Chi-square test is the degree of deviation between the actual observed value and the predicted value of samples. The degree of deviation determines the size of the chi-square value. The larger the chi-square value, the greater the deviation. On the contrary, the deviation between them is smaller; If the two values are completely equal, the chi-square value is 0, indicating that the theoretical value is completely consistent.

Contingency table test is a special case of chi-square test, and it is one of the common methods of independence hypothesis test. 
$$ \chi^2 = \sum_{i=1}^{2} \sum_{j=1}^{2} \frac{(n_{ij} - \hat{\mu_{ij}})^2}{\hat{\mu_{ij}}} $$


Chi-square

Contingency table is used to summarize the relationship between several categorical variables. If individuals in the population can be classified by two attributes, A and B, A has r levels $A_1, A_2 ... A_r$, and B has c levels $B_1, B_2 ... B_c$. Take a sample of size n from the population. Suppose there are $n_{ij}$ individuals whose attributes belong to rank $A_i$ and $B_j$, $n_{ij}$ is called **frequency**. 

University of Michigan - Ann Arbor

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

Statistics References

James, G., Witten, D., Hastie, T., & Tibshirani, R. (2013). An introduction to statistical learning (Vol. 112, pp. 3-7). New York: springer.

An Introduction to Statistical Learning with Applications in R

- Introductory Statistics through OpenStax
- Introduction to Regression Analysis by Golberg & Cho 

Helpful Statistics books

Statistical software are powerful programs that allow us to analyze and interpret data in a concise and organized manner.

Statistical Software

[StatQuest with Josh Starmer]. (2020, February 21). Statistics Fundamentals [Video File]. Retrieved from [www.youtube.com/playlist?list=PLblh5JKOoLUK0FLuzwntyYI10UQFUhsY9](https://www.youtube.com/playlist?list=PLblh5JKOoLUK0FLuzwntyYI10UQFUhsY9) 

StatQuest with Josh Starmer

Contingency Table

- Paper Describing Concept of Parametric vs. Nonparametric Tests:
  Ali, Z., & Bhaskar, S. (2016). Basic statistical tools in research and data analysis. 
  Indian Journal of Anaesthesia, 60(9), 662. doi: 10.4103/0019-5049.190623. 
  Link: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5037948/

- Paper Describing Central Limit Theorem and Its Connection to Normality:
  Kwak, S. G., &amp; Kim, J. H. (2017). Central limit theorem: The cornerstone of 
  modern statistics. Korean Journal of Anesthesiology, 70(2), 144. 
  doi:10.4097/kjae.2017.70.2.144.
  Link: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5370305/

- Paper Describing Path Analysis: 
  Streiner, D. L. (2005). Finding Our Way: An Introduction to Path Analysis. The 
  Canadian Journal of Psychiatry, 50(2), 115-122. 
  doi:10.1177/070674370505000207
  Link: https://journals.sagepub.com/doi/pdf/10.1177/070674370505000207

- Paper Describing Effectiveness vs. Efficacy: 
  Singal, A. G., Higgins, P. D., & Waljee, A. K. (2014). A primer on effectiveness 
  and efficacy trials. Clinical and translational gastroenterology, 5(1), e45. 
  https://doi.org/10.1038/ctg.2013.13
  Link: https://bit.ly/3jaAi9I

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