Designing belowground ®eld experiments with the help
of semi-variance and power analyses
John N. Klironomos
a,*, Matthias C. Rillig
b, Michael F. Allen
c aDepartment of Botany, Fungal and Soil Ecology Lab, University of Guelph, Guelph, Ont., Canada N1G 2W1 bDepartment of Plant Biology, Carnegie Institution of Washington, 260 Panama Street, Stanford, CA 94305, USAcCenter for Conservation Biology, University of California, Riverside, CA 92521-0334, USA
Received 23 March 1998; received in revised form 14 January 1999; accepted 8 February 1999
Abstract
Soil microorganisms mediate below- and aboveground processes, but it is dif®cult to monitor such organisms because of the inherent cryptic nature of the soil. Traditional `blind' sampling methods yield high sample variance. Coupled with low sample size, this results in low statistical power and thus high type II error rates. Consequently, when null hypotheses are rejected they are dif®cult to interpret further (either biologically insigni®cant or biologically signi®cant but statistically insigni®cant). To help alleviate this problem and remove the `blindness' from belowground sampling we suggest researchers perform geostatistical analyses to describe the spatial distribution of the organisms/processes coupled with power analyses to assess required sample sizes. To illustrate this we intensively sampled the soil of a 3 m10 m plot from a southern Californian chaparral ecosystem and spatially-described a series of biological and chemical parameters. We then sampled again and strati®ed the data in relation to plant location and evaluated the probability of detecting a 30% increase in abundance for each variable. Overall, we found that soil organisms do not all function at similar scales, and preliminary spatial analyses help determine which organisms are suitable for study under the scales of interest. Furthermore, the results predict that required sample sizes and type II error rates will be signi®cantly reduced for many belowground variables parameters when using a strati®ed sampling design. An understanding of how this spatial structure changes over time is also required to properly design strati®cations and avoids bias. Thus, a priori spatial- and power-analyses can be useful tools in constructing sampling-strategies for belowground ®eld studies.#1999 Elsevier Science B.V. All rights reserved.
Keywords:Soil ecosystem; Geostatistics; Power analysis; Microarthropods; Nematodes; Fungi; Bacteria; Arbuscular mycorrhiza; Soil nutrients
1. Introduction
Soil organisms strongly in¯uence nutrient cycles and primary productivity (Coleman and Crossley, 1996), and thus are important at regulating ecosystems
(Anderson, 1995). It is therefore crucial that soil organisms be evaluated in experiments that assess effects of perturbation on ecosystem functioning. Yet, our understanding of the functioning of soil organisms under ®eld settings has not advanced at the same rate as other aboveground organisms, mostly because of their cryptic nature which makes them dif®cult to monitor (Dindal, 1990).
Applied Soil Ecology 12 (1999) 227±238
*Corresponding author. Tel.: +1-519-824-4120 x6007; fax: +1-519-767-1991; e-mail: jklirono@uoguelph.ca
Soil organisms are highly aggregated (Allen and MacMahon, 1985; Klironomos et al., 1993; Smith et al., 1994; Klironomos and Kendrick, 1995). Predicting the locations of such clusters (hot spots) has not been an easy task. More recently, geostatistics have been used to describe the spatial distribution of soil organ-isms (Robertson and Freckman, 1995; Boerner et al., 1996; Robertson et al., 1997), illustrating that they are structured at various spatial scales. This is taken for granted in aboveground studies where we can visua-lize the organisms of interest; in the belowground we rarely have an a priori idea of where the organisms live within the de®ned experimental areas. As a result, researchers either sample in a random fashion, which leads to an underestimation of population sizes, increased population variance, and low statistical power, or they develop strati®ed sampling schemes based on erroneous strata weights, which leads to bias (Van Noordwijk et al., 1985).
In laboratory pot experiments, it is not uncommon to ®nd moderate to high changes in the abundance of soil organisms (: 20±100%) in response to altered
environmental conditions such as elevated atmo-spheric CO2 or nutrient deposition (Klironomos et
al., 1996, 1997; Rillig et al., 1997). However, the detection of such responses under ®eld conditions has been less frequent (O'Neill, 1994). Typically in ®eld experiments, soil ecologists take 5±10 random samples per treatment, and are often left reporting non-signi®cant treatment effects. It has traditionally been dif®cult to make any hard conclusions from such results. The combination of low sample size and highly clustered spatial distributions (lognormal data) often lead to low statistical power in below-ground monitoring programs, and thus high Type II error rates (Peterman, 1990). The classical conclusion is that soils are `well buffered' and `stable' and thus their functioning are not affected by mild or even moderate perturbation (Lucier and Haines, 1990).
This study is part of a larger study whose objectives are to assess the effects of elevated CO2on microbial
activity belowground in a Mediterranean-type ecosys-tem (Klironomos et al., 1996; Rillig et al., 1997). The CO2 study is ongoing and utilizes a Free-Air CO2
Enrichment (FACE) ring (Hendrey, 1993) to manip-ulate the atmospheric CO2 concentration within the
14-m diam ring. Using a plot next to the FACE ring, the objectives of the present study were to (1) describe
the spatial structure of various below-ground biotic and abiotic parameters, (2) relate this spatial structure to the location of the vegetation, and stratify the soil in relation to areas of high and low activity, and (3) using a separate dataset, test whether a strati®ed sampling design is potentially better than a fully randomized sampling design (i.e. reduce type II error rates) at detecting changes in the belowground.
2. Materials and methods
2.1. Study site
This study was conducted at Sky-Oaks Biological Field Station (338230N, 116
8370W) in San Diego
County, California. The site is located approximately 75 km from the Paci®c Ocean to the west and 20 km from the desert ¯oor to the east, at a 1300 m elevation. The vegetation type is chaparral, and the soil is a sandy loam (Ultic Haploxeroll). Over most of the chaparral area at Sky-Oaks Adenostoma fasciculatum,A. spar-sifoliumandCeanothus greggiiaccount for more than 70% of the plant cover. This study was performed at a portion of the site dominated by A. fasciculatum, which was burned 2 years earlier, in order to monitor above- and belowground responses to CO2
fertiliza-tion during succession in a Mediterranean-type eco-system. This study is part of a preliminary analysis prior to CO2monitoring in a Free-Air CO2Enrichment
(FACE) ring system.
2.2. Experimental design and sampling
On June 15, 1994, a 3 m10 m plot was estab-lished next to a FACE ring. Forty-four samples were taken at 1-m intervals throughout the plot and then 27 samples around each of threeA. fasciculatumshrubs at 0.125-m intervals (total of 125 soil samples). Samples were taken to a depth of 15 cm using a 10-cm diameter corer. All samples were bagged and returned to the laboratory for immediate analysis.
Roots were separated from the soil by dry sieving. They were then oven-dried at 808C for 24 h to deter-mine dry mass. Total organic matter was estimated as loss on ignition at 5508C for 6 h (Jenny, 1980).
described in Morris et al. (1997). Stained material was then viewed under UV light (620 nm), where active cells were detected as red ¯uorescence. Images under the microscope were analyzed using a computer image-analysis program. Preliminary analyses suggest that this DFS method stains living bacterial cells and active hyphal tips, making it a good indicator of microbial activity (Morris et al., 1997).
Roots were stored in formalin acetic acid alcohol (FAA) for at least 24 h. Roots were then cleared by autoclaving for 15 min in 10% potassium hydroxide, acidi®ed in FAA for 3 min and stained using Trypan Blue. Fungal infection was quanti®ed using the mag-ni®ed intersections method (McGonigle et al., 1990) by inspecting intersections between the microscope eyepiece cross-hair and roots at 200magni®cation. The proportion of root length containing arbuscules, vesicles, and hyphae was determined.
Mycorrhizal spore abundance was calculated directly by extracting AM fungal spores from soil using (a) a wet-sieving technique (Klironomos et al., 1993), and (b) differential centrifugation technique (Ianson and Allen, 1986). They were then sorted at the genus level.
Nematodes were extracted by the same wet-sieving/ sucrose centrifugation technique used for mycorrhizal spores (Klironomos et al., 1993). Microarthropods were extracted onto dishes containing ethylene glycol, using a high ef®ciency canister-type soil-arthropod extractor (Lussenhop, 1971). Collembolans and mites were sorted and counted.
Soil nutrients measured included total Kjeldahl nitrogen (Bremner and Mulvaney, 1982), ammonium, and nitrate (Keeney and Nelson, 1982); total and bicarbonate-extractable phosphorous (Olsen and Som-mers, 1982). pH was determined electrometrically (McLean, 1982). Soil moisture was calculated after heating soil in an oven at 68C for 24 h.
2.3. Statistical analyses
Parametric descriptive statistics were calculated for each of the biotic and abiotic parameters using the SPSS software (version 6.1.1, SPSS, Chicago, IL). These include the sample mean, standard deviation, as well as the skewness and kurtosis of each set of distributions. Where appropriate, data was log(x1) transformed prior to analysis.
The geostatistics software MGAP (Version 1.0, RockWare, Wheat Ridge, CO) was used to calculate semi-variograms from the ®eld data and to ®t various models. Using unweighted least squares analysis, the spherical models showed the best ®t in all cases. For variables with skewed distributions, the data was log(x1) transformed prior to analysis. Spatial dependence was de®ned as the proportion of the model sample variance (CCo) explained by the structural
variance (C). Contour maps were created using Mac-Gridzo (Version 3.3, RockWare, Wheat Ridge, CO) following ordinary block kriging.
With this spatial information, it may be possible to stratify the soil by abundance of each parameter. If the abundance of each parameter is predictable over space and time, then a strati®ed sampling design may lead to higher statistical power in ®eld experiments. On the other hand, if these distributions are not predictable, then strati®cation may lead to increased bias. It was not possible in this study to test the predictability of distributions over time (seasons throughout the year), but we could test for predictability over space. Dis-tributions for each parameter were drawn of abun-dance Vs distance from the base of the nearest A. fasciculatumshrub. With these distributions we then described the boundary conditions for each parameter using a form of boundary line analysis (Maller et al., 1983). This better illustrates the maximum perfor-mance of each parameter at different distances. Hot-spots and locations adjacent to hot-spots were identi®ed using these distributions. We considered hot-spots to be areas of peak activity with adjacent areas of similar size-interval having a reduction in activity by at least 25%. If these criteria were not met, then hot-spots were not identi®ed and strati®cation was not performed for that parameter since this would lead to bias. On June 30, 1994, we collected another 125 samples from the same site, processed them as described above, and tested the probability of detect-ing a 30% increase in abundance with strati®ed versus fully random sampling. A 30% increase relates to changes in abundance of soil organisms previously found in CO2-enrichment experiments (Klironomos et
al., 1996). Twenty random samples were chosen either from the predicted hot-spots, adjacent to the hot-spots or from the total dataset. Power analyses were per-formed on a two-meant-test using NCSS/PASS (Ver-sion 6.0, NCSS, Kaysville, UT).
Table 1
Below-ground parameters in the 3 m10 m plot at Sky-Oaks Biological Field Station
Soil variable Mean SD Min. Max. Skewness Kurtosis
Root biomass (mg kgÿ1soil) 120.3 39.9 25.0 185.0
ÿ0.639 ÿ0.367
Arbuscular infection (%) 5.6 6.0 0 22.0 0.746 ÿ0.693
Vesicular infection (%) 16.5 14.5 0 61.0 0.483 ÿ0.526
Hyphal infection (%) 54.8 24.3 7.0 99.0 0.044 ÿ1.040
Acaulosporaspores (mm3gÿ1soil) 0.017 0.029 0 0.143 2.241 4.960
Glomusspores (mm3gÿ1soil) 0.004 0.006 0 0.027 2.291 4.664
Scutellosporaspores (mm3gÿ1soil) 0.035 0.064 0 0.376 2.741 8.405
Total spores (mm3gÿ1soil) 0.056 0.069 0 0.384 2.171 5.277
Bacteria (mg kgÿ1soil) 5.9 3.4 0.9 17.7 0.945 0.791
Fungi (mg kgÿ1soil) 11.1 5.5 1.7 26.8 0.652 ÿ0.243
Microbial biomass (mg kgÿ1soil) 17.1 6.3 3.8 31.6 0.171 ÿ0.709
Nematodes (# kgÿ1soil) 279.4 227.6 0.0 956.0 0.548 ÿ0.092
Collembola (# kgÿ1soil) 100.2 317.4 0.0 1784.0 3.965 15.700
Mites (# kgÿ1soil) 207.1 458.7 0.0 2291.0 3.033 9.487
pH 6.86 0.14 6.44 7.21 ÿ0.228 0.228
% Water 1.74 0.64 0.39 5.28 1.946 7.591
% Organic matter 5.98 3.44 0.92 17.73 0.945 0.791
Total N (mg gÿ1soil) 1.22 0.44 0.73 5.20 6.040 52.861
NH4(mg kg
ÿ1soil) 2.24 0.59 1.00 3.60 ÿ0.019 ÿ0.739
NO3(mg kg
ÿ1soil) 6.05 3.51 1.00 21.50 1.070 2.057
Avail P (mg kgÿ1soil) 12.96 2.49 7.20 18.20
ÿ0.186 ÿ0.438 Total P (mg kgÿ1soil) 251.24 86.98 104.00 451.60 0.409
ÿ0.660
Table 2
Variogram model parameters for soil variables in the 3 m10 m plot at Sky-Oaks Biological Field Station
Soil variable Lag Sill Nugget Range r2 C/(CCo) s2 Sill/s2
Root biomass 13 1200 650.0 100 0.521 0.458 1669.2 0.719
Arbuscular infection 13 37.0 37.0 ± 0.316 0.0 50.3 0.735
Vesicular infection 13 0.22 0.22 ± 0.117 0.0 0.30 0.733
Hyphal infection 13 630.0 630.0 ± 0.596 0.0 655.6 0.961
Acaulosporaspores 7 111 5.0 30 0.516 0.957 125.6 0.884
Glomusspores 5 215 168.0 33 0.318 0.219 488.6 0.441
Scutellosporaspores 15 42 23.0 120 0.553 0.452 71.9 0.584
Total spores 6 750 100.0 30 0.362 0.867 848.6 0.884
Bacteria 13 8 6.0 120 0.436 0.250 10.5 0.762
Fungi 15 15 8.0 80 0.516 0.467 23.7 0.633
Microbial biomass 10 298 35.0 39 0.708 0.883 347 0.859
Nematodes 13 50000 10000.0 50 0.828 0.800 51983.6 0.962
Collembola 18 59000 30000.0 100 0.725 0.492 99897 0.591
Mites 13 120000 55000.0 80 0.494 0.542 208641 0.575
pH 13 0.02 0.01 65 0.912 0.500 0.02 1.000
% Water 13 1.52 1.52 ± 0.698 0.0 10.6 0.143
% Organic matter 52 2.80 1.45 425 0.394 0.482 7.0 0.400
Total N 10 0.38 0.12 39 0.936 0.684 0.40 0.950
NH4 15 0.19 0.10 50 0.711 0.474 0.34 0.559
NO3 52 3.00 2.57 350 0.341 0.143 12.31 0.244
Avail P 15 780 630.0 38 0.508 0.192 1045 0.746
3. Results
The biological and chemical variables analyzed in this study were not randomly distributed in the soil, and each had a unique distribution pattern (Table 1). The distributions varied greatly, from the close-to-normally distributed `soil pH', to others such as mycorrhizal spores, the two animal groups, and total soil nitrogen which were highly skewed with high levels of kurtosis. Soil nutrients and microbial popula-tions varied across 1±2 orders of magnitude through-out the plot, but soil animal populations varied across more than 3 orders of magnitude.
The proportion of the total model variance attribu-table to spatial structure [C/(CCo)] was high for
some variables but low for others (Table 2). It ranged from 0.0 for all root infection parameters and water content to > 0.80 forAcaulosporaspores, total spores, microbial biomass, and nematodes.
Some of the above variables that exhibited spatial dependence were further explored by kriged maps (Figs. 1±4). Mycorrhizal spores (Fig. 2) and micro-arthropods (Fig. 3) exhibited spatial gradients from distinctive areas of low to high activity. Therefore, samples taken close together would be more likely to be correlated than samples taken farther apart. These variables are very strongly structured spatially. Other variables such as root biomass (Fig. 1), fungal and bacterial biomass (Fig. 1), soil nutrients (Fig. 4), and nematodes (Fig. 3) showed small isolated areas of either very high or very low activity, within a larger matrix of intermediate activity. Most of the isolated points with low activity were associated with sites of concentration of rocks (data not shown).
To determine if shrub location is important in structuring biotic soil properties, we graphed the respective activity versus the distance away from the nearest A. fasciculatum shrub (Figs. 5±7). Some
Fig. 1. Population isopleths for (a) root biomass, (b) bacterial biomass, and (c) fungal biomass in the 3 m10 m plot. Black filled circles represent locations for Adenostoma fasciculatum shrubs. Biomass intervals for roots>0, >30, >60, >90, >120, >150 mg gÿ1 soil;
bacteria>0, >3, >6, >9, >12, >15 mg kgÿ1soil; fungi>0, >4.2, >8.4, >12.6, >16.8, >21 mg kgÿ1soil.
variables had distinct peaks of activity in relation to shrub location, i.e., root biomass and Acaulospora
spores, which peaked at a distance of 30±40 cm from the base of the nearest shrub. Peaks for some other parameters were not as sharp, but based on the criteria mentioned in the methods, there were distinct peaks in bacterial and nematode biomass near the base of the shrubs and fungal biomass at a distance of 50 cm. For each parameter we then attempted to stratify the data based on distance and peak activity. ``Hot-spot'' (area with peak activity) and ``adjacent-to-hotspot'' (area next to peak activity) were de®ned for the ®ve para-meters that showed peak activity at speci®c distances (root biomass, bacterial biomass, fungal biomass,
Acaulospora spores and nematodes). For the other parameters no attempt was made to stratify the data. Using strati®ed and non-strati®ed datasets, we per-formed a power analysis on a two-samplet-test, where
one mean was the sample mean, and the other was a 30% increase of that mean (Table 3). Twenty random samples were chosen to perform the test, and this was repeated three times and then averaged. With all ®ve groups, a strati®ed approach resulted in higher statis-tical power in detecting a 30% change in either the hot-spot or the area adjacent to the hot-spot, compared to using 20 random samples from the entire dataset (Table 3). Results from the power analyses were used to generate power curves for each parameter (Fig. 8). It is clear that for root biomass, strati®ed sampling does not greatly increase statistical power for any particular sample size. This is because variability is low, so power is high regardless of sample size. However, for bacterial biomass, fungal biomass,
Acaulosporaspores and nematodes, higher statistical power can be reached with a signi®cantly smaller sample size provided strati®ed sampling is used. If
Fig. 2. Population isopleths for (a)Acaulospora, (b)Glomus, and (c)Scutellosporaspore levels in the 3 m10 m plot. Black filled circles represent locations forAdenostoma fasciculatumshrubs. Spore number intervals forAcaulospora>0, >4, >8, >12, >16, >20 spores gÿ1soil;
Fig. 3. Population isopleths for (a) collembolan, (b) mite, and (c) nematode levels in the 3 m10 m plot. Black filled circles represent locations forAdenostoma fasciculatum shrubs. Intervals for collembolans>0, >100, >200, >300, >400, >500 individuals kgÿ1 soil;
mites>0, >100, >200, >300, >400, >500 individuals kgÿ1soil; nematodes>0, >110, >220, >330, >440, >550 individuals kgÿ1soil.
Table 3
Probability of detecting a 30% increase in abundance when sampling predicted hot-spots, areas next to hot-spots, and total area at random (N20)
Mean SD 30% increase of the mean Power
Root biomass
Hot-spot 162.31 15.11 211.00 1.000
Adjacent to hot-spot 114.28 21.70 148.56 0.999
Total dataset 130.02 37.28 169.03 0.911
Acaulospora biovolume
Hot-spot 0.046 0.021 0.060 0.559
Adjacent to hot-spot 0.002 0.010 0.0026 0.054
Total dataset 0.014 0.028 0.0182 0.076
Bacterial biomass
Hot-spot 7.33 3.41 9.53 0.532
Adjacent to hot-spot 5.61 2.26 7.29 0.652
Total dataset 6.20 3.38 8.06 0.413
Fungal biomass
Hot-spot 18.91 5.41 24.58 0.912
Adjacent to hot-spot 9.25 4.73 12.03 0.460
Total dataset 14.36 5.98 18.67 0.625
Nematodes
Hot-spot 450.21 139.23 585.27 0.866
Adjacent to hot-spot 313.16 122.68 407.11 0.678
we target for power0.75, then using the total dataset one would need more than 200 samples for Acaulos-poraspores, 60 samples for nematodes, 50 samples for bacterial biomass, and 35 samples for fungal biomass. Using hot-spots, and provided that the spatial distribu-tion remains constant, this is reduced to 35, 20, 35, and 20 respectively.
4. Discussion
The results from this study clearly illustrate that many biotic and abiotic variables in soil are spatially structured at scales of less than one meter. This spatial information, particularly if it can be described over time, can be used to design more statistically-powerful experiments in the ®eld. The spatial organization of soil micro-organisms and chemical cycles are linked
to that of primary producers (Allen and MacMahon, 1985; Jackson and Caldwell, 1993; Schlesinger et al., 1996; Robertson et al., 1997) so plants can be used as reference points to predict below-ground hot spots. Different groups of organisms do not co-occur within this relatively small area. Each is reacting to environ-mental conditions in different ways.
Nutrient and biotic variables were not randomly distributed in this study. As a result, random sampling without any a priori knowledge of distribution will typically lead to high sample variance and low statis-tical power and thus high type II error rates (failure to reject a null hypothesis that is false). For example, mycorrhizal fungi have been shown to respond greatly to increases in the concentration of atmospheric CO2
under laboratory conditions (O'Neill, 1994; Klirono-mos et al., 1996). Yet, this response is rarely statis-tically signi®cant in the ®eld (O'Neill, 1994). The
Fig. 4. Isopleths for (a) ammonium, (b) nitrate and (c) total nitrogen levels in the 3 m10 m plot. Black filled circles represent locations for Adenostoma fasciculatumshrubs. Intervals for ammonium>0, >0.5, >1, >1.5, >2, >2.5 mg kgÿ1 soil; nitrate>0, >3, >6, >9, >12,
present study indicates that spatial and power analyses need to be performed as a preliminary study prior to designing ®eld experiments. This may seem to be a large amount of extra work, but without this preli-minary information experiments may be doomed from the start. Furthermore, a ``spatially structured'' sam-pling design will likely reduce sample size and
increases statistical power, which would then be less labor intensive in the long run.
The present analysis also demonstrates that most of the soil variables measured here are spatially struc-tured, but the scale at which each group functions is different. As a result, independent sampling methods need to be devised for each variable. Furthermore, some variables are less suitable for ®eld monitoring because they function at scales much smaller than
Fig. 5. (a) Root, (b) bacterial and (c) fungal abundance in relation toAdenostomashrub location. Hhot-spot; AAdjacent to hot-spot. The line represents the upper boundary of maximum performance for each variable.
Fig. 6. (a)Acaulospora, (b)Glomusand (c) Scutellosporaspore abundance in relation toAdenostomashrub location. Hhot-spot; AAdjacent to hot-spot. The line represents the upper boundary of maximum performance for each variable.
those typically used by researchers. An example of this is mycorrhizal root infection. This parameter is highly dependent on root phenology (Allen, 1991; Smith and Read, 1997) and its variance was not explained by spatial autocorrelation. Thus, it was spatially expressed at scales below the minimum average separation distance used in this analysis. Yet this is the most widely used parameter to track
mycorrhizal functioning in plants (Allen, 1991; Smith and Read, 1997). Mycorrhizal fungal spores, on the other hand, are more appropriate with regards to scale, but even with a priori spatial analysis, they are still quite variable and require high sample sizes. More-over, there is still considerable debate on how to interpret mycorrhizal spores levels in soil (Allen, 1991; Klironomos et al., 1993). At the present site, fungal and bacterial biomass and nematodes proved to be very good candidates. They had relatively low variability in their distributions, and when combined with hot-spots, less than 35 samples were required to achieve adequate statistical power.
Other issues that need to be resolved are: does this spatial heterogeneity change with season? And how predictable is it among years? These questions need to be addressed if we are going to use this method to its fullest potential. It must be remembered that this study, although sampling was intensive, reports data from one location (one plot) and one point in time. Thus, the results are speci®c to the Sky-Oaks site, and furthermore it is certainly dangerous to extrapolate to other seasons and locations. Robertson and Freckman (1995) found autocorrelation with nematode trophic groups, but this was in an agronomic system, and the spatial heterogeneity was de®ned by tillage structure at a larger spatial resolution than in our present study. Also, Schlesinger et al. (1996) found very high spatial autocorrelation structure in southwestern US desert systems, but this was dependent on the present-day shrub distribution of those areas as well as the grass-lands that were present a century ago prior to invasion by the shrubs.
The below-ground system has traditionally been regarded as `out of sight, out of mind', and when investigators took samples they tended to do so hap-hazardly. We need an a priori understanding of the spatial organization of any variable in studies invol-ving below-ground systems, if there is any hope for powerful conclusions and interpretation of results. We have shown that such an approach can increase sta-tistical power and at the same time reduce sample size. The strati®ed hot-spot approach may prove very use-ful. Hot-spots may be important for ecosystem func-tion, but we are interested in changes, and these may not be most important at hot-spots, but rather at locations next to hot-spots. A small increase may stimulate areas next to hot-spots proportionally more
than at hot-spots. By knowing the location of hot-spots it is then possible to design a study where one can use random, strati®ed sampling (Krebs, 1989).
Acknowledgements
We wish to thank A. Harizanos, F. Edwards, J. Verfaille, K. Conners, and T. Zink for technical
assis-tance. This study was supported by a grant to MFA by the Department of Energy, Program for Ecosystem Research.
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