13.2 Simple linear regression

Simple linear regression models the relationship between two variables, like advertising spend and sales revenue. It uses an equation to predict how changes in one variable affect the other, helping businesses make data-driven decisions.

The model's effectiveness is evaluated using measures like R-squared and statistical tests. These tools help determine if the relationship between variables is significant and how well the model fits the data, guiding marketers in understanding the impact of their strategies.

Simple Linear Regression

Simple linear regression models

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• Statistical method models linear relationship between two variables
  • Independent (explanatory) variable denoted as $x$ (advertising expenditure)
  • Dependent (response) variable denoted as $y$ (sales revenue)
• Simple linear regression model represented by equation: $y = \beta_0 + \beta_1x + \epsilon$
  • $\beta_0$ is y-intercept, value of $y$ when $x$ is zero (baseline sales without advertising)
  • $\beta_1$ is slope, change in $y$ for one-unit increase in $x$ (sales increase per advertising dollar)
  • $\epsilon$ is random error term, accounts for variability in $y$ not explained by linear relationship with $x$ (other factors affecting sales)
• Goal is to estimate $\beta_0$ and $\beta_1$ using least squares method
  • Minimizes sum of squared residuals (differences between observed and predicted values)
  • Finds line of best fit that minimizes overall prediction error (ordinary least squares regression)

Interpretation of regression coefficients

• Slope coefficient $\beta_1$ represents change in dependent variable $y$ for one-unit increase in independent variable $x$
  • Positive slope indicates positive linear relationship (increasing advertising increases sales)
  • Negative slope indicates negative linear relationship (increasing price decreases demand)
• Intercept coefficient $\beta_0$ represents value of dependent variable $y$ when independent variable $x$ is zero
  • May or may not have meaningful interpretation depending on research context (sales with no advertising)
• Estimated regression coefficients used to make predictions for dependent variable given specific value of independent variable
  • Predict sales revenue for a given level of advertising expenditure
• Interpret coefficients in context of research problem
  • Consider units of measurement (dollars, units sold)
  • Assess practical significance of relationship (small vs large effect on sales)

Model Evaluation and Inference

Goodness-of-fit in regression

• Coefficient of determination $R^2$ measures proportion of total variation in dependent variable $y$ explained by linear relationship with independent variable $x$
  • $R^2$ ranges from 0 to 1, higher values indicate better model fit (more variation explained)
  • $R^2 = 1$ indicates model perfectly explains all variability in $y$ (perfect fit)
  • $R^2 = 0$ indicates model does not explain any variability in $y$ (no relationship)
• Adjusted R-squared modifies $R^2$ to account for number of independent variables in model
  • Penalizes addition of unnecessary independent variables that do not significantly improve model fit
  • Adjusted R-squared always lower than or equal to $R^2$ (more conservative measure)
• High $R^2$ or adjusted R-squared suggests good model fit, low value suggests model may not be appropriate or other factors influence dependent variable
  • High $R^2$ (0.8) indicates strong linear relationship, model explains most variation in sales
  • Low $R^2$ (0.2) indicates weak linear relationship, model explains little variation in sales

Statistical tests for regression coefficients

• Hypothesis tests assess statistical significance of regression coefficients
  • Null hypothesis ($H_0$): coefficient equals zero, no linear relationship
  • Alternative hypothesis ($H_a$): coefficient not equal to zero, significant linear relationship
• t-test used to test significance of regression coefficients
  • Test statistic calculated as ratio of estimated coefficient to its standard error
  • p-value associated with test statistic determines statistical significance
  • If p-value < significance level (0.05), reject null hypothesis, coefficient considered statistically significant
• Confidence intervals provide range of plausible values for true population coefficients
  • 95% confidence interval indicates 95% probability true coefficient lies within calculated interval
  • Narrow intervals suggest precise estimates, wide intervals indicate greater uncertainty
• Statistical significance and confidence intervals help determine reliability and generalizability of simple linear regression model
  • Significant slope coefficient (p < 0.05) and narrow confidence interval indicate reliable relationship between advertising and sales
  • Non-significant slope coefficient (p > 0.05) and wide confidence interval suggest relationship may not be generalizable to larger population