Regression Modelling in Concrete and Material Science - CHAPTER FOUR Step-by-Step Derivations and Worked Examples for Multi-Predictor Systems

CHAPTER FOUR

Step-by-Step Derivations and Worked Examples for Multi-Predictor Systems

By Owus M. Ibearugbulem – Light for the Living

4.1 Overview

In Chapters 1–3, we built the conceptual and mathematical foundation of regression and introduced Ibearugbulem’s matrix-based framework. In this chapter, we apply that framework to real numerical examples, beginning with a simple two-predictor model and progressing through three-, four-, and five-predictor systems. By the end of this chapter, students will:

Understand how to set up and solve regression models with any number of predictors.
Learn to compute the model coefficients manually or in spreadsheet form.
Appreciate the practical use of Ibearugbulem’s approach in material mix prediction.

4.2 The Universal Regression Equation

For any number of predictors, the general regression model is expressed as:

$$Y = a_0 + a_1 X_1 + a_2 X_2 + a_3 X_3 + a_4 X_4 + a_5 X_5 + \cdots + a_k X_k \tag{4.1}$$

where $ k $ = number of predictors, and $ a_i $ = model coefficients. The coefficients are determined by Ibearugbulem’s matrix equation:

$$[a] = ([H]^T [H])^{-1} ([H]^T [Y]) \tag{4.2}$$

This equation is valid for all cases, from 2 to 5 predictors — only the matrix dimensions change.

4.3 Case 1: Two-Predictor System

4.3.1 Example Context

A simple cement–water paste mix is studied to predict compressive strength $Y$ based on the quantities of cement ($X_1$) and water ($X_2$).

Mix	X₁ (Cement kg/m³)	X₂ (Water kg/m³)	Y (Strength MPa)
1	410	185	35
2	380	175	31
3	360	170	28

4.3.2 Model Formulation

$$Y = a_0 + a_1 X_1 + a_2 X_2 \tag{4.3}$$

Design matrix:

$$[H] = \begin{bmatrix} 1 & 410 & 185 \\ 1 & 380 & 175 \\ 1 & 360 & 170 \end{bmatrix}, \quad [Y] = \begin{bmatrix} 35 \\ 31 \\ 28 \end{bmatrix}$$

Compute: $$[CC] = [H]^T [H]; \quad [R] = [H]^T [Y]$$ $$[a] = [CC]^{-1} [R]$$

After computation (using spreadsheet or calculator):

$$a_0 = -10; \quad a_1 = 0.20; \quad a_2 = -0.20$$

Thus,

$$Y = -10 + 0.2X_1 - 0.2X_2 \tag{4.4}$$

4.3.3 Interpretation

$a_1 > 0$: Increasing cement increases strength.
$a_2 < 0$: Increasing water reduces strength.

This agrees with fundamental mix design principles.

EXPLANATORY BOX – “The See-Saw of Strength”
Imagine strength as a see-saw. Cement pushes one side up, water pushes the other down. Regression finds the exact weights on both sides that explain the balance seen in your experiments.

4.4 Case 2: Three-Predictor System

4.4.1 Example Context

Predicting mortar strength ($Y$) from cement ($X_1$), water ($X_2$), and sand ($X_3$).

Mix	X₁	X₂	X₃	Y
1	420	180	621	36.65
2	399	180	582	33.79
3	378	179	542	27.27
4	357	178	503	19.33
5	336	176	464	15.23

Model:

$$Y = a_0 + a_1 X_1 + a_2 X_2 + a_3 X_3 \tag{4.5}$$

Solving gives:

$$a_0 = 51.8325; \quad a_1 = -3.38571; \quad a_2 = 1.1775; \quad a_3 = 1.925$$

Hence:

$$Y = 51.8325 - 3.38571X_1 + 1.1775X_2 + 1.925X_3 \tag{4.6}$$

EXPLANATORY BOX – “The Three Cooks”
Cement, water, and sand are like three cooks preparing a meal. Each adds a unique flavour. Regression tells you who’s seasoning too much and who’s balancing the taste.

4.5 Case 3: Four-Predictor System

4.5.1 Example Context

Ordinary concrete with four main ingredients: Water ($X_1$), Cement ($X_2$), Sand ($X_3$), and Coarse Aggregate ($X_4$).

Model:

$$Y = a_0 + a_1 X_1 + a_2 X_2 + a_3 X_3 + a_4 X_4 \tag{4.7}$$

Computation yields:

$$a_0 = 1914.221; \; a_1 = -0.58841; \; a_2 = -0.86673; \; a_3 = -0.80222; \; a_4 = -0.80222$$

Therefore:

$$Y = 1914.221 - 0.58841X_1 - 0.86673X_2 - 0.80222X_3 - 0.80222X_4 \tag{4.8}$$

EXPLANATORY BOX – “Balancing the Four Drummers”
Think of each predictor as a drummer. If they all play too loudly (too much material), the song (the concrete) becomes noisy. Regression finds the right rhythm that keeps the mix in harmony.

4.6 Case 4: Five-Predictor System

Extended concrete system including admixture or pozzolan (e.g., fly ash or silica fume). Predictors: Water ($X_1$), Cement ($X_2$), Sand ($X_3$), Coarse Aggregate ($X_4$), Admixture ($X_5$).

$$Y = a_0 + a_1 X_1 + a_2 X_2 + a_3 X_3 + a_4 X_4 + a_5 X_5 \tag{4.9}$$

Spreadsheet calculation gives:

$$a_0 = -7616; \; a_1 = 6; \; a_2 = 3.6641; \; a_3 = 2.4375; \; a_4 = 3.627; \; a_5 = 0.625$$

Hence:

$$Y = -7616 + 6X_1 + 3.6641X_2 + 2.4375X_3 + 3.627X_4 + 0.625X_5 \tag{4.10}$$

EXPLANATORY BOX – “Five Cooks in the Kitchen”
When five people are cooking, too many hands can spoil the soup. Regression helps identify who adds value and who adds confusion — keeping only the right balance of flavours.

4.7 Validation Snapshot

For all systems, validation follows statistical tools:

$$R^2 = 1 - \frac{SSR}{TSS}; \quad SER = \sqrt{\frac{SSR}{n - k - 1}}; \quad F = \frac{MSR}{MSE}$$

A good model typically shows $R^2 > 0.9$ and $F_{computed} > F_{critical}$.

4.8 Summary Table of Model Forms

Predictors	Equation	Example
2	Y = a₀ + a₁X₁ + a₂X₂	Cement–Water Paste
3	Y = a₀ + a₁X₁ + a₂X₂ + a₃X₃	Cement–Water–Sand Mortar
4	Y = a₀ + a₁X₁ + a₂X₂ + a₃X₃ + a₄X₄	Concrete (no admixture)
5	Y = a₀ + a₁X₁ + a₂X₂ + a₃X₃ + a₄X₄ + a₅X₅	Concrete (with admixture)

4.9 Chapter Summary

Ibearugbulem’s regression model works seamlessly for 2–5 predictors.
All cases use the same matrix equation — only matrix size changes.
The coefficients are physically meaningful and easily interpretable.
Model accuracy should always be verified using $R^2$, SER, and F-test.
The method empowers students and engineers to construct reliable predictive models from small datasets.

Test Your Understanding (Chapter 4)

Write the general regression equation for five predictors.
What is the significance of the term $[H]^T [H]$ in regression computation?
In the four-predictor example, why are most coefficients negative?
Describe in your own words how Ibearugbulem’s approach differs from Scheffé’s mixture model.
EXPLANATORY Challenge: If you were designing the “perfect soup,” which ingredients would represent predictors that increase or decrease taste ($Y$)?

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