# Multivariate Regression Analysis for Identifying Key Drivers of Harmful Algal Bloom in Lake Erie
## 2 Materials and Methods
### Multivariate Regression Analysis  
Multivariate analysis is essential for categorizing environmental drivers (i.e., HABs) with similar traits and summarizing related multivariate patterns, offering valuable insights for creating targeted mitigation strategies [58; 59]. Multivariate regression analysis was performed to develop a regression model between the dependent variable and different independent variables. The Chl-a and TSS were set as the dependent variables for our study. To avoid the influence of collinearity on the regression analysis, independent variables were selected by comparing the relative importance values. A simple correlation analysis was performed to estimate the correlations between the dependent variable and independent variables. Then, multivariate regression analysis was performed using the least squares  
\begin{table}
\begin{tabular}{l l l} \hline
**Variable** & **Unit** & **Definition** \\ \hline Secchi Depth & m & Penetration depth of sunlight through the water \\ CTD Temperature & \({}^{\circ}\)C & Water temperature at site \\ CTD Specific Conductivity & \(\mu\)S/cm & Conductivity value of water at site \\ CTD Dissolved Oxygen & mg/L & Concentration of dissolved oxygen at site \\ Turbidity & NTU & Cloudiness of a fluid caused by suspended solids \\ Total Phosphorus & \(\mu\)g/L & Concentration of the sum of all phosphorus compounds that occur in \\ Total Dissolved Phosphorus & \(\mu\)g/L & Concentration of the portion of phosphorus that is dissolved at site \\ Ammonia & \(\mu\)g/L & Concentration of Ammonia at site \\ Nitrate + Nitrite & mg N/L & Concentration of NOx at site \\ Particulate Organic Carbon & mg/L & Concentration of organic carbon particles suspended in water at site \\ Particulate Organic Nitrogen & mg/L & Concentration of organic nitrogen particles suspended in water at site \\ Total Suspended Solids & mg/L & Concentration of both organic and inorganic particles suspended in \\ Chlorophyll-a & \(\mu\)g/L & Indicator of HABs \\ \hline \end{tabular}
\end{table}
Table 1: Summary of the variables used in this study.  
method. When there are \(n\) independent variables (\(X_{i}\)), the dependent variable (\(Y\)) can be described under the form of the below equation:  
\[Y=b_{0}+b_{1}X_{1}+b_{2}X_{2}+\ldots+b_{n}X_{n}+\epsilon \tag{2}\]  
From Equation (2), it can be seen that the regression coefficient, or slope, \(b_{i}\) (unbiased estimate), represents the change in \(Y\) per unit change in the (\(X_{i}\)) variable after the adjustment for simultaneous linear change; and the y-intercept, \(b_{0}\), also called the multi-regression constant, standing for the y value where the regression line crosses the y-axis. Hence, it is the value of y when the value of x is equal to 0. The last parameter \(\epsilon\) in Equation (2) represents the residuals (error term). Equation (2) can be very helpful in predicting the value of the dependent variable (\(Y\)) from the given value of the independent variables (\(X_{i}\)). It also may predict \(Y\) from the outer given ranges of (\(X_{i}\)), but such extrapolation is not highly recommended [60]. We have also herein the ANOVA-F test in our regression model to nullify our hypothesis that there is no relationship between the independent variable (\(X\)) and dependent (\(Y\)) variable, i.e., all regression coefficients equal to zero (\(b_{1}=b_{2}=\ldots=b_{n}=0\)). From the ANOVA-F test, the significant \(p\)-value (<0.05 at 95% confidence interval) suggests that the relationship between (\(X_{i}\)) and \(Y\) is crucial. The independent variables (\(X_{i}\)) can reliably predict the dependent variable \(Y\).