A Field Guide to Experimental 
Designs

RCB repeated in time

Depending upon the specific effect of time, either split-plot or regression analyses could be appropriate for these experiments. However, advances in statistics and computer applications allow the researcher to further explore the effect of time and insure that the appropriate hypotheses are tested.

Other designs (such as CRD or Latin) can also be repeated in time, but are not included at this time in the field guide.

Field marks:

  • Multiple measurements of the same experimental subjects are made in time.
  • Treatments are assigned at random within blocks of adjacent subjects, each treatment once per block.
  • The number of blocks is the number of replications.

 

Sample layout:
There are 3 blocks (I-III) and 6 treatments (A-F) measured 3 times (First-Third) through the season in this example.
RCB sample layout

  First   Block I    A  B  C  D  E  F
          Block II   F  A  E  B  D  C
          Block III  C  B  F  A  D  E

       Second  Block I    A  B  C  D  E  F
               Block II   F  A  E  B  D  C
               Block III  C  B  F  A  D  E

Third   Block I    A  B  C  D  E  F
        Block II   F  A  E  B  D  C
        Block III  C  B  F  A  D  E

ANOVA table format:

The appropriate table format will depend upon the analysis and the effect of time. The following takes it as a more traditional split-plot in time and assumes that there is no more correlation between samples taken closer in time than those taken further apart in time. The SAS System provides tests for this assumption.

Source of
variation
Degrees of
freedoma
Sums of
squares (SSQ)
Mean
square (MS)
F
Blocks (B) b-1 SSQB SSQB/(b-1) MSB/MSEm
Treatment (Tr) t-1 SSQTr1 SSQTr/(t-1) MSTr/MSEm
Error-main (Em) (b-1)*(t-1) SSQEm SSQEm/((b-1)*(t-1))  
Time (Ti) (s-1) SSQTi SSQTi/(s-1) MSTi/MSE
Time X Blocks (TxB) (s-1)*(b-1) SSQTxB SSQTxB/((s-1)*(b-1)) MSTxB/MSE
Time X Treatments (TxT) (s-1)*(t-1) SSQTxT SSQTxT/((s-1)*(t-1)) MSTxT/MSE
Error (E) (s-1)*(t-1)*(b-1) SSQE SSQE/((s-1)*(t-1)*(b-1))  
Total (Tot) f*s*b-1 SSQTot    
awhere t=number of treatments, s=number of times measurements are taken, and b=number of blocks or replications.

Sample ANOVA table:

Source of
variation
Degrees of
freedom
Sums of
squares (SSQ)
Mean
square (MS)
F
Blocks 2 205.01 102.50 22.21a
Treatments 5 122.95 24.59 5.33b
Error-main 10 46.16 4.61  
Timec 2 4167.78 2083.89 2166.83d
Time X Blocks 4 1.23 0.31 0.32e
Time X Treatments 10 20.48 2.05 2.13f
Error 20 19.23 0.96  
Total 53 4582.85    
aF test with 2,10 degrees of freedom at P=0.05 is 4.10
bF test with 5,10 degrees of freedom at P=0.05 is 3.33
cChi-square for assumption about correlations between times is 2.62; chi-square with 2 degrees of freedom at P=0.05 is 5.99
dF test with 2,20 degrees of freedom at P=0.05 is 3.49
eF test with 4,20 degrees of freedom at P=0.05 is 2.87
fF test with 10,20 degrees of freedom at P=0.05 is 2.35

Sample SAS GLM statements:

PROC GLM;
  CLASS BLOCKS TREAT;
  MODEL TIME1 TIME2 TIME3 = BLOCKS TREAT;
  REPEATED TIME /PRINTE;
RUN;

Note: the PRINTE options produces tests for assumptions concerning time.

Compare with:

 

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Friday, August 18, 2000