The significance test is perhaps the most used statistical procedure in the world, though has never been without its detractors. This is the second of two posts exploring Neyman's frequentist arguments against the significance test; if you have not read Part 1, you should do so before continuing (“The frequentist case against the significance test, part 1”).

Neyman offered two major arguments against the significance test:

**The significance test fails as an epistemic procedure.**There is no relationship between the \(p\) value and rational belief. More broadly, the goal of statistical inference is tests with good error properties, not beliefs.**The significance test fails as a test.**The lack of an alternative means that a significance test can yield arbitrary results.

First, it must be said that Neyman did not think that significance tests were useless or misleading, all the time. He said “The [significance test procedure] has been applied since the invention of the first systematically applied test, the Pearson chi-square of 1900, and has worked, on the whole, satisfactorily. However, now that we have become sophisticated we desire to have a theory of tests.” Obviously, he is not making a blanket statement that significance tests are, generally, good science; he was making an empirical statement about the applications of significance tests in the first half of the twentieth. It is debatable whether he would say the same about the significance test since then.

Of course, we should not evaluate a procedure by its purported results; we can be misled by results, and even worse, this involves an inherent circularity (how do we determine whether the procedure

*actually*performed satisfactorily? Another test?). However, this was merely an informal judgment of Neyman's; we should not over-interpret it either way. After all: he will show that the foundation of the significance test is flawed, and he clearly thought this was important.

### An example: Cushney and Peebles' soporific drugs

Suppose that we are interested in the effect of sleep-inducing drugs. Cushney and Peebles (1905) reported the effects of two sleep-inducing drugs on 10 patients in a paired design. Conveniently, R has the data for these 10 patients built-in, as the sleep data set; the data comprise 10 participants' improvements over baseline hours of sleep, for each drug. If we wished to compare the two drugs, we might compute a difference score for each participant and subject these difference scores to a one-sample \(t\) test.The null hypothesis, in this case, is that the population mean of the difference scores, \(\mu=0\). Making the typical assumptions of normality and independence, we know that, under the null hypothesis,

\[ t = \frac{\bar{x}}{\sqrt{s^2/N}} \sim t_{N-1} \] where \(\bar{x}\) and \(s^2\) are the difference-score sample mean and variance.

The figure below shows the distribution of the \(t\) statistic assuming that the null hypothesis is true, with the corresponding \(p\) values on the top axis. Increasingly red areas show increasing evidence against the null hypothesis, according to the Fisherian view.

If we decided to use the \(t\) statistic to make a decision in a significance test, we would decide on a criterion: say, \(|t|>2.26\), which would lead to \(\alpha=0.05\). Repeating the logic of the significance test, as Neyman put it:

When an “improbable sample” was obtained, the usual way of reasoning was this: “Were the hypothesis \(H\) true, then the probability of getting a value of [test statistic] \(T\) as or more improbable than that actually observed would be (e.g.) \(p = 0.00001\). It follows that if the hypothesis \(H\) be true, what we actually observed would be a miracle. We don't believe in miracles nowadays and therefore we do not believe in \(H\) being true.” (Neyman ,1952)In the case of our sample, we can perform the \(t\) test in R:

```
##
## Paired t-test
##
## data: sleep$extra[1:10] and sleep$extra[11:20]
## t = -4.0621, df = 9, p-value = 0.002833
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -2.4598858 -0.7001142
## sample estimates:
## mean of the differences
## -1.58
```

In a typical significance test scenario, this would lead to a rejection of the null hypothesis, because \(|t|>2.26\).### Neyman's second argument: Significance testing can be arbitrary

Remember that at this point, we have not considered anything about what we would expect if the null hypothesis were*not*true. In fact, Fisherian significance testing does not need to consider any alternatives to the null. The pseudo-falsificationist logic of the significance test means that we only need consider the implications for the data under the null hypothesis.

Neyman asks: why use the \(t\) statistic for a significance test? Why use the typical \(\bar{x}\) and \(s^2\)? Neyman then does something very clever: he defines two new statistics, \(\bar{x}_1\) and \(s^2_1\), that have

*precisely*the same distribution as \(\bar{x}\) and \(s^2\) when the null hypothesis is true, and shows that using these two statistics leads to a

*different*test, and

*different*results:

\[ \begin{eqnarray*} \bar{x}_1 &=& \frac{x_1 - x_2}{\sqrt{2N}},\ s^2_1 &=& \frac{\sum_{i=1}^N x_i^2 - N\bar{x}^2_1}{N-1}, \end{eqnarray*} \]

where \(x_i\) is the difference score of the $i$th participant (assuming the samples are in arbitrary order).

Neyman proves that these statistics have the same joint distribution as \(\bar{x}\) and \(s^2\), but we can verify Neyman's proof using R (code available here). The top row of the plot below shows the histogram of 100,000 samples of \(\bar{x}\), \(s^2\), and the \(t\) statistic for \(N=10\) and \(\sigma^2=1\), assuming the null hypothesis is true; the bottom row shows the same 100,000, but computing \(\bar{x}_1\), \(s^2_1\), and \(t_1\), the \(t\) statistic computed from \(\bar{x}_1\) and \(s^2_1\). The red line shows the theoretical distributions. The distributions match precisely.

We now have two sets of statistics that have the same distributions, and will thus produce a significance test with precisely the same properties when the null hypothesis is

*true*. Which should we choose? Fisher might object that \(\bar{x}_1\) and \(s^2_1\) are not sufficient, but this only pushes the problem onto sufficiency: why sufficiency?

The figure below shows that this matters for the example at hand. The figure shows 100,000 simulations of \(t\) and \(t_1\) plotted against one another; when \(t\) is large, \(t_1\) tends to be small; when \(t\) is small, \(t_1\) tends to be large. The red dashed lines show the \(\alpha=0.05\) critical values for each test, and the blue curves show the limits of bounds within which \((t,t_1)\) has to be contained.

The red point shows \(t\) and \(t_1\) for the Cushny and Peebles' data set; \(t\) would lead to a rejection of the null, while \(t_1\) would not.

Examining the definitions of \(\bar{x}_1\) and \(s^2_1\), it isn't difficult to see what is happening; when the null is true, these statistics will have identical distributions to \(\bar{x}\) and \(s^2\). However, when the null is

*false*, they will not. The distribution of \(\bar{x}_1\) will continue to have a mean of 0 (instead of \(\mu\)), while the distribution of \(s^2_1\) will become more spread than \(s^2\). The effect of this is that the power of the test based on \(t_1\) will

*decrease*as the true effect size increases!

A consideration of both Type I and Type II errors makes it obvious which test to choose; we should choose the test that yields the higher power (this is, incidentally, closely related to the Bayesian solution to the problem through the Neyman-Pearson lemma). The use of \(t_1\) would lead to a

*bad*test, when both Type I error rates and Type II error rates are taken into account. A significance test, which does not consider Type II error rates, has no account of

*why*\(t\) is better than \(t_1\).

### More problems

The previous development is bad for a significance test; it shows that there can be two statistics that lead to different answers, yet have the same properties from the perspective of significance testing. Following this, Neyman proves something even better: we can always find a statistic that will have the same long-run distribution under the null as \(t\), yet will yield an arbitrarily high test statistic for*our sample*. This means that we cannot simply base our choice of test statistic on what would yield a more or less conservative test statistic for our sample.

Neyman defines some constants \(\alpha_i\) using the obtained samples \(x_i\): \[ \alpha_i = \frac{x_i}{\sqrt{\sum_{i = 1}^N x_i^2}} \] then for future samples \(y_i\), \(i=1,\ldots,N\) defines \[ \begin{eqnarray*} \bar{y}_2 &=& \frac{\sum_{i=1}^N \alpha_iy_i}{\sqrt{N}},\ s^2_2 &=& \frac{\sum_{i=1}^N y_i^2 - N\bar{y}_2^2}{N-1}, \end{eqnarray*} \] and of course we can compute a \(t\) statistic \(t_2\) based on these values. If we use our observed \(x_i\) values for \(y_i\), this will yield a \(t_2=\infty\), because \(s^2_2 = 0\), exactly! However, if we check the long-run distribution of these statistics under the null hypothesis, we again find that they are exactly the same as \(\bar{x}\), \(s^2\), and \(t\):

If we considered the power of the test based on \(t_2\), we would find that it is worse than the power based on \(t\). The significance test offers no reason why \(t\) is better than \(t_2\), but a consideration of the frequentist properties of the test does. Neyman has thus shown that we

*must*consider an alternative hypothesis in choosing a test statistic, otherwise we can select a test statistic to give us any result we like.

### Conclusion: The importance of power

At the risk of belaboring a point that has been made over and over, power is not a mere theoretical concern for a frequentist. Neyman and Pearson offer an account of why some tests are better than others, and also, in some cases, an account of which test is the optimal; however, just because a test is optimal, does not mean it is*good*.

We might always manage to avoid Type I errors at the same rate (assuming the null hypothesis is true), but as Neyman points out, this is not enough; one needs to consider power, and how one wants to treat both Type I error and power. A good frequentist test may balance Type I and Type II error rates; a good frequentist test may control the Type I error rate while having a power that is above a certain probability. From a frequentist perspective these are decisions that must be made prior to an experiment; none of them can be addressed within the significance testing framework.

To recap both posts, Neyman makes clear why significance testing, as commonly deployed in the scientific literature, does not offer a good theory of inference: it is fails epistemically by allowing arbitrary “rational” beliefs, and it fails on statistical grounds by allowing arbitrary results.

From a frequentist perspective, what might a significance test be useful for? Neyman allows that before a critical set of experiments is performed, exploratory research must be undertaken. Generating a test or a confidence procedure requires some assumptions. Neyman does not offer an account of the process of choosing these assumptions, and seems content to leave this up to substantive researchers. Once a formal inference is needed, however, it is clear that from a frequentist perspective the significance test is inadequate.

*[the source to this blog post, including R code to reproduce the figures, is available here: https://gist.github.com/richarddmorey/5806aad7191377dcbf4f]*

To me the main problem with confidence interval or hypothesis testing is the following but I don't see discussions such as yours addressing this practical issue :

ReplyDeleteConfidence interval or hypothesis testing results depend on the model and the assumptions made. Often there are substantial uncertainties about the model and the assumptions but these are often ignored in performing a hypothesis test. You might prescribe to use robust inference or to put priors on these elements but I don't think these suggestions are often practical.

On another matter:

I have tried to use Bayes Factor in a number of occasions but I have found it very sensitive to the priors and in many interesting cases I run into Bartlett paradox.

A Reader

I like how people describe such things

ReplyDeleteCheap London Escorts

ReplyDeleteالرائد افضل شركات تنظيف خزانات المياه يسعدنا ان نقدم لكم افضل خدمات

شركة غسيل خزانات بالمدينة المنورة تنظيف خزانات بالمدينة المنورة

افضل شركة تنظيف منازل بالمدينة المنورة شركة تنظيف بيوت بالمدينة المنورة

Bayesian insights are a type of inductive derivation, that is case based surmising. You separate however much data from each case as could be expected and join that data. buy assignment.

ReplyDeleteWe are an exceptionally rumored cleaning organization conveying probably the best cleaning and expert jack of all trades administrations. We are basically outstanding amongst other neighborhood cleaning organizations in Dubai. In addition to the fact that we cater to private customers, we further spotlight on corporate clients too.

ReplyDeletehttps://www.cleanandshine.ae/

Want to know what are the parts of an essay? Your final essay draft should contain these elements in the following order: title, outline, introduction, several body paragraphs, and conclusion.

ReplyDeleteexperimental articulation about the utilizations of hugeness tests in the main portion of the twentieth. It is easily proven wrong whether he would say the equivalent regarding the centrality test from that point forward.Research Proposal Help

ReplyDeleteTo fabricate a vocation in this field, one ought to have proficient abilities in these three offices: programming, space information and investigation. In programming you have to have a solid information in Python, R, Scala, Julia and Java. You should have the option to comprehend a few expository issues. In SQL, you have to have hands-on understanding. What's more, a short information in Machine Learning.

ReplyDeleteoffice cleaning

A ton of understudies nowadays don't give English the significance it merits and the essential explanation is that they believe that they are very knowledgeable with the language and there is nothing that they could learn. The issue is that learning English sentence structure and upgrading your jargon are two extremely vital necessities all together for an individual to be completely outfitted with the arms stockpile of English.ISO 14001 Saudi Arabia

ReplyDeleteAt the point when you study the subjects like material science, science you get increasingly specialized and inside and out comprehension of the world you live in. It permits you to create logical information and hypothesis about everything on the planet that you go over. Science venture like assignments drive you to consider potential arrangements that can occur inside the setting of this world and even past.

ReplyDeletecontinental car hire lahore

Understudies who are shown this subject flee from it in light of the fact that the educators may be inept, yet whenever watched, science is an extremely fun subject and empowers the understudy to think carefully control and mental quality. That is the reason it is supposed to be a mandatory subject of primary school so understudies can familiarize themselves with the subject at an early age. It challenges the current standards and customs of the world and causes the understudies to have a positive point of view of the world with a superior comprehension and information.

ReplyDeleteFrozen truck for rent in Dubai

Ecommerce website design, cheap website development & Integration company in the UK.Expert PrestaShop, Open Cart, Shopify, Magento and WordPress developers. visit : https://www.budgetweb.ae

ReplyDeleteSince we realize how to get the entirety of odd arrangements we have to figure what we're including. On the off chance, website design dubai price that we investigate the gatherings we see that for bunch number z, we have z continuous odd digits starting with the [(z^2 - z + 2)/2]th odd digit. This arrangement has z continuous digits so it would end at the [z^2 - z + 2)/2 + z - 1]th odd digit.

ReplyDeleteThese basic advances are the establishments for variable based math, geometry, trigonometry, analytics, direct polynomial math and measurable examination. The more noteworthy, website design dubai price australia dominance you accomplish at the lower levels, the more instinctive you will be at the more elevated levels. When you perceive the examples, math does itself..

ReplyDeleteAmazing work. Appreciate it for this grand post.Such a nice information.

ReplyDeleteWrite My Finance Homework

great article amazing

ReplyDeleteNursing Writing Help

ReplyDeleteI was reading your contents and found a lot of information. Keep posting such contents for us.

Car Rental Islamabad

ReplyDeleteVery nice and informative blog .Keep posting such contents for us.We are also providing the Best Rent a Car in Islamabad without driver.

Custom Homework Services. It is important to choose the best one among the many. Before choosing the one go for that website reviews first and then check the website like their services.

ReplyDeleteRent a car Islamabad,

ReplyDeleteWe are providing rent a car in Rawalpindi/Islamabad for all your transport needs.When it comes to Car Rental Islamabad, our prime focus is to provide reliable car rental services. Being the best Car Rental Company in Islamabad we provide insured cars

https://rent-car-islamabad.pk/

uzairhassan13@yahoo.com

Thanks for sharing such a wonderful article.car Rental Islamabad

ReplyDelete

ReplyDeleteRent Car Islamabad offers one of the most professional car rental services throughout Islamabad.

Rent a Car in IslamabadTravel & Tour masters is a leading CAR RENTAL SERVICES in Pakistan.

ReplyDeleteCar Rental Islamabad,Rent a Car in Islamabad,Rent a Car Islamabad

Bridge City is a Reputable Cabinetry & Renovations Contractor. We Provide Services in IKEA Kitchen Design and Construction. We are a Quality Renovation Contractor that Renews and Brings Life back to Existing Properties no Matter what kind of Space you are Looking to Renovate: Bathrooms, Living Rooms, Kitchens, or Other Living Spaces. ikea kitchen designer

ReplyDeleteRamma foundation is a company and a foundation that provide crack repair snow remove and gutter clean service. We are the crack repair foundation in Alberta Edmonton. We provide services like crack repair, gutter clean services, show removal, in short we are the foundation repair in Edmonton. If you are worried about the cracks on your home do not worry we are here to get it repair. If you are worried about snow on your roof, we can remove it. We are here to clean you gutter. crack repair foundation in Alberta Edmonton

ReplyDeleteWe are a Quality Renovation Contractor that Renews and Brings Life back to Existing Properties regardless . can someone do my assignment

ReplyDeleteLocal Cabs Service bring the best in class ride-hailing services right at your doorstep. Now ride to your destination in luxury and comfortable rides. Download the app.

ReplyDeleteRide with B4U Cabs

Drive with B4U Cabs

Buy Franchise

About B4U

Cab Service Lahore

Cab Service Karachi

Cab Service Hyderabad

Download as Customer

Download as Boss

I truly cheerful discovered this site in the long run. Truly educational and defective, Thanks for the post and exertion! It would be ideal if you continue sharing all the more such blog.

ReplyDeleteProbability And Statistics Assignment

Drive Cool Transport is one of the distinguished companies that are engaged in offering high quality freezer trucks for rent in UAE. For clients who are engaged in some trading or distribution business, we have the best Freezer Truck Dubai. Our fleet of freezer vehicles is inclusive of Freezer Truck Dubai, Refrigerator Truck Dubai, Refrigerator Van Rental Dubai, Refrigerator Truck With Tail Lift, Refrigerated Truck With Partition, chiller van Van Rental Dubai, Chiller Truck Rental Dubai, and various other types of refrigerated transport.

ReplyDeleteGreat Job! Thanks For sharing the post.

ReplyDeleteRent a Car Islamabad

https://falconrentcar.com/

ReplyDeleteFalcon rent car is a leading CAR RENTAL SERVICES in Pakistan.

Car Rental Islamabad,Rent a Car in Islamabad,Rent a Car Islamabad,Insured Car Rental Islamabad

https://www.yourcar.pk/

ReplyDeleteYour car is a leading CAR RENTAL SERVICES in Pakistan.

Car Rental Islamabad,Rent a Car in Islamabad,Rent a Car Islamabad,Insured Car Rental Islamabad

ReplyDeleteSpecial Thanks for Sharing such a good content to aurthor ...But Nowdays rent a car facility available in Islamabad & Rawalpindi One Of My Best Experience with Falcon Rent a car in Islamabad guys you can also avail this opportunity at lowest Rates.

Rent a car Islamabad services are just one call away. The latest model cars are available for rent in Islamabad with & without driver.

https://falconrentcar.com/

We are providing the best car rental service to you people if anyone wants a car for rent then rent your car from Rently

ReplyDeletewww.amanathospital.com

ReplyDelete958, the year when Amanat Eye Hospital opened its doors for public, at Rawalpindi. The first private eye hospital of the city.

As we moved step by step, staying abreast with the latest technology of the field, we marked our presence and our foot print for others to follow. Today we stand in our realm with a crown on our head, cherishing every accomplishment with a pride and with humility for we owe it to our patients.

amanathospital.com

ReplyDelete958, the year when Amanat Eye Hospital opened its doors for public, at Rawalpindi. The first private eye hospital of the city.

As we moved step by step, staying abreast with the latest technology of the field, we marked our presence and our foot print for others to follow. Today we stand in our realm with a crown on our head, cherishing every accomplishment with a pride and with humility for we owe it to our patients.

Taxis in Kent island Maryland

ReplyDeleteIf you’ve crossed the Chesapeake Bay Bridge, you’ve gone right by Kent Island. Located off the west coast of Queen Anne’s County, Kent Island sits at the base of the Chesapeake Bay Bridge, connecting Maryland’s eastern and western shores. Public transportation through this area is limited, and Annapolis Taxi Cabs can alleviate difficult transportation situations. Annapolis Taxi Cabs offers reliable Taxi Cab Service in Kent Island Maryland.

Taxis in stevensville Maryland

ReplyDeleteIf you’ve crossed the Chesapeake Bay Bridge, you’ve gone right by Kent Island. Located off the west coast of Queen Anne’s County, Kent Island sits at the base of the Chesapeake Bay Bridge, connecting Maryland’s eastern and western shores. Public transportation through this area is limited, and Annapolis Taxi Cabs can alleviate difficult transportation situations. Annapolis Taxi Cabs offers reliable Taxi Cab Service in stevensvill Maryland.

Honeywell Led Light

ReplyDeleteCecon Engineering being one of the top LED brands in Pakistan to provides you one of the best-LED lights in Pakistan to make your life lighten with our cheapest LED lights.

An interesting discussion is worth a comment. I think that you should write more on this topic, it might not be a taboo subject but generally, people are not enough to speak on such topics. To the next. Cheers Rent a Car Service in Islamabad

ReplyDelete

ReplyDeleteI was diagnosed as HEPATITIS B carrier in 2013 with fibrosis of the

liver already present. I started on antiviral medications which

reduced the viral load initially. After a couple of years the virus

became resistant. I started on HEPATITIS B Herbal treatment from

ULTIMATE LIFE CLINIC (www.ultimatelifeclinic.com) in March, 2020. Their

treatment totally reversed the virus. I did another blood test after

the 6 months long treatment and tested negative to the virus. Amazing

treatment! This treatment is a breakthrough for all HBV carriers.

Thank you admin for sharing this post with us. I really appreciate your hard work in writing this post. Keep uploading these types of information. Rent a Car Islamabad With Driver Rates

ReplyDeleteI like your post. It is good to see you verbalize from the heart and clarity on this important subject can be easily observed...Van for rent in Islamabad

ReplyDeleteIf you are looking for Rent a Car in Islamabad or Rent a Car Rawalpindi then you are on right place.

ReplyDeletePakistan Qureshi Tours offer all types of cars for rent

Rent A Car Islamabad

We are located in Dubai and provide chiller and freezer van rental service for day, week and month not only in Dubai but also in UAE. We provide our professional and secure services to our valuable customer. For more details please visit our website https://vanchiller.com/

ReplyDeleteVery good article thank you for sharing energetic information. I really appreciate you.

ReplyDeletekeep it up,write and upload on regularly. I read the blog on a daily basis.

Get the best home tutor in Karachi At your Doorstep. Best Tutor Academy in Karachi For All Classes and Subject

Jinnah Tutor Academy In Karachi is a trusted tutor academy by parents and children, well-known for its compatibility in a variety of subjects, extensive experienced Home tutor in Karachi to cover loads of curriculum, and determination to guide students on the career-changing path through home tuition.

We are a team of high-qualified, award-winning, top-rated & professional teachers that not just help students study in crucial hours butHome Tutors also prepare them for future academic challenges.

thanks for share this information with us CHAUFFEUR SERVICES

ReplyDeleteThank you for sharing such an informative blog. Chiller vans for rent in dubai

ReplyDelete