10.5 Inference for regression parameters

Inference for regression parameters is a crucial aspect of statistical analysis. It allows us to draw conclusions about the relationship between variables in a population based on sample data. This topic builds on the foundations of simple linear regression, extending our understanding to more complex scenarios.

In this section, we'll explore how to estimate regression parameters, test hypotheses, and construct confidence intervals. We'll also examine the assumptions underlying these methods and learn how to check model adequacy, ensuring our inferences are valid and reliable.

Simple linear regression model

• Simple linear regression is a statistical method used to model the relationship between two variables, where one variable (the independent variable) is used to predict the other variable (the dependent variable)
• The model assumes a linear relationship between the variables, meaning that the change in the dependent variable is proportional to the change in the independent variable
• The goal of simple linear regression is to find the line of best fit that minimizes the sum of squared differences between the observed values and the predicted values

Population regression line

• The population regression line represents the true relationship between the independent and dependent variables in the entire population
• It is a theoretical concept, as it is usually not possible to observe the entire population
• The population regression line is defined by the equation $y = \beta_0 + \beta_1x + \epsilon$, where $\beta_0$ is the intercept, $\beta_1$ is the slope, and $\epsilon$ is the random error term

Least squares line

• The least squares line is an estimate of the population regression line based on a sample of data
• It is the line that minimizes the sum of squared residuals, which are the differences between the observed values and the predicted values
• The least squares line is defined by the equation $\hat{y} = b_0 + b_1x$, where $b_0$ is the estimated intercept and $b_1$ is the estimated slope

Residuals

• Residuals are the differences between the observed values and the predicted values from the regression line
• They represent the portion of the dependent variable that is not explained by the independent variable
• Residuals are used to assess the goodness of fit of the regression model and to check for violations of the model assumptions (homoscedasticity, linearity, independence)

Influential observations

• Influential observations are data points that have a disproportionate effect on the regression line
• They can be identified by examining the residuals and leverage values (a measure of how far an observation is from the mean of the independent variable)
• Influential observations can distort the regression results and should be carefully examined to determine if they are valid data points or outliers that should be removed

Inference for regression parameters

• Inference for regression parameters involves using sample data to make conclusions about the population parameters (intercept and slope) of the regression model
• This includes estimating the parameters, testing hypotheses about the parameters, and constructing confidence intervals for the parameters
• Inference allows us to determine the statistical significance of the relationship between the variables and to quantify the uncertainty in our estimates

Assumptions

• The linear regression model relies on several assumptions about the data and the relationship between the variables
• These assumptions include linearity (the relationship between the variables is linear), independence (the observations are independent of each other), homoscedasticity (the variance of the residuals is constant), and normality (the residuals are normally distributed)
• Violations of these assumptions can lead to biased or inefficient estimates and invalid inference

Estimating parameters

• The parameters of the regression model (intercept and slope) can be estimated using the method of least squares
• The formulas for the estimated intercept and slope are:
  • $b_0 = \bar{y} - b_1\bar{x}$
  • $b_1 = \frac{\sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y})}{\sum_{i=1}^n (x_i - \bar{x})^2}$
• These estimates are unbiased and have the smallest variance among all linear unbiased estimators (BLUE property)

Hypothesis tests for slope

• Hypothesis tests can be used to determine if there is a significant linear relationship between the variables
• The null hypothesis is usually that the slope is equal to zero ($H_0: \beta_1 = 0$), which implies no linear relationship
• The alternative hypothesis can be two-sided ($H_a: \beta_1 \neq 0$) or one-sided ($H_a: \beta_1 > 0$ or $H_a: \beta_1 < 0$)
• The test statistic is calculated as $t = \frac{b_1 - 0}{SE(b_1)}$, where $SE(b_1)$ is the standard error of the estimated slope
• The test statistic follows a t-distribution with $n-2$ degrees of freedom under the null hypothesis

Confidence intervals for slope

• Confidence intervals provide a range of plausible values for the population slope parameter
• A 95% confidence interval for the slope is given by $b_1 \pm t_{0.025, n-2} \cdot SE(b_1)$, where $t_{0.025, n-2}$ is the critical value from a t-distribution with $n-2$ degrees of freedom
• The confidence interval can be interpreted as the range of values within which the true population slope lies with 95% confidence

Inference for intercept

• Inference for the intercept parameter is similar to inference for the slope
• Hypothesis tests and confidence intervals can be constructed for the intercept using the estimated intercept ($b_0$) and its standard error ($SE(b_0)$)
• However, the intercept is often less of interest than the slope, as it represents the predicted value of the dependent variable when the independent variable is zero, which may not be meaningful or realistic in many contexts

Coefficient of determination

• The coefficient of determination ($R^2$) is a measure of the proportion of variance in the dependent variable that is explained by the independent variable
• It is calculated as the ratio of the explained sum of squares (ESS) to the total sum of squares (TSS): $R^2 = \frac{ESS}{TSS} = 1 - \frac{RSS}{TSS}$, where RSS is the residual sum of squares
• $R^2$ ranges from 0 to 1, with higher values indicating a stronger linear relationship between the variables
• However, $R^2$ should be interpreted cautiously, as it can be inflated by adding more independent variables to the model, even if they are not truly related to the dependent variable

Checking model adequacy

• After fitting a regression model, it is important to assess whether the model assumptions are met and whether the model provides an adequate fit to the data
• This can be done through various diagnostic plots and tests, which can help identify potential issues with the model and suggest ways to improve it
• Checking model adequacy is crucial for ensuring that the model results are reliable and can be used for prediction and inference

Residual plots

• Residual plots are scatter plots of the residuals against the predicted values or the independent variable
• They can reveal patterns or trends in the residuals that indicate violations of the model assumptions
• For example, a funnel-shaped pattern in the residuals suggests heteroscedasticity (non-constant variance), while a curved pattern suggests non-linearity in the relationship between the variables
• Ideally, the residuals should be randomly scattered around zero with no apparent patterns

Outliers and influential points

• Outliers are observations that have unusually large residuals, indicating that they are poorly fit by the regression model
• Influential points are observations that have a disproportionate effect on the regression results, due to their extreme values on the independent variable or their large residuals
• Outliers and influential points can be identified using various diagnostic measures, such as standardized residuals, leverage values, and Cook's distance
• It is important to carefully examine these observations and determine whether they are valid data points or errors that should be corrected or removed from the analysis

Assessing linearity assumption

• The linearity assumption states that the relationship between the independent and dependent variables is linear
• This can be assessed by examining the residual plots for any curvature or non-linear patterns
• If the linearity assumption is violated, the model may need to be modified by transforming the variables or adding higher-order terms (such as quadratic or interaction terms)

Assessing constant variance assumption

• The constant variance (homoscedasticity) assumption states that the variance of the residuals is constant across all levels of the independent variable
• This can be assessed by examining the residual plots for any funnel-shaped or wedge-shaped patterns, which indicate heteroscedasticity
• If the constant variance assumption is violated, the model estimates may be inefficient and the inference may be invalid
• Remedies for heteroscedasticity include using weighted least squares, transforming the variables, or using robust standard errors

Assessing normality assumption

• The normality assumption states that the residuals are normally distributed with mean zero and constant variance
• This can be assessed using a normal probability plot (Q-Q plot) of the residuals, which compares the observed residuals to the expected values under a normal distribution
• If the normality assumption is violated, the inference based on t-tests and F-tests may be invalid, especially for small sample sizes
• Remedies for non-normality include using non-parametric tests, bootstrapping, or robust regression methods

Transformations to improve model fit

• If the model assumptions are violated or the model fit is poor, transforming the variables can sometimes improve the model
• Transformations can help stabilize the variance, linearize the relationship between the variables, or make the residuals more normally distributed
• Common transformations include log, square root, reciprocal, and Box-Cox transformations

Variance stabilizing transformations

• Variance stabilizing transformations are used to make the variance of the residuals more constant across the levels of the independent variable
• Examples include taking the square root or the logarithm of the dependent variable
• These transformations are useful when the residual plot shows a funnel-shaped pattern, indicating that the variance increases with the level of the independent variable

Linearizing transformations

• Linearizing transformations are used to make the relationship between the variables more linear
• Examples include taking the logarithm or the reciprocal of the independent variable
• These transformations are useful when the residual plot shows a curved pattern, indicating that the relationship between the variables is non-linear
• Linearizing transformations can also help reduce the influence of outliers and make the model more robust

Inference for prediction

• In addition to inference for the regression parameters, we can also use the regression model to make predictions for new observations and to quantify the uncertainty in those predictions
• This involves constructing prediction intervals and confidence intervals for the mean response, as well as using inverse regression to estimate the value of the independent variable corresponding to a given value of the dependent variable

Prediction intervals

• A prediction interval is a range of values within which a new observation of the dependent variable is likely to fall, given a specific value of the independent variable
• The prediction interval takes into account both the uncertainty in the estimated regression line and the variability of the individual observations around the line
• A 95% prediction interval for a new observation $y_0$ at a given value of $x_0$ is given by $\hat{y}0 \pm t{0.025, n-2} \cdot \sqrt{MSE \cdot (1 + \frac{1}{n} + \frac{(x_0 - \bar{x})^2}{\sum_{i=1}^n (x_i - \bar{x})^2})}$, where $MSE$ is the mean squared error from the regression model

Confidence intervals for mean response

• A confidence interval for the mean response is a range of values within which the true mean value of the dependent variable is likely to fall, given a specific value of the independent variable
• The confidence interval takes into account the uncertainty in the estimated regression line, but not the variability of the individual observations around the line
• A 95% confidence interval for the mean response $\mu_{y|x_0}$ at a given value of $x_0$ is given by $\hat{y}0 \pm t{0.025, n-2} \cdot \sqrt{MSE \cdot (\frac{1}{n} + \frac{(x_0 - \bar{x})^2}{\sum_{i=1}^n (x_i - \bar{x})^2})}$

Inverse regression

• Inverse regression is used to estimate the value of the independent variable corresponding to a given value of the dependent variable
• This can be useful in situations where we want to determine the level of the independent variable needed to achieve a desired outcome on the dependent variable
• To perform inverse regression, we first construct a confidence interval for the mean response at the desired value of the dependent variable, and then solve the confidence interval equation for the independent variable
• However, inverse regression should be used with caution, as it can be sensitive to model misspecification and extrapolation beyond the range of the observed data

Multiple regression

• Multiple regression is an extension of simple linear regression that allows for more than one independent variable to be used in predicting the dependent variable
• Multiple regression can help capture more complex relationships between the variables and can improve the predictive power of the model
• However, multiple regression also introduces new challenges, such as multicollinearity and variable selection, which need to be addressed to ensure the validity and interpretability of the model

Multiple regression model

• The multiple regression model is defined by the equation $y = \beta_0 + \beta_1x_1 + \beta_2x_2 + ... + \beta_px_p + \epsilon$, where $y$ is the dependent variable, $x_1, x_2, ..., x_p$ are the independent variables, $\beta_0, \beta_1, ..., \beta_p$ are the regression coefficients, and $\epsilon$ is the random error term
• The regression coefficients represent the change in the dependent variable for a one-unit change in the corresponding independent variable, holding all other variables constant
• The multiple regression model can be estimated using the method of least squares, similar to simple linear regression

Partial regression plots

• Partial regression plots are used to visualize the relationship between each independent variable and the dependent variable, while controlling for the effects of the other independent variables
• They are created by plotting the residuals from regressing the dependent variable on all the other independent variables against the residuals from regressing the independent variable of interest on all the other independent variables
• Partial regression plots can help identify the strength and direction of the relationship between each independent variable and the dependent variable, as well as detect any non-linearity or outliers

Adjusted R-squared

• The adjusted R-squared is a modified version of the coefficient of determination ($R^2$) that takes into account the number of independent variables in the model
• It is calculated as $1 - \frac{(1 - R^2)(n - 1)}{n - p - 1}$, where $n$ is the sample size and $p$ is the number of independent variables
• The adjusted R-squared penalizes the addition of unnecessary independent variables to the model and provides a more conservative estimate of the proportion of variance explained by the model
• It is useful for comparing models with different numbers of independent variables and for avoiding overfitting

Multicollinearity

• Multicollinearity refers to the presence of high correlations among the independent variables in a multiple regression model
• It can cause problems in estimating the regression coefficients and interpreting their significance, as the effects of the correlated variables can be confounded
• Multicollinearity can be detected using the variance inflation factor (VIF), which measures the extent to which the variance of each regression coefficient is inflated due to the correlations among the independent variables
• Remedies for multicollinearity include removing one of the correlated variables, combining them into a single variable, or using regularization methods such as ridge regression or lasso regression

Variable selection methods

• Variable selection methods are used to identify the subset of independent variables that are most relevant for predicting the dependent variable
• They can help simplify the model, improve its interpretability, and reduce overfitting
• Common variable selection methods include:
  • Forward selection: starting with no variables in the model and adding the most significant variable at each step
  • Backward elimination: starting with all variables in the model and removing the least significant variable at each step
  • Stepwise selection: a combination of forward and backward selection, adding or removing variables based on their significance at each step
• Other methods, such as best subset selection and regularization methods, can also be used for variable selection
• The choice of variable selection method depends on the goals of the analysis, the sample size, and the complexity of the relationships among the variables