Real Data Case Studies

Regression Diagnostics

Should You Trust That Table?

The previous lesson's naive regression said marketing spend has no measurable effect on sales (p≈0.96) — but it ignored a documented marketing→sales lag and never checked whether OLS's assumptions hold. This lesson reruns the regression with the lagged predictor (which flips the marketing result to significant), then runs lmda regdiag's diagnostics on that same lagged model — and finds a residual-autocorrelation problem that means even the "corrected" result shouldn't be trusted at face value either.

Correcting the model: use the lagged predictor

data/sales.csv (same 156-week retail dataset as the previous lesson) also carries marketing_lag2 — marketing spend shifted 2 weeks earlier, matching the delay the dataset's design documents between spend and its effect on sales. Swapping marketing for marketing_lag2:

lmda regress data/sales.csv --y units_sold --x price,marketing_lag2,holiday --format json

{
  "adjusted_r_squared": 0.7837039477843158,
  "coefficients": [
    {
      "estimate": 1595.4813610748583,
      "p_value": 1.2487294651448963e-10,
      "std_error": 230.90216821705334,
      "t_statistic": 6.909772105626437,
      "term": "intercept"
    },
    {
      "estimate": -18.01801664577129,
      "p_value": 0.03913916340141923,
      "std_error": 8.659594922541006,
      "t_statistic": -2.080699710198941,
      "term": "price"
    },
    {
      "estimate": 0.20788142385582084,
      "p_value": 0.04522991766503192,
      "std_error": 0.10295517026688451,
      "t_statistic": 2.0191450639821418,
      "term": "marketing_lag2"
    },
    {
      "estimate": 396.8395821484956,
      "p_value": 2.3871191779506156e-34,
      "std_error": 24.83716128981042,
      "t_statistic": 15.97765451204366,
      "term": "holiday"
    }
  ],
  "dropped": 0,
  "method": "ols",
  "n": 156,
  "r_squared": 0.7878903229884904,
  "x": [
    "price",
    "marketing_lag2",
    "holiday"
  ],
  "y": "units_sold"
}

With the lag, marketing_lag2's coefficient flips from the previous lesson's p≈0.96 to p≈0.045 — significant at the conventional 0.05 threshold. Lagging really did change the naive conclusion: this model finds marketing spend from 2 weeks earlier does predict sales. Before trusting that p≈0.045, run the same model through diagnostics.

Diagnosing the lagged model

lmda regdiag data/sales.csv --y units_sold --x price,marketing_lag2,holiday --format json

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  "dropped": 0,
  "durbin_watson": 0.6776686876158092,
  "heteroskedasticity": null,
  "n": 156,
  "r_squared": 0.7878903229884904,
  "reset": null,
  "residual_degrees_of_freedom": 152,
  "residual_standard_error": 68.5540684346531,
  "robust": null,
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  "studentization": "internal",
  "vif": [
    {
      "predictor": "price",
      "vif": 1.0793450019089978
    },
    {
      "predictor": "marketing_lag2",
      "vif": 1.8526576539594635
    },
    {
      "predictor": "holiday",
      "vif": 1.779603376774197
    }
  ],
  "x": [
    "price",
    "marketing_lag2",
    "holiday"
  ],
  "y": "units_sold"
}

Reading the diagnostics

Multicollinearity (VIF): all 3 predictors have variance inflation factors under 2 (well below the usual concern threshold of 5-10) — price, marketing_lag2, and holiday are not dangerously entangled with each other, so their individual coefficient estimates are trustworthy on that front. Influence (Cook's distance): most rows have Cook's distance well under 0.1; one row (week 88, a non-holiday week) stands out at ≈0.068 — driven by a large residual (studentized residual ≈-4.91) on below-average leverage (≈0.011 vs. the ≈0.026 average for 3 predictors), not by an unusual combination of predictor values. With no single row extreme enough to be dictating the fit by itself, no single week is silently driving the whole regression. Residual autocorrelation (Durbin-Watson): this is the real finding. durbin_watson is the statistic to check here — a value near 2.0 means residuals are independent; values close to 0 mean strong positive autocorrelation (each week's error resembles the previous week's). The reported value is well below 1, deep in positive-autocorrelation territory. That matches the dataset's documented yearly seasonal wave and trend, which this model's 3 predictors don't account for at all — the regression is leaving a systematic, week-to-week-correlated pattern in its residuals.

Honesty

price (p≈0.039) and marketing_lag2 (p≈0.045) are real, checked p-values from a real fit on n=156 realistically simulated weeks, both barely under the conventional 0.05 threshold — but the Durbin-Watson result means those p-values should not be taken at face value. OLS's standard-error formula assumes independent residuals; strong positive autocorrelation like this typically makes reported standard errors too small, which can make an estimate look more statistically significant than it really is. A p-value that only just clears 0.05 is exactly the kind of result a modest amount of standard-error deflation could flip back to non-significant. So the honest conclusion here is not "lagging fixed it, marketing has a confirmed real effect" but "lagging changed the naive answer, but this model is still misspecified (it's missing a seasonal/trend component) and needs to be fixed — with something like decompose on the residuals or a time-series-aware model — before either lesson's p-values can be trusted." That a model can go from clearly non-significant to marginally significant just by adding a lag, while its diagnostics quietly disqualify that same p-value, is the point of this lesson: diagnostics are not optional after a regression, they're part of it. As with the rest of this chapter, sales.csv is a realistic simulated retail dataset (lmda's own case-study-4 worked example, seeded RNG), not an actual store's live data — the fits and diagnostics are real, the store is not.

Implementation notes

  • Both commands above are re-run against the committed CSV on every validate_source.py pass and must byte-for-byte match the checked-in outputs/regress-sales-lagged.json and outputs/regdiag-sales-lagged.json.
  • regdiag reports one row per observation (fitted value, residual, leverage, standardized/studentized residual, Cook's distance) plus the aggregate fields (r_squared, durbin_watson, vif) read above — shown in full, not excerpted, so nothing is cherry-picked out of the real output.
  • Cross-reference: corr-regress-marketing-sales (this chapter) for the naive unlagged model this corrects; residuals-and-sse (ch08) for the residual concept this diagnostic extends to a real multi-predictor fit.