Inference is to a hypothetical population, all possible measurements with this measurement protocol. Statistical inference cannot be made to the population of all lakes in the Experimental Lakes Area in Ontario. Sample consists of 52 measurement of annual production in 16 lakes (10 treated, 5 untreated, 1 treated in 1 year and untreated in 2 years, lake 228 omitted).
Setting Up the Analysis of Uncertainty Via Hypothesis Testing (Steps 4-6 in Table 15.3) for Analysis of Primary Production Scaled to Epilimnion Area in Treated and Untreated Lakes Data from Fee (1979) The residuals are homogeneous ( Figure 15.5b) and acceptably normal ( Figure 15.5c). The data (with one lake dropped) are consistent with the linearized power law, as indicated by absence of bowls or arches in the residual versus fit plot ( Figure 15.5b). This gives revised estimates of the scaling parameters ( Figure 15.5a), with new ANOVA table and diagnostic plots ( Figures 15.5b,c). The analysis is executed again (return to Step 2).
Dropping this lake strengthens the analysis but at the same time restricts the conclusions to a small range of lake sizes. However, the conclusion is suspect because of the undue influence of a single lake. It is interesting to note that with lake 228 in the analysis, the difference in exponents ( β Tr = 1.0801−0.74021 = 0.3399) is almost significant ( F 1,49 = 3.71, p = 0.0599), where the assumptions of homogeneous ( Figure 15.4b) and normal ( Figure 15.4c) errors were met. Because lake 228 differs substantially from the others in epilimnion area and because there is no untreated lake of comparable size, the lake was dropped from the analysis. Because of this tilting of the line, the data points (inside the box) do not follow the line for treated lakes. This single value, because it is far from the mean value of production and area, tilts the regression line upward on the right side in Figure 15.4a. This is the result of the undue influence of a single large lake with an epilimnion area 30 times that of the next largest lake. The residual versus fitted value graph ( Figure 15.4b) shows a pattern where most of the residuals (inside the box drawn round them) trend downward from left to right. The command generates parameter estimates, standard errors, and diagnostic plots to evaluate the model (Step 3). Once the model is written (Step 1, Box 15.8) the calculations are readily executed (Step 2) in any statistical package with a GLM command. (15.8) β A = β T r = Y e s β A + β T r * A = β T r = N o However, if we are originally on the vaporization curve and change both the temperature and pressure of the system (e.g., decrease T and increase P), we move off the vaporization curve and change the number of phases in the system (from water + steam to steam in this example).
In each case we independently varied one intensive parameter ( T or P) and the other intensive parameter responded to maintain phase equilibrium. Likewise, if we are originally on the water vaporization curve at a pressure of 150 bars and decrease the pressure of the system to 50 bars, more steam condenses to water and the system moves back onto the vaporization curve at a lower temperature and lower vapor pressure of steam. If we are originally on the vaporization curve of water and increase the temperature of the system (liquid water + steam) from 400 K to 450 K, more water vaporizes to steam and the system moves back onto the vaporization curve at a higher temperature and higher vapor pressure of steam. For example, there is only one degree of freedom at any P- T point along the vaporization curve of water in Figure 7-1. The degrees of freedom of a system are the number of intensive state variables (e.g., pressure, temperature, concentration) of the components that can be arbitrarily and independently varied without altering the number of phases in the system. In Practical Chemical Thermodynamics for Geoscientists, 2013 I Degrees of freedom