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Concept Overview
- Linearity and homoscedasticity are key assumptions in regression diagnostics, fundamental for assessing the validity of the linear regression model.

Residual Plot: A graphical representation of the residuals against predicted values.
- Types of Residuals: Pure, Standardized, Studentized, Jackknife.
- Jackknife Residuals: Most effective in diminishing the influence of extreme observations.
- Creating a Residual Plot in R:
1. Save predictions using: mlb$pred = predict(model).
2. Plot residuals against predictions: plot(mlb $pred, mlb$ jackknife, pch = 19, xlab = "Predicted Wins", ylab = "Jackknife Residual").
3. Add horizontal line at zero: abline(h=0).

Criteria:
- Satisfied: No pattern in the residual plot; observations randomly scattered around zero.
- Violated: Emergence of a clear nonlinear pattern in the residual plot.
- Types of Relationships:
- Satisfied: Randomly scattered residuals indicate a linear relationship.
- Violated: Shows potential quadratic or cubic relationships or an undefined relationship between response and predictor.

Criteria:
- Satisfied: No distinct pattern in residual spread; random scattering around zero.
- Violated: Increasing spread of residuals as predicted values grow, leading to a funnel shape.
Examples:
- Satisfied: Residuals remain stable across predicted values.
- Violated: Residuals spread increases with larger predicted values.

Scenario: Analyzing runs scored and allowed to predict wins. Given residual plot of jackknife residuals vs. predicted values.
- Question: Are linearity and homoscedasticity satisfied?
- Answer: Linearities assessed aside from a few extreme observations.
- Findings:
- Residuals are evenly scattered.
- Spread of residuals remains consistent across predicted win totals.

Definition: A plot of the residuals arranged in increasing order (Y-axis) versus the percentile they would occupy if normally distributed (X-axis).
- Interpretation of Curves:
- Concave down: Indicates left-skewed residuals.
- Straight Line: Suggests normally distributed residuals.
- Concave up: Indicates right-skewed residuals.
R Code for Creating Normal Probability Plot:
- Generate plot: qqnorm(mlb$jackknife, pch = 19).
- Add diagonal for perfect normality: qqline(mlb$jackknife).

Example:
- 2019 Detroit Tigers:
- Location: Circled point represents a notable residual.
- Sample Quantile: Tigers’ jackknife residual deviated significantly from predicted value.
- Theoretical Quantile: Expected residual if perfectly normal would be around a specific calculated z-score.
Results from R: R calculates the percentile and Z-score of the residual. Similar process applied to 2021 Seattle Mariners, highlighting skewness and its implications on regression assumptions.

Visual Method: Normal probability plot insights.
- Straight line signifies normal distribution.
- Deviations may indicate skewness (left or right).
Statistical Test: Shapiro-Wilk Test for normality.
- Hypotheses:
- Null Hypothesis (H0): Data is normally distributed.
- Alternative Hypothesis (H_A): Data is not normally distributed.
- Test Invocation in R: Execute through shapiro.test(mlb$jackknife).
- Failing to Reject H0: Supports normality condition.
Example of Assessing Normality:
- Scenario: Analyzing normality with histogram and normal probability plot for jackknife residuals.
- Conclusion drawn from visual inspection and statistical testing.

Influential Point: An observation strongly affecting the regression model’s predictions due to its presence or absence.
- High leverage and outlying residuals signify potential influence on the model.
Measures of Influence:
- Cook's Distance: Reflects model prediction changes when an observation is removed.
- Formula: $di = \frac{1}{k+1}hi(1-hi)ri^2$
- Influential if Cook's distance exceeds the threshold of $\frac{4}{n}$ .
- DFBETAS: Measures the change in regression coefficients upon exclusion of an observation.
- Definition: $DFBETAS(-i)j = \frac{\hat{\beta}j - \hat{\beta}{-i,j}}{S{\hat{\beta}_{-i,j}}}$
- Influential when condition holds for coefficients.

Concept: Identifies observations whose exclusion results in substantial changes to regression model predictions.
- Notably influential if both outlier and high leverage.
- General rule: observations with Cook’s distance greater than $\frac{4}{n}$ are flagged as influential.

Question: Are 2019 Detroit Tigers or 2021 Seattle Mariners influential based on Cook's distance?
- Calculate Cook’s distances for both and compare against established cutoff.
- Outcomes: Identification of influential observation with respect to model behavior using Cook's metric.

Provides a standardized perspective of how much each regression coefficient shifts with the exclusion of an observation.
Criteria for Influence:
- If $DFBETAS(-i)_j > 2 \cdot \frac{n}{n}$ , it is influential.
- Conversely, $DFBETAS(-i)_j \le 2 \cdot \frac{n}{n}$ indicates no significant influence.

Scenario: Remove a specific observation and compute DFBETAS to evaluate impacts on coefficients.
Provide calculated values and discuss their implications based on the model adjustments made.

Examine the validity and origin of influential points.
Assess linearity, homoscedasticity, and normality of residuals.
- If any of these are violated, consider that the model may lack appropriateness for the data.
- Explore possible transformations or inclusion of additional predictors.
Model re-evaluation: Retain or remove influential points based on their performance post-adjustment.

Summarize the scenario involving model adjustments from removing influential observations, and report on the rationale behind the modifications made to enhance model fidelity.

Observations may show a disconnect between Cook’s distances and DFBETAS.
- Important to ensure both measures align in indicating influence.
- A noticeable change in one coefficient may be countered by changes in others, maintaining overall model predictions.