import numpy as np
import pandas as pd
import statsmodels.formula.api as smf

df = pd.read_csv("dataset.csv")
formula = (
    "G3 ~ G1 + G2 + studytime + failures + absences + age "
    "+ C(school) + C(sex) + C(address)"
)

def pinball_loss(actual, predicted, tau):
    residual = actual - predicted
    return np.where(
        residual >= 0,
        tau * residual,
        (1 - tau) * (-residual)
    )

rows = []
predictions = pd.DataFrame({"row": np.arange(1, len(df) + 1), "observed": df["G3"]})

for tau in (0.25, 0.50, 0.75):
    fit = smf.quantreg(formula, data=df).fit(q=tau, max_iter=10000)
    fitted = fit.predict(df)
    loss = pinball_loss(df["G3"].to_numpy(), fitted.to_numpy(), tau)

    null_quantile = df["G3"].quantile(tau, interpolation="linear")
    null_pred = np.repeat(null_quantile, len(df))
    null_loss = pinball_loss(df["G3"].to_numpy(), null_pred, tau).sum()

    predictions[f"q{int(tau * 100)}"] = fitted
    predictions[f"loss_q{int(tau * 100)}"] = loss

    rows.append({
        "tau": tau,
        "n": int(fit.nobs),
        "pinball_loss": float(loss.sum()),
        "null_loss": float(null_loss),
        "pseudo_r_squared": 1 - float(loss.sum()) / float(null_loss),
        "iterations": fit.iterations,
    })

    fit.summary2().tables[1].to_csv(
        f"quantile_coefficients_tau_{tau:.2f}.csv"
    )

pd.DataFrame(rows).to_csv("quantile_fit_summary.csv", index=False)
predictions.to_csv("quantile_predictions.csv", index=False)

library(quantreg)

dat <- read.csv("dataset.csv")
dat$school <- factor(dat$school)
dat$sex <- factor(dat$sex)
dat$address <- factor(dat$address)

form <- G3 ~ G1 + G2 + studytime + failures + absences + age +
  school + sex + address

taus <- c(0.25, 0.50, 0.75)
fits <- lapply(taus, function(tau) {
  rq(form, tau = tau, data = dat, method = "br")
})

pinball <- function(actual, predicted, tau) {
  u <- actual - predicted
  ifelse(u >= 0, tau * u, (1 - tau) * (-u))
}

summary_rows <- lapply(seq_along(taus), function(i) {
  tau <- taus[i]
  fit <- fits[[i]]
  pred <- as.numeric(predict(fit, newdata = dat))
  fitted_loss <- sum(pinball(dat$G3, pred, tau))
  null_value <- as.numeric(quantile(dat$G3, probs = tau, type = 7))
  null_loss <- sum(pinball(dat$G3, rep(null_value, nrow(dat)), tau))

  data.frame(
    tau = tau,
    n = nrow(dat),
    pinball_loss = fitted_loss,
    null_loss = null_loss,
    pseudo_r_squared = 1 - fitted_loss / null_loss
  )
})

write.csv(do.call(rbind, summary_rows),
          "quantile_r_fit_summary.csv", row.names = FALSE)

for (i in seq_along(taus)) {
  write.csv(
    as.data.frame(summary(fits[[i]], se = "nid")$coefficients),
    sprintf("quantile_r_coefficients_tau_%.2f.csv", taus[i])
  )
}

* Quantile Regression command availability depends on SPSS version.
* Confirm the installed procedure and preserve the tau label.

* Example outline when QUANTILE REGRESSION is available.
QUANTILE REGRESSION G3 WITH
 G1 G2 studytime failures absences age
 BY school sex address
 /QUANTILE=0.25 0.50 0.75
 /PRINT=PARAMETER COVB
 /CRITERIA=CILEVEL(95).

* Export fitted quantiles and calculate pinball loss separately.
* Do not substitute ordinary linear regression coefficients
* for a requested conditional quantile model.

Excel quantile-regression audit

1. Residual for one tau
=ObservedCell-PredictedQuantileCell

2. Pinball loss
=IF(ResidualCell>=0,
    TauCell*ResidualCell,
    (1-TauCell)*ABS(ResidualCell))

3. Total fitted loss
=SUM(PinballLossRange)

4. Null fitted quantile
=PERCENTILE.INC(ObservedRange,TauCell)

5. Null loss
Apply the same pinball formula using the null quantile
as the predicted value for every row.

6. Pseudo R-squared
=1-(FittedLossCell/NullLossCell)

7. Crossing flag
=IF(OR(Q25Cell>Q50Cell,Q50Cell>Q75Cell),
    "Crossing","Ordered")

Excel can audit predictions and losses. Estimating all coefficients
requires Solver or imported software results and must preserve the
factor coding and tau-specific objective.

Quantile Regression estimates a percentile of the outcome distribution after conditioning on the predictor pattern. The target at tau 0.25 is therefore the lower conditional quartile for each covariate profile, not the quartile calculated once from all 649 observed grades.

Report the tau label beside every equation and compare this conditional target with the unconditional definitions in Percentiles and Quartiles. Omitting tau removes the identity of the parameter.

All 649 observations contribute to each objective. The lower model is not estimated from the lowest 25 percent of students, and the upper model is not estimated from the highest 25 percent.

The asymmetric weights determine the fitted location. Subgroup restriction would answer a different, selection-biased question and would not reproduce the supplied coefficients or pinball losses.

The check-loss function assigns different weights to positive and negative residuals. At tau 0.25, an overprediction receives weight 0.75 and an underprediction receives weight 0.25. The pattern reverses at tau 0.75.

At tau 0.50 the two weights are equal, so Quantile Regression minimizes one-half of the total absolute error. The scaling constant does not change the coefficient minimizer.

The reported pseudo R-squared is one minus fitted pinball loss divided by intercept-only pinball loss at the same tau. It measures improvement in the selected asymmetric loss.

It is not the variance decomposition used by Ordinary Least Squares Regression. The values 0.686417, 0.685453 and 0.715500 must not be described as percentages of variance explained.

G3 is bounded and integer-valued. Tied outcomes create many residuals that can be exactly zero at a candidate solution and can produce flat portions of the piecewise-linear objective.

This geometry helps explain the extremely simple Python lower- and upper-quartile coefficient vectors. It also makes optimizer sensitivity and crossing diagnostics more important than they would be with a smooth continuous response.

A convex objective can have more than one minimizing coefficient vector when the design and tied outcomes create a flat optimum. Numerical convergence therefore does not guarantee a unique scientific allocation among correlated predictors.

Keep each coefficient vector attached to its own software, objective value, inference method and fitted predictions. Do not average Python and R estimates to manufacture a solution that neither program fitted.

G1 and G2 contain strongly overlapping grade-progression information. At the lower and upper Python optima, G2 receives a slope near one while G1 is near zero; the median fit distributes information across both.

This redistribution is conditional and model-dependent. Review the same design from a mean-model perspective in Multiple Linear Regression without treating one model as a significance test for the other.

Independent fitting does not enforce q25 less than or equal to q50 less than or equal to q75. The supplied prediction ledger has 65 q25-above-q50 rows and 15 q50-above-q75 rows, for 80 observations with at least one ordering violation.

Crossing does not erase the fitted objectives, but it prevents the three equations from being advertised as an automatically coherent interval. Constrained estimation or post-estimation rearrangement would be a new analysis.

A fitted conditional tau quantile should place the outcome below and above the surface in proportions compatible with tau, subject to sampling variation and ties. Integer outcomes can produce a large equality group.

Report counts below, equal to and above the fitted value rather than forcing a continuous-response calibration rule. Equality is substantive information in this grade dataset.

Quantile standard errors depend on how the conditional density or sparsity near the target quantile is estimated. Tied outcomes can make asymptotic estimates sensitive to bandwidth and covariance choices.

The interval must remain attached to the corresponding coefficient solution. General interpretation of significance can be reviewed in Regression P Value, but OLS standard errors must never be copied onto Quantile Regression slopes.

The Python fits report 48, 506 and 40 iterations at tau 0.25, 0.50 and 0.75. Iteration count describes the numerical path, not predictive quality or scientific importance.

Preserve the software version, tolerance, maximum iterations and convergence message. A successful status should be accompanied by an independent recomputation of fitted pinball loss.

The R workflow is an independent cross-check with its own optimization route and standard-error implementation. Its median loss of 245.068163 is close to the Python value 244.875001 but not identical.

Close objectives support similar overall fit, while coefficient differences disclose numerical or identification sensitivity. The R result should not be relabeled as a replication of every Python decimal.

Raw pinball losses at different tau values use different residual weights and different intercept-only benchmarks. The smallest raw loss does not identify the best quantile model.

Compare fitted and null loss within the same tau, then interpret pseudo R-squared and calibration. Cross-tau model choice is primarily a scientific-target decision, not a competition for one minimum number.

OLS remains useful because it estimates the conditional mean from the same predictors. The supplied OLS R-squared of 0.852222 and the quantile pseudo R-squared values use different objectives and denominators.

Use Ordinary Least Squares Regression to understand the mean equation. Do not rank it against a quantile fit by comparing the two R-squared labels directly.

Median Quantile Regression estimates the conditional 50th percentile. Robust mean procedures may reduce the impact of large residuals while continuing to estimate a conditional mean.

The estimands remain different even when both appear less sensitive to outliers than OLS. See Robust Regression for the mean-focused alternative.

The current equations are linear in the supplied design columns. Curvature in G1, G2, age or absences would require polynomial terms, splines or another nonlinear quantile specification.

A Generalized Additive Model illustrates flexible conditional-mean modeling. Flexible quantiles require their own validated basis functions and cannot be inferred from the current linear fits.

Changing spread is one reason to examine multiple conditional locations. Different slopes across tau can describe distributional changes that one mean equation and one residual-variance summary cannot show.

A White Test evaluates heteroscedasticity in an OLS comparison, whereas Quantile Regression directly estimates selected conditional locations.

The supplied analysis treats rows as independent. When observations are clustered by classroom, school, family or repeated measurement, the coefficient target and uncertainty calculation must reflect that dependence.

Cluster-robust covariance or multilevel quantile methods may be needed. A large row count does not compensate for an incorrectly specified sampling unit.

Quantile Regression does not square residuals, but high-leverage design points can still alter a fitted conditional surface substantially. Influence can vary with tau.

Review Influence Diagnostics and compare fitted equations after justified sensitivity checks. Do not delete observations solely because they change a coefficient.

The worked analysis uses 649 complete rows. Complete-case estimation is transparent, but its validity depends on why other records are missing and whether the retained sample represents the intended population.

Any imputation model must be fitted without leaking validation outcomes and should preserve the bounded, discrete grade structure. Quantile Regression is not itself a missing-data method.

School, sex and address are treatment-coded. Their coefficients compare one category with the declared reference while every numeric predictor is held fixed.

Changing a reference category changes the intercept and displayed contrasts but not fitted values when the same column space is preserved. Every report should state GP, female and rural as the references used here.

The distance between fitted q25 and q75 describes an estimated conditional interquartile span. It is not automatically a 50 percent prediction interval with guaranteed coverage.

Crossing, estimation uncertainty and model misspecification affect coverage. The descriptive Interquartile Range guide provides context but does not replace validation of conditional curves.

The supplied metrics are in-sample optimization summaries. Deployment claims require a protected validation design that fits the complete preprocessing and coefficient estimation process on training data.

Evaluate tau-specific pinball loss, calibration, crossing and subgroup behavior on unseen observations. New-data validation is more informative than reporting additional training-set decimals.

Tau 0.25, 0.50 and 0.75 have reasonable central support. More extreme levels such as 0.05 or 0.95 rely on fewer effective observations and can produce substantially greater uncertainty.

Extreme-tail analyses require larger samples, stronger diagnostics and often specialized methods. The current central-quartile evidence should not be extrapolated automatically.

A complete result names the software, estimator, optimization method, tau, standard-error method and factor coding. These details are part of the statistical result, not optional technical notes.

Python and R outputs should be presented side by side when they differ. SPSS and Excel should be described according to what was actually estimated or audited rather than assumed capability.

Choose Quantile Regression because the research question concerns a conditional percentile or changing distributional location. Prespecify tau, coding, functional form and inference method before comparing results.

Fit each model, verify the objective, inspect crossing and calibration, compare with Regression Diagnostics, and reserve substantive conclusions for patterns supported across estimates, uncertainty and validation.