## 5 Methodology
### Problem Setup
#### 5.2.5 Total RE Installed Capacity/Technology vs. Total No. of Patents/Technology  
These relationships depict the connection between hydro, wind, solar, and bio installed capacities (dependent variables) and the corresponding number of patents for each technology (independent variables). The simple linear regression method is used to illustrate such relationships individually (for example, the relation between the total installed wind capacity versus the total number of wind patents and so on). The solar energy results in Brazil are excluded because all the data are reported as zero in the IRENA database over the period 2000-2016.  
All the regression analyses provided in this study were performed using the SPSS-20 software. Several tests/criteria are used to show the success of each model developed for each case, such as:  
1. The \(R^{2}\) value (coefficient of determination), which is a statistical tool that measures how the regression line fits the data and ranges from 0 to 1, with 0 representing the worst model and 1 the best model.
2. The adjusted \(R^{2}\), which is also known as \(R^{2}_{adj}\), is the rescaling of \(R^{2}\) by a degree of freedom and gives a more realistic indication of its predictive power.
3. The \(p\)-value, which is the probability of the observed result to show how the changes in the independent variables affect the dependent variable. The \(p\)-value must be \(<\)0.05 for a significant result.
4. The standard error, which is the mean distance that the observed values fall from the regression line [41].
5. The F-stats, which compare the joint effect of all the variables together. The probability must be less than 0.05 for a significant result [42].
6. Autocorrelation refers to the degree of correlation between the values of the same variables across different observations in the data. To test for the data auto correlation, the Durbin-Watson test is used. If there is no autocorrelation, the Durbin-Watson factor lies between the values 1.5 and 2.5 [43].
7. Multicollinearity in regression analysis occurs when two or more predictor variables are highly correlated with each other, such that they do not provide unique or independent information in the regression model [43]. The variance inflation factor (VIF) measures the correlation between independent variables. If the degree of correlation between variables is high enough (VIF \(>\) 5) [43], it can cause problems in the model fit [43].
8. Normality, which means that the prediction errors are normally distributed in the population [43].
9. Homoscedasticity, which means that the variance of the errors is constant in the population [43].  
In the following section, the results of the five relations (Section 5.2.1, Section 5.2.2, Section 5.2.3, Section 5.2.4 to Section 5.2.5), given above are presented and discussed. The related statistical tests are carried out to validate the established models. Only the important results, such as the model coefficients, adjusted \(R^{2}\), \(p\)-value, and F stats, are provided in the results. The application of the normality, autocorrelation, multicollinearity, and homoscedasticity tests is illustrated in one of the multiple regression cases (Section 6.3.1) only, due to space requirements.