Regression analysis is a statistical method used to model the relationship between a dependent variable and one or more independent variables. The primary goal of regression analysis is to find the best-fitting model that describes the relationship between the variables.
There are several types of regression models, including linear regression, logistic regression, Poisson regression, Cox regression, and polynomial regression. Each of these models has its own specific formula and assumptions.
Linear Regression
Linear regression is used to model the linear relationship between a dependent variable and one or more continuous independent variables. The general formula for linear regression is:
y = β0 + β1×1 + β2×2 + … + βnxn + ε
where y is the dependent variable, x1, x2, …, xn are the independent variables, β0, β1, β2, …, βn are the coefficients, and ε is the error term.
In R, you can fit a linear regression model using the lm() function. Here’s an example:
#Create example data
x <- 1:10
y <- 2*x + rnorm(10)
#Fit linear regression model
model <- lm(y ~ x)
#Print model summary
summary(model)
Logistic Regression
Logistic regression is used to model the relationship between a binary dependent variable (i.e., one that takes values of 0 or 1) and one or more independent variables. The general formula for logistic regression is:
p = 1 / (1 + e^(-z))
where p is the probability of the dependent variable being 1, and z is the log-odds of the probability, defined as:
z = β0 + β1×1 + β2×2 + … + βnxn
In R, you can fit a logistic regression model using the glm() function. Here’s an example:
#Create example data
x <- rnorm(100)
y <- rbinom(100, 1, plogis(0.5 + 2*x))
#Fit logistic regression model
model <- glm(y ~ x, family = binomial())
#Print model summary
summary(model)
Poisson Regression
Poisson regression is used to model the relationship between a count dependent variable and one or more independent variables. The general formula for Poisson regression is:
log(μ) = β0 + β1×1 + β2×2 + … + βnxn
where μ is the expected count of the dependent variable, and log is the natural logarithm.
In R, you can fit a Poisson regression model using the glm() function. Here’s an example:
#Create example data
x <- rnorm(100)
y <- rpois(100, exp(0.5 + 2*x))
#Fit Poisson regression model
model <- glm(y ~ x, family = poisson())
#Print model summary
summary(model)
Cox Regression
Cox regression, also known as the proportional hazards model, is used to model the relationship between a survival time dependent variable and one or more independent variables. The general formula for Cox regression is:
h(t|x) = h0(t) * exp(β1×1 + β2×2 + … + βnxn)
where h(t|x) is the hazard function at time t for a set of covariates x, h0(t) is the baseline hazard function, and exp is the exponential function.
The survival package in R provides functions for fitting Cox regression models. Here’s an example:
#Load the survival package
library(survival)
#Load example data
data(lung)
#Fit Cox regression model
model <- coxph(Surv(time, status) ~ age + sex + ph.karno, data = lung)
#Print model summary
summary(model)
Polynomial Regression
Polynomial regression is used to model the relationship between a dependent variable and one independent variable with a polynomial function. The general formula for a polynomial regression model is:
y = β0 + β1x + β2x^2 + … + βnx^n + ε
where y is the dependent variable, x is the independent variable, β0, β1, β2, …, βn are the coefficients, x^n is the nth power of x, and ε is the error term.
In R, you can fit a polynomial regression model using the lm() function. Here’s an example:
#Create example data
x <- 1:10
y <- 2*x^2 + rnorm(10)
#Fit polynomial regression model
model <- lm(y ~ poly(x, 2, raw = TRUE))
#Print model summary
summary(model)
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