Nand,
I appreciate your interest in the various traditions (if I can call them that) within statistics, including decision analysis. Someone & I took the decision analytic "bicycle" example outside of the group; in that discussion I provided the following info (in >>><<<) & also the attachment:
>>>> The attached exemplifies what I’m talking about
> (“bicycle” changed to “vehicle”). To decide whether
> to perform the experiment, one compares Cells b7 & b24;
> in this case, the risk (expected loss) of performing the
> experiment & then making a decision exceeds that of
> simply making the decision not to buy a new vehicle, so we
> minimize the risk by simply retaining & maintaining the
> old vehicle.
>
> If you tell me whether you understand how to derive
> everything up to Cell b7, which is the easier,
> pre-experimental portion of the analysis, then I will start
> discussing the remainder with you.
<<<
I'd be happy to discuss the details with anyone, though I hesitate to crowd the mailboxes of everyone with what might turn into lengthy discussions.
In any case, I didn't respond to that question about portfolio analysis because--assuming that refers to portfolios of financial products--I don't know much about stocks/bonds/options/hedging etc.
On the topic of "worth of an experiment:" Based on your note below, I gather that you're asking here about what SN Goodman calls the "evidential value" of an experiment. Bayesians take the likelihood ratio (LR) to indicate evidential value, & Goodman's second paper you mention discusses the LR (also called the "bayes factor"). Yes I certainly could have mentioned both papers, but sometimes less is more (one tends to ignore mails that are just too long).
At the risk of violating my own objective of brevity, I'll just add that the perennial question of "whether the p-value measures evidence" is meaningless until one defines evidence (People can argue endlessly about whether a gradna can fly, but they will not reach agreement until they define a gradna). Frequentists say p measures evidence; bayesians say it does not, but many times each party is right in its own way. That admission notwithstanding, I will say that bayesians, at least, can formalize evidence into something specific enough to be measurable, to wit: evidence = the amount by which beliefs should change as the result of data.
Hopefully, that addresses all your concerns; if not, please contact me again.
--- On Tue, 9/29/09, nand kishore <nk_singh1@...> wrote:
From: nand kishore <nk_singh1@...>
Subject: [Statisticians_group] Discussion on Basics of Statistics
To: Statisticians_group@...
Date: Tuesday, September 29, 2009, 9:49 AM
In last three weeks there were good discussions around basics of statistics. Some attracted a lot of discussion while others could not. Radu raised basic question "whether random variable exist?" why assumption linearity is required to use correlation? What is meaning of correlation for historical (time series data)? How to interpret statistical results without longer (towards infinity) sequence of trials?
Gowri raised relevance of Binomial distribution in medical data. Nishant raised question related with Cronbach's Alpha
Before discussing these questions, first I would like to discuss my own questions
(1) Whether p-value reflects worth of experiment itself?
(2) Why concept of countable union is so important in mathematics for representing real life phenomena through set?
Perhaps second question is too theoretical for drawing attention of group members. I would like to close this issue with simple comment that idea of countable union is related with infinite set. It is as interesting as concept of infinity as number. Many interesting discussions (like Russell's paradox ) generated due to complexity of infinite set and generated new branch
of mathematics called axiomatic set theory. Since set is one of fundamental framework though which human mind thinks to explain worldly affair, concept of infinite set is described as related with god by mathematician like Georg Cantor.
In reference to my first question "Whether p-value reflects worth of experiment itself", Andrew wanted to know "whether worth is the amount by which the experiment increases the expected utility (or decreases the expected loss) of a decision?" . He suggested to follow decision theoretic approach to get worth of experiment. I am impressed by his example (to explain things by examples from general experiences) but also surprised why Andrew did not try to give same type of solution to Radu (although he touched this aspect in his long reply) for getting relative worth of portfolio by using decision theoretic approach (may be Bayesian decision where one can combine your prior information, loss or utility function and data to take decision on basis of current data and no need long sequence of trial for interpretation) .
I hope further discussion between Andrew and Radu will generate appreciation for Bayesian decision theory in portfolio management. Although I have very limited knowledge of portfolio management, I would like to participate in it.
Now again on my question "Whether p-value reflects worth of experiment itself". Certainly my intension of using `worth' is different what Andrew understood. Before writing `worth', I explored many words like power, evidence, index etc but most of them were reserved for statistician. In paper "…The P value fallacy" (mentioned by Andrew), Fisher describes P value as informal index between null hypothesis and data (experiment) (page 997 in section of P value). In fact, unlike type1 error, which is
constant and fixed by user, P value is an index and based on data through experiment. My intention with `worth' was near to `informal index' used by Fisher.
Paper mentioned by Andrew is very useful. I do not know why he did not mentioned second paper by some Author. In first paper Author has raised question on P value and significance test while in second paper he has tried to provide solutions in Bayesian framework.
Radu has raised question on framework of statistics. He asked question "Whether random variable exhibits". Perhaps he could not search better word than "random variable" to express his feeling nearer to "unpredictable" . In fact "random variable" is reserve word in statistics as mentioned by Indrajt Sengupta. He correctly wrote that "random variable" is function from event set to set of real numbers such that …". Here nothing is "random" or "variable". It is random variable which makes our works easy by providing platform to use mathematical tools for real numbers in place of set. Certainly it has cost also. We study "induced probability" in place of real probability on event set.
In answer of "whether randomness exist?", I can say what exist is combination of known (rule) and known (error) as in framework of regression. I do not know how one can declare any thing "unpredictable" as Radu tried to explain with toothbrush example. Statisticians (frequentist) start works with assumption of randomness (sequence of uncertain events, not haphazard, with certain conditions). There are interesting discussion around concept of "Randomness" (see also) and
"Unpredictable" I think statistics is not a tool to find the truth (like toothbrush) but should be used to find most suitable answer under partial certainty. In toothbrush example we should first think how toothbrush (parameter) exhibiting information about itself through data. If there is framework to associate parameter and data then we can proceed to search the toothbrush either in Classical or Bayesian (more suitable) setup.
Radu also raised question on assumption of linearity to measure relationship through Karl Pearson correlation coefficient. In case of non linear relationship, one can use Correlation Ratio which is nearer to Cronbach's Alpha which is used for measuring internal consistency between different factors.
Above discussion shows representation of any phenomena in world of statistics is as important as phenomena in itself (may be movements of price of asset or searching a toothbrush). Question raised by Gowri for use of Binomial distribution in biological data may also be evaluated in this light. Base of Binomial distribution is Bernoulli trial and it based on two opposites (yes/no). Binomial distribution is number of successes in a sequence of prefix number of trial. If we can represent a non dichotomous phenomena (like cured, partially cured, get worse …) according to number of some basic (dichotomous) events. I am not very precise to give practical example,
but hope some of my friends (who has experience on medical data) may help in this regard.
On conclusion, I can say, statistics provides tools under its interrelated frameworks (like Statistical Inference, Decision Theory, Probablity Theory, Stochastic Processes, Design of Experiments, Design of Survey etc.) in two measure setups Classical and Baysian. To represent a real life phenomena so that it may be fitted in particular framework (and setup) is very crucial and is not guarantied that all real life phenomena (collection of information and believes) can be translated suitably in existing frameworks and setup. Hope discussion for getting a confluence of Classical (P value and confidence interval) and Baysian (Bayes factor and credible interval) will be continued. I hope, through this exploration we will identify role of milestones (like exploratory analysis, confirmatory analysis).
Nand Kishore
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