#hypothesis testing

[jadagul]

I am spectacularly offended by this Matt Levine reader email about using astrology in consumer finance prediction.

This was a machine learning model – the job of the data scientist was, put everything in, see what's significant, of that discard everything that's discriminatory, the rest is your model. Ultimately with twelve astrological signs it's over 50/50 that one will come out significant at 95%. I thought it was elegant. "Astrological signs? Do you believe that?" my boss said. I said it wasn't a question of belief, I was a statistician and was going to follow the numbers rather than letting anyone's preexisting theories about the stars and planets influence the data science. I think he believed that meant I'd agreed to take it out.

Like, the guy literally said "We're very likely to have a false positive here by chance, but since we got one we have to take it seriously. I'm a statistician."

He's fully aware that he's p-hacking and garden-pathing. He's fully aware of the multiple comparisons problem. And then he endorses the conclusion anyway!

(And, as a side note, it's not over 50/50; If you do twelve tests the chance of one coming out significant by chance is about 46%. So he fucked up the arithmetic too!)

[omegaphilosophia]

The Philosophy of Statistics

The philosophy of statistics explores the foundational, conceptual, and epistemological questions surrounding the practice of statistical reasoning, inference, and data interpretation. It deals with how we gather, analyze, and draw conclusions from data, and it addresses the assumptions and methods that underlie statistical procedures. Philosophers of statistics examine issues related to probability, uncertainty, and how statistical findings relate to knowledge and reality.

Key Concepts:

Probability and Statistics:

Frequentist Approach: In frequentist statistics, probability is interpreted as the long-run frequency of events. It is concerned with making predictions based on repeated trials and often uses hypothesis testing (e.g., p-values) to make inferences about populations from samples.

Bayesian Approach: Bayesian statistics, on the other hand, interprets probability as a measure of belief or degree of certainty in an event, which can be updated as new evidence is obtained. Bayesian inference incorporates prior knowledge or assumptions into the analysis and updates it with data.

Objectivity vs. Subjectivity:

Objective Statistics: Objectivity in statistics is the idea that statistical methods should produce results that are independent of the individual researcher’s beliefs or biases. Frequentist methods are often considered more objective because they rely on observed data without incorporating subjective priors.

Subjective Probability: In contrast, Bayesian statistics incorporates subjective elements through prior probabilities, meaning that different researchers can arrive at different conclusions depending on their prior beliefs. This raises questions about the role of subjectivity in science and how it affects the interpretation of statistical results.

Inference and Induction:

Statistical Inference: Philosophers of statistics examine how statistical methods allow us to draw inferences from data about broader populations or phenomena. The problem of induction, famously posed by David Hume, applies here: How can we justify making generalizations about the future or the unknown based on limited observations?

Hypothesis Testing: Frequentist methods of hypothesis testing (e.g., null hypothesis significance testing) raise philosophical questions about what it means to "reject" or "fail to reject" a hypothesis. Critics argue that p-values are often misunderstood and can lead to flawed inferences about the truth of scientific claims.

Uncertainty and Risk:

Epistemic vs. Aleatory Uncertainty: Epistemic uncertainty refers to uncertainty due to lack of knowledge, while aleatory uncertainty refers to inherent randomness in the system. Philosophers of statistics explore how these different types of uncertainty influence decision-making and inference.

Risk and Decision Theory: Statistical analysis often informs decision-making under uncertainty, particularly in fields like economics, medicine, and public policy. Philosophical questions arise about how to weigh evidence, manage risk, and make decisions when outcomes are uncertain.

Causality vs. Correlation:

Causal Inference: One of the most important issues in the philosophy of statistics is the relationship between correlation and causality. While statistics can show correlations between variables, establishing a causal relationship often requires additional assumptions and methods, such as randomized controlled trials or causal models.

Causal Models and Counterfactuals: Philosophers like Judea Pearl have developed causal inference frameworks that use counterfactual reasoning to better understand causation in statistical data. These methods help to clarify when and how statistical models can imply causal relationships, moving beyond mere correlations.

The Role of Models:

Modeling Assumptions: Statistical models, such as regression models or probability distributions, are based on assumptions about the data-generating process. Philosophers of statistics question the validity and reliability of these assumptions, particularly when they are idealized or simplified versions of real-world processes.

Overfitting and Generalization: Statistical models can sometimes "overfit" data, meaning they capture noise or random fluctuations rather than the underlying trend. Philosophical discussions around overfitting examine the balance between model complexity and generalizability, as well as the limits of statistical models in capturing reality.

Data and Representation:

Data Interpretation: Data is often considered the cornerstone of statistical analysis, but philosophers of statistics explore the nature of data itself. How is data selected, processed, and represented? How do choices about measurement, sampling, and categorization affect the conclusions drawn from data?

Big Data and Ethics: The rise of big data has led to new ethical and philosophical challenges in statistics. Issues such as privacy, consent, bias in algorithms, and the use of data in decision-making are central to contemporary discussions about the limits and responsibilities of statistical analysis.

Statistical Significance:

p-Values and Significance: The interpretation of p-values and statistical significance has long been debated. Many argue that the overreliance on p-values can lead to misunderstandings about the strength of evidence, and the replication crisis in science has highlighted the limitations of using p-values as the sole measure of statistical validity.

Replication Crisis: The replication crisis in psychology and other sciences has raised concerns about the reliability of statistical methods. Philosophers of statistics are interested in how statistical significance and reproducibility relate to the notion of scientific truth and the accumulation of knowledge.

Philosophical Debates:

Frequentism vs. Bayesianism:

Frequentist and Bayesian approaches to statistics represent two fundamentally different views on the nature of probability. Philosophers debate which approach provides a better framework for understanding and interpreting statistical evidence. Frequentists argue for the objectivity of long-run frequencies, while Bayesians emphasize the flexibility and adaptability of probabilistic reasoning based on prior knowledge.

Realism and Anti-Realism in Statistics:

Is there a "true" probability or statistical model underlying real-world phenomena, or are statistical models simply useful tools for organizing our observations? Philosophers debate whether statistical models correspond to objective features of reality (realism) or are constructs that depend on human interpretation and conventions (anti-realism).

Probability and Rationality:

The relationship between probability and rational decision-making is a key issue in both statistics and philosophy. Bayesian decision theory, for instance, uses probabilities to model rational belief updating and decision-making under uncertainty. Philosophers explore how these formal models relate to human reasoning, especially when dealing with complex or ambiguous situations.

Philosophy of Machine Learning:

Machine learning and AI have introduced new statistical methods for pattern recognition and prediction. Philosophers of statistics are increasingly focused on the interpretability, reliability, and fairness of machine learning algorithms, as well as the role of statistical inference in automated decision-making systems.

The philosophy of statistics addresses fundamental questions about probability, uncertainty, inference, and the nature of data. It explores how statistical methods relate to broader epistemological issues, such as the nature of scientific knowledge, objectivity, and causality. Frequentist and Bayesian approaches offer contrasting perspectives on probability and inference, while debates about the role of models, data representation, and statistical significance continue to shape the field. The rise of big data and machine learning has introduced new challenges, prompting philosophical inquiry into the ethical and practical limits of statistical reasoning.

[raffaellopalandri]

Statistical Tools

Daily writing promptWhat was the last thing you searched for online? Why were you looking for it?View all responses Checking which has been my most recent search on Google, I found that I asked for papers, published in the last 5 years, that used a Montecarlo method to check the reliability of a mathematical method to calculate a team’s efficacy. Photo by Andrea Piacquadio on Pexels.com I was…

[schoolseducation]

Evaluating After-School Programmes through Hypothesis Testing

India is fortunate to have a well-established educational system that places significant emphasis on theoretical knowledge. However, society has evolved, and to address current challenges, a generation of students equipped with a holistic approach is essential. Mere academic knowledge no longer suffices for students’ personal growth.

[juliebowie]

Exploring the Various Types of Hypothesis Testing: A Comprehensive Overview

Summary: This blog explains the importance of various types of hypothesis testing in statistical analysis, highlighting its role in evaluating assumptions about population parameters. It discusses the significance of null and alternative hypotheses, types of hypothesis tests, and common pitfalls, emphasizing the value of hypothesis testing for making informed, data-driven decisions across various fields.

Introduction to Hypothesis Testing

Hypothesis testing is a fundamental aspect of statistical analysis that allows researchers to make informed decisions based on data. It provides a structured framework for evaluating assumptions about a population parameter, enabling scientists, statisticians, and decision-makers to determine the validity of their hypotheses. 

This process is crucial in various fields, including medicine, social sciences, and business, where data-driven conclusions can significantly impact outcomes.

At its core, hypothesis testing involves formulating two competing statements: the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis typically posits that there is no effect or difference, while the alternative hypothesis suggests that there is a significant effect or difference. 

By collecting and analyzing sample data, researchers can assess the likelihood of observing the data under the null hypothesis, ultimately leading to a decision about whether to reject or fail to reject H₀.

This blog will explore the various types of hypothesis testing, their importance in statistical analysis, and the common pitfalls associated with the process. Understanding these concepts is essential for anyone looking to apply statistical methods effectively in their research or decision-making processes.

Importance in Statistical Analysis and Research

Hypothesis testing is crucial in statistical analysis and research, providing a systematic framework for making data-driven decisions, quantifying uncertainty, and validating scientific theories through empirical evidence and rigorous evaluation.

Data-Driven Decision Making

In an era where data is abundant, hypothesis testing provides a systematic approach to make decisions based on evidence rather than assumptions or intuition. This is particularly important in fields like medicine, where clinical trials rely on hypothesis testing to determine the efficacy of new treatments.

Quantifying Uncertainty

Hypothesis testing allows researchers to quantify the uncertainty associated with their conclusions. By calculating p-values and confidence intervals, researchers can assess the strength of their evidence and the likelihood of making errors in their conclusions.

Facilitating Scientific Inquiry

The process of hypothesis testing is central to the scientific method. It encourages researchers to formulate clear, testable hypotheses and to seek empirical evidence to support or refute them. This iterative process fosters a deeper understanding of phenomena and contributes to the advancement of knowledge.

Guiding Policy and Strategy

In business and public policy, hypothesis testing can guide strategic decisions. For example, companies can use A/B testing to evaluate marketing strategies, while policymakers can assess the impact of interventions based on statistical evidence.

Identifying Relationships and Effects

Hypothesis testing helps researchers identify significant relationships between variables, allowing for a better understanding of causal mechanisms and the development of theories.

Null and Alternative Hypotheses

Null and alternative hypotheses form the foundation of hypothesis testing, representing competing statements about a population parameter. Understanding these hypotheses is essential for conducting rigorous statistical analyses and drawing valid conclusions. 

Null Hypothesis (H₀)

The null hypothesis is a statement that indicates no effect, no difference, or no relationship between variables. It serves as the default position that researchers seek to test against. For example, if a researcher is studying the effect of a new drug on blood pressure, the null hypothesis might state that the mean blood pressure of patients taking the drug is equal to that of those not taking it.

Mathematically, the null hypothesis is often expressed as:

 ​where μμ is the population mean, and μ0μ0​ is a specific value (e.g., the mean blood pressure of the control group).

Alternative Hypothesis (H₁)

The alternative hypothesis represents the statement that researchers aim to support. It posits that there is a significant effect, difference, or relationship between variables. Continuing with the drug example, the alternative hypothesis might state that the mean blood pressure of patients taking the drug is different from that of those not taking it.

The alternative hypothesis can be one-tailed or two-tailed:

One-Tailed Alternative Hypothesis: This specifies the direction of the effect. For instance, if the researcher believes the drug lowers blood pressure, the alternative hypothesis would be H1:μ<μ0H1​:μ<μ0​.

Two-Tailed Alternative Hypothesis: This does not specify the direction of the effect, only that there is a difference. In this case, it would be expressed as H1:μ≠μ0H1​:μ=μ0​.

Types of Hypothesis Tests

There are several types of hypothesis tests, each suited for different data types and research questions. Here are some of the most commonly used tests:

Z-Test

The Z-test is used when the sample size is large (typically n>30) and the population standard deviation is known. It assesses whether the sample mean differs from a known population mean. The formula for the Z-test statistic is:

where x‾x is the sample mean, μμ is the population mean, σσ is the population standard deviation, and nn is the sample size.

T-Test

The T-test is appropriate for smaller sample sizes (typically n<30) or when the population standard deviation is unknown. It compares the sample mean to a known value or another sample mean. The formula for the one-sample T-test statistic is:

where s is the sample standard deviation.

There are different types of T-tests, including:

One-Sample T-Test: Compares the sample mean to a known population mean.

Independent Two-Sample T-Test: Compares the means of two independent groups.

Paired Sample T-Test: Compares means from the same group at different times.

Chi-Square Test

The Chi-square test is used to assess relationships between categorical variables. It evaluates whether the observed frequencies in a contingency table differ significantly from expected frequencies. The formula for the Chi-square statistic is:

ANOVA (Analysis of Variance)

ANOVA is used to compare means across three or more groups. It tests the null hypothesis that all group means are equal. If the ANOVA test indicates significant differences, further post-hoc tests can identify which specific groups differ. The F-statistic is used to determine the ratio of between-group variance to within-group variance.

Non-parametric Tests

When data do not meet the assumptions of parametric tests (e.g., normality), non-parametric tests can be used. Examples include the Mann-Whitney U test (for two independent samples) and the Kruskal-Wallis test (for three or more independent samples).

Read More: Crucial Statistics Interview Questions for Data Science Success

Choosing the Right Hypothesis Test

Selecting the appropriate hypothesis test is crucial for obtaining valid results. Here are some factors to consider:

Data Type: Determine whether your data is continuous or categorical. For continuous data, consider T-tests or ANOVA; for categorical data, consider Chi-square tests.

Sample Size: The size of your sample influences the choice of test. Larger samples may allow for Z-tests, while smaller samples may require T-tests.

Distribution: Assess whether your data follows a normal distribution. If not, consider using non-parametric tests.

Number of Groups: If comparing means across multiple groups, ANOVA may be appropriate. For two groups, use T-tests or non-parametric alternatives.

Hypothesis Direction: Determine whether your hypothesis is one-tailed or two-tailed, as this will influence the test selection.

Errors, Common Misinterpretations, and Pitfalls

Hypothesis testing is not without its challenges. Researchers must be aware of common errors and misinterpretations:

Type I and Type II Errors

Type I Error (α): This occurs when the null hypothesis is rejected when it is actually true. The significance level (α) represents the probability of making a Type I error, commonly set at 0.05.

Type II Error (β): This occurs when the null hypothesis is not rejected when it is false. The power of a test (1 - β) indicates the probability of correctly rejecting a false null hypothesis.

Common Misinterpretations

P-Value Misunderstandings: A common misconception is that a p-value indicates the probability that the null hypothesis is true. Instead, it reflects the probability of observing the data given that the null hypothesis is true.

Overemphasis on Statistical Significance: Researchers may focus solely on p-values, neglecting the practical significance of their findings. A statistically significant result may not always translate to a meaningful effect in real-world applications.

Ignoring Assumptions: Each hypothesis test comes with underlying assumptions (e.g., normality, independence). Violating these assumptions can lead to incorrect conclusions.

Sample Size Effects: Small sample sizes can result in unreliable estimates, while large sample sizes can lead to statistically significant results even for trivial effects. Researchers should balance sample size with practical significance.

Conclusion

Hypothesis testing is a cornerstone of statistical analysis, providing a structured approach to evaluate assumptions about population parameters. By understanding the various types of hypothesis tests, the importance of null and alternative hypotheses, and the common pitfalls associated with the process, researchers can make informed decisions and draw valid conclusions from their data.

As data continues to play a pivotal role in decision-making across various fields, mastering hypothesis testing will empower researchers and practitioners to navigate uncertainties and enhance the robustness of their findings.

Frequently Asked Questions

What is the Purpose of Hypothesis Testing?

Hypothesis testing aims to evaluate assumptions about a population parameter based on sample data. It helps researchers determine whether to reject or fail to reject the null hypothesis, guiding data-driven decision-making.

What are the Types of Hypothesis Tests?

Common types of hypothesis tests include Z-tests, T-tests, Chi-square tests, ANOVA, and non-parametric tests. The choice of test depends on factors such as data type, sample size, and distribution.

What is a P-value?

A p-value is a statistical measure that indicates the probability of obtaining results as extreme as those observed, assuming the null hypothesis is true. It helps researchers assess the strength of evidence against the null hypothesis.

[promise-of-soup]

CRYING OVER THE YURI WEDDING CARD BEING CALLED "HAPPINESS EXPERIMENT" LMFAO IMAGINE HIM GETTING MARRIED AS AN EXPERIMENT THAT'S SO FUNNY! ahdasdsakd

[birchtreecat]

theory time I may be grasping at straws and insane. who knows.

OBLIGATORY DISCLAIMER THIS IS AN AU ANALYSIS AND NOT CANNON TO PJSK also cws for mentions of rot, unreality, body horrors, etc etc yall know the drill!!

Character interpretations are relative to THIS AU ONLY and i will be disregarding how Airi usually is in cannon-universe for the most part. please don't come at me i know how my beloved acts in cannon pjsk <3

for those of you who dont know the blog and enjoy arg/unreality/weirdcore horror please check @daily-airimomoi-vitamins out!!! this entire yap session is sponsored a theory of their lore!!! :333!

TLDR as the paragraphs might be slightly incoherent- We, the user are either losing our memories and sense of self to Vitairi. Or, we, the user, are Airi, trying to cope with her early death. More explaination and thought process below cutoff. Feel free to ask any questions/add to the discussion in the comments, reblogs, tags or my asks !!!

Considering the last airi text only lore post had the pink text presumably be us- anon, the viewers, the user- the center of this post and the main thing I will touch on is the concept of Airi Momoi. The differences between her and her as well as if we, the audience, are her.

The proof of my claims are linked as we go. I'll be using similar text color styles to differentiate between pharmacist and the other.

Putting out what we know first-

- The vitamins cause loss of self + memory, as seen in VitaAiri’s pinned intro as of posting. They also cause regained memory, the question is if they’re our own.

- Vitamins are made of human meat? As hinted at in this ask chain, I don’t want to say anything for sure though

- Death is a common concept in this pharmacy- as well as rotting bodies, blood, etc.

With how things are and the certain side blog that was around (@backof-yourmind ) I want to say the naming it of “your mind” and not “her mind” was intentional, implying that either-

A) The vitamins handed out by vitairi slowly make the user lose their sense of self, turning them into an amalgamation mentally of Airi Momoi and the user, traumas and all.

Or, B)- Airi died. That much is true. In dying, she slowly lost parts of herself, and eventually became the pharmacist she is now. User is Airi and the pharmacist represents the cold truths of her death and how traumatic it was, as well as potentially her feelings towards this denial. (the arguably more vague, more exciting option)

A and B both contradict eachother in their own ways, but here’s my logic behind both!!

With option A, it would explain the recent confusion of who the user is, how they cannot remember much if anything about themself, why they continue to trust vitairi + the side effects of regaining memories (that may not be our own) and our loss of self that is pinned at the top of Vitairi’s blog. User was most likely more aware of themself, but upon taking airi vitamins lost that. Question is if it was intentional or not, but in the user’s confusion current story I can’t deduct that well. My thinking is with the back of-your mind blog, User was aware of who they were as seen in this post, but it was foggy and vague, something they didn’t give full thought to until Vitairi brought it up for us. I don't know if that side blog has been sent any more asks, nor if she will answer. Either that, or Vitairi is able to control what memories user has access to for themselves, but subsequently leaks her traumas into them. Actions have consequences after all, even for the pharmacist. The back of your mind blog would be what it literally is in that case, what seems to be the back of the pharmacy.

We don’t know if user is alive or dead, if Vitairi is just saying Airi was dead/never real to fuck with us or if she was telling the truth. She may very well be somehow absorbing or otherwise removing the user’s memories through vitamins along with keeping them complacent- as in the last poll post, we were apparently more pushy than usual and she hinted at it being due to withdrawal. Combined with the most recent lore post as of making this theory, she may be using the vitamins to stay in business and keep afloat. It semi ties in with option B, as I want to keep the idea of the two hers. Part of Airi wanting to warn User and save herself, the other wanting to keep her own denial and stopping at nothing to forget their deaths. Maybe parts of Airi transfer into the User as they forget more and more of themselves and as the Pharmacist takes over Airi entirely.

For option b- her death and losing parts of herself as she did- the pharmacist represents the cold truths of her death and how traumatic it was, as well as potentially her feelings towards this denial. (I.e how it must be better to forget if it’s so painful, and everyone would have their reasons)

Meanwhile Airi, or User would be her true feelings about the death. What she missed, how wonderful life was and how her dreams were cut short. User is a sort of perspective beyond it all- maybe Airi’s aware of both feelings, and her softer side is just more willing to move on, while her harsher or colder side is stubborn and overpowering when she tries to remember their death. This denial and constant attempts to forget all the trauma cause Airi to "forget" parts of herself, the pharmacist acting as a guardian to all these memories and further pushing the want to forget everything. The rot surrounding the pharmacy could very well be Airi's own corpse in the real world as her spirit mourns what could have been.

Today's vitamin is

a memory not remembered

but someone who never even lived

...

You don't have to phrase it like that

Oh, but I do ♡

hm.

//tw: eyes, slight blood

[teaboot]

Wait wait wait

So sweat glands don't grow back the same when scar tissue forms

Tattoos are, essentially, scars over pigment

If tattoos don't sweat as much as the rest of your skin

And if sweat leaves visible dark spots on dry fabrics

Could you lay fabric over a sweating, tattooed area and produce a silhouette of the tattoo

[malewifenjoyer]

Studio BONES has released a new illustration featuring the hosts in suits, which will be used for future merchandise preorders starting January 10.

[deliriousblue]

gmmtv really said we are pivoting fully into QL and ok fine one het show to keep nanon employed

[quasi-normalcy]

(I'm testing a hypothesis, please reblog)

[fairie-bread]

mr shelton unfortunately u have become the latest victim of my pookification beam

[whatsfourteenupto]

Donna: I can’t find the Doctor, has anyone seen them?

Rose: He’s in the living room listening to music and staring dramatically out the window

Donna: Again? It’s not even raining

Rose: I turned the sprinkler toward the house to see what they’d do

[alexandritefang]

I'm probably just being dumb but I don't understand some of your posts...

The ones where you talk about making a sub all twitchy and desperate with pleasure... But you don't mention what he'd be doing for you to earn the pleasure... Like I don't understand how there can be pleasure without doing something for you to get the pleasure😶

Pleasure is what you get from serving and obeying...

I'm sorry for being dumb I just wanted to be able to learn so I can be better please 🥺

🌹

That may be because I'm a hellish pleasure dom, so I adore to write about subs receiving pleasure. But there's no problem angel, I'll help you with that.

As you said, pleasure is something you get from serving and obeying. But the thing with my brain is that a sub being good (even not within the sexual setting) is already a excuse to treat them good. I'll give you some examples:

Are you getting hard by wearing a collar? My brain: Do you think he drools if he gets teased enough? I bet he does, he for sure would look sooo pretty...

Are you putting on a suit to go to wherever?

My brain: Hot. How would he look like if he got his arms tied up by his necktie sitting on a chair with his clothes still but with his belt unbuckled? Oh, I bet his thghs look yummy as he buckles his hips up...

Are you exercising?

My brain: If he gets pegged, would his moans get high-pitched or turn into messy depper growls? Would he arch his back to get spanked or his body would give up just like it did finishing this set?

Are you studying or very focused on a task?

My brain: So diligent. Would he keep that composure if I slutted him out? I bet he would beg like a needy whore...

Of course I do love when he's a sub within sexual interaction. Using pretty clothing and accessories to show off just for me, listen to my orders and adoring me are definitely a turn on but that's not necessarily the only thing that gets me aroused.

That's because a sub's pleasure is what's turn me on, is the thing my brain seeks the most. I wanna watch him aroused, because he's the prettiest letting go of things, because his reactions turn me on.

You deserve the pleasure not just because you have been good, but mostly because I need to see how it looks on you, how it's provoked on you.

[mirrorball-leclerc]

mark webber what fucking curse did you place on the 2nd red bull seat and why was the only driver that didn’t fall victim to it daniel ricciardo??