Appendix A. More methods and results of regression, correlation, and ANOVA.
In this Appendix, we present results of the pond and mesocosm experiments, more methods for the mesocosm experiment, and randomizationbased methods to deal with potential autocorrelation issues. In support of the results described in the text and here in Appendix A, we included four tables containing relevant statistical results. Three of them (Tables A1, A2, and A3) have been already described in the text and therefore receive more brief attention here. We also include three figures describing experimental findings from the perspective of accompanying ANOVA models (Table A4).
Methods: Light extinction
In the mesocosm experiment, we measured light just below water surface (I_{0}) and at 0.25 m depth (I_{0.25}); the coefficient for vertical light extinction, k(A), was then estimated as:
(A.1) 
where depth z equals 0.25 m and ln is the natural logarithm. Using this index of light extinction, k(A), we then modified Sterner et al.’s (1997) index of light (I_{M}) supply to estimate the incident light supply to these tanks; this modified index is the fraction of the sun’s light reaching the pond surface times extinction of light with depth:
(A.2) 
where I_{0} is light reaching the tank surface and I_{S} is the potential environmental light supply, i.e., that from the sun. We used the ratio, I_{0}/I_{S}, as an index of relative light supply for data from the pond survey because we were unable to measure light extinction within all ponds. For the cattle tanks, we assumed that I_{0}/I_{S} was 0.1 for shaded and 1.0 for unshaded systems.
Methods: Biovolume calculations
We received phytoplankton data from a commercial firm as counts, not biovolume. We converted these counts into fairly crude estimates of biovolume using average figures for each species found in published datasets. The main dataset stemmed from the Baltic Sea Environment Proceedings (Olenina et al. 2006). We chose intermediate size classes for each species when a range of values were reported. This dataset was supplemented with mean sizes of species collected by the Patrick Center for Environmental Research at The Academy of Natural Sciences, Philadelphia, Pennsylvania, USA (http://diatom.acnatsci.org/autecology/) and intermediate values chosen from a standard algal taxonomy guide (Prescott 1973).
Methods: “Spurious” correlation issues and ANOVAs
In analysis of pond and tank data, a difficult but important issue arose: lack of independence between variables of interest could affect the interpretation of some correlations. This “spurious” correlation issue arises in general when analyzing ratios which contain shared variables (Jackson and Somers 1991). Although some authors urge ecologists to worry less about these issues (Prairie and Bird 1989), we complement our traditional correlation analysis here with a more conservative approach, promoted by Jackson and Somers (1991). In that approach, one tests the observed correlation coefficient, r_{obs}, against the expected correlation arising from relationships between variables with shared terms. In other words, the expected correlation in some cases may not be zero, an assumption made by traditional tests of correlation coefficients. If the expected correlation, E(r), differs from zero, then the traditional test might inflate the statistical significance of the observed statistic. This problem is rectified by estimating the expected correlation coefficient using randomizations (Jackson and Somers 1991).
This randomizationbased remedy for spurious correlation sounds easier to implement in theory than reality. Some cases from our pond survey dataset were more straightforward. Tests of correlations involving patterns like X:Y vs. Y (TN:TP vs. TP, light:TP vs. TP) involve the shared term Y, while those following a X:Y vs. X (light:TP vs. light) share the X term. In both cases, we randomized (9999 times) the nonshared variable while keeping the shared variable constant (e.g., for the X/Y vs. Y relationship, we randomly shuffled X, not Y). We then estimated the expected correlation, E(r) as the average of these randomizations. The Pvalue associated with this more conservative test required tallying the number of randomizations, r_{rand,j} producing a difference from the expected correlation, , that is larger in magnitude than the difference between the observed correlation and the expected correlation, . For the correlations with shared denominators having the pattern X:Z vs. Y:Z (e.g., TN:TP vs. light:TP), we generated expected correlations and related statistics by randomly shuffling the X and Y variables.
The more challenging case involved the seston C:P vs. light:TP ratios, the core pattern of the stoichiometric light:nutrient hypothesis. In these cases, seston P is part of the whole TP (for ponds: mean proportion = 0.40, standard deviation = 0.24; for algaeonly tanks: mean = 0.79, SD = 0.30; for tanks with grazers: mean = 0.25, SD = 0.23). What is the proper term to randomize here for this partwhole problem? We chose a very conservative option: we randomly shuffled seston C and light while keeping the partwhole relationship between seston P and TP intact. This option was conservative because it completely links the two phosphorus fractions. We interpreted correlations between seston C:P and light:TP that remained significant after this procedure as particularly robust. Correlations which became “insignificant” after this test implied that at least some of the observed correlation statistic was likely driven by an autocorrelated, partwhole relationship between seston P and TP (e.g., pond data). This possibility arose for the pond dataset, where the relationship was weaker, but not for the experimental data, where the patterns were robust even to this more conservative test.
As a complementary level of analysis, we also analyzed mesocosm data by fitting ANOVA models using the design of the experiment. Significance of each factor in the ANOVA models was determined using 9999 randomizations (following Anderson 2001).
Results
Indeed, some correlations that were significant using the standard null hypothesis of no relationship became nonsignificant after accounting for autocorrelation, implying that the observed pattern may be due largely to the nonindependence of variables (Tables A1, A2, A3). An example is the negative relationship between TN:TP ratio and TP. This relationship could have arisen by chance alone given the shared variable (TP; Table A2). One should not dismiss the intuition behind this relationship too quickly, however, since the relationship between log_{10}(TP) and log_{10}(TN; slope = 0.43, 95% CIs: 0.33, 0.54; R^{2} = 0.60, P < 0.0001 for all ponds) indicated that TN accrued more slowly in ponds than TP (suggesting a drop in N:P ratio). Therefore, again, the tests for spurious correlations should be seen as providing conservative bounds on interpretation of relationships. In other words, strong relationships that may have also arisen spuriously were not necessarily meaningless (Sterner and Elser 2002).
The mesocosm experiment highlights the importance of crustacean grazers as drivers of the link between relative light:nutrient supply and algal stoichiometry. This role can be further dissected by examining both Yaxis (C:P ratios) and Xaxis (light:phosphorus ratios) components separately, using ANOVA (Table A4; Figs. A1, A2). While average C:P ratio remained flat with increases in nutrients and light in systems without crustacean grazers, it dropped in systems with those herbivores that were shaded (Fig. A1.A) or enriched with nutrients (Fig. A1.B). Additionally, a Nutrient Ratio × Nutrient Supply interaction (Fig. A1.C) revealed a complicated role for nutrient ratios. Specifically, overall C:P content of algae dropped with nutrient enrichment at lower N:P supply ratios (5:1, 14:1), but this rank order switched at the highest N:P ratio (50:1).
Grazer Composition and Nutrient Ratio treatments also interacted with Nutrient Supply and Light supply to shape relative light:nutrient ratio reaching in these systems (Table A4). These complex interactions are challenging to summarize, so we emphasize only the key points. Grazed systems had higher light:nutrient ratios than ungrazed systems in some environments: in fully lit systems (Light × Grazer Composition interactions; Fig. A2.A) and especially in highly enriched systems with 50:1 N:P ratios (Nutrient Ratio × Nutrient Supply × Grazer Composition interaction; Fig. A2.B). These grazerdriven differences in light:nutrient ratio almost certainly involved total phosphorus (TP), not the light index. We found less accumulation of TP in systems grazed by crustacean zooplankton than not grazed by them, especially at high light supply (Light × Nutrient Supply × Grazer Composition interaction; Fig. A2.C), at low nutrient supply and 5:1 and 50:1 N:P ratios (Nutrient Ratio × Nutrient Supply × Grazer Composition interaction), and in enriched, 50:1 systems (Fig. A2.D). Interestingly, in those highly enriched systems with lower N:P ratios (5:1, 14:1), TP accumulated in grazed tanks compared to ungrazed tanks (Fig. A2.D). However, the light index mainly reflected the shading treatment. The exception involved fully lit, enriched, ungrazed tanks, where selfshading by dense algal blooms dropped the light index (and created the Light × Nutrient Supply × Grazer Composition interaction; Fig. A2.E). Otherwise, shelfshading was not intense enough to alter the light index notably in these shallow systems.
Along gradients of resource supply and grazer composition, we also found shifts in species composition of algae. One can see these shifts even at the aggregate level of greens (Chlorophyta), cyanobacteria (Cyanophyta), and diatoms (Chromophyta). In general, green algae dominated most systems (Fig. A3, Table A4). Nutrient Supply × Grazer Composition and Light × Grazer Composition interactions indicated that diatoms were particularly abundant with crustacean grazers at low light and high nutrients (Fig. A3, Table A4). Meanwhile, without grazing by crustaceans, greens were particularly abundant in shaded systems (Light × Grazer Composition interaction), yet cyanobacteria were more abundant in systems receiving high rather than low nutrient supply (Nutrient Supply × Grazer Composition interaction; Fig. A3, Table A4). Finally, as nutrient ratio increased from 5:1 to 50:1, we saw increases in relative biovolume of cyanobacteria and decreases in that of greens (Nutrient Ratio effect; Fig. A3, Table A4).
TABLE A1. Results of fits of linear regression models (using ordinary least squares) to data collected from Michigan ponds (see Fig. 2). Independent variables included canopy openness (CO, dimensionless), total phosphorus (TP, mg P/m^{3}), total crustacean zooplankton biomass (g dry weight/m^{3}), and total nitrogen:total phosphorus ratio (TN:TP mg N/mg P). For each regression, the dependent variable was seston carbon:phosphorus ratio (C:P, mg C/mg P).
Indep.^{a} 
Pond 
Regression Statistics^{c} 
Correlation statistics^{d} 

Var. 
Set^{b} 
Inter. 
95% CI 
Slope 
95% CI 
r 
P 
E(r) 
P 
CO:TP 
All 
1.84 
(1.64, 2.12) 
0.13 
(0.04, 0.23) 
0.40 
0.0080 
0.40 
0.97 
< 50 
1.95 
(1.69, 2.24) 
0.18 
(0.06, 0.30) 
0.48 
0.0088 
0.46 
0.87 

CO 
All 
1.70 
(1.59, 1.81) 
0.36 
(0.16, 0.56) 
0.48 
0.0009 
0 
(N/A) 
< 50 
1.70 
(1.59, 1.83) 
0.44 
(0.18, 0.69) 
0.53 
0.0022 
0 
(N/A) 

TP 
All 
1.76 
(1.48, 2.03) 
0.11 
(0.23, 0.02) 
0.24 
0.12 
0.48 
0.0094 
< 50 
1.85 
(1.53, 2.19) 
0.16 
(0.33, 0.01) 
0.32 
0.086 
0.53 
0.030 

ZBM 
All 
1.85 
(1.63, 2.11) 
0.15 
(0.27.0.05) 
0.35 
0.023 
0 
(N/A) 
< 50 
1.97 
(1.73, 2.23) 
0.21 
(0.33,0.09) 
0.47 
0.0092 
0 
(N/A) 

TN:TP 
All 
1.28 
(1.06, 1.52) 
0.21 
(0.03, 0.40) 
0.32 
0.0320 
0.27 
0.72 
< 50 
1.23 
(0.90, 1.59) 
0.25 
(0.01, 0.50) 
0.33 
0.077 
0.25 
0.62 
^{a} Independent variables (Indep. Var.) in regression models. All variables were log_{10} transformed.
^{b} Pond set: all ponds (All, N = 44), and ponds with less than 50 percent cover by duckweeds (< 50, N = 30).
^{c} Parameter estimates of intercepts (Int.) and slopes of the regression models are accompanied by biascorrected 95% confidence intervals (95% CIs) calculated with 10,000 bootstraps.
^{d} Correlation statistics include observed Pearson correlation coefficient (r) with traditional P value (after 9999 randomizations), and expected coefficient, E(r), for relationships with shared variables (see Methods: “Spurious” correlation issues in this appendix for details) and P values testing difference between observed and expected coefficients (again, estimated with 9999 randomizations). For relationships without shared variables, the expected coefficient is zero, so no P value was calculated (“N/A” = not applicable).
TABLE A2. Correlation matrices of potential drivers of variation in algal carbon:phosphorus content in small Michigan ponds. Upper subdiagonal: standard Pearson correlation coefficients with associated probability values (in parentheses), based on 9999 randomizations (uncorrected for multiple comparisons). Lower subdiagonal: expected correlation coefficients for relationships with shared variables, calculated as the mean of 9999 randomizations, and associated P values testing the difference between observed and expected correlation coefficient (see Methods: “Spurious” correlation issues in this appendix for details). For relationships without shared variables, the expected coefficient is zero, so no P value was calculated (“N/A” = not applicable). Abbreviations: CO = canopy openness; TP = total phosphorus; ZBM = biomass of crustacean zooplankton; TN = total nitrogen. All variables were log_{10}transformed prior to analysis.
A) All ponds
CO:TP 
CO 
TP 
ZBM 
TN:TP 

CO:TP 
0.70 (<0.0001) 
0.91 (<0.0001) 
0.58 (<0.0001) 
0.81 (<0.0001) 

CO 
0.50 (0.049) 
0.33 (0.03) 
0.38 (0.0099) 
0.36 (0.016) 

TP 
0.86 (0.021) 
0 (N/A) 
0.54 (0.0001) 
0.85 (<0.0001) 

ZBM 
0 (N/A) 
0 (N/A) 
0 (N/A) 
0.49 (0.0003) 

TN:TP 
0.75 (0.22)^{*} 
0 (N/A) 
0.87 (0.17)^{*} 
0 (N/A) 
B) Ponds with less than 50% duckweed cover
CO:TP 
CO 
TP 
ZBM 
TN:TP 

CO:TP 
0.71 (< 0.0001) 
0.91 (< 0.0001) 
0.67 (0.0001) 
0.77 (< 0.0001) 

CO 
0.50 (0.044) 
0.34 (0.071) 
0.42 (0.024) 
0.35 (0.061) 

TP 
0.86 (0.014) 
0 (N/A) 
0.63 (0.0002) 
0.82 (< 0.0001) 

ZBM 
0 (N/A) 
0 (N/A) 
0 (N/A) 
0.55 (0.0019) 

TN:TP 
0.75 (0.28)^{*} 
0 (N/A) 
0.87 (0.11)^{*} 
0 (N/A) 
^{*} Results that were significant are no longer so after conservative correction for potential “spurious” correlation.
TABLE A3. Regression and correlation statistics relating algal carbon:phosphorus ratio (dependent, mg C/mg P) to the ratio of an index of light supply ( , dimensionless) to total phosphorus (mg P/m^{3}) measured in pond mesocosms supplied different nitrogen:phosphorus (N:P, mg N/mg P) ratios and that either contained or did not contain crustacean grazers.
Regression statistics^{a} 
Correlation statistics^{b} 

N:P 
Inter. (95% CI) 
Slope (95% CI) 
r_{obs} 
P 
E(r) 
P 
N^{c} 
Crustacean grazers absent 

All 
2.09 (1.84, 2.34) 
0.044 (0.06, 0.14) 
0.14 
0.41 
0.35 
0.11 
36 
5:1 
2.27 (1.92, 2.65) 
0.16 (0.02, 0.30) 
0.54 
0.078 
0.46 
0.69 
12 
14:1 
2.14 (1.82, 2.54) 
0.05 (0.09, 0.021) 
0.20 
0.54 
0.35 
0.56 
12 
50:1 
1.89 (1.48, 2.23) 
0.07 (0.22, 0.09) 
0.20 
0.52 
0.23 
0.092 
12 
Crustacean grazers present 

All 
2.23 (2.04, 2.42) 
0.24 (0.16, 0.32) 
0.80 
<0.0001 
0.44 
0.0015 
36 
5:1 
2.22 (2.06, 2.37) 
0.24 (0.18, 0.30) 
0.91 
0.0004 
0.48 
0.012 
12 
14:1 
2.60 (2.30, 2.90) 
0.35 (0.23, 0.47) 
0.86 
0.0006 
0.39 
0.030 
12 
50:1 
1.91 (1.49, 2.34) 
0.10 (0.11, 0.32) 
0.55 
0.083 
0.19 
0.24 
12 
^{a} Intercept (Inter.)and slope coefficients fit with ordinary least squares and accompanied by biascorrected, 95% confidence intervals, estimated with 10,000 bootstraps.
^{b} Observed correlation coefficient (r_{obs}) with associated P value determined using 9999 randomizations assuming no shared relationships; expected correlation, E(r) calculated by randomizing light and particulate carbon while keeping potentially shared relationship of particulate phosphorus and total phosphorus intact, with associated P values testing difference between observed r and expected r (estimated using 9999 randomizations). See Methods: “Spurious” correlation issues in this appendix for details.
^{c} Number of tank mesocosms.
TABLE A4. Summary of fits of ANOVA models to data collected from the experimental mesocosm experiment. Dependent variables involved in the light:nutrient hypothesis included log_{10}(carbon:phosphorus ratio of “edible” seston, mg C/mg P), log_{10}(ratio of light:total phosphorus, :TP, m^{3}/mg P), and index of light supply ( , dimensionless), and log_{10}(total phosphorus, TP, mg P/m^{3}). Algal groups included greens (Chlorophyta), cyanobacteria (Cyanos, Cyanophyta), and diatoms (Chromophyta). All relative abundance data (based on biovolume) were arcsinesquare root transformed prior to analysis . Results are F values where bolding and stars indicate statistical significance (*, **, and *** correspond to P < 0.05, 0.01, and 0.001, respectively) and underlining indicates P < 0.10. P values were determined using 9999 randomizations.
Treatment 
Light:nutrient variables 
Algal groups 

(df)^{a} 
Log_{10}(C:P) 
Log_{10}( :TP) 

Log_{10}(TP) 
Greens 
Cyanos 
Diatoms 
Ratio, R (2) 
5.20^{**} 
35.95^{***} 
0.01 
47.87*** 
3.77* 
5.05* 
0.04 
Supply, S (1) 
7.63^{**} 
522.1^{***} 
71.25^{***} 
515.56^{***} 
0.39 
1.02 
1.09 
Light, L (1) 
21.53^{***} 
650.39^{***} 
5615^{***} 
2.05 
0.86 
5.57* 
10.73** 
Grazing, G (1) 
45.74^{***} 
8.45^{**} 
38.45^{***} 
3.07 
0.13 
0.00 
9.86** 
R × S (2) 
8.37^{***} 
9.02^{***} 
0.44 
13.84^{***} 
0.79 
2.93 
1.75 
R × L (2) 
0.78 
0.51 
0.07 
0.29 
0.15 
0.35 
0.07 
R × G (2) 
0.43 
13.52^{***} 
3.69 
13.00^{***} 
0.70 
0.70 
0.83 
S × L (1) 
0.47 
1.25 
45.53^{***} 
1.37 
1.80 
0.04 
13.23*** 
S × G (1) 
11.25^{**} 
2.20 
24.45^{***} 
9.48^{**} 
1.20 
6.82* 
18.03*** 
L × G (1) 
9.63^{**} 
17.32^{***} 
31.36^{***} 
17.99^{***} 
7.60** 
0.08 
18.75*** 
R × S × L (2) 
0.21 
0.67 
0.20 
0.38 
1.51 
1.46 
0.23 
R × S × G (2) 
1.01 
13.05^{***} 
3.31^{*} 
10.36^{***} 
2.95 
3.89 
0.17 
R × L × G (2) 
1.73 
2.57 
3.16 
2.71 
0.54 
1.78 
0.28 
S × L × G (1) 
0.25 
3.42 
17.02^{**} 
4.23^{*} 
4.07 
1.01 
2.88 
R × S × L × G (2) 
0.58 
0.06 
1.76 
0.00 
0.28 
0.45 
0.29 
SSE^{b} (48) 
1.65 
1.53 
0.089 
1.28 
6.45 
5.54 
1.50 
R^{2} 
0.73 
0.97 
0.99 
0.94 
0.66 
0.71 
0.79 
^{a} df = degrees of freedom.
^{b} SSE = sumofsquares error.
FIG. A1. Deeper understanding of the response of carbon:phosphorus ratio of algal seston (C:P, mg C/mg P) in the mesocosm experiment, guided by significant ANOVA terms (see corresponding Table A3 in Online Appendix A for details). As seen in interactions between (A) Light supply and Grazer Composition (L × G) treatments and (B) Nutrient Supply and Grazer Composition (S × G) treatments, algal C:P decreased with more nutrients or less light in systems with zooplankton grazers, but remained comparatively unchanged in the absence of crustacean grazers. (C). A Nutrient Ratio (nitrogen:phophorus, mg N/mg P) by Nutrient Supply (R × S) interaction showed how C:P ratio tended to drop with nutrient enrichment in 5:1 and 14:1 N:P treatments but actually increased with nutrient enrichment in 50:1 N:P treatments. Points are means ± one standard error. 
FIG. A2. Further dissection of drivers of the light:phosphorus supply response in the mesocosm experiment. (A) and (B). Light:phosphorus supply; (C) and (D): total phosphorus (mg P/m^{3}); (E) light index, . Two and threeway interactions involving Light (L), Nutrient Supply (S), Nutrient Ratio (R), and Grazer Composition (G) shaped the light:phosphorus ratio (m^{3}/mg P) and its two components, total phosphorus (mg P/m^{3}) and the modified light index, dimensionless). In panels (B)–(E), “Present” and “Absent” refer to levels of the Grazer Composition factor, while “Low” and “High” refer to those of the Nutrient Supply factor. Points are means ± one standard error. 
FIG. A3. Response of the three dominant taxonomic groups of phytoplankton in the mesocosm experiment. (A). Light × Grazer Composition interactions showed that diatoms (Chromophyta) were more abundant in the presence of grazers in shaded environments (present, “Pres.”), while greens (Chlorophyta) dominated even more in shaded tanks without crustacean grazers (absent, “Abs.”). (B). Nutrient Supply × Grazer Composition interactions show a similar pattern for diatoms: diatoms were particularly abundant with crustacean grazers at high nutrient supply and particularly rare in high nutrient environments without these grazers. Meanwhile, in systems without crustacean grazing, cyanobacteria (Cyanophyta) reached highest abundance in high nutrient environments. (C). Significant effects of the nutrient ratio treatment showed increasing abundance of cyanobacteria but declining abundance of greens with higher N:P ratios. See Table A4 for supporting statistics. Points are means ± one standard error. 
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