372 The UMAP Journal 22.4( 2001) Actual and Model growth Rates for an"Average Year 分30000 25000 15000 10000 Time(fraction of average year) Figure 3. The derivative of the population growth model, along with data for an average year at Lake A (at site 2, the most populous site). The peak height of 38,000 is the quantity that best characterizes the populations success, so it is used in the regression analysis Influence of the Environment: Multiple Regression Analysis To determine the effect of environmental conditions on growth rates, we lust correlate the peak growth rates in the logistic model with the chemical concentrations at each site. To this end, we perform a multiple regression with peak growth rate as the dependent variable and some or all of the chemical concentrations as independent variables There are only 10 data points, far fewer than needed to separate the effects of all 1l variables. Fortunately, the literature provides guidance in selecting which variables to use. The dominant factors influencing the success of a Dreissena population are the concentration of calcium and the pH. Although alkalinity seems to be somewhat important, it is included in only the first data set; more- over, it also appears to be closely correlated with calcium concentration,so we exclude it. Another marginally important factor, dissolved oxygen,was not measured in the first data set. According to the literature other chemical perform the regression on just two variables: calcium concentration andpt o factors are negligible as long as they are present in trace amounts. Thus, w The equation we obtain is maximum rate= 1687 [Ca2+]+55703 pH-454995 where the maximum growth rate is in juveniles settling per day and [Ca2+
372 The UMAP Journal 22.4 (2001) Actual and Model Growth Rates for an “Average Year” Growth Rate (juveniles/day) 0.2 0.4 0.6 0.8 1 5000 10000 15000 20000 25000 30000 35000 Time (fraction of average year) Figure 3. The derivative of the population growth model, along with data for an average year at Lake A (at site 2, the most populous site). The peak height of 38,000 is the quantity that best characterizes the population’s success, so it is used in the regression analysis. Influence of the Environment: Multiple Regression Analysis To determine the effect of environmental conditions on growth rates, we must correlate the peak growth rates in the logistic model with the chemical concentrations at each site. To this end, we perform a multiple regression with peak growth rate as the dependent variable and some or all of the chemical concentrations as independent variables. There are only 10 data points, far fewer than needed to separate the effects of all 11 variables. Fortunately, the literature provides guidance in selecting which variables to use. The dominant factors influencing the success of a Dreissena population are the concentration of calcium and the pH. Although alkalinity seems to be somewhat important, it is included in only the first data set; moreover, it also appears to be closely correlated with calcium concentration, so we exclude it. Another marginally important factor, dissolved oxygen, was not measured in the first data set. According to the literature, other chemical factors are negligible as long as they are present in trace amounts. Thus, we perform the regression on just two variables: calcium concentration and pH. The equation we obtain is maximum rate = 1687 [Ca2+] + 55703 pH − 454995, (1) where the maximum growth rate is in juveniles settling per day and [Ca2+]
A Multiple regression Model 373 is in mg/L. Thus, by measuring the concentration of Ca2+ and the ph of the water, we can predict the population growth rate Tests and refinements The population growth model fits the data surprisingly well, considering its simplicity. Although in some cases the model could be strengthened by allowing two peaks of different heights, doing so would introduce at least one more degree of freedom and thus make it difficult to perform a meaningful regression with just 10 sites. Because we are interested in the overall success or failure of the population, we accept some inaccuracy in the population model n order to set up a better regression As a first check on the model, we use it to predict the growth rates at sites 1-10 in Lake a and compared the predictions to the actual rates(fable 1) Table 1 ctual growth rates in Lake A(first data set)vs. predicted growth rates, in thousands per day Site Actua Model 12 123456789 8600 0.003 100.001 Although far from perfect, the agreement gave us confidence that the model can give at least a qualitative idea of how well a Dreissena population will do in a given calcium concentration and ph For a second test of the model, we use it to predict the minimum ph and calcium concentration tolerable to Dreissena. At a pH of 7. 7, which is typical of the data available for Lake A, the regression equation predicts that the lowest tolerable concentration of Ca2+ would be 15.4 mg/L-very close to the accepted value of 15 mg/L[McMahon 1996]. At a calcium concentration of 25 mg/L, also typical of freshwater lakes, the model predicts a minimum pH of 7. 4; this is only slightly higher than the literature value of about 7.3 Having established some confidence in our model, we test it against th econd data set for Lake A. Because this data set does not include ph, we assume that the values reported in the first data set are accurate and use them in concert with the new calcium concentrations to predict growth rates(Table 2) Although this agreement is coincidentally somewhat better than that with the first data set, we perform a new regression on both data sets at once to see
A Multiple Regression Model 373 is in mg/L. Thus, by measuring the concentration of Ca2+ and the pH of the water, we can predict the population growth rate. Tests and Refinements The population growth model fits the data surprisingly well, considering its simplicity. Although in some cases the model could be strengthened by allowing two peaks of different heights, doing so would introduce at least one more degree of freedom and thus make it difficult to perform a meaningful regression with just 10 sites. Because we are interested in the overall success or failure of the population, we accept some inaccuracy in the population model in order to set up a better regression. As a first check on the model, we use it to predict the growth rates at sites 1–10 in Lake A and compared the predictions to the actual rates (Table 1). Table 1. Actual growth rates in Lake A (first data set) vs. predicted growth rates, in thousands per day. Site Actual Model 1 12 18 2 38 28 3 15 6 4 1 10 5 30 20 6 0.002 −100 7 0.003 0.2 8 0.2 9 9 3 14 10 0.001 3 Although far from perfect, the agreement gave us confidence that the model can give at least a qualitative idea of how well a Dreissena population will do in a given calcium concentration and pH. For a second test of the model, we use it to predict the minimum pH and calcium concentration tolerable to Dreissena. At a pH of 7.7, which is typical of the data available for Lake A, the regression equation predicts that the lowest tolerable concentration of Ca2+ would be 15.4 mg/L—very close to the accepted value of 15 mg/L [McMahon 1996]. At a calcium concentration of 25 mg/L, also typical of freshwater lakes, the model predicts a minimum pH of 7.4; this is only slightly higher than the literature value of about 7.3. Having established some confidence in our model, we test it against the second data set for Lake A. Because this data set does not include pH, we assume that the values reported in the first data set are accurate and use them in concert with the new calcium concentrations to predict growth rates (Table 2). Although this agreement is coincidentally somewhat better than that with the first data set, we perform a new regression on both data sets at once to see