McElreath, R. (2020). Statistical Rethinking: A Bayesian Course with Examples in R and STAN (2nd ed.). Chapman and Hall/CRC.

Statistical Rethinking: A Bayesian Course with Examples in R and STAN (2nd edition)

- Statistical Golems
- Plausibility
- Posterior Distribution
- Sampling the Imaginary
- Building a Bayesian Model
- Gaussian Distribution
- Correlation vs. Causation
- Linear Models
- Multiple Regression Models
- Perils of Multiple Regression
- Underfitting vs Overfitting 
- Entropy and Accuracy
- Interactions
- Markov Chain Monte Carlo
- Generalized Linear Models



Concepts in Bayesian Statistics

When using Bayesian statistics, what must a researcher explicitly specify before conducting a study?

In an approach to inferential statistics that begins by assigning initial probabilities to competing hypotheses, those probability estimates remain fixed and do not change once new data from a study are collected.

Bayesian statistics is an approach to inferential statistics where a researcher specifies the prior probability that the null hypothesis and alternative hypotheses are true before conducting a study. Once the study is conducted, the probabilities are updated based on the newly collected data. This approach offers a potential alternative to traditional null hypothesis testing by providing a logical framework for quantifying uncertainty using probability distributions.

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Statistics is the process of gathering, analyzing, and interpreting information in order to help people make decisions when faced with uncertainty.

Statistics

Statement that says there is no relationship between the two measured items

Null Hypothesis

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KPU Research Methods in Psychology - 4th American Edition

The list of helpful broad references about Statistics is categorized as follows:
- Statistics books
- Statistical Software 
- Statistics online courses
- Statistics research papers

Statistics References

The marriage of Statistics, Computer Science, and Domin knowledge built Data Science.

Data science is interdisciplinary

A few of the common statistical paradoxes are:"

- Monty Hall's paradox 
- Berkson's paradox
- Simpson's paradox

Common statistical paradoxes

Probability is the branch of math concerned with the likelihood of an event occuring, or the likelihood that an event is true. 

Probability Theory

Bayesian Statistics

Components of Statistical Analysis

Descriptive statistics are a set of quantitative techniques used to summarize, organize, and visually display the data collected from a research sample. Important examples include measures of central tendency (such as the mean, median, and mode), which describe average scores, and measures of dispersion (such as the range, standard deviation, and variance), which indicate how spread apart the scores are.

Descriptive Statistics

The statistical power of a research design is the probability that the study will reject the null hypothesis, given a specific sample size and an expected relationship strength. For instance, if a design has a power of $$.80$$, it means there is an $$80\%$$ chance of correctly rejecting the null hypothesis when the expected relationship exists in the population. It is essentially the complement of the probability of making a Type II error, so researchers strive for high statistical power to avoid missing a real effect.

Statistical Power

Critics of null hypothesis testing argue that the strict convention of rejecting the null hypothesis only when $$p < .05$$ is logically flawed. This rigid threshold creates an artificial divide where a result of $$p = .04$$ is deemed significant and worthy of publication, while a nearly identical result of $$p = .06$$ is dismissed as not significant. This arbitrariness can prevent valuable research from being published and exacerbates the file drawer problem by suppressing non-significant findings.

Arbitrariness of the p-value Threshold

A Type I error is a false positive in null hypothesis testing. It occurs when a researcher mistakenly concludes that their results are statistically significant—thereby rejecting the null hypothesis—when, in reality, the null hypothesis is true and there is no actual relationship in the population. These errors happen because random sampling error can occasionally produce extreme results even when no true effect exists. The predetermined alpha level dictates the probability of committing this error when the null hypothesis is true.

Type I Error

A Type II error is a false negative in null hypothesis testing. It occurs when a researcher mistakenly concludes that their results are not statistically significant—thereby retaining the null hypothesis—when, in reality, the null hypothesis is false and a true relationship does exist in the population. In practical research, Type II errors typically happen because a study lacks adequate statistical power to detect the underlying relationship, often due to an insufficiently large sample size.

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