Strip Plot Design

A Strip Plot Design (also called a Criss-Cross Design) is an experimental layout used when two factors are both difficult or costly to randomise at the level of individual plots. It is a natural extension of the split plot concept but treats both factors symmetrically — neither is nested within the other.

In a strip plot:

• Factor A (the row factor) is applied to horizontal strips running across each block
• Factor B (the column factor) is applied to vertical strips running down each block
• The intersection of a row strip and a column strip forms the experimental unit for the interaction A×B

Key idea: Both factors are randomised within blocks in perpendicular directions. The interaction is estimated at the smallest unit — the intersection cell — which typically has the highest precision.

Comparison with Related Designs

Feature	CRD / RCBD	Split Plot	Strip Plot
Hard-to-change factors	0	1 (whole plot)	2 (row + column)
Nesting structure	None	Subplot nested in whole plot	No nesting — crossed
Number of error terms	1	2	3
Precision: main effect A	High	Low	Medium
Precision: main effect B	High	High	Medium
Precision: A×B interaction	High	High	Highest
Typical application	Lab / fully randomisable	One machine setting	Two large-scale operations

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When to Use a Strip Plot Design

Use a strip plot design when:

1. Both factors are hard or expensive to change at the plot level (e.g., large machinery, irrigation systems, field operations).
2. The interaction A×B is of primary scientific interest.
3. You can accept somewhat lower precision on both main effects relative to a CRD.
4. Factors are naturally applied in perpendicular directions across a field or experimental area.

Typical applications:

Field	Row Factor (A)	Column Factor (B)
Agronomy	Tillage method	Irrigation type
Food processing	Oven temperature	Packaging material
Textile	Dye bath	Fabric weave
Manufacturing	Machine line	Raw material supplier
Horticulture	Row spacing	Fertiliser formulation

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Structure of the Strip Plot Design

Layout Diagram (1 Block, $a = 3$ rows, $b = 4$ columns)

          ← Column Factor B →
          B1     B2     B3     B4
        ┌──────┬──────┬──────┬──────┐
  A1    │A1B1  │A1B2  │A1B3  │A1B4  │  ← Row strip for A1
        ├──────┼──────┼──────┼──────┤
  A2    │A2B1  │A2B2  │A2B3  │A2B4  │  ← Row strip for A2
        ├──────┼──────┼──────┼──────┤
  A3    │A3B1  │A3B2  │A3B3  │A3B4  │  ← Row strip for A3
        └──────┴──────┴──────┴──────┘
           ↑      ↑      ↑      ↑
         Col    Col    Col    Col
        strip  strip  strip  strip
        for B1 for B2 for B3 for B4

Each cell is an intersection plot receiving one level of A and one level of B.

Randomisation

Within each block:

• Levels of Factor A are randomised among the row strips independently
• Levels of Factor B are randomised among the column strips independently
• The cell values are determined by which row and column strip intersect — no further randomisation at the cell level

Multi-Block Layout ($r$ blocks)

Block 1            Block 2            Block 3
┌──┬──┬──┬──┐      ┌──┬──┬──┬──┐      ┌──┬──┬──┬──┐
│  │  │  │  │      │  │  │  │  │      │  │  │  │  │
├──┼──┼──┼──┤      ├──┼──┼──┼──┤      ├──┼──┼──┼──┤
│  │  │  │  │      │  │  │  │  │      │  │  │  │  │
├──┼──┼──┼──┤      ├──┼──┼──┼──┤      ├──┼──┼──┼──┤
│  │  │  │  │      │  │  │  │  │      │  │  │  │  │
└──┴──┴──┴──┘      └──┴──┴──┴──┘      └──┴──┴──┴──┘
  A randomised        A randomised        A randomised
  B randomised        B randomised        B randomised
  independently       independently       independently

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Statistical Model

\[\begin{aligned} Y_{ijk} =\;& \mu + \rho_i + \alpha_j + \delta_{ij} \\ &+ \beta_k + \gamma_{ik} + (\alpha\beta)_{jk} \\ &+ \varepsilon_{ijk} \end{aligned}\]

Term	Description
$\mu$	Overall mean
$\rho_i$	Effect of block $i$   ($i = 1, \ldots, r$)
$\alpha_j$	Main effect of row factor A, level $j$   ($j = 1, \ldots, a$)
$\delta_{ij}$	Row-strip error — block × A interaction; error for testing A
$\beta_k$	Main effect of column factor B, level $k$   ($k = 1, \ldots, b$)
$\gamma_{ik}$	Column-strip error — block × B interaction; error for testing B
$(\alpha\beta)_{jk}$	A × B interaction effect
$\varepsilon_{ijk}$	Cell (intersection) error — error for testing A×B

Three separate error terms are used in a strip plot ANOVA — one for each stratum of the design.

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ANOVA Table

Source of Variation	df	MS	F-ratio	Error Term Used
Block Stratum
Blocks	$r - 1$	$MS_{Blk}$	—	—
Row-Strip Stratum
Factor A (rows)	$a - 1$	$MS_A$	$MS_A \,/\, MS_{EA}$	Row-strip error
Row-strip error ($E_A$)	$(r-1)(a-1)$	$MS_{EA}$	—	—
Column-Strip Stratum
Factor B (columns)	$b - 1$	$MS_B$	$MS_B \,/\, MS_{EB}$	Column-strip error
Column-strip error ($E_B$)	$(r-1)(b-1)$	$MS_{EB}$	—	—
Intersection Stratum
A × B	$(a-1)(b-1)$	$MS_{AB}$	$MS_{AB} \,/\, MS_{EC}$	Cell error
Cell error ($E_C$)	$(r-1)(a-1)(b-1)$	$MS_{EC}$	—	—
Total	$rab - 1$

Degrees of Freedom Summary

\[df_{EA} = (r-1)(a-1) \qquad df_{EB} = (r-1)(b-1)\] \[df_{EC} = (r-1)(a-1)(b-1)\]

Precision Hierarchy

\[\text{Interaction (A×B)} > \text{Main effects (A, B)}\]

The interaction is tested against the smallest (cell) error, giving it the highest precision — an important practical advantage of the strip plot design.

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Relative Efficiency

The efficiency of estimating each effect relative to a completely randomised design (CRD) depends on the magnitudes of the three error variances:

\[\sigma^2_{EA} \geq \sigma^2_{EB} \geq \sigma^2_{EC} \quad \text{(typically)}\]

• Both main effects have lower efficiency than in a CRD (larger error denominators)
• The interaction has higher efficiency than in a CRD (smallest error denominator)
• This trade-off is acceptable when the interaction is the primary research question

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Worked Example

Experiment: Effect of tillage method (Factor A: conventional, reduced, zero) and irrigation system (Factor B: furrow, sprinkler, drip) on wheat yield (t/ha), in $r = 4$ blocks.

Parameters

\[r = 4,\quad a = 3,\quad b = 3\] \[N = 4 \times 3 \times 3 = 36\]

Mean Yield Table (t/ha)

	B: Furrow	B: Sprinkler	B: Drip	Row Mean
A: Conventional	3.8	4.6	5.2	4.53
A: Reduced	4.2	5.0	5.9	5.03
A: Zero	3.5	4.1	4.8	4.13
Col Mean	3.83	4.57	5.30	4.57

Degrees of Freedom

Source	df
Blocks	$4 - 1 = 3$
A (Tillage)	$3 - 1 = 2$
Row-strip error	$(4-1)(3-1) = 6$
B (Irrigation)	$3 - 1 = 2$
Column-strip error	$(4-1)(3-1) = 6$
A × B	$(3-1)(3-1) = 4$
Cell error	$(4-1)(3-1)(3-1) = 12$
Total	35

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Analysis in R

Using lme() (nlme package)

library(nlme)
library(emmeans)

# Data must have columns: block, row_factor, col_factor, yield
data <- read.csv("strip_plot_data.csv")
data$block      <- factor(data$block)
data$tillage    <- factor(data$tillage)
data$irrigation <- factor(data$irrigation)

# Fit strip plot model — three random effects strata
model <- lme(yield ~ tillage * irrigation,
             random = list(block = pdBlocked(list(
               pdIdent(~ 1),
               pdIdent(~ tillage - 1),
               pdIdent(~ irrigation - 1)
             ))),
             data = data,
             method = "REML")

summary(model)
anova(model)

Using aov() with Error strata

# Traditional aov approach with explicit Error() strata
model_aov <- aov(yield ~
                   tillage * irrigation +
                   Error(block +
                         block:tillage +
                         block:irrigation),
                 data = data)

summary(model_aov)

Post-hoc Comparisons

# Marginal means and pairwise comparisons
emmeans(model_aov, pairwise ~ tillage,    adjust = "tukey")
emmeans(model_aov, pairwise ~ irrigation, adjust = "tukey")

# Simple effects of B at each level of A
emmeans(model_aov, pairwise ~ irrigation | tillage, adjust = "tukey")

# Interaction plot
interaction.plot(
  x.factor     = data$irrigation,
  trace.factor  = data$tillage,
  response      = data$yield,
  col           = c("steelblue", "tomato", "forestgreen"),
  lwd           = 2,
  xlab          = "Irrigation System",
  ylab          = "Mean Yield (t/ha)",
  trace.label   = "Tillage"
)

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Analysis in SAS

/* Strip plot design using PROC MIXED */
proc mixed data=strip_plot;
  class block tillage irrigation;
  model yield = tillage irrigation tillage*irrigation / ddfm=satterth;
  /* Three random effects — one per stratum */
  random block;
  random block*tillage;
  random block*irrigation;
  /* Interaction comparisons */
  lsmeans tillage*irrigation / pdiff slice=tillage adjust=tukey;
run;

/* Interaction plot */
proc sgplot data=strip_plot;
  series x=irrigation y=yield / group=tillage markers lineattrs=(thickness=2);
  xaxis label="Irrigation System";
  yaxis label="Mean Yield (t/ha)";
  keylegend / title="Tillage Method";
run;

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Assumptions

1. Normality — Residuals within each stratum are approximately normally distributed.
2. Homogeneity of variance — Equal variance within row strips, column strips, and cells.
3. Independence — Blocks are independent; randomisation is carried out correctly within each block.
4. Correct error terms — Factor A tested against row-strip error; Factor B against column-strip error; A×B against cell error. Using a single pooled error is incorrect and leads to biased F-tests.
5. Additivity of block effects — Blocks affect all treatment combinations equally (no block × treatment interaction beyond the defined strata).

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Advantages and Disadvantages

Advantages ✓

• Accommodates two hard-to-change factors in the same experiment
• Provides maximum precision for the interaction A×B — the effect most relevant when both factors are of interest
• Operationally efficient — Factor A applied in strips, Factor B applied in perpendicular strips, reducing factor-level changes
• Straightforward field layout — rows and columns are natural physical divisions
• Reduces total operational cost compared to a fully randomised two-factor experiment

Disadvantages ✗

• Lower precision for both main effects compared to CRD or RCBD
• Three error terms complicate the analysis; standard ANOVA software must be used carefully
• Small degrees of freedom for row-strip and column-strip errors, especially with few blocks
• Missing data are difficult to handle without mixed-model software
• Less familiar than split plot; risk of misidentifying the error structure

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Comparison: Split Plot vs Strip Plot

Aspect	Split Plot	Strip Plot
Factor A randomisation	Among whole plots	Among row strips within blocks
Factor B randomisation	Within each whole plot	Among column strips within blocks
Nesting	B nested within A	A and B crossed (not nested)
Error terms	2	3
Precision for A	Low	Medium
Precision for B	High	Medium
Precision for A×B	High	Highest
Use when	Only A is hard to change	Both A and B are hard to change

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Extensions

Extension	Description
Strip-Split Plot	A third factor added as subplots within intersection cells
Replicated Strip Plot	Multiple blocks increase df for row- and column-strip errors
Unbalanced Strip Plot	Missing cells handled via REML mixed model
Strip Plot in Space–Time	One factor varied across space, another across time (repeated measures analogue)
Strip Plot with Covariates	ANCOVA model includes plot-level covariates to reduce residual error

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Glossary

Term	Definition
Row factor	Factor applied to horizontal strips spanning the full width of a block
Column factor	Factor applied to vertical strips spanning the full height of a block
Intersection plot	The experimental unit formed at the crossing of one row strip and one column strip
Row-strip error	Variability among row strips within a block; denominator for testing Factor A
Column-strip error	Variability among column strips within a block; denominator for testing Factor B
Cell error	Residual variability at the intersection level; denominator for testing A×B
Criss-cross design	Alternative name for the strip plot design
Stratum	A level of the hierarchical error structure (block, row strip, column strip, cell)

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References

1. Montgomery, D.C. (2017). Design and Analysis of Experiments (9th ed.). Wiley.
2. Cochran, W.G. & Cox, G.M. (1957). Experimental Designs (2nd ed.). Wiley.
3. Federer, W.T. (1955). Experimental Design: Theory and Application. Macmillan.
4. Littell, R.C., Milliken, G.A., Stroup, W.W., Wolfinger, R.D., & Schabenberger, O. (2006). SAS for Mixed Models (2nd ed.). SAS Institute.
5. Piepho, H.P., Büchse, A., & Emrich, K. (2003). A Hitchhiker’s Guide to Mixed Models for Randomized Experiments. Journal of Agronomy and Crop Science, 189(5), 310–322.

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