Strip-Plot Designs Sometimes called split-block design For experiments involving factors that are

difficult to apply to small plots Three sizes of plots so there are three

experimental errors The interaction is measured with greater

precision than the main effects

For example: Three seed-bed preparation methods Four nitrogen levels Both factors will be applied with large scale machinery

S3 S1 S2

N1

N2

N0

N3

S1 S3 S2

N2

N3

N1

N0

Advantages --- Disadvantages

Advantages– Permits efficient application of factors that would be

difficult to apply to small plots

Disadvantages– Differential precision in the estimation of interaction

and the main effects– Complicated statistical analysis

Strip-Plot Analysis of Variance

Source df SS MS FTotal rab-1 SSTotBlock r-1 SSR MSR A a-1 SSA MSA FA

Error(a) (r-1)(a-1) SSEA MSEA Factor A error B b-1 SSB MSB FB

Error(b) (r-1)(b-1) SSEB MSEB Factor B error AB (a-1)(b-1) SSAB MSAB FAB

Error(ab) (r-1)(a-1)(b-1) SSEAB MSEAB Subplot error

Computations

SSTot

SSR

SSA

SSEA

SSB

SSEB

SSAB

SSEAB SSTot-SSR-SSA-SSEA-SSB-SSEB-SSAB

There are three error terms - one for each main plot and interaction plot

2i j k ijkY Y

2..kkab Y Y

2i..irb Y Y

2i.ki kb Y Y SSA SSR

2. j.jra Y Y

2. jkj ka Y Y SSB SSR

2ij.i jr Y Y SSA SSB

F Ratios F ratios are computed somewhat differently

because there are three errors

FA = MSA/MSEA tests the sig. of the A main effect

FB = MSB/MSEB tests the sig. of the B main effect

FAB = MSAB/MSEAB tests the sig. of the AB interaction

Standard Errors of Treatment Means

AMSErb

BMSEra

ABMSEr

Factor A Means

Factor B Means

Treatment AB Means

SE of Differences for Main Effects

Differences between 2 A means

with (r-1)(a-1) df

Differences between 2 B means with (r-1)(b-1) df

2 A* MSErb

2 B* MSEra

SE of Differences Differences between A means at same level of B

Difference between B means at same level of A

Difference between A and B means at diff. levels

For sed that are calculated from >1 MSE, t tests and df are approximated

2 1 AB A* b MSE MSErb

2 1 AB B* a MSE MSEra

2 AB A B* ab a b MSE a * MSE b * MSErab

Interpretation

Much the same as a two-factor factorial: First test the AB interaction

– If it is significant, the main effects have no meaning even if they test significant

– Summarize in a two-way table of AB means

If AB interaction is not significant– Look at the significance of the main effects– Summarize in one-way tables of means for factors

with significant main effects

Numerical Example

A pasture specialist wanted to determine the effect of phosphorus and potash fertilizers on the dry matter production of barley to be used as a forage– Potash: K1=none, K2=25kg/ha, K3=50kg/ha– Phosphorus: P1=25kg/ha, P2=50kg/ha– Three blocks– Farm scale fertilization equipment

K3 K1 K2

K1 K3 K2

K2 K1 K3

P1

P2

P2

P1

P2

P1

56 32 49

67 54 58

38 62 50

52 72 64

54 44 51

63 54 68

Raw data - dry matter yields

Treatment I II IIIP1K1 32 52 54P1K2 49 64 63P1K3 56 72 68P2K1 54 38 44P2K2 58 50 54P2K3 67 62 51

Construct two-way tables

K I II III Mean1 43.0 45.0 49.045.672 53.5 57.0 58.556.333 61.5 67.0 59.562.67Mean 52.67 56.33 55.6754.89

Potash x BlockP I II III Mean1 45.67 62.67 61.6756.672 59.67 50.00 49.6753.11Mean 52.67 56.33 55.6754.89

Phosphorus x Block

P K1 K2 K3 Mean1 46.00 58.67 65.33 56.672 45.33 54.00 60.00 53.11Mean 45.67 56.33 62.67 54.89

Potash x Phosphorus

SSEA =2*devsq(range) – SSR – SSA

SSR=6*devsq(range)

SSA=6*devsq(range)Main effect of Potash

Construct two-way tables

K I II III Mean1 43.0 45.0 49.045.672 53.5 57.0 58.556.333 61.5 67.0 59.562.67Mean 52.67 56.33 55.6754.89

P I II III Mean1 45.67 62.67 61.6756.672 59.67 50.00 49.6753.11Mean 52.67 56.33 55.6754.89

P K1 K2 K3 Mean1 46.00 58.67 65.33 56.672 45.33 54.00 60.00 53.11Mean 45.67 56.33 62.67 54.89

Potash x Block Phosphorus x Block

Potash x Phosphorus

SSEB =3*devsq(range) – SSR – SSB

SSB=9*devsq(range)Main effect of Phosphorus

SSAB=3*devsq(range) – SSA – SSB

ANOVA

Source df SS MS FTotal 17 1833.78Block 2 45.78 22.89Potash (K) 2 885.78 442.89 22.64**Error(a) 4 78.22 19.56Phosphorus (P) 1 56.89 56.89 0.16nsError(b) 2 693.78 346.89KxP 2 19.11 9.56 0.71nsError(ab) 4 54.22 13.55

See Excel worksheet calculations

Interpretation

Only potash had a significant effect on barley dry matter production

Each increment of added potash resulted in an increase in the yield of dry matter (~340 g/plot per kg increase in potash

The increase took place regardless of the level of phosphorus

Potash None 25 kg/ha 50 kg/ha SEMean Yield 45.67 56.33 62.67 1.80

Repeated measurements over time We often wish to take repeated measures on experimental units to

observe trends in response over time. – Repeated cuttings of a pasture– Multiple harvests of a fruit or vegetable crop during a season– Annual yield of a perennial crop– Multiple observations on the same animal (developmental responses)

Often provides more efficient use of resources than using different experimental units for each time period.

May also provide more precise estimation of time trends by reducing random error among experimental units – effect is similar to blocking

Problem: observations over time are not assigned at random to experimental units.– Observations on the same plot will tend to be positively correlated– Violates the assumption that errors (residuals) are independent

Analysis of repeated measurements The simplest approach is to treat sampling times

as sub-plots in a split-plot experiment. – Some references recommend use of strip-plot rather

than a split-plot Univariate adjustments can be made Multivariate procedures can be used to adjust for

the correlations among sampling periods Mixed Model approaches can be used to adjust

for the correlations among sampling periods

Split-plot in time In a sense, a split-plot is a specific case of repeated

measures, where sub-plots represent repeated measurements on a common main plot

Analysis as a split-plot is valid only if all pairs of sub-plots in each main plot can be assumed to be equally correlated– Compound symmetry– Sphericity

When time is a sub-plot, correlations may be greatest for samples taken at short time intervals and less for distant sampling periods, so assumptions may not be valid– Not a problem when there are only two sampling periods

Formal names for required assumptions

Univariate adjustments for repeated measures

Fit a smooth curve to the time trends and analyze a derived variable– average– maximum response– area under curve– time to reach the maximum

Use polynomial contrasts to evaluate trends over time (linear, quadratic responses) and compare responses for each treatment

Reduce df for subplots, interactions, and subplot error terms to obtain more conservative F tests

Multivariate adjustments for repeated measures

In PROC GLM, each repeated measure is treated like an additional variable in a multivariate analysis:model yield1 yield2 yield3 yield4=variety/nouni;repeated harvest / printe;

MANOVA approach is very conservative– Effectively controls Type I error– Power may be low

• Many parameters are estimated so df for error may be too low• Missing values result in an unnecessary loss of available

information No real benefit compared to a Mixed Model approach

Covariance Structure for Residuals

 

1 2 1 2 1 2

2 2 2Y Y Y Y Y ,Y2 sed2 se2 se2 covariance

1 2

2 2Y Y

1 2Y ,Y 0 1 2

2Y Y

2MSEr

Covariance Structure for Residuals

No correlation (independence)– 4 measurements per subject– All covariances = 0

Compound symmetry (CS)– All covariances (off-diagonal elements) are the same– Often applies for split-plot designs (sub-plots within main plots are equally

correlated)

2

22

2

2

1 0 0 00 0 00 1 0 00 0 00 0 1 00 0 00 0 0 10 0 0

2

11

11

Covariance Structure for Residuals

Autoregressive (AR)– Applies to time series analyses– For a first-order AR(1) structure, the

within subject correlations drop off exponentially as the number of time lags between measurements increases (assuming time lags are all the same)

Unstructured (UN)– Complex and computer intensive– No particular pattern for the

covariances is assumed– May have low power due to loss of df

for error

2 3

22

2

3 2

11

11

12 13 14

12 23 242

13 23 34

14 24 34

11

11

Mixed Model adjustment for error structure

Stage one: estimate covariance structure for residuals1. Determine which covariance structures would make sense for

the experimental design and type of data that is collected2. Use graphical methods to examine covariance patterns over time3. Likelihood ratio tests of more complex vs simpler models4. Information content

= (-2 res log likelihood)simple model

minus (-2 res log likelihood)complex model

df = difference in # parameters estimated

AIC, AICC, BIC – information contentadjust for loss in power due to loss of df in more complex models

Null model - no adjustment for correlated errors

2

Mixed Model adjustment for error structure

Stage two:– include appropriate covariance structure in the model– use Generalized Least Squares methodology to evaluate

treatment and time effects Computer intensive

– use PROC MIXED or GLIMMIX in SAS