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 freedom^a	Sums of squares (SSQ)	Mean square (MS)	F
Blocks (B)	b-1	SSQ_B	SSQ_B/(b-1)	MS_B/MS_Em
Treatment (Tr)	t-1	SSQ_Tr1	SSQ_Tr/(t-1)	MS_Tr/MS_Em
Error-main (Em)	(b-1)*(t-1)	SSQ_Em	SSQ_Em/((b-1)*(t-1))
Time (Ti)	(s-1)	SSQ_Ti	SSQ_Ti/(s-1)	MS_Ti/MS_E
Time X Blocks (TxB)	(s-1)*(b-1)	SSQ_TxB	SSQ_TxB/((s-1)*(b-1))	MS_TxB/MS_E
Time X Treatments (TxT)	(s-1)*(t-1)	SSQ_TxT	SSQ_TxT/((s-1)*(t-1))	MS_TxT/MS_E
Error (E)	(s-1)(t-1)(b-1)	SSQ_E	SSQ_E/((s-1)(t-1)(b-1))
Total (Tot)	fsb-1	SSQ_Tot
^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	22.21^a
Treatments	5	122.95	24.59	5.33^b
Error-main	10	46.16	4.61
Time^c	2	4167.78	2083.89	2166.83^d
Time X Blocks	4	1.23	0.31	0.32^e
Time X Treatments	10	20.48	2.05	2.13^f
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:

Two-way Factorial arrangement on a RCB assigned treatments are combinations of treatments from two sets
Split Plot on a RCB: one set of treatments is assigned at random to experimental units; a second set is assigned to at random to sub-units.
Split-block design: one set of treatments is randomized in one directios; a second set is randomized in a second direction.
Regression--comparing trends from different treatments: a set of treatments from a quantitative range are crossed on treatments from a qualitative set.

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Friday, August 18, 2000