# Conflicting results of Type III sum of squares in ANOVA in SAS and R

I'm analyzing data from an unbalanced factorial experiment both with SAS and R. Both SAS and R provide similar Type I sum of squares but their Type III sum of squares are different from each other. Below are SAS and R codes and outputs.

DATA ASD;
INPUT Y T B;
DATALINES;
20 1 1
25 1 2
26 1 2
22 1 3
25 1 3
25 1 3
26 2 1
27 2 1
22 2 2
31 2 3
;

PROC GLM DATA=ASD;
CLASS T B;
MODEL Y=T|B;
RUN;


Type I SS from SAS

Source  DF       Type I SS     Mean Square    F Value    Pr > F
T       1     17.06666667     17.06666667       9.75    0.0354
B       2     12.98000000      6.49000000       3.71    0.1227
T*B     2     47.85333333     23.92666667      13.67    0.0163


Type III SS from SAS

Source  DF     Type III SS     Mean Square    F Value    Pr > F
T       1     23.07692308     23.07692308      13.19    0.0221
B       2     31.05333333     15.52666667       8.87    0.0338
T*B     2     47.85333333     23.92666667      13.67    0.0163


R Code

Y <- c(20, 25, 26, 22, 25, 25, 26, 27, 22, 31)
T <- factor(x=rep(c(1, 2), times=c(6, 4)))
B <- factor(x=rep(c(1, 2, 3, 1, 2, 3), times=c(1, 2, 3, 2, 1, 1)))
Data <- data.frame(Y, T, B)
Data.lm <- lm(Y~T*B, data = Data)
anova(Data.lm)
drop1(Data.lm,~.,test="F")


Type I SS from R

Analysis of Variance Table

Response: Y
Df Sum Sq Mean Sq F value  Pr(>F)
T          1 17.067  17.067  9.7524 0.03543 *
B          2 12.980   6.490  3.7086 0.12275
T:B        2 47.853  23.927 13.6724 0.01629 *
Residuals  4  7.000   1.750
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1


Type III SS from R

Single term deletions

Model:
Y ~ T * B
Df Sum of Sq    RSS     AIC F value  Pr(>F)
<none>               7.000  8.4333
T       1    28.167 35.167 22.5751 16.0952 0.01597 *
B       2    20.333 27.333 18.0552  5.8095 0.06559 .
T:B     2    47.853 54.853 25.0208 13.6724 0.01629 *
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1


Am I missing something here? If not which one is correct Type III SS?

Type III SS depend on the parameterization used. If I set

  options(contrasts=c("contr.sum","contr.poly"))


before running lm() and then drop1() I get exactly the same type III SS as SAS does. For the R-community dogma on this issue, you should read Venables' Exegeses on linear models.

• @Peter If you think it can fit in a comment, why not. I don't think so, so why not asking a new question (and link to this one)? – chl Feb 20 '12 at 22:40
• @chl My basic point is that main effects do have meaning in the presence of interactions - they are the effect when the other variable is 0. Often this is meaningful. Not sure it's worth a whole thread. – Peter Flom Feb 21 '12 at 0:20
• I agree that there are situations where the main effects can be interpreted -- Venables takes a very strong line -- but there are lots of situations where they are difficult. I think "don't do this unless you know what you're doing" is a reasonable default setting ... – Ben Bolker Feb 21 '12 at 1:51
• Will the following reset the contrasts to the R standard? options(contrasts=c("contr.treatment", "contr.poly")) – Rasmus Larsen Oct 17 '17 at 11:17
• yes ........... – Ben Bolker Oct 17 '17 at 11:47