10.3 Residuals and Model Appropriateness

10.3 Residuals and Model Appropriateness

Quick Answer: A residual measures the vertical prediction error of a regression model, calculated as $\text{residual} = \text{observed } y - \text{predicted } y = y - \hat{y}$ (Actual minus Predicted). A positive residual indicates that the model underpredicted the observation, while a negative residual indicates an overprediction. In a residual plot ($x$ vs. residual), a random scatter of points around the horizontal zero line proves that the chosen model is appropriate. A distinct curved pattern (such as a U-shape) proves that a linear model is inappropriate and a non-linear model should be used. The correlation coefficient ($r$) measures linear strength and direction ($-1 \le r \le 1$), but correlation never implies causation.

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1. Defining and Calculating Residuals

[!NOTE] Scope note. Under the Next Generation standards, the residual-plot standard (S-ID.6b) and the correlation-coefficient standard (S-ID.8) sit in Algebra I, not on the Algebra II blueprint, which lists only AII-S.ID.6a for this domain. Residual reasoning is still worth your time in Algebra II for two reasons: it is how you decide which model to fit under AII-S.ID.6a, and interpreting numerical summaries and critiquing statistical claims are exactly AII-S.IC.6a and AII-S.IC.6b. Expect this material to show up as justification rather than as a standalone "make a residual plot" prompt.

Once a regression line or curve is constructed, statisticians must evaluate how accurately the model reflects the observed sample data. The foundational diagnostic tool for this assessment is the residual.

The Residual Formula

For each paired observation $(x_i, y_i)$ in a dataset:

$\text{Residual} = y - \hat{y} = \text{Observed } y - \text{Predicted } y$

Where:

• $y$ is the actual, observed response measurement from the data table.
• $\hat{y}$ (pronounced "$y$-hat") is the predicted response value obtained by substituting the explanatory value $x$ into the regression equation.

[!TIP] The "AP" Memory Aid: Remember the letters AP — Actual minus Predicted ($y - \hat{y}$). Never subtract in reverse order ($\hat{y} - y$), as that inverts the sign of every residual.

Directional Interpretation of Residuals

The Zero-Sum Property of Least-Squares Regression

In any ordinary least-squares linear regression line, the sum of all individual residuals is mathematically guaranteed to equal zero:

$\sum_{i=1}^n (y_i - \hat{y}_i) = 0$

Positive prediction errors cancel negative prediction errors exactly.

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2. Constructing and Interpreting Residual Plots

A residual plot is a specialized diagnostic graph where the horizontal axis represents the explanatory variable $x$ (or the predicted values $\hat{y}$), and the vertical axis represents the computed residuals ($y - \hat{y}$). A horizontal reference line is drawn at $\text{residual} = 0$.

Appropriate Model (Random Scatter)      Inappropriate Model (Curved Pattern)
   Residual ^                             Residual ^
            |    *       *                         |   *               *
          + |        *                           + |     *           *
       ---0-+----+-------+----+---> x         ---0-+-------+-------+-----> x
          - |  *       *                           - |       *   *
            |      *                               |           *

Diagnostic Decision Rules

1. Random Scatter $\implies$ Model is Appropriate: If the points in the residual plot are distributed randomly above and below the zero line across the entire domain—with approximately equal spread and no discernible geometric shape—the chosen regression model is appropriate.
2. Curved Pattern $\implies$ Linear Model is Inappropriate: If the residual plot exhibits a distinct curved shape (such as a U-shape, inverted U-shape, or sinusoidal wave), the linear model is inappropriate. A curved residual plot reveals systematic curvature in the original data that the straight line failed to capture, proving that a non-linear model (e.g., quadratic or exponential) is necessary.
3. Funnel / Fan Shape $\implies$ Non-Constant Variance: If residuals fan out or pinch inward as $x$ increases, the prediction error variability is not constant across the domain (heteroscedasticity).

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3. The Linear Correlation Coefficient ($r$) and Determination ($r^2$) (Algebra I standard, used here as supporting evidence)

While residual plots evaluate model appropriateness, the Pearson correlation coefficient ($r$) quantifies the strength and direction of a linear relationship between two quantitative variables.

Mathematical Properties of $r$

• Bounded Interval: $-1 \le r \le 1$.
• Direction:
  • If $r > 0$, there is a positive linear association (as $x$ increases, $y$ tends to increase).
  • If $r < 0$, there is a negative linear association (as $x$ increases, $y$ tends to decrease).
• Strength:
  • $r = 1$: Perfect positive linear correlation.
  • $r = -1$: Perfect negative linear correlation.
  • $r = 0$: No linear relationship between the variables.
  • General benchmarks: $|r| \ge 0.8$ represents strong correlation; $0.5 \le |r| < 0.8$ represents moderate correlation; $|r| < 0.5$ represents weak correlation.

[!IMPORTANT] Enabling Diagnostics on TI-84: By default, TI-84 calculators do not display $r$ and $r^2$ after running LinReg. You must activate diagnostics once: Press 2nd $\to$ 0 (CATALOG), scroll down to DiagnosticOn, press ENTER, and press ENTER again until Done appears.

Coefficient of Determination ($r^2$)

The coefficient of determination ($r^2$) is the square of the correlation coefficient, satisfying $0 \le r^2 \le 1$:

$r^2 = (r)^2$

• Contextual Interpretation: $r^2$ represents the proportion (or percentage) of the total variation in the response variable $y$ that is explained by the linear regression relationship with $x$.
• For example, if $r = -0.92$, then $r^2 = (-0.92)^2 \approx 0.8464$. Thus, approximately $84.6%$ of the variability in the response variable can be explained by the linear model, while $15.4%$ is due to unexplained random variation or lurking variables.

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4. The Critical Principle: Correlation Does Not Imply Causation

A high correlation coefficient ($r \approx 1$ or $r \approx -1$) proves the existence of a strong statistical association, but it never establishes that changes in $x$ cause changes in $y$.

Why Association Does Not Mean Causation

1. Lurking / Confounding Variables: A third unmeasured variable may simultaneously drive changes in both $x$ and $y$.
   • Classic Example: Ice cream sales and swimming pool drownings are strongly positively correlated ($r \approx 0.85$). Buying ice cream does not cause drownings; both variables increase during hot summer weather (the lurking variable: ambient outdoor temperature).
2. Reverse Causation: It may be that $y$ influences $x$, rather than $x$ causing $y$.
3. Establishing Causality: Causation can only be established through a randomized controlled experiment where subjects are randomly assigned to treatment and control groups to isolate the explanatory variable. Observational regression data cannot prove cause and effect.

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5. Worked Examples

Worked Problem 1: Tabular Residual Calculation

Problem: A researcher models the relationship between hours of flight training ($x$) and simulated flight errors ($y$) using the linear equation $\hat{y} = -1.8x + 24.5$. The data table includes an observation of a pilot with $5\text{ hours}$ of training who committed $13\text{ errors}$. Calculate the residual and interpret its meaning.

• Step 1: Calculate the predicted value $\hat{y}$ for $x = 5$. $\hat{y} = -1.8(5) + 24.5 = -9.0 + 24.5 = 15.5\text{ errors}$
• Step 2: Calculate the residual using $y - \hat{y}$. $\text{Residual} = 13 - 15.5 = -2.5\text{ errors}$
• Step 3: Interpret the residual in context. The residual is $-2.5$. Because the residual is negative, the pilot committed $2.5$ fewer errors than the model predicted. The linear regression equation overpredicted the number of flight errors for this pilot.

Worked Problem 2: Evaluating Residual Plots for Model Appropriateness

Problem: A student fits a linear regression model to the trajectory of a rocket, obtaining $r = 0.94$. However, the residual plot displays points below the zero line for small and large values of $x$, and points above the zero line for intermediate values of $x$, forming an inverted U-shape. Is the linear model appropriate? Explain your reasoning.

• Step 1: Inspect the residual plot for pattern. The residuals follow a distinct non-random, inverted U-shaped curved pattern.
• Step 2: Formulate the verdict. No, the linear model is inappropriate.
• Step 3: Provide mathematical justification. Even though the correlation coefficient $r = 0.94$ indicates a strong linear association, model appropriateness is determined strictly by the residual plot. The presence of a clear curved pattern indicates that the underlying relationship is non-linear (specifically quadratic), and a non-linear regression model must be used instead.

Worked Problem 3: Interpreting $r$ and $r^2$ in Context

Problem: A study of $200$ high school seniors reports a correlation of $r = 0.80$ between hours of physical exercise per week and hours of sleep per night. Calculate $r^2$, interpret it in context, and determine whether the study proves that exercising more causes students to sleep longer.

• Step 1: Calculate the coefficient of determination $r^2$. $r^2 = (0.80)^2 = 0.64 = 64\%$
• Step 2: Interpret $r^2$ in context. Approximately $64%$ of the variation in nightly sleep duration can be explained by the linear relationship with weekly exercise hours.
• Step 3: Evaluate the causal claim. The study does not prove causation. Because this was an observational study, confounding variables (such as overall stress levels, extracurricular workload, or dietary habits) may influence both exercise and sleep. Correlation alone cannot establish causality.

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6. Common Regents Pitfalls & Exam Strategies

• Pitfall 1: Relying Solely on $r$ to Judge Linearity. A high correlation coefficient ($r = 0.95$) does not guarantee that a line is appropriate. Curved datasets often yield high $r$ values. Always examine the residual plot.
• Pitfall 2: Confusing Overprediction and Underprediction. If $\text{residual} = -4$, the actual value is $4$ units lower than predicted, meaning the model overpredicted the outcome.
• Pitfall 3: Claiming $r^2$ is a Probability. The coefficient of determination $r^2$ is the proportion of explained variance in the response variable, never the probability that a prediction is correct.