Choosing the Right Regression Analysis in Minitab

1. Learning objectives

By the end of this lesson, you should be able to:

Distinguish a Fitted Line Plot from a full multiple regression model.
Explain the difference between None, Stepwise, Forward Selection, Backward Elimination, Forward Information Criteria, and Forward Selection with Validation.
Explain how Best Subsets Regression searches the model space.
Understand why validation selects the best model size rather than the next term in Minitab’s forward-selection-with-validation procedure.
Recognize the difference between a curved linear model and a genuinely nonlinear model.
Choose a regression procedure based on the response type, number of predictors, purpose of the model, and engineering knowledge.

2. Regression foundations

2.1 What regression is trying to do

Regression relates a response variable, usually called Y, to one or more predictor variables, usually called X. A regression model can be used for explanation, prediction, process understanding, optimization, or screening variables.

Predicted response = systematic model component + unexplained random error

For a simple linear regression:

ŷ = b₀ + b₁X

For multiple regression:

ŷ = b₀ + b₁X₁ + b₂X₂ + ··· + bₖXₖ

Important: “Linear regression” does not necessarily mean the graph must be a straight line. It means the model is linear in its unknown coefficients.

2.2 Prediction versus explanation

Explanatory modeling

Focuses on effects, coefficient signs, confidence intervals, p-values, engineering interpretation, and causal caution.

Predictive modeling

Focuses on performance on new data, predicted R², validation RMSE, model stability, and avoiding overfitting.

2.3 One predictor versus many predictors

Situation	Typical Minitab tool	Main purpose
One continuous X and one continuous Y	Stat → Regression → Fitted Line Plot	Visualize and model one relationship
One or many predictors	Stat → Regression → Regression → Fit Regression Model	Fit a specified linear model
Many possible predictors	Stepwise methods or Best Subsets	Select a smaller model
Known nonlinear scientific equation	Stat → Regression → Nonlinear Regression	Estimate nonlinear parameters

3. Fitted Line Plot

A Fitted Line Plot is the simplest regression tool in Minitab. It answers: “How does one X variable relate to one Y variable?”

3.1 Minitab path

Stat → Regression → Fitted Line Plot

3.2 What it produces

A scatterplot of Y versus X
A fitted line or curve
The regression equation
R² and often adjusted R²
ANOVA and coefficient results
Residual plots when requested

3.3 Example

Suppose the response is Tensile Strength and the predictor is Cooling Rate.

Tensile Strength = 420 + 8.5(Cooling Rate)

The slope means that a 1°C/s increase in cooling rate is associated with an estimated 8.5 MPa increase in tensile strength, within the observed data range and under the assumptions of the model.

Do not interpret the intercept automatically. If a cooling rate of zero is outside the observed range or physically impossible, the intercept is mainly a mathematical anchor.

3.4 Straight line, quadratic, or cubic

Minitab’s Fitted Line Plot can also fit polynomial curves. A quadratic model is:

ŷ = b₀ + b₁X + b₂X²

Even though the graph is curved, this is still a linear regression model because the unknown coefficients b₀, b₁, and b₂ enter linearly.

3.5 When to use it

Exploring one predictor at a time
Teaching simple regression
Checking whether a relationship appears linear or curved
Showing a relationship visually to stakeholders

3.6 When not to use it

When several predictors jointly influence Y
When confounding is likely
When interactions matter
When the response is binary, ordinal, nominal, or a count

4. Fit Regression Model

Fit Regression Model is Minitab’s main workhorse for ordinary linear regression. It can fit one predictor or many predictors, continuous terms, categorical factors, interactions, and polynomial terms.

4.1 Minitab path

Stat → Regression → Regression → Fit Regression Model

4.2 What “None” means in the Stepwise dialog

When the stepwise method is set to None, Minitab fits exactly the model you specify. It does not add or remove variables automatically.

Use None when engineering knowledge, theory, a study protocol, or a designed experiment already defines the terms that must be in the model.

4.3 Example full model

Tensile Strength = β₀ + β₁(Carbon) + β₂(Rolling Temperature) + β₃(Cooling Rate) + β₄(Thickness) + β₅(Furnace Time) + β₆(Line Speed)

4.4 Interactions

An interaction means the effect of one predictor depends on the level of another predictor.

ŷ = b₀ + b₁Carbon + b₂Cooling + b₃(Carbon × Cooling)

If the interaction is significant, the effect of cooling rate should not be interpreted as one constant number for every carbon level.

4.5 Polynomial terms

Use polynomial terms when the response bends smoothly.

ŷ = b₀ + b₁Temperature + b₂Temperature²

A polynomial model belongs in Fit Regression Model, not necessarily Nonlinear Regression.

5. Variable-selection methods in the Stepwise dialog

The methods in Minitab’s Stepwise dialog differ in three fundamental ways: where the search starts, whether terms can enter or leave, and what rule controls the next move. The examples below use six candidate predictors—A, B, C, D, E, and F—so you can see exactly how each method proceeds.

Method	Starts with	Can enter?	Can leave?	Main selection rule	Typical use
None	User-defined model	No	No	No automatic selection	Predefined engineering model
Stepwise	Usually intercept or initial terms	Yes	Yes	Entry and removal p-values	Flexible screening
Forward Selection	Small model	Yes	No	Smallest qualifying p-value	Simple sequential selection
Backward Elimination	Full model	No	Yes	Largest nonqualifying p-value removed	Reduce a plausible full model
Forward Information Criteria	Small model	Yes	Typically no	AIC/AICc-type improvement	Fit-complexity balance
Forward Selection with Validation	Small model	Yes	No along forward path	Stepwise rule builds path; validation selects step	Prediction-focused model-size selection

Common setup for the examples: assume the candidate predictors are A, B, C, D, E, and F. For p-value-based methods, suppose the alpha-to-enter and alpha-to-remove thresholds are both 0.15 unless stated otherwise.

5.1 None — fit exactly the model you specify

With None, Minitab performs no variable-selection search. You decide the model before fitting it. If you place A, B, C, and the interaction A×B in the model, Minitab fits exactly that equation:

Y = β₀ + β₁A + β₂B + β₃C + β₄(A×B)

Minitab does not ask whether D, E, or F should enter, and it does not remove A, B, C, or A×B automatically.

When this is the correct choice

The model was specified by engineering theory or prior studies.
A designed experiment defines the required main effects and interactions.
You are validating a published or regulated model.
You need stable coefficient interpretation rather than exploratory screening.

Mental model: “I am telling Minitab what equation to fit.”

5.2 Forward Selection — variables enter one at a time and stay

Forward Selection begins with the intercept-only model:

Model 0: Y = β₀

Step 1 — test every possible first predictor

Minitab temporarily evaluates:

Y = β₀ + A
Y = β₀ + B
Y = β₀ + C
Y = β₀ + D
Y = β₀ + E
Y = β₀ + F

Suppose the candidate p-values are:

Candidate	Entry p-value
A	0.001
B	0.028
C	0.190
D	0.350
E	0.110
F	0.420

A has the smallest p-value and satisfies alpha-to-enter = 0.15, so A enters:

Model 1: Y = β₀ + A

Step 2 — test the remaining predictors conditional on A

Minitab evaluates A+B, A+C, A+D, A+E, and A+F. Suppose B has the smallest qualifying p-value.

Model 2: Y = β₀ + A + B

Step 3 — continue from the current model

Minitab tests A+B+C, A+B+D, A+B+E, and A+B+F. Suppose C enters.

Model 3: Y = β₀ + A + B + C

The sequence might continue:

A → A+B → A+B+C → A+B+C+D

Once A enters, ordinary Forward Selection does not remove it later, even if A becomes less useful after other terms are added. That is the method’s main limitation.

How it stops

The procedure stops when none of the remaining candidates has an entry p-value at or below alpha-to-enter.

Mental model: “Start empty, add the strongest qualifying term, and never go backward.”

5.3 Backward Elimination — start full and remove weak terms

Backward Elimination starts with all candidate predictors:

Model 0: Y = β₀ + A + B + C + D + E + F

Step 1 — inspect all included terms

Suppose the p-values in the full model are:

Included term	p-value
A	0.004
B	0.031
C	0.082
D	0.410
E	0.170
F	0.690

F has the largest p-value and exceeds alpha-to-remove = 0.15, so F is removed:

Model 1: Y = β₀ + A + B + C + D + E

Step 2 — refit before removing anything else

Minitab refits the reduced model because p-values can change when a correlated predictor is removed. Suppose D now has the largest p-value, 0.38. D is removed:

Model 2: Y = β₀ + A + B + C + E

Step 3 — continue until all remaining terms qualify

Suppose E now has p = 0.09 and all remaining terms are at or below 0.15. The procedure stops with A, B, C, and E.

Backward Elimination requires the full model to be estimable. If the sample is small relative to the number of terms, or if predictors are perfectly aliased, the method may be inappropriate.

Mental model: “Start with everything, remove the weakest term, refit, and repeat.”

5.4 Stepwise — variables can enter and later leave

Stepwise regression combines forward entry with backward checking. It begins like Forward Selection, but after each addition Minitab re-examines terms already in the model.

Step 1 — A enters

From the intercept-only model, A has the smallest qualifying p-value:

Model 1: Y = β₀ + A

Step 2 — B enters

Conditional on A, B has the smallest qualifying p-value:

Model 2: Y = β₀ + A + B

Step 3 — C enters

Model 3: Y = β₀ + A + B + C

Step 4 — Minitab checks whether A and B still deserve to stay

Suppose adding C reveals that B was mostly acting as a proxy for C. After refitting, the p-values are:

Included term	p-value after C enters
A	0.006
B	0.220
C	0.012

B now exceeds alpha-to-remove = 0.15, so B leaves:

Model 4: Y = β₀ + A + C

Step 5 — forward entry resumes

Minitab again evaluates the remaining candidates B, D, E, and F conditional on A and C.

A possible path is:

A → A+B → A+B+C → A+C → A+C+D

A term can enter, leave, and potentially re-enter later because its p-value depends on the other terms currently in the model.

Stepwise results can be unstable when predictors are highly correlated. Small changes in the data may change the selected path.

Mental model: “Move forward, but after every move, look backward and clean up.”

5.5 Forward Information Criteria — add terms using fit-versus-complexity

Forward Information Criteria also builds a model from small to large, but the next term is selected using an information criterion, such as AIC or AICc, rather than only a significance threshold.

Step 1 — compare one-predictor candidates

Candidate model	AICc
A	412.4
B	418.9
C	415.7
D	426.1
E	421.5
F	429.0

The lower information criterion is preferred, so A enters:

Model 1: Y = β₀ + A

Step 2 — compare additions to A

Candidate model	AICc
A+B	398.5
A+C	392.8
A+D	409.4
A+E	404.7
A+F	411.6

C enters because A+C gives the largest improvement, represented here by the lowest AICc.

Step 3 — stop when no addition improves the criterion enough

Suppose A+C+D has AICc = 391.9, while the current A+C model has AICc = 392.8. Depending on the exact rule and software implementation, D may enter because it improves the criterion.

A retained term may have an individual p-value above 0.05. That is not necessarily a contradiction: information criteria optimize the model’s fit-complexity balance, not a sequence of individual hypothesis tests.

Mental model: “Add the term that gives the best trade-off between improved fit and added complexity.”

6. Forward Selection with Validation

Critical correction: the next term is selected primarily by the forward stepwise significance rule. Validation is then used to identify the best model size or step. It is not a brute-force search that directly tests every possible equation and adds whichever term gives the largest validation improvement.

6.1 Two separate jobs

Job 1
Build a forward sequence using the stepwise entry rule

Job 2
Evaluate the completed models on validation data

Decision
Select the model size with the best validation performance

6.2 Job 1 — build the forward-selection path

Assume the potential predictors are A, B, C, D, E, and F. The starting model is:

Model 0: Y = β₀

Step 1 — choose the first term by the forward significance rule

Minitab temporarily evaluates six one-predictor models:

A, B, C, D, E, and F.

Candidate term	Entry p-value
A	0.001
B	0.028
C	0.190
D	0.350
E	0.110
F	0.420

A has the smallest qualifying p-value, so A enters:

Model 1: Y = β₀ + A

Validation has not chosen A. A entered because of the forward stepwise significance rule.

Step 2 — choose the next term conditional on A

Minitab compares A+B, A+C, A+D, A+E, and A+F.

Candidate addition	Entry p-value given A
B	0.004
C	0.080
D	0.310
E	0.160
F	0.270

B enters:

Model 2: Y = β₀ + A + B

Step 3 — continue from A+B

Minitab tests A+B+C, A+B+D, A+B+E, and A+B+F. Suppose C enters:

Model 3: Y = β₀ + A + B + C

Step 4 and beyond

Suppose D enters next, followed by E:

Intercept → A → A+B → A+B+C → A+B+C+D → A+B+C+D+E

That entire sequence is created by the forward-selection rule. It is only one path through the possible model space.

6.3 Job 2 — evaluate each completed model using validation data

Step	Model	Validation RMSE
0	Intercept only	18.4
1	A	12.9
2	A+B	8.7
3	A+B+C	6.8
4	A+B+C+D	7.5
5	A+B+C+D+E	8.4

Although D and E were statistically eligible to enter along the forward path, the three-predictor model A+B+C has the lowest validation RMSE. Therefore, the preferred model size is Step 3:

Selected model: Y = β₀ + A + B + C

6.4 What Minitab is not doing

Minitab is not performing this brute-force loop:

Try every possible equation → calculate validation error → add whichever term improves validation most → repeat.

Instead, it performs:

Forward significance selection → create one ordered sequence → calculate validation performance for each step → choose the best stopping point.

6.5 Why this is not Best Subsets

With six predictors, there are:

2⁶ − 1 = 63 possible non-empty subsets

Forward Selection with Validation does not test all 63 subsets. It may evaluate only the six nested models on its selected path. A strong model such as A+C+E can be missed if B entered earlier and the path never visits A+C+E.

6.6 Why validation does not choose the next term

Validation results contain sampling noise. Suppose:

Candidate	Entry p-value	Validation RMSE
A	0.0001	8.72
B	0.0800	8.70

The RMSE difference of 0.02 may be random. If the software repeatedly chased tiny validation improvements, it could overfit the validation sample. Separating entry selection from model-size validation reduces that risk.

Mental model: “Forward Selection first, Validation second.”

7. Best Subsets Regression

Best Subsets Regression evaluates many or all combinations of predictors and identifies the strongest candidate model for each model size.

7.1 Minitab path

Stat → Regression → Regression → Best Subsets

7.2 How it works with six predictors

For six candidate predictors, the number of non-empty subsets is:

2⁶ − 1 = 63 models

Model size	Number of possible models
1 predictor	6
2 predictors	15
3 predictors	20
4 predictors	15
5 predictors	6
6 predictors	1
Total	63

7.3 Why it differs from forward selection

Forward Selection

Follows one path:

A → A+B → A+B+C → A+B+C+D

It does not revisit every alternative subset of the same size.

Best Subsets

Compares competing combinations:

A+B+C, A+B+D, A+C+D, and so on.

It is much more exhaustive for a moderate number of predictors.

7.4 What “best” means

Minitab commonly compares models using:

R²
Adjusted R²
Predicted R²
Mallows’ C_p
S, the residual standard deviation

The model with the highest ordinary R² is usually the largest model, because R² never decreases when predictors are added. Do not select a model from R² alone.

7.5 Model count calculator

Number of candidate predictors

For 6 predictors: 63 possible non-empty models.

7.6 When Best Subsets is useful

You have a moderate number of candidate predictors.
You want to compare alternative models of the same size.
You want a transparent shortlist rather than one automatic final answer.
You plan to use engineering judgment after statistical screening.

7.7 Limitations

The search grows exponentially as predictors increase.
Searching many models increases the chance of finding patterns that do not reproduce.
Results can be unstable with correlated predictors.
Best Subsets does not replace residual analysis, validation, or subject-matter judgment.

7.8 Best Subsets and interaction terms

Major limitation to understand: Best Subsets Regression does not automatically create interaction terms. It only searches among the numeric predictor columns that you supply.

What Minitab can and cannot see

Suppose your worksheet contains only three predictors:

A, B, and C

Then Best Subsets can compare models such as:

A
B
C
A + B
A + C
B + C
A + B + C

It cannot automatically evaluate:

A × B
A × C
B × C
A²
B²
C²

Those terms do not exist from Best Subsets’ point of view unless you first create them as separate worksheet columns.

Mental model: Best Subsets searches the columns you give it. It does not invent new terms.

Example engineering situation

Suppose:

A = Carbon content
B = Cooling rate
C = Thickness
Y = Tensile strength

Engineering knowledge suggests that the effect of cooling rate changes with carbon content. The intended model includes an interaction:

Y = β₀ + β₁A + β₂B + β₃C + β₁₂(A×B)

If the worksheet contains only A, B, and C, Best Subsets will never evaluate the A×B interaction. The interaction is invisible until you create it as its own column.

How to create interaction columns in Minitab

Go to Calc → Calculator.

In Store result in variable, enter a new worksheet column name such as A_B.

In Expression, enter A * B.

Click OK.

Repeat for any additional interactions you want to evaluate, such as A_C = A*C and B_C = B*C.

Worksheet structure after creating interactions

Response	Main-effect columns	Interaction columns
Y	A, B, C	A_B, A_C, B_C

Now Best Subsets can compare models containing any of the supplied columns, including:

A + B + A_B
A + C + A_C
B + C + B_C
A + B + C + A_B
A + B + C + A_B + A_C

How to set up Best Subsets

Go to Stat → Regression → Regression → Best Subsets.

Place Y in Response.

Place the main-effect columns and the manually created interaction columns in the predictor list.

Run the analysis and compare the leading models using predicted R², adjusted R², S, and Mallows’ C_p.

7.9 The hierarchy principle

Best Subsets treats an interaction column as an ordinary numeric predictor. It does not automatically understand that A_B represents A×B or that A and B should accompany it.

This means Best Subsets can return a mathematically possible but structurally undesirable model such as:

Y = β₀ + β₁(A×B)

In most engineering and statistical applications, prefer the hierarchical form:

Y = β₀ + β₁A + β₂B + β₃(A×B)

This is called the principle of hierarchy. When an interaction is retained, the associated lower-order main effects should ordinarily remain in the model.

Why hierarchy matters

The interaction describes how the effect of A changes with B.
Without A and B, the coefficient of A×B becomes difficult to interpret.
Dropping the main effects can make the result depend strongly on how the variables are centered or coded.
Hierarchical models are usually easier to communicate and defend scientifically.

Example interpretation

For the hierarchical model:

Y = β₀ + β₁A + β₂B + β₃(A×B)

The effect of A is not just β₁. It depends on B:

Effect of A = β₁ + β₃B

Similarly, the effect of B depends on A:

Effect of B = β₂ + β₃A

Because the main effects are conditional when an interaction is present, interpret them at meaningful reference values. Centering continuous predictors before creating interaction columns often makes the coefficients easier to interpret.

7.10 Creating polynomial terms for Best Subsets

The same limitation applies to curvature. Best Subsets does not automatically create squared or cubic terms. If you want it to consider a quadratic effect of A, first create:

A_sq = A * A

Then supply both A and A_sq as candidate predictors.

Respect polynomial hierarchy as well:

If A² is retained, ordinarily retain A.

7.11 Recommended Best Subsets workflow with interactions

Use engineering knowledge, prior studies, process maps, or DOE principles to identify plausible interactions and curved terms.

Create those interaction and polynomial columns using Calc → Calculator.

Run Best Subsets using the main effects and the manually created derived terms.

Review the strongest candidate models, but reject nonhierarchical combinations unless there is a compelling technical reason.

Rebuild the chosen candidate in Fit Regression Model.

Check hierarchy, VIF, residual plots, influential observations, predicted R², and engineering plausibility.

Validate the final model on new or held-out data before using it for decisions or prediction.

Best practice: use Best Subsets to generate candidate models, then finalize the model in Fit Regression Model, where you can explicitly control main effects, interactions, polynomial terms, hierarchy, diagnostics, and interpretation.

7.12 Best method when your model contains interaction terms

Best Subsets is useful for comparing prepared predictor columns, but when your analysis contains several interaction terms, categorical factors, or polynomial terms, Fit Regression Model is usually the better Minitab workflow. It can generate model terms automatically and can protect model hierarchy during selection.

Why Best Subsets becomes awkward with many interactions

Suppose your candidate terms are:

A, B, C, A_B, A_C, and B_C.

Best Subsets treats all six worksheet columns as unrelated numeric predictors. It does not inherently understand that A_B represents A×B. Therefore, it can select:

Y = β₀ + β₁₂(A×B)

without selecting A or B. That model is mathematically estimable, but it is normally undesirable because it violates hierarchy and makes the interaction difficult to interpret.

Recommended approach: Fit Regression Model

Minitab path:

Stat → Regression → Regression → Fit Regression Model

Place the response variable in Response. For example, use Tensile_Strength_MPa.

Place the continuous predictors in Continuous predictors. For example, use Carbon, Rolling Temperature, Cooling Rate, and Thickness.

Place categorical predictors, such as grade, rolling practice, production line, or furnace, in the appropriate categorical or factor field. Do not manually multiply text-coded categories to create interaction columns.

Click Model....

Highlight the predictors that should participate in the interaction structure.

Choose Interactions through order 2 to generate all two-way interactions among the selected predictors.

Click Add. Minitab adds the requested main effects and interaction terms to the model specification.

Return to the main dialog and click Stepwise... if automatic term selection is required.

Choose Stepwise, Forward Selection, Backward Elimination, Forward Information Criteria, or Forward Selection with Validation, depending on your modeling purpose.

Use the hierarchy controls so lower-order terms remain whenever a higher-order interaction or polynomial term is retained.

What Minitab creates automatically

If A, B, and C are selected and you request interactions through order 2, Minitab can generate the following terms within the model:

A
B
C
A*B
A*C
B*C

You do not have to create separate worksheet columns for A×B, A×C, and B×C when using Fit Regression Model.

How hierarchy protects the model

Suppose the selection procedure retains:

A*B

With hierarchy enforced, Minitab also retains:

The resulting structure is:

Y = β₀ + β₁A + β₂B + β₃(A×B)

This is easier to interpret, more stable under recoding or centering, and usually more defensible scientifically.

Example with engineering variables

Suppose:

A = Carbon content
B = Cooling rate
C = Thickness
Y = Tensile strength

Requesting two-way interactions produces:

Carbon
Cooling Rate
Thickness
Carbon × Cooling Rate
Carbon × Thickness
Cooling Rate × Thickness

If Carbon × Cooling Rate is retained, hierarchy keeps both Carbon and Cooling Rate in the model. The interaction then means that the effect of cooling rate depends on carbon content.

Best Subsets versus Fit Regression Model for interactions

Feature	Best Subsets	Fit Regression Model
Creates interaction terms automatically	No	Yes, through the Model dialog
Requires manual worksheet interaction columns	Yes	No
Understands the parent main effects of an interaction	No	Yes
Can enforce hierarchy	Not inherently	Yes, through hierarchy controls
Handles categorical-factor interactions naturally	Awkward and error-prone	Yes
Best use	Compare prepared subsets of mainly main-effect predictors	Build structured engineering models with interactions and curvature

When to use Best Subsets

You mainly have continuous main-effect predictors.
You want to compare many competing subsets of a moderate predictor set.
You have only a few carefully prepared derived terms.
You are willing to manually review hierarchy afterward.

When to use Fit Regression Model

You have several interaction terms.
You need to preserve hierarchy.
You have categorical factors.
You are analyzing DOE, response-surface, or structured engineering models.
You want Minitab to create interaction and polynomial terms automatically.
You plan to use a stepwise method while maintaining model structure.

Recommended engineering workflow

Use process knowledge to identify plausible main effects, interactions, and curved terms.

Open Fit Regression Model and define the response, continuous predictors, and categorical factors.

Use the Model dialog to generate the required interaction and polynomial terms.

Apply hierarchy so interactions retain their component main effects and squared terms retain their linear terms.

If screening is necessary, use Stepwise, Forward Selection, Forward Information Criteria, or Forward Selection with Validation within the structured model.

Refit and review the selected model using coefficient tables, VIF, residual plots, influential-observation diagnostics, adjusted R², predicted R², and validation performance.

Confirm that the selected interactions make physical and process sense before using the model for optimization or prediction.

Practical rule: Best Subsets is excellent for comparing prepared subsets. Fit Regression Model is usually better for building and selecting models that contain many interactions, categorical factors, or polynomial terms.

8. Nonlinear Regression

Nonlinear Regression is used when the expected response follows a known equation that is nonlinear in its unknown parameters.

8.1 Minitab path

Stat → Regression → Nonlinear Regression

8.2 Curved does not automatically mean nonlinear

Model	Curved?	Linear in parameters?	Minitab tool
Y = β₀ + β₁X	No	Yes	Fitted Line Plot or Fit Regression Model
Y = β₀ + β₁X + β₂X²	Yes	Yes	Fitted Line Plot or Fit Regression Model
Y = a·exp(bX)	Yes	No	Nonlinear Regression
Y = aX/(b+X)	Yes	No	Nonlinear Regression

8.3 Why Minitab asks for an expectation function

In Nonlinear Regression, you provide the scientific or engineering equation. Minitab then estimates the unknown parameters. Examples include:

Exponential decay

C = C₀e^−kt

Useful for chemical decay, cooling, and degradation.

Saturating response

Y = aX/(b+X)

Useful for rate processes that rise and level off.

Logistic growth

Y = K/[1 + Ae^−rt]

Useful for S-shaped growth with a limiting capacity.

Asymptotic rise

Y = a(1 − e^−bX)

Useful when the response approaches a maximum.

8.4 How nonlinear estimation works

Supply the response and nonlinear expectation function.

Provide starting values for the unknown parameters when required.

Minitab calculates predictions and residuals.

An iterative algorithm updates the parameter estimates.

The process repeats until the objective function changes negligibly or a stopping rule is reached.

8.5 Risks and cautions

Poor starting values can prevent convergence or lead to an inferior local solution.
Parameters may be highly correlated or weakly identified.
The equation should have scientific meaning, not merely fit the current data.
Extrapolation beyond the observed range can be especially dangerous.

9. Which method should I choose?

What is your main situation?

Your question	Recommended Minitab analysis	Why
How does one X relate to one continuous Y?	Fitted Line Plot	Provides the clearest visual and simple regression output.
What is the adjusted effect of several X variables?	Fit Regression Model, method None	Fits the model you specify.
Which variables may be useful?	Stepwise, Forward, or Backward	Provides automatic screening.
Which combinations are strongest?	Best Subsets	Compares alternative subsets across model sizes.
Which model size predicts new data best?	Forward Selection with Validation	Uses validation to choose among models along the forward path.
Is a curved polynomial relationship adequate?	Fitted Line Plot or Fit Regression Model	Polynomial regression is still linear in coefficients.
What are the parameters in a known nonlinear equation?	Nonlinear Regression	Uses iterative estimation for nonlinear parameters.
Is the response yes/no?	Binary Logistic Regression	Models probability rather than an unbounded continuous response.
Is the response an ordered category?	Ordinal Logistic Regression	Respects the order of categories.
Is the response an unordered category?	Nominal Logistic Regression	Handles multiple unordered outcomes.
Is the response a count?	Poisson Regression	Models count outcomes and rate structures.

10. How to interpret model-selection statistics

R²Variation explained in the training data

Adjusted R²R² penalized for unnecessary predictors

Predicted R²Estimated performance on new observations

STypical residual size in response units

10.1 R²

R² measures the percentage of observed response variation explained by the fitted model. It never decreases when predictors are added, so it can reward unnecessary complexity.

10.2 Adjusted R²

Adjusted R² penalizes the addition of terms that do not improve the model enough. It is more useful than R² for comparing models with different numbers of predictors.

10.3 Predicted R²

Predicted R² estimates how well the model predicts observations that were not used to estimate the coefficients. A large gap between adjusted R² and predicted R² can indicate overfitting, influential observations, or instability.

10.4 Mallows’ C_p

A desirable candidate often has a relatively small C_p that is close to the number of model parameters, including the intercept. C_p should be interpreted alongside predicted R², S, residual plots, and subject-matter logic.

10.5 S

S is the residual standard deviation. It represents the typical distance between observed responses and fitted values, in the original units of the response.

10.6 P-values

A coefficient p-value tests whether the data provide evidence that the adjusted coefficient differs from zero, given the other terms in the current model. It does not establish causality, practical importance, or predictive value by itself.

10.7 VIF

Variance Inflation Factors diagnose multicollinearity. Large VIF values mean that a coefficient is being estimated with greater uncertainty because predictors overlap in the information they provide.

11. Recommended practical workflow

Define the response and purpose. Is the goal explanation, prediction, optimization, or screening?

Confirm the response type. Continuous, binary, ordinal, nominal, or count?

Plot the data. Use scatterplots, matrix plots, time plots, and fitted line plots.

Use engineering knowledge. Identify required terms, likely interactions, physical bounds, and confounders.

Fit a reasonable full model. Review coefficient signs, p-values, VIF, residual plots, leverage, and influential points.

Use selection methods as screening tools. Compare Forward, Stepwise, Backward, and Best Subsets rather than blindly trusting one method.

Validate. Use validation data, predicted R², PRESS-type results, or cross-validation where available.

Choose the simplest defensible model. Prefer a model that is stable, interpretable, and nearly as predictive as larger alternatives.

Confirm assumptions. Linearity, constant variance, residual normality where relevant, independence, and absence of severe influence.

Document the final equation and its limits. State the data range, intended use, uncertainty, and conditions under which predictions are valid.

12. Guided practice using the practice dataset

Download the practice dataset below and work through the exercises in Minitab. It is a fictitious steel-coil dataset with 50 rows and 10 columns: an identifier, one response (Tensile Strength), and eight candidate predictors. It was built on purpose to contain useful predictors, one mostly irrelevant predictor, mild collinearity, a Carbon × Cooling Rate interaction, Rolling Temperature curvature, and random noise — so every selection method has something to reveal.

⇩ Download the practice dataset (Excel .xlsx)

Column	Role	What it does in the data
Coil ID	Identifier	Row label only — not a predictor.
Tensile Strength (MPa)	Response (Y)	The variable you model and predict.
Carbon (wt%)	Predictor	Strong, useful predictor.
Manganese (wt%)	Predictor	Useful, but mildly collinear with Carbon.
Rolling Temperature (°C)	Predictor	Curved (quadratic) effect; strength peaks mid-range.
Cooling Rate (°C/s)	Predictor	Useful; interacts with Carbon.
Thickness (mm)	Predictor	Useful; more thickness lowers strength.
Furnace Time (min)	Predictor	Weak-to-moderate; mildly collinear with Rolling Temperature.
Line Speed (m/min)	Predictor	Very weak effect.
Ambient Humidity (%)	Predictor	Mostly irrelevant noise — a good test for selection methods.

Before you start. The workbook has two tabs: Practice Data (load this into Minitab) and Data Dictionary (documents every column and the structure built into the data). When you choose predictors, ignore Coil ID — it is only a row label.

Exercise 1 — Eight Fitted Line Plots

Create Tensile Strength versus each candidate predictor separately. Record slope direction, R², p-value, and any visual curvature. Expect Rolling Temperature to look curved and Ambient Humidity to look essentially flat.

Exercise 2 — Full multiple regression

Use Fit Regression Model with all eight candidate predictors and method None. Compare coefficient p-values, VIF (watch the Carbon/Manganese and Rolling Temperature/Furnace Time pairs), adjusted R², predicted R², and residual plots.

Exercise 3 — Forward Selection

Use alpha-to-enter = 0.15. Record the order in which variables enter. Then refit the final selected model using method None and inspect full diagnostics.

Exercise 4 — Stepwise

Use alpha-to-enter = 0.15 and alpha-to-remove = 0.15. Observe whether any variable enters and later leaves.

Exercise 5 — Backward Elimination

Start with all predictors. Record the removal order and compare the final model with Forward and Stepwise.

Exercise 6 — Forward Information Criteria

Compare the selected model with the p-value-based selections. Note whether a term is retained despite a p-value above 0.05.

Exercise 7 — Forward Selection with Validation

Split the data into training and validation samples. Confirm that the entry sequence follows the forward-selection path, while the validation statistic identifies the preferred step.

Exercise 8 — Best Subsets

Compare the best one- through eight-predictor models. Look for the smallest model with strong predicted R², low S, and a reasonable C_p. Confirm that Ambient Humidity is rarely selected.

Exercise 9 — Interaction

Fit Carbon, Cooling Rate, and Carbon × Cooling Rate. Interpret the interaction before interpreting the main effects.

Exercise 10 — Polynomial regression

Fit Rolling Temperature and Rolling Temperature². Compare residual plots and predicted R² against the straight-line model.

Exercise 11 — Nonlinear thought exercise

Suppose theory says strength approaches an upper limit according to Strength = a*(1-exp(-b*CoolingRate)). Explain why this belongs in Nonlinear Regression and what starting values for a and b would mean.

13. Excel companion functions

Minitab is the primary tool for this lesson, but Excel can reproduce or support several calculations.

Purpose	Excel function or tool	Comment
Slope of simple regression	`SLOPE(known_y, known_x)`	Returns the fitted slope.
Intercept	`INTERCEPT(known_y, known_x)`	Returns the fitted intercept.
R²	`RSQ(known_y, known_x)`	Simple linear regression only.
Regression coefficients and statistics	`LINEST(...,TRUE,TRUE)`	Supports multiple predictors and additional statistics.
Prediction from a simple line	`FORECAST.LINEAR(x, known_y, known_x)`	Predicts Y for a given X.
Correlation	`CORREL(array1,array2)`	Measures linear association, not adjusted regression effects.
Full regression dialog	Data → Data Analysis → Regression	Requires the Analysis ToolPak.
Polynomial terms	Create X², X³ columns	Then include them as predictors.
Interaction term	Create X1*X2 column	Then include main effects and interaction.

14. Common mistakes and how to avoid them

Mistake 1: Treating variable selection as proof

Automatic selection is exploratory. It does not prove that selected terms are causal or that omitted variables are unimportant.

Mistake 2: Believing forward selection searches every model

Forward selection follows one path. Best Subsets is the more exhaustive procedure for moderate predictor counts.

Mistake 3: Believing validation chooses the next variable in Minitab’s forward-validation method

The forward stepwise rule determines the sequence. Validation chooses the preferred model size along that sequence.

Mistake 4: Choosing the largest R²

R² always favors larger models. Compare adjusted R², predicted R², S, C_p, validation error, and practical simplicity.

Mistake 5: Calling every curve “nonlinear regression”

A polynomial curve is still linear regression if the unknown coefficients enter linearly.

Mistake 6: Ignoring hierarchy

If an interaction or squared term is included, retain the associated lower-order terms unless there is a strong, defensible reason not to.

Mistake 7: Ignoring residual diagnostics

A selected model can still violate assumptions, contain influential points, or show unmodeled curvature.

Mistake 8: Extrapolating beyond the data

Regression predictions outside the observed predictor range can be physically unrealistic and highly uncertain.

15. Predicting a response at specific predictor settings

Once you have chosen a model, a common next step is to predict the response at particular predictor values — for example, “what tensile strength should I expect at these settings?” You could substitute the values into the fitted equation by hand, but the reliable approach is to let Minitab predict from the stored model, because it also reports the uncertainty around the prediction. This section shows the official workflow, a worked example, and how to read the output.

15.1 First make sure the model is stored

Best Subsets Regression only identifies candidate models — it does not leave behind a fitted model that Minitab can predict from. So if you are working from Best Subsets, refit your chosen subset with Fit Regression Model first.

Stat → Regression → Regression → Fit Regression Model

In the Best Subsets output, identify the model you selected using the X marks in its row.

Enter your response (Y).

Enter the selected predictors as continuous or categorical terms.

Under Model, add the exact interaction and polynomial terms that the chosen subset includes.

Check the residual plots and model assumptions, then click OK.

This creates a stored regression model. Everything in the Predict step below uses that stored equation, so the model must exist before you can predict from it.

15.2 Predict the response

Stat → Regression → Regression → Predict

Select the response/model you just fitted.

Choose Enter individual values.

Enter the desired setting for every predictor in the model.

Click Options and normally keep Confidence level 95% and Interval type: Two-sided.

Click OK. Minitab uses the stored equation to calculate the prediction.

To predict at many settings at once, choose Enter columns of values instead and point Minitab at worksheet columns holding the predictor settings — it returns one prediction per row.

15.3 Worked example

Suppose your fitted equation is:

ŷ = 10 + 2A − 1.5B + 0.4(A × B)

and you want the predicted response at A = 3 and B = 4. Minitab computes:

ŷ = 10 + 2(3) − 1.5(4) + 0.4(3 × 4) = 10 + 6 − 6 + 4.8 = 14.8

So the predicted response is ŷ = 14.8.

Entering interaction terms correctly. If A×B was added through the Model dialog, enter only A = 3 and B = 4 — Minitab computes the interaction for you. If instead you built an A_B worksheet column manually before running Best Subsets, Minitab will ask for that column too, and you must enter its value yourself:

Predictor	Value to enter
A	3
B	4
A_B	3 × 4 = 12

15.4 Understanding the prediction output

Output	What it means
Fit	The best point estimate of the response at those settings.
SE Fit	The uncertainty in that estimated mean response.
95% CI	A range for the average response at those settings.
95% PI	A range for a single individual future result at those settings.

For our example the output might read:

Fit	95% CI	95% PI
14.8	13.9 – 15.7	11.2 – 18.4

Read that as:

The estimated average response at these settings is 14.8.
That average is likely between 13.9 and 15.7 (the 95% confidence interval).
One new result could reasonably fall between 11.2 and 18.4 (the 95% prediction interval).

Confidence interval vs. prediction interval. When you care about a single future coil, heat, slab, or test result, the prediction interval is the one that matters. It is wider than the confidence interval because an individual outcome carries more variation than an estimated average.

15.5 A caution on extrapolation

Avoid predicting far outside the predictor ranges used to build the model. Minitab may flag such settings as unusual (an X in the output), and extrapolated predictions can be unreliable because the fitted relationship was never verified in that region.

Exam tip (CSSBB/CQE). Be ready to distinguish the confidence interval for the mean response from the prediction interval for an individual value, and to pick the prediction interval when a question asks about one future outcome. Knowing the Predict menu path and that interactions/polynomials must be present in the stored model is frequently tested in applied items.

16. Final summary

Method	Best concise description
Fitted Line Plot	Visual simple regression for one continuous X and one continuous Y.
Fit Regression Model — None	Fits exactly the linear model you specify.
Forward Selection	Adds qualifying predictors one at a time and usually does not remove them.
Backward Elimination	Starts full and removes weak terms.
Stepwise	Allows predictors to enter and leave according to significance rules.
Forward Information Criteria	Adds predictors according to a fit-versus-complexity criterion.
Forward Selection with Validation	Builds one forward path, then uses validation to select the preferred step.
Best Subsets	Compares many or all predictor combinations and presents leading models by size.
Nonlinear Regression	Fits a user-specified scientific equation that is nonlinear in its parameters.

Best overall principle: use selection procedures to generate candidate models, not to replace engineering judgment. The final model should be statistically sound, physically credible, interpretable, stable, validated, and appropriate for the intended decision.