💻Intro to Programming in R Unit 16 – Linear Regression Models

What's Linear Regression?

• Linear regression is a statistical modeling technique used to examine the relationship between a dependent variable and one or more independent variables
• Aims to find the best-fitting linear equation that describes the relationship between the variables
• The equation takes the form $y = \beta_0 + \beta_1x_1 + \beta_2x_2 + ... + \beta_nx_n + \epsilon$, where:
  • $y$ is the dependent variable
  • $\beta_0$ is the y-intercept (value of y when all independent variables are 0)
  • $\beta_1, \beta_2, ..., \beta_n$ are the coefficients for each independent variable
  • $x_1, x_2, ..., x_n$ are the independent variables
  • $\epsilon$ is the error term (represents the variation in y not explained by the model)
• Can be used for both simple linear regression (one independent variable) and multiple linear regression (two or more independent variables)
• Helps predict the value of the dependent variable based on the values of the independent variables
• Useful for identifying the strength and direction of the relationship between variables (positive or negative correlation)

Setting Up R for Linear Regression

• Install and load the necessary packages for linear regression analysis, such as

  stats

  (included in base R) and

  lm.beta

• Ensure your data is in a suitable format for analysis, typically a data frame with columns representing variables
• Use the

  read.csv()

  or

  read.table()

  functions to import your data into R from a CSV or text file
• Check for missing values in your dataset using functions like

  is.na()

  or

  complete.cases()

  • Handle missing values by removing rows with missing data (

    na.omit()

    ) or imputing values (e.g., using the mean or median)
• Explore your data using summary statistics and visualizations to gain insights and identify potential issues
  • Use

    summary()

    to view descriptive statistics for each variable
  • Create scatterplots using

    plot()

    to visualize the relationship between the dependent and independent variables
• If needed, transform variables to meet the assumptions of linear regression (e.g., log transformations for skewed data)

Key Concepts in Linear Regression

• Dependent variable (response variable) is the variable you want to predict or explain
• Independent variables (predictor variables) are the variables used to predict or explain the dependent variable
• Coefficients represent the change in the dependent variable for a one-unit change in the corresponding independent variable, holding other variables constant
• R-squared ($R^2$) measures the proportion of variance in the dependent variable explained by the independent variables
  • Ranges from 0 to 1, with higher values indicating a better fit
• Adjusted R-squared adjusts the R-squared value based on the number of independent variables in the model
  • Useful for comparing models with different numbers of predictors
• P-values indicate the statistical significance of each coefficient in the model
  • A small p-value (typically < 0.05) suggests that the coefficient is significantly different from zero
• Residuals are the differences between the observed values of the dependent variable and the predicted values from the model
  • Used to assess the model's assumptions and goodness of fit

Building Your First Linear Model

• Use the

  lm()

  function to create a linear regression model in R
• Specify the model formula in the form

  dependent_variable ~ independent_variable_1 + independent_variable_2 + ...

  • Example:

    model <- lm(sales ~ advertising + price, data = sales_data)

• Include the

  data

  argument to specify the data frame containing the variables
• Assign the model to an object (e.g.,

  model

  ) for later use
• View the model summary using

  summary(model)

  to see the coefficients, p-values, and other key statistics
• Interpret the coefficients as the change in the dependent variable for a one-unit change in the corresponding independent variable, holding other variables constant
• Use the

  predict()

  function to make predictions based on the model
  • Example:

    predictions <- predict(model, newdata = new_sales_data)

• Assess the model's performance by comparing the predicted values to the actual values of the dependent variable

Interpreting Model Results

• Examine the model summary output to interpret the results
• Check the p-values for each coefficient to determine if they are statistically significant
  • A p-value less than 0.05 indicates that the coefficient is significantly different from zero at the 5% level
• Interpret the coefficients as the change in the dependent variable for a one-unit change in the corresponding independent variable, holding other variables constant
  • Example: If the coefficient for advertising is 0.5, a one-unit increase in advertising is associated with a 0.5-unit increase in sales, holding other variables constant
• Look at the R-squared and adjusted R-squared values to assess the model's goodness of fit
  • Higher values indicate that the model explains a larger proportion of the variance in the dependent variable
• Examine the residual standard error to understand the average deviation of the observed values from the predicted values
• Check the F-statistic and its associated p-value to determine if the overall model is statistically significant
• Use the

  confint()

  function to calculate confidence intervals for the coefficients
  • Example:

    confint(model, level = 0.95)

    provides 95% confidence intervals

Checking Model Assumptions

• Linear regression relies on several assumptions that must be met for the model to be valid and reliable
• Linearity assumes a linear relationship between the dependent variable and independent variables
  • Check linearity using scatterplots of the dependent variable against each independent variable
  • Look for a roughly linear pattern in the plots
• Independence of errors assumes that the residuals are not correlated with each other
  • Check for autocorrelation using the Durbin-Watson test (

    durbinWatsonTest()

    from the

    car

    package)
  • Values close to 2 indicate no autocorrelation, while values close to 0 or 4 suggest positive or negative autocorrelation, respectively
• Homoscedasticity assumes that the variance of the residuals is constant across all levels of the independent variables
  • Check homoscedasticity using a scatterplot of the residuals against the predicted values
  • Look for a roughly even spread of residuals across the range of predicted values
• Normality assumes that the residuals are normally distributed
  • Check normality using a histogram or QQ plot of the residuals
  • Look for a roughly bell-shaped distribution in the histogram or a straight line in the QQ plot
• Multicollinearity occurs when independent variables are highly correlated with each other
  • Check for multicollinearity using the variance inflation factor (VIF) for each independent variable
  • VIF values greater than 5 or 10 indicate potential multicollinearity issues

Improving Your Model

• Identify and remove outliers that may be influencing the model results
  • Use scatterplots and residual plots to visually identify potential outliers
  • Consider removing or adjusting extreme values that are not representative of the overall pattern
• Handle missing data appropriately to avoid bias in the model
  • Use techniques like listwise deletion (removing rows with missing values) or imputation (replacing missing values with estimated values)
• Transform variables if necessary to meet the assumptions of linear regression
  • Apply log transformations to variables with skewed distributions
  • Standardize or scale variables to ensure they are on a similar scale
• Consider adding interaction terms to capture the combined effect of two or more independent variables
  • Example:

    model <- lm(sales ~ advertising + price + advertising:price, data = sales_data)

• Use feature selection techniques to identify the most important variables for the model
  • Stepwise regression (forward, backward, or both) can help select a subset of variables based on their contribution to the model
  • Regularization methods like lasso or ridge regression can shrink the coefficients of less important variables towards zero
• Validate the model using techniques like cross-validation or holdout validation
  • Split the data into training and testing sets to assess the model's performance on unseen data
  • Use metrics like mean squared error (MSE) or root mean squared error (RMSE) to evaluate the model's predictive accuracy

Real-World Applications

• Predicting house prices based on features like square footage, number of bedrooms, and location
  • Example:

    house_price_model <- lm(price ~ sqft + bedrooms + location, data = housing_data)

• Analyzing the impact of advertising expenditure on product sales
  • Example:

    sales_model <- lm(sales ~ tv_ads + radio_ads + newspaper_ads, data = advertising_data)

• Examining the relationship between student performance and factors like study hours and attendance
  • Example:

    performance_model <- lm(grade ~ study_hours + attendance, data = student_data)

• Investigating the effect of customer demographics on purchasing behavior
  • Example:

    purchase_model <- lm(amount_spent ~ age + gender + income, data = customer_data)

• Predicting stock prices based on economic indicators and company performance metrics
  • Example:

    stock_price_model <- lm(price ~ gdp_growth + inflation + revenue + profit, data = stock_data)

• Modeling the relationship between employee salaries and factors like education, experience, and job title
  • Example:

    salary_model <- lm(salary ~ education + experience + job_title, data = employee_data)