World's Information: March 2020

Pearson Correlation

EXAMPLES

An example of a positive correlation is the relationship between the speed of a wind turbine and the amount of energy it produces. As the turbine speed increases, electricity production also increases.

An example of a negative correlation is the relationship between outdoor temperature and heating costs. As the temperature increases, heating costs decrease.

Correlation is used to represent the linear relationship between two variables. On the contrary, regression is used to fit the best line and estimate one variable on the basis of another variable

“Covariance” indicates the direction of the linear relationship between variables. “Correlation” on the other hand measures both the strength and direction of the linear relationship between two variables.

Chi Square

Independence of observations.

There is no relationship between the subjects in each group.

The categorical variables are not "paired" in any way (e.g. pre-test/post-test observations).

H1: "[Variable 1] is not independent of [Variable 2]"
H1: "[Variable 1] is associated with [Variable 2]"

χ^{2} = \sum_{i = 1}^{R} \sum_{j = 1}^{C} \frac{(o_{i j} - e_{i j})^{2}}{e_{i j}}

χ^{2} = \sum_{i = 1}^{R} \sum_{j = 1}^{C} \frac{(o_{i j} - e_{i j})^{2}}{e_{i j}}

χ^{2} = \sum_{i = 1}^{R} \sum_{j = 1}^{C} \frac{(o_{i j} - e_{i j})^{2}}{e_{i j}}

χ^{2} = \sum_{i = 1}^{R} \sum_{j = 1}^{C} \frac{(o_{i j} - e_{i j})^{2}}{e_{i j}}

χ^{2} = \sum_{i = 1}^{R} \sum_{j = 1}^{C} \frac{(o_{i j} - e_{i j})^{2}}{e_{i j}}

χ^{2} = \sum_{i = 1}^{R} \sum_{j = 1}^{C} \frac{(o_{i j} - e_{i j})^{2}}{e_{i j}}

χ^{2} = \sum_{i = 1}^{R} \sum_{j = 1}^{C} \frac{(o_{i j} - e_{i j})^{2}}{e_{i j}}

χ^{2} = \sum_{i = 1}^{R} \sum_{j = 1}^{C} \frac{(o_{i j} - e_{i j})^{2}}{e_{i j}}

χ^{2} = \sum_{i = 1}^{R} \sum_{j = 1}^{C} \frac{(o_{i j} - e_{i j})^{2}}{e_{i j}}

χ^{2} = \sum_{i = 1}^{R} \sum_{j = 1}^{C} \frac{(o_{i j} - e_{i j})^{2}}{e_{i j}}

χ^{2} = \sum_{i = 1}^{R} \sum_{j = 1}^{C} \frac{(o_{i j} - e_{i j})^{2}}{e_{i j}}

χ^{2} = \sum_{i = 1}^{R} \sum_{j = 1}^{C} \frac{(o_{i j} - e_{i j})^{2}}{e_{i j}}

χ^{2} = \sum_{i = 1}^{R} \sum_{j = 1}^{C} \frac{(o_{i j} - e_{i j})^{2}}{e_{i j}}

χ^{2} = \sum_{i = 1}^{R} \sum_{j = 1}^{C} \frac{(o_{i j} - e_{i j})^{2}}{e_{i j}}

χ^{2} = \sum_{i = 1}^{R} \sum_{j = 1}^{C} \frac{(o_{i j} - e_{i j})^{2}}{e_{i j}}

χ^{2} = \sum_{i = 1}^{R} \sum_{j = 1}^{C} \frac{(o_{i j} - e_{i j})^{2}}{e_{i j}}

χ^{2} = \sum_{i = 1}^{R} \sum_{j = 1}^{C} \frac{(o_{i j} - e_{i j})^{2}}{e_{i j}}

χ^{2} = \sum_{i = 1}^{R} \sum_{j = 1}^{C} \frac{(o_{i j} - e_{i j})^{2}}{e_{i j}}

χ^{2} = \sum_{i = 1}^{R} \sum_{j = 1}^{C} \frac{(o_{i j} - e_{i j})^{2}}{e_{i j}}

χ^{2} = \sum_{i = 1}^{R} \sum_{j = 1}^{C} \frac{(o_{i j} - e_{i j})^{2}}{e_{i j}}

χ^{2} = \sum_{i = 1}^{R} \sum_{j = 1}^{C} \frac{(o_{i j} - e_{i j})^{2}}{e_{i j}}

χ^{2} = \sum_{i = 1}^{R} \sum_{j = 1}^{C} \frac{(o_{i j} - e_{i j})^{2}}{e_{i j}}

χ^{2} = \sum_{i = 1}^{R} \sum_{j = 1}^{C} \frac{(o_{i j} - e_{i j})^{2}}{e_{i j}}

χ^{2} = \sum_{i = 1}^{R} \sum_{j = 1}^{C} \frac{(o_{i j} - e_{i j})^{2}}{e_{i j}}

χ^{2} = \sum_{i = 1}^{R} \sum_{j = 1}^{C} \frac{(o_{i j} - e_{i j})^{2}}{e_{i j}}

χ^{2} = \sum_{i = 1}^{R} \sum_{j = 1}^{C} \frac{(o_{i j} - e_{i j})^{2}}{e_{i j}}

χ^{2} = \sum_{i = 1}^{R} \sum_{j = 1}^{C} \frac{(o_{i j} - e_{i j})^{2}}{e_{i j}}

χ^{2} = \sum_{i = 1}^{R} \sum_{j = 1}^{C} \frac{(o_{i j} - e_{i j})^{2}}{e_{i j}}

o_{i j}

o_{i j}

o_{i j}

o_{i j}

is the observed cell count in the ith row and jth column of the table

e_{i j}

e_{i j}

e_{i j}

e_{i j}

is the expected cell count in the ith row and jth column of the table, computed as

e_{i j} = \frac{row i total * col j total}{grand total}

e_{i j} = \frac{row i total * col j total}{grand total}

e_{i j} = \frac{row i total * col j total}{grand total}

e_{i j} = \frac{row i total * col j total}{grand total}

e_{i j} = \frac{row i total * col j total}{grand total}

e_{i j} = \frac{row i total * col j total}{grand total}

e_{i j} = \frac{row i total * col j total}{grand total}

e_{i j} = \frac{row i total * col j total}{grand total}

e_{i j} = \frac{row i total * col j total}{grand total}

e_{i j} = \frac{row i total * col j total}{grand total}

r_{i j}

r_{i j}

r_{i j}

r_{i j}

What is your idea of a great holiday (Beach, Cruise) by Genders (Men, Women)

Respondents were asked their gender and whether or not they were a cigarette smoker. There were three answer choices: Nonsmoker, Past smoker, and Current smoker. Suppose we want to test for an association between smoking behavior (nonsmoker, current smoker, or past smoker) and gender (male or female) using a Chi-Square Test of Independence (we'll use α = 0.05).

The null hypothesis (H0) and alternative hypothesis (H1) of the Chi-Square Test of Independence can be expressed in two different but equivalent ways:

H0: "[Variable 1] is independent of [Variable 2]"

H0: "[Variable 1] is not associated with [Variable 2]"

The test statistic for the Chi-Square Test of Independence is denoted Χ2, and is computed as:

where

o_{i j}

o_{i j}

o_{i j}

e_{i j}

e_{i j}

e_{i j}

The quantity (oij - eij) is sometimes referred to as the residual of cell (i, j), denoted

r_{i j}

r_{i j}

r_{i j}

The calculated Χ2 value is then compared to the critical value from the Χ2 distribution table with degrees of freedom df = (R - 1)(C - 1) and chosen confidence level. If the calculated Χ2 value > critical Χ2 value, then we reject the null hypothesis.

EXAMPLES

Great Reference and refresher on Stat and p values: https://www.mathsisfun.com/data/chi-square-test.html

Pages

An Easy Overview of Statistical Tests - 2 Inferential Statistics for Association

The Technical Knowhow of Web!