Soil Phosphorus Discussion

Transcription

1 Solution: Soil Phosphorus Discussion Summary This analysis is ambiguous: there are two reasonable approaches which yield different results. Both lead to the conclusion that there is not an independent effect of organic P on available P, but for different reasons. One approach starts with a first-order model including all three explanatory variables. Plots of residuals from this model do not indicate a need for higher-order terms. In this model the two forms of organic P have clearly non-significant relationships with available P, whether tested individually (sequentially) or together (as a chunk ). This obviously yields a negative answer to the question: there is no evidence of an effect of soil organic P on available P after accounting for the effect of inorganic P. The second reasonable approach to the analysis fits a model including the three two-way interaction terms, or at least the two including an organic form and the inorganic P. Analyzing interactions could be justified a priori (i.e. without graphical evidence) simply from a desire to fully describe the relationship of the response variable to the explanatory variables, together with realization that residual plots can fail to show significant interactions. A better argument for considering interactions arises from the question to be answered by the analysis, which concerns whether there is an effect of organic P independent of inorganic P: an interaction between organic and inorganic would be an explicitly dependent effect. In this analysis each and all of the terms including nonhydrolyzable organic P are found to be very non-significant, but the interaction of hydrolyzable organic P and inorganic P is statistically significant (at α=0.05). The conclusion in this case is not that there is no effect of organic P, but that the effect is not independent of inorganic P, but rather depends on the amount of inorganic P. Data Exploration The distributions of the four variables show no major outliers or other likely problems. (Remember that normality of these variables is not an assumption of regression analysis.) The matrix of all the pairwise scatterplots primarily shows a strong positive association between available and inorganic P. There is also a weak positive association between inorganic and hydrolyzable-organic P. Most other relationships are weak, as well as not very linear. These patterns can be quantified by pairwise correlations, as in the table below.

2 available inorg hydro_org inorg hydro_org nonhyd_org Analysis General Considerations Regression or correlation? Arguments could be made for using correlation rather than regression to analyze these data, since the study was observational and the soil variables probably had substantial measurement error. On the other hand, the question s wording ( effect of ) clearly suggests treating available P as the response variable and the forms of soil P as explanatory variables, i.e. a regression approach; regression also more easily deals with terms such as interactions. General assumptions Some assumptions must be assessed for a specific model, but a couple are general. First, we have no information about independence of the observations but it seems reasonable to assume the samples, and the measurements of the variables for each sample, were taken in ways giving independence. Second, as mentioned in the previous paragraph, the assumption of no measurement error in explanatory variables is likely not exactly true, but we have no way of knowing how badly it is violated. Analysis First-order Model Inference Regression Residual Error S = R-Sq = 52.8% The model is statistically significant: the response variable is associated with one or more of the independent variables. Constant inorg hydro_org nonhyd_org The effect of inorganic P is highly significant even after accounting for the effects of the organic forms. At least one of the latter, however, clearly can be dropped from the model; I would first drop the most nonsignificant of them, the nonhydrolyzable, and then re-test the hydrolyzable. (Remember that the t tests, being added-last tests, cannot be used to declare more than one variable nonsignificant at a single step.) Given the order in which the variables were specified in the model, the hydrolyzable term can be tested in the absence of the nonhydrolyzable term by its added-in-order SS in the preceding model.

3 Source DF Seq SS inorg hydro_org nonhyd_org The F* for hydrolyzable organic P, in a model with inorganic P but not nonhydrolyzable organic P, is (29.9/1) / = 0.186, which is very nonsignificant. The hydrolyzable organic term therefore also can be removed from the model. Alternatively, the two organic forms could be tested together in a multiple-partial ( chunk ) F test, in the context of a model also including the inorganic term. The added SS for these terms is the sum of the added-in-order SSs above: = 44.0, with 2 df. The F* is (44.0/2) / = This test again is very nonsignificant. Diagnostics Plots of residuals from the first-order model against the fitted values or the explanatory variables show no particular patterns of nonlinearity or major unevenness of variability; the smoothers do bounce up and down some, particularly in the plot with inorganic P, but I think this is to be expected with a fairly small data set such as this. (Note that I have increased the degree of smoothing in these and all other plots above the Minitab default.) The distribution of the residuals is moderately skewed, as shown below. This mild nonnormality will not have severely affected the analysis.

4 Finally, plots of the residuals against terms representing the three two-way interactions do not show any apparent trends: the smoothers bounce around some but show no distinct nonlinear trends, and the regression fits are nearly flat. Overall there do not appear to be any major problems with this model, or any suggestions of interactions. The results of the analysis therefore seem valid. Conclusion There is no relationship between available P and either form of organic P, after the effect of inorganic P has been accounted for.

5 Analysis Model with Interactions Inference Regression Residual Error S = R-Sq = 70.0% The model is statistically significant: the response variable is associated with one or more of the independent variables. Constant inorg hydro_org nonhyd_org inorgxhydro inorgxnonhyd hydroxnonhyd The interaction of inorganic and hydrolyzable-organic P is statistically significant (at α=0.05) even after accounting for all the other terms in this model. This result could be taken as answering the question, with no further analysis needed (other than diagnostic assessment of the adequacy of the model). Alternatively, it would be reasonable to try to simplify the model, to possibly gain precision in the inferences by increasing the error degrees of freedom (and possibly reducing the MSE). As a first step in reducing the model, the preceding added-last t tests show that at least one of the interaction terms involving hydrolyzable organic P can be removed; it would be sensible to start with the most nonsignificant one, the interaction between the two organic forms. After removing that term, the interaction of nonhydrolyzable organic with inorganic could be tested, either by running the simplified model, or using the added-in-order SS from the preceding model, as given below: Source DF Seq SS inorg hydro_org nonhyd_org inorgxhydro inorgxnonhyd hydroxnonhyd The F* for the interaction of nonhydrolyzable organic and inorganic P is F* = (86.7/1) / = 0.65, which is clearly nonsignificant. This interaction therefore also can be removed from the model. The remaining interaction, between hydrolyzable organic P and inorganic P, however, is statistically significant in the model obtained by removing the other two interactions. From the preceding added-in-order SSs, its partial F* is (671.9/1) / = With 1 and 10 df, this gives a P-value of This and the preceding F* test, however, used the MSE from the full

6 model above. Since the main reason for simplifying the model was to try to obtain a better estimate of the error variance, it would make sense to incorporate the two excluded terms into the error term; this could be done by hand using the added-in-order SSs but it is easier to re-run the model. Regression Residual Error S = R-Sq = 68.0% The model now is considerably more significant, and the MSE is indeed a good bit smaller, so that the added-last test for the interaction (which uses the same added-ss as in the preceding F* test) now has a smaller P-value. Constant inorg hydro_org nonhyd_org inorgxhydro A reasonable final step in simplifying the model would be to remove nonhydrolyzable organic P, since it is clearly nonsignificant in this model, and not involved in the interaction. Results for the model with only inorganic P, hydrolyzable organic P, and their interaction, are as follows. Regression Residual Error Constant inorg hydro_org inorgxhydro The model is highly significant. We conclude from it that there is a highly significant positive effect of soil inorganic P on available P, and that the strength of that effect lessens with increasing soil hydrolyzable organic P (or equivalently, that there is a positive effect of hydrolyzable organic P on available P when there is little inorganic P but that this effect decreases with higher levels of inorganic P). Diagnostics The validity of the full model used at the start of this analysis, with all three explanatory variables and all three two-way interactions, will be assessed; if it is acceptable, the reduced models derived from it will be also.

7 Plots of the residuals against the fitted values and the explanatory variables (including interaction terms), as shown below, do not show any important patterns of nonlinearity or uneven variance. The distribution of the residuals is moderately skewed, but this degree of nonnormality would not substantially affect, or invalidate, the conclusions. This analysis therefore appears to be valid: the model is an adequate representation of the data. Conclusion The amount of hydrolyzable organic P in the soil does affect the available P, but not independently of inorganic P: the effects of the two forms interact, such that the effect of either one is greater at lower levels of the other, and decreases as the other increases. Perhaps they are somewhat substitutable for each other? Overall the effect of inorganic P appears to be much stronger than that of hydrolyzable organic P.