400, you wrote: (This note is largely in support of points made by Rich Ulrich and Paul Swank.) snip Now consider estimation of a proportion. Using the information that the data consist only of 0's and 1's, and an approximate value of the proportion, we can calculate an approximate standard error more accurately (for p near 1/2) than we could without this information. The interval based on the usual variance formula p(1-p) and the z distribution is therefore better than the one based on the t distribution. This is why (as Paul pointed out) everybody uses z tests in comparing proportions, not t tests. The same applies to generalizations of tests of proportions as in logistic regression. snip Alan Zaslavsky Harvard Med School = 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: ANCOVA vs. sequential regression
I agree. ANCOVA is often said to be used when the groups are not experimental. It is a linear model of the same nature as ACOVA but technically ANCOVA refers to the experimental situation. The general linear model would allow comparing the groups while controlling for X but as William says, you should check for the interaction first to make sure the simpler model is really appropriate. At 04:16 PM 4/20/01 -0400, William B. Ware wrote: >On Fri, 20 Apr 2001, William Levine wrote: > >> A study was conducted to assess whether there were age differences in memory >> for order independent of memory for items. Two preexisting groups (younger >> and older adults - let's call this variable A) were tested for memory for >> order information (Y). These groups were also tested for item memory (X). >> >> Two ways of analyzing these data came to mind. One was to perform an ANCOVA >> treating X as a covariate. But the two groups differ with respect to X, >> which would make interpretation of the ANCOVA difficult. Thus, an ANCOVA did >> not seem like the correct analysis. > >Here's my take on it... The ANCOVA model can be implemented with >sequential/hierarchical regression as you note below... however, ANCOVA >has at least two assumptions that your situation does not meet. First, it >assumes that assignment to treatment condition is random. Second, it >assumes that the measurement on the covariate is independent of >treatment. That is, the covariate should be measured before the treatment >is implemented. Thus, I believe that you should implement the >hierarchical regression... but I'm not certain what question you are >really answering... > >I guess it is whether there is variabilty in memory for order that is >related to age, that is independent of variability in memory for >items... So, I would not call it an ANCOVA... You might also consider the >possibiltiy of interaction... That is, is the relationship between memory >for order and memory for items the same for younger and older >participants... > >WBW > >__ >William B. Ware, Professor and Chair Educational Psychology, >CB# 3500 Measurement, and Evaluation >University of North Carolina PHONE (919)-962-7848 >Chapel Hill, NC 27599-3500 FAX: (919)-962-1533 >http://www.unc.edu/~wbware/ EMAIL: [EMAIL PROTECTED] >__ > >> >> A second analysis option (suggested by a friend) is to perform a sequential >> regression, entering X first and A second to >> test if there is significant leftover variance explained by A. >> >> This second option sounds to me like the same thing as the first option. In >> an ANCOVA, variance in Y that is predictable by X is removed from the total >> variance, and then variance due to A (adjusted) is tested against variance >> due to S/A (adjusted). In >> the sequential regression, variance in the Y that is predictable by X is >> removed from the total variance, and then the leftover variance in Y is >> regressed on A. Aren't these two analyses identical? If not, what is it that >> differs? Finally, does anyone have any suggestions? >> >> Many thanks! >> -- >> William Levine >> Department of Psychology >> University of North Carolina-Chapel Hill >> Chapel Hill, NC 27599-3270 >> [EMAIL PROTECTED] >> http://www.unc.edu/~whlevine >> >> >> >> >> = >> 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/ >= > 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 soon to be moving to the Department of Pediatrics UT Houston School of Medicine = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
No Subject
Dear Eric, I'm writing my summer school course outline, and would like to know what the budget is for outside speakers before approaching anyone. The outline should be finished by the end of next week. best wishes, janeh application/ms-tnef
Re: Student's t vs. z tests
alan and others ... perhaps what my overall concern is ... and others have expressed this from time to time in varying ways ... is that 1. we tend to teach stat in a vacuum ... 2. and this is not good the problem this creates is a disconnect from the question development phase, the measure development phase, the data collection phase, and THEN the analysis phase, but finally the "what do we make of it" phase. this disconnect therefore means that ... in the context of our basic stat course(s) ... we more or less have to ASSUME that the data ARE good ... because if we did not, like you say we would go dig ditches ...at this point, we are not in much of a position to question the data too much since, whether it be in a book we are using or, some of our own data being used for illustrative examples ... there is NOTHING we can do about it at this stage. it is not quite the same as when a student comes in with his/her data to YOU and asks for advice ... in this case, we can clearly say ... your data stink and, there is not a method to "cleanse" it but in a class about statistical methods, we plod on with examples ... always as far as i can tell making sufficient assumptions about the goodness of the data to allow us to move forward bottom line: i guess the frustration i am expressing is a more general one about the typical way we teach stat ... and that is in isolation from other parts of the question development, instrument construction, and data collection phases ... what i would like to see .. which is probably impossible in general (and has been discussed before) ... it a more integrated approach to data collection ... WITHIN THE SAME COURSE OR A SEQUENCE OF COURSES ... so that when you get to the analysis part ... that we CAN make some realistic assumptions about the quality of the data, quality of the data collection process, and make sense of the question or questions being investigated At 02:01 PM 4/20/01 +1000, Alan McLean wrote: All of your observations about the deficiencies of data are perfectly valid. But what do you do? Just give up because your data are messy, and your assumptions are doubtful and all that? Go and dig ditches instead? You can only analyse data by making assumptions - by working with models of the world. The models may be shonky, but they are presumably the best you can do. And within those models you have to assume the data is what you think it is. = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: Student's t vs. z tests
nice note mike Impossible? No. Requiring a great deal of effort on the part of some cluster of folks? Definitely! absolutely! There is some discussion of this very possibility in Psychology, although I've yet to see evidence of fruition. A very large part of the problem, in my mind, is breaking out of established stereotypes of what a Stats and Methods sequence should look like, and then finding the materials to support that vision. i think it may ONLY be possible within a large unit that requires their students to take their methods courses ... design, testing, statistics, etc. i think it will be very hard for a unit that PROVIDES SUBSTANTIAL cross unit service courses ... to do this for example, in our small edpsy program at penn state, most of the courses in research methods, measurement, and stat ... are for OTHERS ... even though our own students take most of them too. if we redesigned a sequence that would be more integrative ... for our own students, students from outside would NOT enroll for sure ... because they are looking for (or their advisors are) THE course in stat ... or THE course in research methods ... etc. they are not going to sit still for say a two/3 course sequence If I could find good materials that were designed specifically to support the integrated sequence, I might be able to get others to go along with it. i think the more serious problem would be agreeing what should be contained in what course ... that is, the layout of this more integrative approach if that could be done, i don't think it would be that hard to work on materials that fit the bill ... by having different faculty write some modules ... by finding good web links ... and, gathering a book of readings what you want is NOT necessarily a BOOK that does it this way but, a MANUAL you have developed over time that accomplishes the goals of this approach It can be done, but it will require someone with more energy and force of will than I. i doubt i have the energy either ... Mike *** Michael M. Granaas Associate Professor[EMAIL PROTECTED] Department of Psychology University of South Dakota Phone: (605) 677-5295 Vermillion, SD 57069 FAX: (605) 677-6604 *** All views expressed are those of the author and do not necessarily reflect those of the University of South Dakota, or the South Dakota Board of Regents. _ dennis roberts, educational psychology, penn state university 208 cedar, AC 8148632401, mailto:[EMAIL PROTECTED] http://roberts.ed.psu.edu/users/droberts/drober~1.htm = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Introducing inference using the binomial (was: Student's t vs. z
On 19 Apr 2001, Paul Swank wrote: I agree. I normally start inference by using the binomial and then then the normal approximation to the binomial for large n. It might be best to begin all graduate students with nonparametric statistics followed by linear models. Then we could get them to where they can do something interesting without taking four courses. At 01:28 PM 4/19/01 -0500, you wrote: Why not introduce hypothesis testing in a binomial setting where there are no nuisance parameters and p-values, power, alpha, beta,... may be obtained easily and exactly from the Binomial distribution? Jon Cryer I concur with Jon and Paul. (I'll refrain from making a crack about Ringo.) When I was an undergrad, the approach was z-test, t-test, ANOVA, simple linear regression, and if there was time, a bit on tests for categorical data (chi-squares) and rank-based tests. I got great marks, but came away with very little understanding of the logic of hypothesis testing. The stats class in 1st year grad school (psychology again) was different, and it was there that I first started to feel like I was achieving some understanding. The first major chunk of the course was all about simple rules of probability, and how we could use them to generate discrete distributions, like the binomial. Then, with a good understanding of where the numbers came from, and with some understanding of conditional probability etc, we went on to hypothesis testing in that context. One thing I found particularly beneficial was that we started with the case where the sampling distribution could be specified under both the null and alternative hypotheses. This allowed us to calculate the likelihood ratio, and to use a decision rule to minimize the overall probability of error. We could also talk about alpha, beta, and power in this simple context. Then we moved on to the more common case where the distribution cannot be specified under the alternative hypothesis, and came up with a different decision rule--i.e., one that controlled the level of alpha. The other thing I found useful was that all of this was without reference to any of the standard statistical tests--although we found out that the sign test was the same thing when we did get to our first test with a proper name. We followed that with the Wilcoxon signed ranks test and Mann-Whitney U before ever getting to z- and t-tests. By the time we got to these, we already had a good understanding of the logic: Calculate a statistic, and see where it lies in its sampling distribution under a true null hypothesis. An undergrad text that takes a similar approach (in terms of order of topics) is Understanding Statistics in the Behavioral Sciences, by Robert R. Pagano. Not only is the ordering of topics good, but the explanations are generally quite clear. I would certainly use Pagano's book again (and supplement certain sections with my own notes) for a psych-stats class. -- Bruce Weaver New e-mail: [EMAIL PROTECTED] (formerly [EMAIL PROTECTED]) Homepage: http://www.angelfire.com/wv/bwhomedir/ = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
ANCOVA vs. sequential regression
Here is a statistical issue that I have been pondering for a few weeks now, and I am hoping someone can help set me straight. A study was conducted to assess whether there were age differences in memory for order independent of memory for items. Two preexisting groups (younger and older adults - let's call this variable A) were tested for memory for order information (Y). These groups were also tested for item memory (X). Two ways of analyzing these data came to mind. One was to perform an ANCOVA treating X as a covariate. But the two groups differ with respect to X, which would make interpretation of the ANCOVA difficult. Thus, an ANCOVA did not seem like the correct analysis. A second analysis option (suggested by a friend) is to perform a sequential regression, entering X first and A second to test if there is significant leftover variance explained by A. This second option sounds to me like the same thing as the first option. In an ANCOVA, variance in Y that is predictable by X is removed from the total variance, and then variance due to A (adjusted) is tested against variance due to S/A (adjusted). In the sequential regression, variance in the Y that is predictable by X is removed from the total variance, and then the leftover variance in Y is regressed on A. Aren't these two analyses identical? If not, what is it that differs? Finally, does anyone have any suggestions? Many thanks! -- William Levine Department of Psychology University of North Carolina-Chapel Hill Chapel Hill, NC 27599-3270 [EMAIL PROTECTED] http://www.unc.edu/~whlevine = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Statistical Notation
Does anyone know of a resource that lists symbols often used in statistics and probability. What I am looking for is something with the symbol, its name, and some common uses. In particular, I would like web sources, but I would be grateful for any suggestions. Best, Brett = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
