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

df = pd.read_csv("dataset.csv")
df["G2_squared"] = df["G2"] ** 2
df["G2_cubed"] = df["G2"] ** 3

linear_model = smf.ols(
    "G3 ~ G2 + G1 + studytime + failures + C(school) + C(sex)",
    data=df
).fit()

quadratic_model = smf.ols(
    "G3 ~ G2 + G2_squared + G1 + studytime + failures + C(school) + C(sex)",
    data=df
).fit()

cubic_model = smf.ols(
    "G3 ~ G2 + G2_squared + G2_cubed + G1 + studytime + failures + C(school) + C(sex)",
    data=df
).fit()

import numpy as np
from scipy.optimize import curve_fit

def saturation_curve(x, a, b, c):
    return a + b * (1 - np.exp(-c * x))

params, covariance = curve_fit(
    saturation_curve,
    df["G2"].to_numpy(),
    df["G3"].to_numpy(),
    p0=[-0.5, 100, 0.2],
    maxfev=10000
)

df <- read.csv("dataset.csv", stringsAsFactors = TRUE)
df$G2_squared <- df$G2^2
df$G2_cubed <- df$G2^3

linear_model <- lm(
  G3 ~ G2 + G1 + studytime + failures + school + sex,
  data = df
)

quadratic_model <- lm(
  G3 ~ G2 + G2_squared + G1 + studytime + failures + school + sex,
  data = df
)

cubic_model <- lm(
  G3 ~ G2 + G2_squared + G2_cubed + G1 + studytime + failures + school + sex,
  data = df
)

AIC(linear_model, quadratic_model, cubic_model)

COMPUTE G2_squared = G2 ** 2.
COMPUTE G2_cubed = G2 ** 3.
EXECUTE.

REGRESSION
 /MISSING LISTWISE
 /STATISTICS COEFF OUTS R ANOVA COLLIN CI(95)
 /DEPENDENT G3
 /METHOD=ENTER G2 G2_squared G1 studytime failures school_MS sex_M
 /SAVE PRED(Predicted_G3) RESID(Residual_G3).

OUTPUT SAVE
 /OUTFILE='D:\DATA ANALYSIS\H Regression Tests and Models\Nonlinear Regression\SPSS_Output\spv\Nonlinear-Regression.spv'.

OUTPUT EXPORT
 /CONTENTS EXPORT=ALL LAYERS=PRINTSETTING MODELVIEWS=PRINTSETTING
 /PDF DOCUMENTFILE='D:\DATA ANALYSIS\H Regression Tests and Models\Nonlinear Regression\SPSS_Output\pdf\Nonlinear-Regression-SPSS-Output.pdf'.

G2_squared:
=G2^2

Predicted_G3:
=-0.9595220668
 +(1.0780376607*G2)
 +(-0.0101070485*G2_squared)
 +(0.1662554310*G1)
 +(0.0626907228*studytime)
 +(-0.1589263026*failures)
 +(-0.1732881105*school_MS)
 +(-0.2181066706*sex_M)

Residual:
=Observed_G3-Predicted_G3

Squared_Residual:
=Residual^2

Absolute_Residual:
=ABS(Residual)

• A coefficient table can feel abstract to a new student.
• A worked G2 = 12 prediction shows where every number enters.
• The calculation connects the equation with the final fitted value.

• A simpler equation is easier to reproduce, audit and explain.
• Readers can calculate one prediction without specialized nonlinear software.
• Parsimony strengthens transparency when predictive performance is nearly equal.

• It represents a theoretically attractive leveling process.
• Its fit is weaker in this worked comparison.
• A different dataset could favor saturation over a quadratic polynomial.

• Coefficient confidence intervals describe uncertainty in the polynomial terms.
• A confidence band describes uncertainty in the mean fitted curve.
• A prediction interval is wider because it includes individual residual variation.

• The squared term is a statistical representation of shape.
• It does not identify a biological or educational mechanism by itself.
• Theory is still needed to explain why the relationship flattens.

• The workbook exposes the transformed term and every coefficient contribution.
• Students can change G2 and see how the squared penalty becomes larger.
• This directly demonstrates why the slope changes.

• Adjusted R² and AIC comparisons are meaningful only when models use the same outcome and observations.
• Missing values can silently change the analyzed sample.
• Confirm n = 649 for every candidate model.

• Curvature can only be estimated where observations exist.
• Sparse values near the edges make the curve less stable.
• Always inspect the distribution of G2 before interpreting the fitted shape.

• A small average improvement may matter when predictions are used repeatedly.
• However, the article should not exaggerate a change of only a few hundredths in RMSE.
• State both statistical support and practical size.

• Imagine walking up a hill that becomes gradually flatter.
• The direction remains upward, but each step adds less height.
• That is the practical meaning of the positive G2 term and negative G2² term.

• A straight-line model assumes that one predictor unit has the same expected outcome effect everywhere.
• The curved model allows the slope to become stronger, weaker or even change direction.
• The curve should be supported by theory, visualization and diagnostics.

• Quadratic and cubic equations are nonlinear in predictor shape but linear in the estimated coefficients.
• They can therefore be fitted with ordinary least squares after transformed terms are created.
• This makes them easy to reproduce in Python, R, SPSS and Excel.

• Saturation, exponential and logistic growth curves estimate parameters through nonlinear optimization.
• Starting values and convergence become important.
• Parameter meaning often follows the scientific process more directly than a polynomial.

• A positive G2 term with a negative squared term creates diminishing returns.
• The outcome keeps increasing over the observed range, but the increase becomes less steep.
• The shape is substantively different from a simple constant slope.

• For y = b1x + b2x², the instantaneous slope is b1 + 2b2x.
• Here the slope equals 1.0780 – 0.0202141·G2.
• The slope is therefore larger at lower G2 values and smaller at higher values.

• The mathematical turning point is -b1/(2b2).
• With the reported coefficients it lies above the observed G2 range.
• The observed data therefore show flattening rather than a downward reversal.

• Centering G2 before creating G2² reduces the correlation between first- and second-order terms.
• It changes the intercept and first-order coefficient interpretation.
• It does not change fitted values or overall fit when implemented correctly.

• Orthogonal polynomials reduce numerical correlation among powers.
• Their coefficients are less directly interpretable in original units.
• Raw powers are often preferable when the article emphasizes transparent prediction.

• The additional cubic terms are not significant.
• Adjusted R² does not improve and AIC becomes worse.
• The simpler quadratic model therefore has stronger parsimony.

• When G2² is included, G2 should remain in the model.
• When G2³ is included, lower-order terms should also remain.
• Removing lower-order terms usually produces an awkward and unstable interpretation.

• Adjusted R² penalizes extra parameters.
• Read Adjusted R-Squared for the detailed formula.
• It is useful but should not be the only model-selection criterion.

• AIC estimates relative information loss.
• Lower AIC is preferred among comparable models fitted to the same outcome and rows.
• AIC differences should be interpreted with model purpose and scientific plausibility.

• The saturation curve uses a different direct nonlinear parameterization.
• Its reported AIC is on the fitted curve specification used in that analysis.
• The final worked choice is based on the transparent linear–quadratic–cubic comparison and coefficient evidence.

• RMSE gives large errors extra weight because residuals are squared.
• MAE describes the typical absolute prediction error more directly.
• The quadratic model improves both measures relative to the linear benchmark.

• Exact residual normality is most important for small-sample classical inference.
• Large samples can reveal small tail departures.
• Use plots, influence checks and robust sensitivity analysis together.

• Curvature and non-constant variance are different problems.
• A correct mean curve can still have unequal residual spread.
• Review Breusch-Pagan Test and White Test.

• A case can strongly influence the curve when its predictor value is unusual or its residual is large.
• Use Cook’s Distance and Influence Diagnostics.
• Investigate data quality and substantive context before excluding observations.

• A confidence band describes uncertainty in the mean curve.
• A prediction interval describes uncertainty for one new outcome.
• Prediction intervals are wider because they include residual variation.

• In-sample adjusted R² and error can be optimistic.
• Cross-validation estimates how the candidate curves perform on unseen rows.
• All model selection should occur within the training process.

• Higher-order powers can trace random fluctuations.
• The best in-sample curve may not be the best future predictor.
• Parsimony is especially important when the practical gain is small.

• A Generalized Additive Model estimates a smooth shape without fixing it to one polynomial.
• It is useful when the relationship is more complex than one bend.
• Its smoothness penalty controls overfitting.

• A Log Transformation, Square-Root Transformation or Box-Cox Transformation changes the scale.
• A polynomial term changes the mean relationship while retaining the outcome scale.
• Choose the approach that matches the diagnosed problem.

• A curved association is not automatically causal.
• Confounding, measurement error and selection can remain.
• Use causal language only when the design supports it.

• Grades can show ceilings, floors and diminishing returns.
• The example demonstrates how prior performance can predict final performance nonlinearly.
• The exact shape may differ across schools and cohorts.

• Treatment response often increases and then plateaus.
• See Clinical Trial Data Analysis Using R for a broader applied context.
• Mechanistic curve forms may be more meaningful than polynomials in those settings.

• Power for a curve term depends on sample size, residual variation and predictor range.
• Use Statistical Power when planning a new study.
• A narrow predictor range can make real curvature difficult to detect.

• A Type I error retains a false curve term.
• A Type II error misses genuine curvature.
• Read Type I and Type II Error for the inferential trade-off.

• A P-Value addresses evidence against a zero coefficient.
• It does not show prediction quality, practical magnitude or curve usefulness.
• Report confidence intervals, fit and plots as well.

• Use identical rows, transformations, references and formulas.
• Record whether powers were raw or centered.
• Export fitted values and residuals to verify cross-software agreement.

• Create G2_squared and G2_cubed with COMPUTE commands.
• Fit models through Linear Regression and compare model summaries.
• Direct saturation curves require Nonlinear Regression procedures or extensions.

• Create G2² with a cell formula.
• Use fixed coefficients to calculate row-level fitted values.
• Use the workbook to teach the equation, not to hide model-selection assumptions.

• Say that G3 rises with G2 and then gradually flattens.
• Do not describe the negative squared coefficient as a simple negative G2 effect.
• Connect the coefficient combination to the visible curve.