Discussion Solution Mollusks and Litter Decomposition

Transcription

1 Discussion Solution Mollusks and Litter Decomposition. Is the rate of litter decomposition affected by the presence of mollusks? 2. Does the effect of mollusks on litter decomposition differ among the species? If so, how? This work was published as Meyer, WM, R Ostertag & RH Cowie, 2, Influence of terrestrial molluscs on litter decomposition and nutrient release in a Hawaiian rain forest. Biotropica (6): The actual design was blocked spatially, and the analysis relied on permutation tests in Primer software. Summary This experiment produced six independent sets of observations, so the appropriate analysis was by ANOVA or equivalent. The assumptions of equal variances and Normality both are somewhat shaky but I think are acceptable, given the advantages of ANOVA for the follow up analyses needed to answer the questions. Essentially similar results are obtained using nonparametric or resampling methods. The first question could be answered by either of two methods, depending on exactly how you chose to interpret the question. First, a pre planned contrast of the control to the average of the mollusk treatments would answer the question in terms of the average mollusk. This contrast indicates that decomposition was greater on average in the mollusk treatments than the control (P =.8). Alternatively, Dunnett s multiple comparisons could be used to compare each mollusk treatment to the control; these show (at overall =.) that only the L. maximus and O. alliarius treatments were significantly different from the control. The second set of questions could be answered by unplanned multiple comparisons following the ANOVA, or by ANOVA or equivalent with the control treatment excluded; any of these produces the conclusion that there are no statistically significant differences among the species effects. The conclusions, thus, are that yes, presence of mollusks increased litter decomposition, but the effects of the species were not significantly different from each other. More on species differences A few students tested contrasts among the mollusk species, for instance between the native species and the introduced species or between snails and slugs. None of these showed results anywhere close to statistically significant. Similarly, most contrasts based on the biomasses of the mollusks were not statistically significant. As I suggested in discussion, however, the species mean decomposition rates are very strongly related to their biomasses: after log transforming both variables, the correlation is r =.998, which is highly significant (P <.) even with only n =. These analyses especially the one relating mean decomposition rates to biomass were beyond what I expected you to do for the discussion, though I think they would have been appropriate if this had been your study.

2 Preliminary Analysis Design This study was designed as a completely randomized experiment with 6 treatments and 7 replicates of each. The proper analysis therefore is a comparison of multiple samples, i.e. ANOVA or equivalent, followed by some additional analyses to address the two questions more specifically. Due to the completely randomized experimental design, the observations are independent within and between samples. The choice of analysis, between ANOVA and alternatives to it, depends on the assumptions of equal variances and Normal distributions. ANOVA Assumptions Treatment N Mean StDev Minimum Q Median Q Maximum control A. intermedius Succinea D. laeve O. alliarius L. maximus Boxplot of MassLoss vs Treatment A. intermedius control D. laeve 2 MassLoss Frequency L. maximus O. alliarius Succinea 2 2 A. intermedius control D. laeve L. maximus Treatment O. alliarius Succinea MassLoss The main concern about ANOVA assumptions is that the standard deviations differ somewhat more than the text s 2 fold guideline (largest =., smallest = 2.2), though most of the Levene type tests give fairly large P values (>.). The sample sizes are equal, which enhances the robustness of ANOVA to unequal variances. Assessing Normality of the separate samples is difficult with n = 7, but there is no severe non Normality evident in the boxplots and histograms (above) or the Normal probability plots (next page left); the main concern is the low outlier in the D. laeve treatment. The distribution of the ANOVA residuals (below right) is mildly skewed with a long left tail, but the non Normality is mild enough that ANOVA is acceptable.

3 99 A. intermedius control D. laeve 9 Histogram of ANOVA Residuals 9 8 Percent 99 L. maximus O. alliarius Succinea Frequency RESI 8 MassLoss Probability Plot of ANOVA Residuals Normal Panel variable: Treatment Percent RESI Alternatives? Because the standard deviations are weakly negatively correlated with the sample means, a transformation that expands larger values might make the standard deviations more similar while also reducing the skew in the residuals; in fact squaring the loss values does slightly improve both these issues, but the effects, on these assumptions and on the results of the analysis, are so slight I would not choose to use the transformed data. Welch s ANOVA would avoid the assumption of equal variances. Similarly, nonparametric tests (Kruskal Wallis or Mood s median) could be used instead of ANOVA, on the grounds of being more robust to unequal variances. All these alternatives give results similar to the ANOVA for addressing the initial question of whether the treatments differ. Choice of method With a distribution that is only mildly non Normal, standard deviations that only mildly violate the guideline, the generally similar results from the nonparametric tests, and the advantages of the ANOVA approach for the subsequent analyses that will be needed, I feel ANOVA (on the untransformed data) is acceptable for this data.

4 ANOVA The ANOVA is statistically significant (F = 2.82, df =, 6, P value =.), so we conclude the mean loss is not the same for all treatments. Source DF SS MS F P Treatment Error Total 8.2 S =.29 R-Sq = 28.% R-Sq(adj) = 8.7% The interesting part of this question then is deciding how to most informatively compare the treatments to understand how they differ. Is the rate of litter decomposition affected by the presence of mollusks? This question could be answered in several ways, two of which I consider the best. The most powerful method is a contrast comparing the control treatment to the average of all the mollusk treatments. I would adopt a two sided alternative hypothesis in testing this contrast; a one sided test could be justified if it is known (or strongly expected) that mollusks increase decomposition, but to me it seems possible that they might decrease it, so in the absence of additional information a two sided test would be more appropriate. This contrast is statistically significant (t =.9, two sided P value =.8). Alternatively, Dunnett s comparisons could be used to compare each mollusk treatment to the control treatment. The mean decomposition rate in both L. maximus and O. alliarius treatments was significantly greater than in the control at an overall significance level of.; the other species did not differ significantly from the control.

5 An inferior way to address this question would be by Tukey multiple comparisons. Because these compare each treatment to each other, they are less powerful for the subset of pairwise comparisons pertinent to this question. As a result, only the L. maximus treatment is significantly different from the control (at overall =.). The only advantage to this approach is that it simultaneously addresses the second set of questions, about whether and how the species effects differed. Does the effect of mollusks on litter decomposition differ among the species? If so, how? The most straightforward way to address the second pair of questions would be unplanned pairwise comparisons among these treatments following the ANOVA. These could be done by Tukey comparisons among all six treatments (including the control), as shown above, or by Bonferroni comparisons among only the mollusk treatments. The latter are shown at the right, using JMP s Compare Means > Each Pair Student s t which does not adjust for multiple comparisons; to apply a Bonferroni adjustment the reported p Values would be multiplied by (the number of comparisons of interest). Either approach gives the same conclusion: none of the differences among mollusk treatments are statistically significant at an overall significance level of..

6 Alternatively, a new multi sample analysis could be performed excluding the control treatment. This could reasonably be done by ANOVA or its alternatives: Welch s, Kruskal Wallis or randomization tests. The result in all cases matches that of the Tukey or Bonferroni comparisons: there is no statistically significant evidence of a difference among the decomposition rates for the five mollusk species treatments. Choice of approaches Is it legitimate to test a contrast and perform unplanned multiple comparisons, using the same data? Yes, I think it is. The first contrast, of mollusk treatments vs. the control, clearly was preplanned and so would be legitimate no matter what. Unplanned comparisons that address a different question than the contrast also would be legitimate, in the same way that another contrast, independent of the first one, would be acceptable. By this reasoning, the contrast combined with unplanned comparisons among the mollusk treatments (but ignoring the control) would be legitimate. It would not be sensible to use both the contrast and Dunnett (or other) comparisons to address the first question. I would be hesitant to combine the Dunnett and Tukey comparisons, but since they would be addressing distinct a priori questions you might be able to justify it. My preference, from among the preceding analyses, would be the contrast (control vs. the average of the mollusk treatments) combined with unplanned comparisons among the mollusk treatments. Conclusions The conclusions are that () yes, averaging over the five species, the mollusks increased the rate of decomposition, or yes, at least two species of mollusks increased the rate of composition; and (2) no, there were not statistically significant differences among the effects of the mollusk species. Alternatives to ANOVA For those concerned primarily about unequal variances, Welch s ANOVA could be used instead of the standard ANOVA shown above. For those concerned about non Normality as well as unequal variances, either a Kruskal Wallis test followed by nonparametric multiple comparisons would be a reasonable choice. A multi group randomization test followed by two sets of pairwise (two sample) randomization tests, similar to those outlined in the preceding paragraph, also could be done, though the follow up pairwise tests would be both much more tedious and less powerful than the nonparametric multiple comparisons. Welch s ANOVA Welch s ANOVA gives a similar P value to the standard ANOVA above, and thus the same conclusion, that the mean effects of the treatments are not all the same. It is worth noting that the denominator degrees of freedom ( DFDen ) for Welch s ANOVA are much smaller than for the standard ANOVA

7 (6.62 vs. 6); Welch s uses a version of the Satterthwaite formula and the substantial differences among the sample standard deviations lead to this reduction in df. Follow up analysis would have to be by sets of pairwise two sample t tests with Bonferroni adjustments for multiple testing: one set of five comparisons between the control and the different mollusk species (similar to Dunnett comparisons) and one set of comparisons among pairs of species (similar to Tukey comparisons among these treatments). I haven t bothered to do these comparisons; I would instead use nonparametric methods. Kruskal Wallis The overall test result is similar to that of the regular or Welch s ANOVAs, with P value.88. I would use the broad interpretation of the test and conclude that the distributions are not all the same. Since there isn t a rank based analogue to preplanned contrasts, I would follow the Kruskal Wallis test with pairwise comparisons of mollusk treatments to the control treatment. JMP provides two versions of this. One of these ( Steel With Control ) uses ranks based on only the two groups being compared, while the other ( Dunn With Control for Joint Ranks ) uses the rankings of the entire data set, so that other groups, if they have values within the range of values of the two groups being compared, affect that comparison. For these data the results of these methods (top of next page) are somewhat different, but not with one being consistently more powerful than the other: using the Steel method, both L. maximus and O. alliarius are significantly different from the control, while using the Dunn method only L. maximus is, but with a smaller P value than in the Steel comparisons. The second question could be addressed by nonparametric analogues of those discussed for following ANOVA: pairwise comparisons among the five mollusk treatments using either Steel or Dunn methods, or a new Kruskal Wallis test using only the five mollusk groups (i.e. excluding the control group). The results (not shown) are similar to those shown above for ANOVA. The conclusions from the nonparametric analyses are that () yes, at least one species of mollusk increased the rate of composition; and (2) no, there were not statistically significant differences among the effects of the mollusk species.

8 Randomization test A randomization test in StatKey gives a result very similar to those of the preceding analyses, with a P value of about. (see output on next page). We would conclude that the six distributions are not all identical, that at least one tends to have larger values than one or more of the others. Follow up analyses would be by sets of two sample randomization tests with Bonferroni adjustment for multiple testing. I again have not bothered to do these comparisons, seeing no reason to prefer them to the nonparametric analyses above.

9 More on species differences More focused analyses of possible differences in effects among the mollusk species could have been based on the additional information given about them. I didn t particularly expect any such analyses or require them for full credit, but a couple of students in fact did analyze some of the following contrasts, or others similar in intent. First, contrasts could be analyzed comparing different types of mollusks. None of these is close to statistically significant: snails vs. slugs (t =.8, P =.7), native vs. introduced (t =.8, P =.), native snail vs. introduced snail (t =.8, P =.2); all P values are for two sided tests, as there is no apparent justification for one sided alternative hypotheses. Note that these three contrasts are not statistically independent, as some pairs of species have the same contrasting signs in two or more of the contrasts. Analyses also could be based on the hypothesis that the effect would be positively related to the biomass of mollusks in the different microcosms. This could be tested using any of several different sets of ANOVA contrasts. Some but not all of these do support the alternative hypothesis that decomposition rate increases with increasing biomass, but constructing these sets of contrasts was well beyond anything we did this semester so I don t show them here. A simpler approach to exploring whether species effects on the decomposition rate were related to size would be by regression. It would not be legitimate, however, to simply do a regression of mass loss against species biomass using all the observations for the mollusk treatments: these are not independent observations of litter decomposition at various mollusk biomasses, since they are grouped by mollusk species. What would be legitimate would be to calculate the mean mass loss for each mollusk species and relate these to the biomass values. To do this it makes sense first to first separate the decomposition that occurred even in the absence of mollusks from the additional decomposition attributable to the mollusks; I did this simply by subtracting the mean mass loss of the control treatment from each of the values for the other treatments. The relationship between species biomasses and mean (adjusted) decomposition is very strong but also very curvilinear. If the biomasses are log transformed, however, the relationship is remarkably linear. (Marty s comment on seeing this was that it was too good to be true!) 8 8 mean Decomposition 7 6 mean Decomposition 7 6 Biomass (mg) 2 2 log Biomass (mg) From these analyses we would conclude that the mean decomposition rates for the different mollusk species are strongly positively associated with the species biomasses (linearly related to log transformed biomass).