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Lean Six Sigma · Analyze phase

Interaction Effects in Minitab

Interaction effects are what happens when one factor's effect on your response depends on where another factor is set. This lesson shows how to spot them, test them, and explain them in Minitab, and how they differ from two variables simply moving together.

  • Intermediate
  • 60 min
  • Interactive
  • Excel + Minitab
  • CSSBB · CQE

What you will be able to do

  • Recognize an interaction effect on a plot, in a model equation, and in Minitab output.
  • Choose the right Minitab tool for designed experiments, historical data, continuous or categorical factors, and pass/fail responses.
  • Calculate main effects, the interaction effect and coded coefficients of a 2×2 design by hand.
  • Use centering to remove the artificial collinearity an interaction term creates.
  • Explain the difference between association, collinearity and interaction, with counter-examples.
Download the practice dataset (Excel .xlsx) DOE runs (CSV) Historical heats (CSV)

All data in this lesson is fictitious and was generated for teaching. The steel examples show realistic shapes, not plant results.

What an interaction effect is

An interaction exists when changing factor A moves the response Y by a different amount, or in a different direction, depending on where factor B is set. On an interaction plot, that shows up as lines that are not parallel.

No interaction Two parallel lines: Charpy energy falls with finishing temperature by the same amount at low and high Nb. LowHigh Finishing temperature (A) Charpy (Y) High NbLow Nb
Same slope at both Nb levels. A and B act independently, so each main effect tells the whole story.
Interaction Two crossing lines: temperature barely changes Charpy energy at high Nb but drops it sharply at low Nb. LowHigh Finishing temperature (A) Charpy (Y) High NbLow Nb
Temperature hurts toughness a lot at low Nb and barely at high Nb. "The effect of temperature" has no single answer.

The model behind the plot

A two-factor model with an interaction term is

\[ Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_{12} X_1 X_2 + \varepsilon \]

where \(Y\) is the response, \(X_1\) and \(X_2\) are the two factors, \(\beta_0\) is the intercept, \(\beta_1\) and \(\beta_2\) are the main-effect coefficients, \(\beta_{12}\) is the interaction coefficient and \(\varepsilon\) is random error. Group the terms that contain \(X_1\) and the reason the lines stop being parallel becomes clear:

\[ \text{slope of } Y \text{ in } X_1 = \frac{\partial Y}{\partial X_1} = \beta_1 + \beta_{12} X_2 \]

The slope of \(X_1\) is no longer a single number. It changes with \(X_2\). When \(\beta_{12} = 0\), the slope is \(\beta_1\) everywhere and the lines are parallel. Testing the interaction in Minitab is testing whether \(\beta_{12}\) is different from zero.

Three shapes an interaction can take

Synergistic: both lines rise, the B-high line much more steeply. B highB low
Synergistic. A pushes Y the same way at both B levels, but much harder at one of them.
Dampening: one line falls steeply, the other barely moves. B highB low
Dampening (antagonistic). One factor level shields the response from the other factor. This is the Charpy example.
Crossover: the lines cross, so the effect of A reverses direction with B. B highB low
Crossover. The effect of A reverses direction. A main-effects plot alone can make A look useless.

Interaction explorer

This builds the coded model Minitab fits for a 2×2 design, with A and B at \(-1\) and \(+1\):

\[ Y = 50 + a\,A + b\,B + ab\,(A \times B) \]

where \(a\), \(b\) and \(ab\) are the coded coefficients. Move the \(ab\) slider and watch the interaction plot lose its parallel lines. The main-effects plot on the left averages over B, so it cannot show you what the interaction is doing.

Main effect of A

Main effects plot of A A −1A +1 Averaged over B

Interaction plot

Interaction plot of A by B A −1A +1 — B +1- - B −1
Effect of A at B −1
12
Effect of A at B +1
12
Main effect of A (= 2a)
12
AB effect (= 2ab)
0
ParallelNo interaction.

Minitab reports Effect as twice the coded coefficient, because each factor moves two coded units from \(-1\) to \(+1\). Try the crossover preset: the main-effects plot goes flat, while the interaction plot shows A matters a lot in opposite directions.

Calculating 2×2 effects by hand

In a 2×2 design every effect is a difference between two averages. Write \(\bar{y}_{+-}\) for the mean response at A high and B low, and so on for the other three cells. Then

\[ \begin{aligned} \text{Effect}_A &= \frac{\bar{y}_{+-} + \bar{y}_{++}}{2} - \frac{\bar{y}_{--} + \bar{y}_{-+}}{2} \\ \text{Effect}_B &= \frac{\bar{y}_{-+} + \bar{y}_{++}}{2} - \frac{\bar{y}_{--} + \bar{y}_{+-}}{2} \\ \text{Effect}_{AB} &= \frac{\bar{y}_{--} + \bar{y}_{++}}{2} - \frac{\bar{y}_{+-} + \bar{y}_{-+}}{2} \end{aligned} \]

where the first sign in each subscript is the level of A and the second is the level of B. The AB effect compares the two cells where A and B are at the same level with the two cells where they differ, which is the sign of the product \(A \times B\). Minitab's coded coefficient is half the effect:

\[ \text{Coef} = \frac{\text{Effect}}{2} \qquad \text{SE}_{\text{Coef}} = \sqrt{\frac{\text{MSE}}{N}} \]

where \(\text{MSE}\) is the mean square error from the ANOVA and \(N\) is the total number of runs.

Worked example with the practice data

The DOE sheet in the practice dataset has 12 runs: finishing temperature (A) at 820 and 880 °C, Nb (B) at 0.020 and 0.040 wt%, three replicates per cell. The four cell means of Charpy energy are:

CellA (temperature)B (Nb)Mean Charpy (J)
Mean response at A low and B low: \(\bar{y}_{--}\)820 °C0.020 %137.30
Mean response at A high and B low: \(\bar{y}_{+-}\)880 °C0.020 %88.00
Mean response at A low and B high: \(\bar{y}_{-+}\)820 °C0.040 %132.97
Mean response at A high and B high: \(\bar{y}_{++}\)880 °C0.040 %127.07
\[ \begin{aligned} \text{Effect}_A &= \frac{88.00 + 127.07}{2} - \frac{137.30 + 132.97}{2} = 107.53 - 135.13 = -27.60 \\ \text{Effect}_{AB} &= \frac{137.30 + 127.07}{2} - \frac{88.00 + 132.97}{2} = 132.18 - 110.48 = 21.70 \\ \text{Coef}_{AB} &= \frac{21.70}{2} = 10.85 \end{aligned} \]

The B effect works out to 17.37 J. Now look at the simple effects, the effect of temperature inside each Nb level. At low Nb, raising temperature costs \(88.00 - 137.30 = -49.3\) J. At high Nb it costs only \(127.07 - 132.97 = -5.9\) J. The main effect of \(-27.6\) J is the average of those two very different numbers, and describes neither of them.

Effect calculator

Type any four cell means. The calculator applies the formulas above and redraws the interaction plot. It opens with the practice-data means, rounded to two decimals, so a coefficient can differ from Minitab's in the last digit.

Interaction plot of the four cell means A −1A +1 — B +1- - B −1
TermEffectCoef
A
B
AB

Choosing the right Minitab tool

Three questions decide the tool: what type of X's you have (both categorical, both continuous, or one of each), where the data came from (a designed experiment or historical production), and what type of response you measured. Start with the X's.

By the type of your two X's (continuous response)

Your two X'sLook first withDesigned experimentHistorical data
Both categorical
e.g. mill line × shift, supplier × grade		 Interaction Plot, Multi-Vari Chart Full factorial. Two levels each: Stat → DOE → Factorial → Create Factorial Design; more than two levels: the General full factorial design option. Analyze with Stat → DOE → Factorial → Analyze Factorial Design. Two-way ANOVA through Stat → ANOVA → General Linear Model → Fit General Linear Model with A*B. It handles unbalanced cell counts. For perfectly balanced data, Stat → ANOVA → Balanced ANOVA gives the same answer.
Both continuous
e.g. finishing temperature × Nb wt%		 Contour Plot; scatterplot of Y vs. X1 grouped by low and high X2 2-level factorial with center points; Response Surface if there is curvature. Factorial Plots and the Pareto of effects show AB. Regression: Stat → Regression → Regression → Fit Regression Model with X1*X2, predictors centered. Keep the X's continuous; cutting them into bins throws information away.
One categorical, one continuous
e.g. furnace × soak time		 Graph → Scatterplot → With Regression and Groups; different slopes mean an interaction Factorial design with one text factor and one numeric factor; analyze it the same way. GLM with the continuous X as a covariate and a covariate × factor term (analysis of covariance with separate slopes). Fit Regression Model with a categorical predictor fits the same model.

When the response or the number of X's changes

SituationLook first withTest with
Pass/fail responseInteraction Plot of the percent passingBinary Logistic Regression with A*B
Count response (defects per pipe)Interaction Plot of the mean countPoisson Regression with A*B: Stat → Regression → Poisson Regression → Fit Poisson Model
Many X's, structure unknownTreeNet two-predictor dependence plotsConfirm what the trees suggest with a model or a DOE

Where ANOVA fits

Two-way ANOVA is the interaction test for two categorical factors. It splits the total variation in Y into a part for A, a part for B, a part for A×B and the leftover error, then asks whether the A×B part is large compared with the error:

\[ F_{AB} = \frac{\text{MS}_{AB}}{\text{MS}_{E}} = \frac{\text{SS}_{AB} / \text{df}_{AB}}{\text{SS}_{E} / \text{df}_{E}}, \qquad \text{df}_{AB} = (a-1)(b-1) \]

where \(a\) and \(b\) are the numbers of levels of A and B, \(\text{SS}\) is a sum of squares and \(\text{MS}\) is a mean square. In the worked DOE below, \(F_{AB} = 1412.67 / 10.96 = 128.84\).

ANOVA, regression and DOE analysis are the same general linear model with different X's, which is why every one of them prints an ANOVA table:

  • Both X's categorical: the model is called two-way ANOVA. Minitab removed its separate Two-Way ANOVA command in version 17; run it through Fit General Linear Model. In the Model sub-dialog, select both factors, set Interactions through order to 2 and click Add.
  • Both X's continuous: the same F-test appears as the X1*X2 row of the regression ANOVA table, and its p-value equals the p-value of the coefficient's t-test.
  • Designed experiment: Analyze Factorial Design reports both the ANOVA table and the coded coefficients. For a two-level factor, Effect = 2 × Coef, and the F-value equals the T-value squared (\(11.35^2 \approx 128.84\)).

Two things change when a factor has more than two levels

  • The interaction is no longer one number. A 3 × 4 design has \((3-1)(4-1) = 6\) interaction degrees of freedom, so read the interaction plot and compare cell means with Stat → ANOVA → General Linear Model → Comparisons on the A*B term.
  • You need replicates. With one observation per cell, the interaction uses up every remaining degree of freedom and there is nothing left to estimate error, so the A×B F-test cannot be computed.

Minitab methods, step by step

Step 1

Visual screening

Start here

  • Interaction PlotStat → ANOVA → Interaction Plot Non-parallel or crossing lines suggest an interaction. Pair it with Stat → ANOVA → Main Effects Plot to see what averaging hides.
  • Multi-Vari ChartStat → Quality Tools → Multi-Vari Chart Works well on observational plant data before you commit to a model.
  • Scatterplot with groupsGraph → Scatterplot → With Regression and Groups Different slopes across groups indicate an interaction between a continuous X and a categorical X.

Step 2

Formal testing

Is it real?

  • Two-way ANOVA or GLM (categorical X's)Stat → ANOVA → General Linear Model → Fit General Linear Model Add A*B in the Model sub-dialog and read the p-value of the interaction term. Stat → ANOVA → General Linear Model → Factorial Plots draws the fitted interaction plot.
  • Regression with an interaction term (continuous X's)Stat → Regression → Regression → Fit Regression Model Add X1*X2 in Model. Under Coding, choose Subtract the mean to keep VIF under control. Stat → Regression → Regression → Contour Plot then shows the twisted surface.
  • Binary Logistic RegressionStat → Regression → Binary Logistic Regression → Fit Binary Logistic Model Include A*B terms for pass/fail responses such as Charpy acceptance.

Step 3

Designed experiments

Cleanest evidence

  • 2-level factorial DOEStat → DOE → Factorial → Create Factorial Design Analyze the results with Stat → DOE → Factorial → Analyze Factorial Design. Use Stat → DOE → Factorial → Factorial Plots for the interaction panel, and the Pareto or Normal plot of standardized effects to spot a significant AB effect.
  • Response surfaceStat → DOE → Response Surface Captures curvature plus the interaction. Contour Plot and Surface Plot show the twist in the surface, and Response Optimizer finds settings.

Step 4

Exploratory machine learning

Many X's

  • CART, Random Forests and TreeNetPredictive Analytics Module → TreeNet® Regression → Fit Model Tree models pick up interactions without being told, and TreeNet's two-predictor partial dependence plots are useful for messy plant data. These commands need Minitab's Predictive Analytics Module. Treat what they find as a hypothesis to confirm.

Worked example: a 2×2 DOE on Charpy energy

Open DOE_2x2 in the practice workbook (or the DOE CSV). The runs are listed in standard order with their randomized run order.

StdOrderRunOrderTemperature (°C)Nb (wt%)ABCharpy (J)
138200.020−1−1133.1
288800.020+1−187.8
398200.040−1+1135.5
468800.040+1+1128.0
548200.020−1−1141.6
6128800.020+1−188.7
758200.040−1+1136.0
828800.040+1+1127.8
9108200.020−1−1137.2
1078800.020+1−187.5
11118200.040−1+1127.4
1218800.040+1+1125.4

Running it in Minitab

  1. Paste the data into a worksheet. Choose Stat → DOE → Factorial → Define Custom Factorial Design, select the temperature and Nb columns as factors, and set their low and high levels.
  2. Choose Stat → DOE → Factorial → Analyze Factorial Design, enter Charpy_J as the response, and under Terms include A, B and AB.
  3. Under Graphs, tick Pareto and Normal effects plots and the residual four-in-one.
  4. Choose Stat → DOE → Factorial → Factorial Plots and select the interaction plot.

What the output shows

These are the values the practice data produces (Minitab rounds p-values below 0.0005 to 0.000):

TermEffectCoefSE CoefT-ValueP-Value
Constant–121.3330.956126.940.000
A (temperature)−27.600−13.8000.956−14.440.000
B (Nb)17.3678.6830.9569.080.000
A×B21.70010.8500.95611.350.000
SourceDFAdj SSAdj MSF-ValueP-Value
A12285.282285.28208.430.000
B1904.80904.8082.520.000
A×B11412.671412.67128.840.000
Error887.7110.96––
Total114690.47–––

Model summary: S = 3.311, R-sq = 98.13%, R-sq(adj) = 97.43%. Minitab also prints Model, Linear and 2-Way Interactions subtotal rows; they are left out here.

Check the standard error with the formula from the hand-calculation section: \(\sqrt{10.96/12} = 0.956\). Every coefficient has the same standard error because the design is balanced and orthogonal.

Interaction plot of the practice data: at 0.040 percent Nb Charpy falls from 132.97 to 127.07 joules; at 0.020 percent Nb it falls from 137.30 to 88.00 joules. 14012010080 820 °C880 °C 0.040 % Nb0.020 % Nb Mean Charpy (J) by temperature
The lines are far from parallel, which is what the AB p-value says formally.

How to write up the finding

"Finishing temperature and Nb interact (p < 0.001). At 0.020% Nb, raising the finishing temperature from 820 to 880 °C lowers mean Charpy energy by 49.3 J. At 0.040% Nb the same change lowers it by only 5.9 J."

Report the simple effects. "Temperature lowers Charpy by 27.6 J" is the main effect, and it is true of neither Nb level.

Continuous X's: centering and VIF

With continuous predictors in raw units, the product term \(X_1 X_2\) is almost a copy of \(X_1\) and \(X_2\) themselves, because both are far from zero. That creates collinearity even when the two X's are unrelated. The variance inflation factor measures it:

\[ \text{VIF}_j = \frac{1}{1 - R_j^2} \]

where \(R_j^2\) is the R-squared from regressing predictor \(j\) on all the other predictors. A VIF above about 5 to 10 is usually treated as a problem. Centering, which means subtracting each predictor's mean before forming the product, removes the artificial part.

The Historical_Heats sheet holds 40 heats with recorded finishing temperature, Nb and Charpy energy. Temperature and Nb are almost uncorrelated (\(r = -0.07\)). Switch between raw and centered predictors:

VIF of each term (log scale)

VIF by term
TermCoefSE CoefVIF

Two things to notice. First, the VIFs in raw units come from the product term, not from any relationship between temperature and Nb. Second, centering does not change the interaction coefficient (32.21 J per °C per wt% Nb) or its p-value (0.0003). It only stabilizes the main-effect coefficients, whose standard errors shrink dramatically. In Minitab, set this under Coding in Stat → Regression → Regression → Fit Regression Model.

Association vs. interaction

Association is a two-variable idea: X and Y move together. Interaction is a three-variable idea: the effect of \(X_1\) on Y depends on the level of \(X_2\). Interaction always needs a response variable; association doesn't.

Association: 2 variables

The question
When finishing temperature goes up, does Charpy energy tend to go down?
Minitab tools
Correlation, scatterplot, chi-square. Stat → Basic Statistics → Correlation for continuous variables, Stat → Tables → Cross Tabulation and Chi-Square for categorical ones.
The answer
One relationship, with one direction and strength.

Interaction: 3 variables

The question
Does finishing temperature affect Charpy energy differently for high-Nb heats vs. low-Nb heats?
Minitab tools
An A*B term in ANOVA, regression or a DOE.
The answer
A relationship that changes with context: non-parallel lines.

The two are independent of each other

Knowing whether two X's are correlated tells you nothing about whether they interact on Y, and the reverse.

Interaction without association Left: four design points at the corners of a square, correlation zero. Right: crossing lines showing a strong interaction. ABr = 0 A → YB+B−
In a balanced 2×2 DOE, A and B are perfectly uncorrelated by design, yet A×B can be the largest effect in the model.
Association without interaction Left: carbon and manganese rising together, strongly correlated. Right: two parallel lines, no interaction on hardness. C %Mnr ≈ 0.96 C → hardnessMn highMn low
Carbon and Mn are strongly correlated in the practice heats, yet each adds its own effect to hardness the same way regardless of the other. The lines stay parallel.

Association and interaction lab

Set how strongly \(X_1\) and \(X_2\) are correlated, and separately how strongly they interact on Y. The two dials are independent, and so are the two pictures.

X1 vs. X2 (80 points)

Scatterplot of X1 against X2 X1

Effect of X1 on Y

Interaction plot of X1 by X2 — X2 high- - X2 low X1 low → high
Sample correlation r
0.00
Slope of X1 at X2 low
0
Slope of X1 at X2 high
8

The common mix-up: collinearity

When people say "these two variables interact," they often mean the two X's are correlated. That is collinearity, a data problem that inflates VIF and muddies coefficients. Interaction is a real effect on Y that you put in the model on purpose.

The practice heats show both ideas at once. Carbon and manganese are strongly associated (\(r = 0.956\), VIF = 11.6 in a two-predictor model), but the C×Mn term in a hardness model is nowhere near significant (p = 0.80). The two elements are collinear and do not interact. Temperature and Nb are the opposite case: unrelated (\(r = -0.07\)) and strongly interacting on Charpy energy (p = 0.0003).

Practice drill: spot the interaction

Each plot shows four cell means. Decide whether the plot shows an interaction worth testing, then check your answer.

Reading interaction results correctly

  • If \(A \times B\) is significant, don't interpret the main effects on their own. Describe the effect with the interaction plot, the contour plot or the simple effects.
  • Keep the model hierarchical: retain A and B even if they aren't significant on their own.
  • Check residuals and VIF. With historical data, correlated X's can create or hide apparent interactions, which is where a DOE earns its keep.
  • Separate statistical from practical significance. A tiny interaction can be significant with lots of data; ask whether the change in simple effects matters to the process.
  • Interactions can depend on the scale of Y. A multiplicative process can show an interaction in raw units that disappears after a log transform. Say which scale you analyzed.
  • Don't extrapolate. An interaction found between 820 and 880 °C says nothing about 900 °C.

Statistics Implementation

Excel Functions

FunctionSyntaxPurposeWhen to use it
AVERAGEIFSAVERAGEIFS(average_range, criteria_range1, criteria1, …)Mean of the rows that meet every conditionCell means and main-effect averages of a factorial design
SUMPRODUCTSUMPRODUCT(array1, [array2], …)Sum of element-wise productsContrasts: multiply the coded AB column by Y and divide by half the runs
LINESTLINEST(known_y's, [known_x's], [const], [stats])Least-squares coefficients, with regression statistics when stats is TRUEFitting a model with a product column; coefficients come back in reverse column order
INDEXINDEX(array, row_num, [column_num])One value from an arrayPulling a single coefficient or R-squared out of LINEST
CORRELCORREL(array1, array2)Pearson correlation coefficientChecking association between two X's or between X and Y
RSQRSQ(known_y's, known_x's)R-squared of a one-predictor regressionTwo-predictor VIF: =1/(1-RSQ(x1, x2))
T.DIST.2TT.DIST.2T(x, deg_freedom)Two-tailed t probability (x must be non-negative)p-value of a coefficient from its t-value
F.DIST.RTF.DIST.RT(x, deg_freedom1, deg_freedom2)Right-tailed F probabilityp-value of an ANOVA F-value
CHISQ.TESTCHISQ.TEST(actual_range, expected_range)p-value of a chi-square testAssociation between two categorical variables

Excel Use Cases

The Effects_Worksheet tab of the practice workbook contains every formula below, live. On DOE_2x2, columns E, F, G and H hold A, B, AB and Charpy for rows 2 to 13.

  • Cell mean (A low, B low): =AVERAGEIFS(H2:H13, E2:E13, -1, F2:F13, -1) returns 137.30.
  • AB effect: =AVERAGEIFS(H2:H13, G2:G13, 1) - AVERAGEIFS(H2:H13, G2:G13, -1) returns 21.70. The equivalent contrast is =SUMPRODUCT(G2:G13, H2:H13)/6.
  • AB coefficient from regression: =INDEX(LINEST(H2:H13, E2:G13), 1, 1) returns 10.85, half the effect.
  • p-value of the AB F-test: =F.DIST.RT(128.84, 1, 8) returns about 0.000003.
  • Association of C and Mn: on Historical_Heats, =CORREL(D2:D41, E2:E41) returns 0.956 and =1/(1-RSQ(D2:D41, E2:E41)) returns a VIF of 11.6.

Minitab Navigation

  • Stat → ANOVA → Interaction Plot and Stat → ANOVA → Main Effects Plot: first look at any categorical factors.
  • Stat → Quality Tools → Multi-Vari Chart: screening observational data by factor level.
  • Graph → Scatterplot → With Regression and Groups: slope differences between groups.
  • Stat → ANOVA → General Linear Model → Fit General Linear Model: unbalanced or historical data with categorical factors and covariates. Add A*B in Model.
  • Stat → Regression → Regression → Fit Regression Model: continuous X's. Add X1*X2 in Model and choose Subtract the mean in Coding. Follow with Stat → Regression → Regression → Contour Plot.
  • Stat → ANOVA → Balanced ANOVA: two-way ANOVA when every cell has the same number of observations. Stat → ANOVA → General Linear Model → Comparisons on the A*B term compares cell means when factors have more than two levels.
  • Stat → Regression → Binary Logistic Regression → Fit Binary Logistic Model: pass/fail responses.
  • Stat → Regression → Poisson Regression → Fit Poisson Model: count responses such as defects per pipe.
  • Stat → DOE → Factorial → Analyze Factorial Design and Stat → DOE → Factorial → Factorial Plots: designed experiments. Use Define Custom Factorial Design first when the data was not created in Minitab, and the General full factorial design option in Stat → DOE → Factorial → Create Factorial Design when a factor has more than two levels.
  • Predictive Analytics Module → TreeNet® Regression → Fit Model: many predictors; requires the module.
  • Stat → Basic Statistics → Correlation and Stat → Tables → Cross Tabulation and Chi-Square: association only.

Choose the DOE route when you ran the experiment; it gives balanced, uncorrelated estimates. Choose GLM or regression for historical data, and treat any interaction found there as a lead to confirm.

Exam Tips

  • The ASQ CSSBB and CQE bodies of knowledge both cover factorial designs and regression. Expect questions that give four cell means and ask for an effect or the interaction effect.
  • Know the factor of two: Effect = 2 × Coef for coded factors. A common distractor offers the coefficient when the question asks for the effect.
  • Non-parallel lines suggest interaction; parallel lines suggest none. Crossing lines are a stronger form of interaction, not a different concept.
  • Correlation between predictors is multicollinearity, not interaction. Watch for answer options that swap the two.
  • Know the interaction degrees of freedom, \((a-1)(b-1)\), and that testing A×B needs replicates in every cell.
  • In practice, report simple effects whenever the interaction is significant, and confirm interactions from historical data with a designed experiment.

Worked practice questions

Your GLM shows A×B with p = 0.002 and factor A with p = 0.41. Can you drop A from the model?

No. Keep the model hierarchical. And don't conclude A has no effect: a non-significant main effect alongside a significant interaction often means A's effect reverses direction across B levels. Read the interaction plot.

Two X's in your historical data have r = 0.92. Does that mean they interact?

No. That is association between the X's (collinearity). It inflates VIF and makes their separate effects hard to estimate. Whether they interact on Y is a different question, answered by an \(X_1 X_2\) term.

Why is a 2×2 factorial DOE the cleanest way to establish an interaction?

The design is orthogonal: A, B and AB are uncorrelated by construction, so each effect is estimated independently. Randomization also protects against lurking variables that can fake an interaction in historical data.

You fit X1*X2 on continuous predictors and VIF jumps to 25. What do you change?

Center or standardize the predictors (Coding in Fit Regression Model). The product of uncentered variables is highly correlated with X1 and X2 themselves; centering removes most of that artificial collinearity.

In Minitab's DOE output, the coded coefficient for AB is 3.5. What is the AB effect?

7.0. Effect = 2 × coefficient, because each factor moves two coded units from \(-1\) to \(+1\).

Key takeaways

  • An interaction means the effect of one X on Y depends on another X. Look for non-parallel lines; in the model, the slope of \(X_1\) is \(\beta_1 + \beta_{12} X_2\).
  • Screen visually (Interaction Plot, Multi-Vari, grouped scatterplot), then test the \(A \times B\) term in GLM, regression or a DOE.
  • When the interaction is significant, describe effects through simple effects or the interaction plot, not main effects alone.
  • Center continuous predictors before forming a product term. It fixes the VIF without changing the interaction estimate.
  • Association is two variables moving together. Interaction needs a response and a third variable. Correlated X's are collinearity, not interaction.

Knowledge check

Check your understanding

Comprehension quiz

Pick the strongest answer, then submit. Explanations appear after you submit.

1. On a Minitab interaction plot, the two lines are almost parallel. What does that suggest?
2. Minitab's coded coefficient for AB is 3.5. What is the AB effect?
3. A GLM shows A × B with p = 0.002 and A with p = 0.41. What should you do?
4. Two predictors in historical data have r = 0.92. What does this tell you?
5. You add X1*X2 to a regression on temperature (°C) and Nb (wt%), and the VIFs jump into the thousands. What is the best fix?
6. Cell means are 137.30 (A-, B-), 88.00 (A+, B-), 132.97 (A-, B+) and 127.07 (A+, B+). What is the AB effect?
7. Using the same cell means, what is the simple effect of temperature (A) at low Nb?
8. Which source gives the strongest evidence that an interaction is real?
9. You have historical Charpy results for three mill lines and two shifts, with unequal numbers of tests per combination. How do you test the line × shift interaction in Minitab?

References

Related lessons

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