A Field Guide to Experimental 
Designs

Regression--comparing trends from different treatments.

Note: this page documents an arrangement of treatments rather than an experimental design. The design used in the examples is a CRD.

Field marks:

  • A set of levels from a continuous range (quantitative difference) are crossed on treatments with qualitative differences. For example, amount of fertilizer crossed on method of application

Sample layout:
There are 3 (1, 2, 5) levels crossed on 2 (A, B) treatments with 4 replications (a-d) each. Different colors represent different combinations (blue-->green and yellow-->brown gradients).
Regression sample layout

     5Aa   2Aa   5Ba   5Ab   2Ba   1Ba
     1Bb   1Aa   2Bb   2Ab   1Bc   1Ab
     5Ac   2Bc   2Ac   1Ac   5Bb   5Bc
     1Ad   5Bd   5Ad   1Bd   2Bd   2Ad

 

ANOVA table format:

Source of
variation
Degrees of
freedoma
Sums of
squares (SSQ)
Mean
square (MS)
F
Treatments (Tr) t-1 SSQtr SSQTr/(t-1) MSTr/MSE
Regression (R) 1 SSQR SSQR MSR/MSE
Treatments X Regression (TxR) (t-1) SSQTxR SSQTxR/(t-1) MSTxR/MSE
Error (E) t*(q*r-2) SSQE SSQE/(t*(q*r-2))  
Total (Tot) t*q*r-1 SSQTot    
awhere t=number of treatments, q=number of levels and r=number of replications per combination.

Sample ANOVA table:

Source of
variation
Degrees of
freedom
Sums of
squares (SSQ)
Mean
square (MS)
F
Treatmenta 1 224.55 224.55 49.73b
Regressiona 1 777.06 777.06 172.08b
Treatment X Regressiona 1 4.94 4.94 1.09b
Error 20 90.31 4.52  
Total 23 1096.86    
aThe calculated regression line for treatment A is Y=4.87+3.08X and for treatment B is Y=9.56+3.61X.
bF test with 1,20 degrees of freedom at P=0.05 is 4.35

Note: in this example, the regression line intercepts (Treatment) are significantly different and the common slope (Regression) is different than zero, but the different slopes (Treatment X Regression) are not different.

Sample SAS GLM statements:

PROC GLM;
  CLASS TREAT;
  MODEL Y = TREAT X TREAT*X / SOLUTION;
  LSMEANS TREAT;
RUN;

Note: the SOLUTION option returns the regression line parameters. LSMEANS returns the treatment means adjusted for the regression.

Compare with:

 

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Thursday, August 17, 2000