Joshi, J., Matthies, D.& Schmid, B. (2000) Root hemiparasites and plant diversity in experimental grassland communities 88, 634644. Appendix 2 Skeleton analysis of variance/deviance for the measured parameters in the experiment. The treatment model consisted of the effects of the number of species (species richness), the number of functional groups, the presence of legumes and the effect of the Rhinanthus treatment and its interactions with the previously listed factors. The final model was found by backward elimination of factors of minor interest. These were defined as factors that did not reach an Fvalue > 2 . Because the different species assemblages within each diversity level were randomly selected from the species pool , their mean square was used as the appropriate error term to test the effects of the diversity treatments (number of species and number of functional groups). The interaction term Rhinanthus x assemblage served as error term to test the interaction between the effects of Rhinanthus and the diversity treatments. Because the different aspects of diversity, number of species (A) and number of functional groups (B), were partly confounded we fitted these terms in alternative sequences (in generalised linear modelling terminology the sequence A + B corresponds to "A ignoring B" and "B eliminating A" and the sequence B + A corresponds to "B ignoring A" and "A eliminating B"; the interaction AxB was always omitted from the analyses because of its low Fvalues). If the combinations of the partial factorial A crossed with B are considered as a major factor "diversity treatment", the above approach is equivalent to forming different sets of orthogonal contrasts within this major factor . Incidentally, this approach is similar to the typeIV sumofsquares approach in the statistical package SAS where contrasts within a partial factorial design are chosen arbitrarily (SAS, 1990). The preferred sequence of contrasts was in most cases B + A. Thus, for the sake of clarity we only present the results of the analyses in which the number of functional groups (B) preceded the number of species (A). The main effect of the Rhinanthus treatment was tested against the residual variation among the two quadrats within individual plots. The block and subblock effects were used to eliminate spatial variation within the experimental site.


d.f. 
Mean square 
Varianceratio 

Source of variation 
[d.f. 
[Deviance 
[approx. F ] 


change] 
change] 


Block (b) 
1 
MSb 
MSb/MSsb 

Subblock (sb) within b 
14 
MSsb 
MSsb/MSp 
Overall 
Diversity treatments (d) 
6 (9)^{1} 
MSd 
MSd/MSa 
diversity 
Number of Functional groups (f) 
2 
MSf 
MSf/MSa 
effects 
Functional groups (linear) 
1 
MSfl 
MSfl/MSa 

Functional groups (Deviation from linearity) 
1 
MSfq 
MSfq/MSa 
(between  
Species richness (s) 
4 
MSs 
MSs/MSa 
plot 
Species richness (loglinear) 
1 
MSsl 
MSsl/MSa 
analysis) 
Species richness (Deviation from loglinearity) 
3 
MSsd 
MSsd/MSa 

(Functional groups x Species richness 
(3)^{1} 
MSfs 
MSfs/MSa)^{1} 

Assemblage (a) 
25 (22)^{1} 
MSa 
MSa/MSp 

Legume contrast (l) 
1 
MSl 
MSl/MSar 

Remainder (all other species contrasts) 
24 (21)^{1} 
MSar 
MSar/MSp 

Plot (p) within b 
14 
MSp 
MSp/MSq 

Parasite treatment (r) 
1 
MSR 
MSR/MSq 
Diversity 
Diversity treatments x Parasite 
6 (9)^{1} 
MSdR 
MSdR/MSaR 
effects 
Number Functional groups x Parasite 
2 
MSfR 
MSfR/ MSaR 
depending 
Functional groups (linear) x Parasite 
1 
MSflR 
MSflR/ MSaR 
on 
Functional groups (Deviation) x Parasite 
1 
MSfdR 
MSfdR/ MSaR 
parasite 
Species richness x Parasite 
4 
MSsR 
MSsR/ MSaR 
presence 
Species richness (loglinear) x Parasite 
1 
MSslR 
MSslR/ MSaR 

Species richness (Deviation) x Parasite 
3 
MSsdR 
MSsdR/ MSaR 
(within 
(Functional groups x Species richness x Parasite 
(3)^{1} 
MSfsR 
MSfsR/ MSaR)^{1} 
plot 
Assemblage x Parasite 
25 (22)^{1} 
MSaR 
MSaR/MSq 
analysis) 
Legume contrast x Parasite 
1 
MSlR 
MSlR/MSaR 

Remainder x Parasite 
24 (21)^{1} 
MSrR 
MSrR/MSq 

Quadrat (q) within p = Residual 
29 
MSq 


Total 
120 


Notes: random effects (error model) in italic letters, fixed effects (treatment model) in roman letters; effects are always adjusted for effects that precede them.
1 Lines and values in parentheses refer to a full analysis including the interaction "functional groups x species richness". Because these interactions were never significant they were pooled with the random effect below them.