(1) effects of grass species and variety on soil carbon dynamics (Yudai)

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1 Solutions: Discussion 13, Mixed-model Designs (1) effects of grass species and variety on soil carbon dynamics (Yudai) Four sections of the field (blocks); 8 plots per blocks. Two species, four varieties per species. Randomly assigned to plots within blocks. I would treat the blocks as a random factor, though obviously they are not a random sample from some population of possible blocks. Species and variety should be treated as fixed effects, since it is desired to make specific comparisons among them. Variety is nested in species. An important consideration is that the species effects cannot be interpreted as representing these two species in general, but rather as simply the means of these particular high yielding varieties. This could argue for not even including a species term, though I would not do this, since it probably still would be of interest to compare these four Guinea grass varieties, as a group, to these four Napier grass varieties as a group. Yudai will need to decide whether to include a block x species interaction term, or to leave it out and combine all the block interactions into the error term. There is no physical unit corresponding to this interaction, which would argue for not including it. In this case, in contrast to Bikash s below, the possibility of separating the interaction from the error arises not from there being two levels of blocking, but from there being two levels of the fixed effects. Including the interaction therefore does not substantially alter the scope of inference. I would probably not include it in this case, but again do not think the other choice is wrong. With the interaction: block R 3 error 3 species F 1 block x species variety(species) F 6 error block x species R 3 error error (= block x var) R 18 3 In the restricted model block is tested over the error; in the unrestricted model it is tested over the block x species interaction. Without the interaction: block R 3 error 4 species F 1 error variety(species) F 6 error error (= block x var) R 21 4 If there is no species x block interaction, the block term is tested over the error in both restricted and unrestricted models.

2 (2) effects of Reserve closure on fish communities (Yuko) Three sites were selected within the area of the Reserve closed to public use, and three sites within the area still open to the public. At each site four transects (25m x 5m) were randomly placed. Observations of the fish were made on each transect at the time of the closure in 2008, and every 6 months for 2 years thereafter (so at 0, 6, 12, 18, and 24 months after closure). Scope of inference: Conclusions can be drawn only about difference (and changes) at these two sites. There is no way to logically conclude that differences were caused by the closure rather than by other coincidental differences (or changes) between the two parts of the refuge. If, though, the close site changed in the expected direction while the other ( control ) site didn t, this would be circumstantial evidence that the effect was due to the closure. (This is an example of the Before-After-Control-Impact design often used to assess the effect of an unreplicated and nonrandomized environmental impact.) Analysis: Taking the two populations of interest as the two areas of the Reserve, the sites are independent replicates. (If instead we wanted to draw conclusions about closed vs. open sites generally, these would be subsamples pseudoreplicates since they all come from the same site.) Although the sites were not randomly selected from some larger set of possible sites, I think it still is reasonable to treat the site factor as random, if in fact there are other similar places within the protected and unprotected parts of the Reserve. If there are not other such places, or there is no interest in generalizing the conclusions beyond these 6 sites, site would be treated as a fixed factor. I will show the analysis for both these situations. Transects are subsamples nested within sites. Since the focus is on changes over time, time is a fixed factor of interest; the times are not simply repeat observations. Since the same transects are used every time, transect and time are crossed, and the transects in effect are blocks (or subjects) for the time effect. The design thus is repeated measure but with a twist: there are two levels of subjects, sites, and transects nested within sites. Area (closed, open) is an among-subjects factor and time is a within-subjects (repeated-measures) factor. When there are two nested levels of subjects (or of blocks, in a randomized block design), the higher level turns out to be the one that matters, in that the within-subject factor is tested over its interaction with the higher level subject (or block). with site a random factor: * area F 1 site(area) site(area) R 4 composite* transect(site(area)) R 18 error time F 4 time site area time F 4 time site time site(area) R 16 error error [= tr time] R 72 total 119

3 with site a fixed factor: * area F 1 transect site(area) F 4 transect transect(site(area)) R 18 error time F 4 error area time F 4 error time site(area) F 16 error error [= tr time] R 72 total 119 (3) effects of conservation agriculture practices in Nepal (Bikash) Four treatments: maize + millet, conventional tillage; maize + cowpea, conventional tillage; maize + millet cowpea intercropped, conventional tillage; maize + millet cowpea intercropped, strip tillage. Three villages, representing variation among villages of this tribal group. Nine farmers within each village, randomly selected. Each farmer grew one plot of each treatment. Scope of inference: If the village effect is treated as random if the villages are considered random representatives of that type of village, equivalent to a random sample of villages from the population of villages of this sort and since the farmers were randomly chosen within the villages, conclusions can be drawn about differences among treatments for all villages of this type in this region. If instead the village effect is treated as fixed, the conclusions will only refer to these three villages. I think it is reasonable and preferable to treat the villages as random effects. Analysis: This design has two levels of blocking: villages and farmers nested within villages. I would treat both block effects as random, given the stated reason for using multiple villages and the presumed intent to draw conclusions applicable to the region. The treatment factor is crossed with the village and farmer factors, and should be treated as fixed. The four treatments do have a kind of structure, but not all combinations of cover crop (millet and/or cowpea) and type of tillage were used. It therefore is not workable to model the treatments as some sort of factorial design. One choice that would need to be made is whether to include the treatment x village interaction in the model, or to combine it into the error term (which will already include the treatment x farmer interaction). I would include the interaction: for the purpose of generalizing from this study, the villages are the true replication, and asking whether the treatment differences are consistent across the villages as would be done by an F ratio of the treatment MS divided by the variability in treatment effect across villages, i.e. the interaction is an appropriate way to pose the question of interest. village R 2 farmer(village) 1 farmer(village) R 24 error treatment F 3 treatment x village treatment x village R 6 error error (=farmer x trt) R 72

4 1 the F denominator for village is farmer(village) in the restricted model, or a synthetic MS combining the farmer(village), treatment x village and error MSes in the unrestricted model. 2 In this case farmer(village) is the denominator for testing the village effect in the unrestricted model as well as the restricted model. (4) effects of fire and habitat condition on vegetation (Lisa) The study was done at three sites At each site 5 blocks were established, with large buffers between them. Within each block three plots were established Three fire treatments were applied Five permanent transects were placed within each plot. Vegetation surveys were conducted on each transects immediately prior to the first burns and then the summer after the burns (both surveys being in June). Scope of inference: As in Yuko s study above, there is only one site of each type (and these presumably were not randomly chosen out of some set of possible sites of a given type). Any site differences therefore can only be interpreted as differences among these three sites, not necessarily reflecting differences among pristine, grass-invaded, and juniper-invaded sites generally. In contrast, because the fire treatments were imposed, and were assigned to plots randomly, any differences among them can be interpreted as being the direct results of the treatments. Analysis: Because the three sites were selected for comparison, not simply to represent random spatial variability, I would treat the site factor as fixed. Blocks are nested within sites and would be treated as random effects, representing spatial variation within the site. The fire treatments are fixed effects, crossed with site and block, with the plots being the experimental units, with no replication within a block. Within a site, thus, the fire treatments and blocks constitute a standard randomized-block design without replication; the model will not contain a separate plot term, as it could not be distinguished from the fire block term. The transects are subsamples, nested within the plot. The time factor (before / after) is fixed, and crossed with all the other factors. This design could perhaps be called a 3-level repeated measures design (I just made up this n levels repeated-measures label), with two levels of subjects: blocks and transects. Sites are an among-subject factor with respect to both blocks and transects, fire is a repeated measures factor within the block subjects but among the transect subjects, and time is a repeated measures factor within both levels of subjects.

5 source F/R df F* denom site F 2 block(site block(site) R 12 composite 1* fire F 2 fire block(site) site fire F 4 fire block(site) fire block(site) [= plot] R 24 composite 2** transect(fire block(site)) R 180 error time F 1 time block(site) site time F 2 time block(site) fire time F 2 time fire blk(site) site fire time F 4 time fire blk(site) time block(site) R 12 time fire blk(site) time fire block(site) R 24 error error [= transect time] R 180 total 449 * composite 1 = fire blk(s) + time blk(site) time fire blk(s) ** composite 2 = time fire blk(s) + transect(f b(s)) error Note: It would be reasonable to omit the time block and fire time block terms, pooling them into the error; this would simplify the ANOVA somewhat and allow exact tests for block(site) and fire block(site). An alternative would be to calculate the change in a given response variable for each transect, thereby eliminating the time factor (and all its interactions). All the terms in the analysis then would represent interactions with time; since the focus of the study is the treatment time and site treatment time interactions, nothing important would be lost by this simplification. The ANOVA table would be the first 6 rows of the table above, with the transect term being left out to be the error term; the