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regression_warmup.Rmd
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---
title: "Regression Warm-up"
output: github_document
---
```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```
This section is part of the R Data Analysis Workshop and aims to introduce some basic concepts for regression analysis in R. If you want to learn more in details, I would recommend you read the [Introduction to Econometrics with R](https://www.econometrics-with-r.org/index.html), a free e-book.
## Variable Types
- Discrete: A discrete variable is a variable whose value is obtained by counting. Examples: number of students present.
- Continuous: A continuous variable is a variable whose value is obtained by measuring, i.e., one which can take on an uncountable set of values.
- Binary: Binary variables are variables which only take two values. For example, Male or Female, True or False and Yes or No. You can also construct a binary variable for age like Adult ($>18$ yrs old) or Non-Adult ($<18$ yrs old).
- Categorical: Categorical variables represent types of data which may be divided into groups. Examples of categorical variables are race, sex, age group, and educational level.
## Why Regression Analysis?
Regression analysis is a statistical technique for studying linear relationships. Typically, a regression analysis is done for one of two purposes: In order to predict the value of the dependent variable for individuals for whom some information concerning the explanatory variables is available, or in order to estimate the effect of some explanatory variable on the dependent variable.
- Making individual predictions. If we know the value of several explanatory variables for an individual, but do not know the value of that individual’s dependent variable, we can use the prediction equation to estimate the value of the dependent variable for that individual.
- Estimating the effect of an explanatory variable on the dependent variable. In order to estimate the “pure” effect of some explanatory variable on the dependent variable, we want to control for as many other effects as possible. That is, we’d like to see how our prediction would change for an individual if this explanatory variable were different, while all others aspects of the individual were kept the same.
- Determining whether there is evidence that an explanatory variable belongs in a regression model. Given a specific model, one might wonder whether a particular one of the explanatory variables really “belongs” in the model; equivalently, one might ask if this variable has a true regression coefficient different from 0 (and therefore would affect predictions).
- Measuring the explanatory power of a model. [(Reference)](https://www.kellogg.northwestern.edu/faculty/weber/jhu/statistics/regression.htm)
## Key Terms in Regression Model
- Correlation: is a statistic that measures the degree to which two variables move in relation to each other. Or a statistical relationship, whether causal or not, between two random variables or bivariate data.
- Independent/explanatory variable (predictor): the cause. Its value is independent of other variables in the model.
- In experiment, the independent variable is the treatment that varies between groups, e.g., which type of pill the patient receives.
- Dependent/response variable: the effect. Its value depends on changes in the independent variable(s).
- In experiment, the dependent variable is the outcome being measured, e.g., the blood pressure of the patients.
- In other types of research, researchers have to find already-existing examples of the independent variable, and investigate how changes in this variable affect the dependent variable.
- Regression coefficient: in a regression analysis, the weight associated with a unit change in a specific independent (predictor) variable on the dependent (outcome) variable, given the relationship of that predictor to other independent variables already in the model.
## Assumptions of Multiple Linear Regression
Multiple linear regression analysis makes several key assumptions:
- There must be a linear relationship between the outcome variable and the independent variables. Scatterplots can show whether there is a linear or curvilinear relationship.
- Multivariate Normality: Multiple regression assumes that the residuals are normally distributed.
- No Multicollinearity: Multiple regression assumes that the independent variables are not highly correlated with each other. This assumption can be tested with Variance Inflation Factor (VIF) values.
- Homoscedasticity: This assumption states that the variance of error terms are similar across the values of the independent variables. A plot of standardized residuals versus predicted values can show whether points are equally distributed across all values of the independent variables.
Moreover, multiple linear regression requires at least two explanatory variables, which can be nominal, ordinal, or interval/ratio level variables. A rule of thumb for the sample size is at least 20 cases per independent variable in the analysis.
## Regression Tests
- VIF: Variation inflation factor (VIF) quantifies how much the variance is inflated. The variances of the estimated coefficients are inflated when multicollinearity exists and a variance inflation factor exists for each of the predictors in the model. How do we interpret the variance inflation factors for a regression model? A VIF of 1 means that there is no correlation among the jth predictor and the remaining predictor variables, and hence the variance of bj is not inflated at all. The rule of thumb is that VIFs exceeding 4 warrant further investigation, while VIFs exceeding 10 are signs of serious multicollinearity.
- Cook's Distance: Cook's distance measures how much all of the fitted values in the model change when the ith data point is deleted and is useful for identifying influential data points and outliers in the X values. A general rule of thumb is that any point with a Cook's distance over 4/n (where n is the number of observations) is considered to be an outlier. Please note that just because a data point is influential doesn't mean it should necessarily be deleted. More details [here](https://www.statology.org/how-to-identify-influential-data-points-using-cooks-distance/).
- Leverage: Leverage is a measure of how far away the independent variable values of an observation are from those of the other observations. High-leverage points, if any, are outliers with respect to the independent variables. hat is, high-leverage points have no neighboring points in $R^p$ space, where $p$ is the number of independent variables in a regression model. This makes the fitted model likely to pass close to a high leverage observation. Hence high-leverage points have the potential to cause large changes in the parameter estimates when they are deleted i.e., to be influential points. Although an influential point will typically have high leverage, a high leverage point is not necessarily an influential point. [Reference:Wiki](https://en.wikipedia.org/wiki/Leverage_(statistics))
## Regression Performance
- R-squared: R-squared ($R^2$) is a statistical measure that represents the proportion of the variance for a dependent variable that's explained by an independent variable or variables in a regression model.
- In short, larger R-squared means the independent variables can explain more proportion of the variance for the dependent variable.
- Math detail is [here](https://en.wikipedia.org/wiki/Coefficient_of_determination).
- AIC (Akaike Information Criterion): AIC is an estimator of out-of-sample prediction error and thereby relative quality of statistical models for a given set of data. Given a collection of models for the data, AIC estimates the quality of each model, relative to each of the other models. Thus, AIC provides a means for model selection.
- AIC is typically used when you do not have access to out-of-sample data and want to decide between multiple different model types, or for time convenience.
- The preferred model is the one with the lowest AIC value. The AIC methodology attempts to find the model that best explains the data with a minimum of free parameters.
- Mathematical detail is [here](https://en.wikipedia.org/wiki/Akaike_information_criterion).
- BIC (Bayesian Information Criterion): BIC is a criterion for model selection among a finite set of models. It is based, in part, on the likelihood function, and it is closely related
to AIC.
- When fitting models, it is possible to increase the likelihood by adding parameters, but doing so may result in overfitting. The BIC resolves this problem by introducing a penalty term for the number of parameters in the model. The penalty term is larger in BIC than in AIC.
- When picking from several models, ones with lower BIC value is preferred.
- Mathematical detail is [here](https://www.immagic.com/eLibrary/ARCHIVES/GENERAL/WIKIPEDI/W120607B.pdf).
## Hypothesis Testing
- Null hypothesis ($H_0$): The statement being tested in a test of statistical significance is called the null hypothesis. The test of significance is designed to assess the strength of the evidence against the null hypothesis. Usually, the null hypothesis is a statement of 'no effect' or 'no difference'." Ex., $H_0: \mu_1 = \mu_2$
- Alternative hypothesis ($H_1$): The statement that is being tested against the null hypothesis is the alternative hypothesis. Ex., $H_1: \mu_1 \neq \mu_2$
- Test statistic: A test statistic is a statistic (a quantity derived from the sample) used in statistical hypothesis testing. Two widely used test statistics are the t-statistic and the F-test. More details [here](https://en.wikipedia.org/wiki/Test_statistic).
- P-value: **P-value** is the probability of obtaining test results at least as extreme as the results actually observed, under the assumption that the null hypothesis is correct.
- A very small p-value means that such an extreme observed outcome would be very unlikely under the null hypothesis.
- Reporting p-values of statistical tests is common practice in academic publications of many quantitative fields. Since the precise meaning of p-value is hard to grasp, [misuse is widespread](https://en.wikipedia.org/wiki/Misuse_of_p-values) and has been a major topic in [metascience](https://en.wikipedia.org/wiki/Metascience).
- Significance level for the test ($\alpha$): is the probability of rejecting the null hypothesis when it is true. For example, a significance level of 0.05 indicates a 5% risk of concluding that a difference exists when there is no actual difference. More detail is [here](https://blog.minitab.com/en/adventures-in-statistics-2/understanding-hypothesis-tests-significance-levels-alpha-and-p-values-in-statistics).
[Next: Regression Analysis >>>](https://github.com/YuxiaoLuo/r_analysis_dri_2022/blob/main/regression_analysis.md)