A Field Guide to Experimental 
Designs

The Analysis of Covariance (ANCOVA)

Field marks:

  • Controls variation in an experiment by measuring an independent factor on each experimental subject.
  • Otherwise is laid out using a traditional design with traditional arrangements of treatments

Note: the example here is of a CRD

Sample layout:
Different colors represent different treatments. There are 4 (A-D) treatments with 4 replications (1-4) each. An independent factor will be measured on each subject
CRD sample layout

     A1  B1  C1  A2
     D1  A3  D2  C2
     B2  D3  C3  B3
     C4  A4  B4  D4

 

ANOVA table format:

Source of
variation
Degrees of
freedoma
Sums of
squares (SSQ)
Mean
square (MS)
F
Covariate (C) 1 SSQC SSQC MSTr/MSE
Adjusted Treatmentsb (Tr) t-1 SSQTr SSQTr/(t-1) MSTr/MSE
Error (E) t*(r-1)-1 SSQE SSQE/(t*(r-1)-1)  
Total (Tot) t*r-1 SSQTot    
awhere t=number of treatments and r=number of replications per treatment.
bthis is the treatments adjusted for the covariate

Sample ANOVA table:

Source of
variation
Degrees of
freedom
Sums of
squares (SSQ)
Mean
square (MS)
F
Covariate 1 44.32 44.32 48.42a
Treatments 3 157.06 52.35 9.68b
Error 11 10.07 0.92  
Total 15 211.44    
aF test with 1,11 degrees of freedom at P=0.05 is 4.84
bF test with 3,11 degrees of freedom at P=0.05 is 3.59

Sample SAS GLM statements:

PROC GLM;
  CLASS TREATS;
  MODEL WHATEVER = COV TREATS  / SOLUTION ;
  LSMEANS TREATS;
RUN;

Note: the SAS LSMEANS statement produces means adjusted for the covariate. These are the treatment means as if each treatment was applied to average subjects

Compare with:

 

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Thursday, August 17, 2000