Advanced Category: Statistics for Improvement • ~9 min read

Linear & Multiple Regression Analysis

Model transfer functions relating key process input variables (KPIVs) to key process output variables (KPOVs).

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How does multiple linear regression predict process responses from control factors?

Multiple regression fits Y = β₀ + β₁X₁ + ... + β_k X_k + ε using orthogonal Householder QR factorization. It calculates coefficient significance (t-tests, p-values), overall model significance (ANOVA F-test), and coefficient of determination R² and adjusted R².
β̂ = (XᵀX)⁻¹ Xᵀ y via Householder QR Back-Substitution R β̂ = Qᵀ y

Multiple Linear Regression via Householder QR

Data Matrix (Each row: X1, X2, ..., Y [Dependent Variable in last column])

R-Squared (R²)

99.29%

Adj R²: 99.01%

Residual Std Error

2.049

df = 5

F-Statistic

351.29

p = < 0.001

Model Status

Model Significant

Parameter Estimates

Term	Coefficient (β)	Std Error	t-Stat	p-Value
Intercept (β₀)	67.8823	8.3696	8.111	< 0.001
X1	318255199258620.8750	355702209453149.0000	0.895	0.4248
X2	-63651039851722.4844	71140441890629.7656	-0.895	0.4248

Predictive Modeling via Orthogonal Householder QR

Using Householder reflection matrices ensures numerical stability even in the presence of multicollinearity, avoiding the severe catastrophic cancellation errors that plague naive normal equation matrix inversions.

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Multiple Linear Regression via Householder QR

Parameter Estimates

Predictive Modeling via Orthogonal Householder QR

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