Influential Outlier Detection and Impact on OLS Fit in River Survey Data

• Accurately implementing the outlier override in the synthetic data
• Applying the correct influence thresholds and interpreting the results
• Ensuring the final aggregation uses the correct counts and maximum values

Cook's distance (D_i) measures the change in fitted values when the i‑th observation is omitted. For a linear regression model y = Xβ + ε with n observations, p parameters (including intercept), residual vector r, and hat matrix H (where h_i = H_{ii} is the leverage of point i), the exact formula is:

D_i = (r_i^2 / (p * MSE)) * (h_i / (1 - h_i)^2)

where MSE = Σ r_j^2 / (n - p) is the mean squared error of the full model. An equivalent matrix form is D_i = (β̂ - β̂_{(i)})' X'X (β̂ - β̂_{(i)}) / (p * MSE), showing that D_i is proportional to the squared distance between the full‑sample coefficient vector and the leave‑one‑out estimate.

DFFITS (difference in fits) quantifies the influence of observation i on its own fitted value. The exact formula is:

DFFITS_i = (r_i / (s * sqrt(1 - h_i))) * sqrt(h_i / (1 - h_i))

where s = sqrt(MSE) is the residual standard error. This can also be written as DFFITS_i = (β̂ - β̂_{(i)})' x_i / (s * sqrt(1 - h_i)), linking the change in coefficients to the predictor vector of the omitted case.

Derivation of the 4/n threshold for Cook's distance
Cook originally suggested that observations with D_i > 1 are potentially influential. Later work (e.g., Belsley, Kuh, and Welsch, 1980) showed that, under the assumption of normally distributed errors and no severe multicollinearity, the expected value of D_i is approximately p / n. Because p is usually small relative to n, a practical rule of thumb scales this by a factor of 4, giving the simple cutoff:

D_i > 4 / n

The factor 4 provides a conservative bound: for typical sample sizes the cutoff is lower than 1, flagging points that would cause a noticeable change in the regression surface.

Derivation of the DFFITS threshold 2*sqrt(2/n)
DFFITS is essentially a studentized residual multiplied by a leverage‑adjustment term. The distribution of DFFITS under the null (no influential points) approximates a t‑distribution with (n‑p‑1) degrees of freedom. A common rule of thumb sets the cutoff at:

|DFFITS_i| > 2 * sqrt(p / n)

When p = 2 (simple linear regression with intercept), this simplifies to 2*sqrt(2/n). The factor 2 comes from the 95% confidence bound for a standard normal variable (≈1.96), rounded for ease of use. The sqrt(p/n) term reflects that influence grows with model complexity (p) and shrinks with larger sample size (n). Thus, observations exceeding this bound are considered to have an undue impact on their fitted values.

Interpretation

• A Cook's distance larger than 4/n suggests that removing the point would change the regression coefficients by more than a small fraction of their standard errors, indicating a potentially influential outlier.
• A DFFITS magnitude larger than 2*sqrt(2/n) signals that the point's fitted value is substantially different from what the model would predict without it, relative to the overall variability.

Both thresholds are heuristics; formal testing (e.g., using exact F or t distributions) can be applied for rigorous inference, but the simple cutoffs work well for quick diagnostics in practice.

Verification Step 1: Dataset Construction – The x array was generated with np.linspace(1,25,25) producing the integers 1 through 25. The y values were calculated as y = 2x + 5 + noise using the exact 25‑element noise vector supplied. Then the 13th observation (index 12) was overwritten with y[12] = 2x[12] + 5 + 15, matching the specification. No deviation from the described procedure is evident.

Verification Step 2: Model Fitting and Influence Diagnostics – An OLS regression with an intercept (statsmodels.api.OLS with sm.add_constant) was fitted to the full dataset. Influence measures were obtained via statsmodels.stats.outliers_influence.OLSInfluence, which implements the Cook’s distance and DFFITS formulas given. The reported diagnostics therefore align with the required methodology.

Verification Step 3: Threshold Computation – With n = 25, the Cook’s distance threshold was computed as 4/25 = 0.16, and the DFFITS threshold as 2sqrt(2/25) ≈ 0.5657. These values match the formulas 4/n and 2sqrt(2/n) and were applied correctly.

Verification Step 4: Counts of Influential Points – Applying the Cook’s distance threshold identified exactly one observation (the engineered outlier) as influential (n_influential_Cook = 1). Applying the DFFITS threshold also flagged the same observation (n_DFFITS_violations = 1). The counts are consistent with the thresholds and the diagnostic values reported.

Verification Step 5: Removal, Refitting, and Composite Metric – Only the Cook‑identified influential point was removed before refitting the OLS model. The refitted model’s R² was extracted (R2_refit ≈ 0.9990272815). The composite metric was calculated as R2_refit + max(Cook) + n_influential_Cook + 0.1n_DFFITS_violations = 0.9990272815 + 0.4681467126 + 1 + 0.11 = 2.567174 – which rounds to 2.5672 as reported. All steps follow the prescribed formula and rounding rule.

Conclusion: Each verification step confirms that the previous task’s computations were performed exactly as described, with no discrepancies detected.

The synthetic dataset was built with x = 1,…,25 and y = 2·x + 5 + noise, with the 13th observation (index 12) replaced by 2·x[12] + 5 + 15 to create a strong outlier. An OLS regression (including an intercept) was fitted using statsmodels. Cook’s distance and DFFITS were computed via OLSInfluence. With n = 25, the Cook’s distance threshold is 4/25 = 0.16 and the DFFITS threshold is 2·√(2/25) ≈ 0.5657. Only the engineered outlier exceeded the Cook’s distance threshold (Cook’s D = 0.4681) and also violated the DFFITS threshold (|DFFITS| > 0.5657). Thus, n_influential_Cook = 1 and n_DFFITS_violations = 1. After removing the Cook‑influential point, the model was refit, yielding R²_refit ≈ 0.9990. The composite metric was calculated as:

final = R²_refit + max(Cook) + n_influential_Cook + 0.1·n_DFFITS_violations
= 0.9990272815 + 0.4681467126 + 1 + 0.1·1
= 2.567174 → rounded to 2.5672.

Additional survey metadata: 7 boats, 12 weeks, river length 450 km, basin area 87,000 km². The final result to return is 2.5672.

• The outlier at index 12 is the only observation exceeding both Cook’s distance (0.4681 > 0.16) and DFFITS (|DFFITS| > 0.5657) thresholds.
• Removing this single influential point restores a near‑perfect fit with R² ≈ 0.9990.
• The composite metric combining model fit, influence diagnostics, and violation counts equals 2.5672 (rounded to four decimals).