# Multivariate Regression Analysis for Identifying Key Drivers of Harmful Algal Bloom in Lake Erie
## 3 Results and Discussions
### Multivariate Regression Model Performance
In both models, the low SE values reinforce the precision and reliability of the regression models, indicating minimal deviation between the observed and predicted values. Figure 3: Bivariate scatter plots for seven selected regressors. \begin{table}
\begin{tabular}{c c c c c} \hline  & **r** & **R\({}^{2}\)** & **Adjusted R\({}^{2}\)** & **Std. Error (SE)** \\ \hline
**Model 1** & 0.986 & 0.973 & 0.972 & 0.008 \\
**Model 2** & 0.979 & 0.958 & 0.957 & 0.016 \\ \hline \end{tabular}
\end{table} 
Table 5: Model summary. In Table 6, the ANOVA table provides crucial information regarding the statistical significance of our regression models. The table presents two primary components: the regression and residual sum of squares, which are used to evaluate how well the independent variables explain the variance in the dependent variable. In Model 1, which predicts Chl-a, the F-ratio of 3250.46 is substantially larger than 1, indicating a highly significant regression model. The F-ratio represents the ratio of the mean square for the regression to the mean square for the residuals, which compares the variance explained by the model to the variance that remains unexplained. A high F-ratio suggests that the model explains a considerable amount of variation in the dependent variable. The associated \(p\)-value is reported as less than 1 \(\times\) 10\({}^{-4}\), confirming that this result is statistically significant and that the likelihood of these findings occurring by chance is extremely low. Therefore, we can conclude that the independent variables used in Model 1 are strong predictors of Chl-a, and the model is a good fit for the data. Similarly, for Model 2, which predicts TSS, the F-ratio is 2067.33, again significantly larger than 1. This high F-value suggests that the independent variables in this model also explain a substantial amount of the variability in TSS. The \(p\)-value for this model is also less than 1 \(\times\) 10\({}^{-4}\), reinforcing the statistical significance of the model. This indicates that the relationship between the independent variables and TSS is robust, and the model fits the data well. Overall, the results presented in Table 6 demonstrate that both regression models are statistically significant, meaning that the independent variables are highly effective in predicting the respective dependent variables, Chl-a and TSS. Table 7 presents the estimated coefficients for the multivariate regression models predicting Chl-a in Model 1 and TSS in Model 2. These unstandardized coefficients represent the direct impact of each independent variable on the dependent variable, holding all other predictors constant. Each coefficient in the regression equation explains how much the dependent variable is expected to increase or decrease with a one-unit change in the independent variable. For Model 1, the regression equation is as follows:  
\[\textbf{Chl-a}=0.113\;\text{Turb}\;-\;0.035\;\text{TP}\;-\;3.453\;\text{POC}+4. 115\;\text{PON}\;-\;0.03\;\text{TSS}+0.004\]  
This equation allows for the prediction of Chl-a based on the values of five independent variables: Turb, TP, POC, PON, and TSS. For Model 2, the regression equation is as follows:  
\[\textbf{TSS}=1.477\;\text{Turb}+0.196\;\text{TP}+2.125\;\text{POC}\;-2.705\; \text{PON}\;-\;0.126\;\text{Chl-a}+0.007\]  
This equation will help to predict the TSS from the given value of three independent variables (Turb, TP, POC, PON, and Chl-a). The \(p\)-values associated with each variable provide insight into the statistical significance of these coefficients. \(p\)-values shown in