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 seeA 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