Polynomial regression and splines

How does an outcome change with one or more predictors after specifying a defensible mean and error structure? This note turns that question into a practical analysis workflow.

Core idea

Model smooth nonlinear relationships without forcing a single straight-line effect. The method is useful only when its target, data-generating assumptions, and unit of analysis match the scientific question.

$E[Y\mid X]=\beta_0+\sum_k\beta_kB_k(X)$

The equation is a compact statement of the target; it is not a substitute for checking design and data quality. State what each observation represents, how it entered the sample, and which sources of dependence remain.

Practical workflow

1. Write the scientific question and estimand in one sentence.
2. Identify biological units, technical replicates, nesting, repeated measurements, and exclusions.
3. Visualise raw observations and group structure before fitting the method.
4. Check assumptions using design knowledge, diagnostic plots, and sensitivity analyses.
5. Report the estimate, uncertainty, effect magnitude, sample sizes at every level, and limitations.

import numpy as np
from sklearn.linear_model import LinearRegression
from sklearn.metrics import mean_absolute_error, r2_score

X = np.array([[1], [2], [3], [4], [5]])
y = np.array([1.2, 1.8, 3.1, 3.9, 5.2])
model = LinearRegression().fit(X, y)
pred = model.predict(X)
print(model.coef_, mean_absolute_error(y, pred), r2_score(y, pred))

Interpretation and failure modes

Do not read a software output as an automatic scientific conclusion. Ask whether dependence was modeled, whether preprocessing used information from validation data, whether missingness or selection is informative, and whether the reported uncertainty covers every level of sampling. Prefer estimates and intervals to threshold-only language.

Microscopy case study

A spline can represent rapid early growth followed by a plateau while retaining a continuous fitted curve.

Connections

Use the tags above to continue to notes on design, assumptions, uncertainty, diagnostics, and complementary methods. A robust analysis normally combines several concepts rather than selecting one test in isolation.

Continue through the concepts

Related notes

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Logistic regression

Model binary-outcome probabilities through log odds and interpret coefficients on odds or probability scales.

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How random effects let a model respect that cells nested in wells nested in experiments aren't independent, without throwing away the structure by averaging it all flat.