# A Statistical Performance Analysis of Named Data Ultra Dense Networks
## 4 Statistical Methods
### Multiple Linear Regression  
For the performance analysis of the ultra-dense network, this paper employs the MLR method [29]. MLR can often explain the relationship between a response variable and predictors by fitting a linear model to the observed experimental data. The response variable is often referred to as a dependent variable, whereas the predictor variables may be referred to as independent variables. The rationale for using MLR for the statistical analysis of an ultra-dense network is that (1) this reveals the impact magnitude of the predictor variables, such as NN, NI, RB, and RD on the response variable ISR; (2) how much the response variable changes according to the predictor variables; and (3) it predicts the future values of the response variable. For example, it may help to predict the values of ISR for different values of the factors not presented in Table 1. The linear regression model with \(n\) predictor variables \(x_{1}\), \(x_{2}\), \(\ldots\), \(x_{n}\) can be defined as follows:  
\[y_{i}=\beta_{0}+\beta_{1}x_{1i}+\beta_{2}x_{2i}+\cdots+\beta_{n}x_{ni}+\epsilon _{i}\text{ for }i\text{ }=1,2,\ldots,n \tag{4}\]  
where \(y_{i}\) is a response variable (ISR), \(x_{1i}i=1,2,\ldots,\ n\) are the predictor variables, and the \(\epsilon_{i}\) are random errors in the model. The coefficient \(\beta_{0}\) is the y-intercept at Time 0. \(\beta_{1,2,\ldots,n}\) are coefficients that measure the change in the response variable when \(x_{i(1,2,\ldots,n)}\) changes by a unit. Equation (4) treats the difference between levels as a continuous; hence, it estimates the coefficients that can be interpreted as slopes.  
In the analysis of variance (ANOVA) table for the MLR, the individual \(p\)-values were less than the significance level of 0.05, indicating that the corresponding predictor was significant. The magnitude and sign of the coefficient describe the size of the positive or negative relationship that exists between the predictor variables and the response variable. A positive sign of the coefficient estimate shows that as the value of the predictor variable increases, the mean value of the response variable also increases. A negative sign of the coefficient estimate indicates that as the predictor variable increases, the mean of response variable decreases. Moreover, the coefficient estimates can also be used to analyze the effect of an individual predictor on the response variable by holding all other predictors constant. The standard error measures the accuracy of the coefficient estimates. A small standard error shows that the observations are close to the fitted line.  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Factors** & **Levels** \\ \hline Number of Interests (NI) & 250, 500, 750, 1000 \\ Number of Nodes (NN) & 9, 25, 41, 49 \\ Router Bandwidth in Mbps (RB) & 1, 5, 10, 20 \\ Router Delay in ms (RD) & 10, 50, 100, 200 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Factors and the levels used in simulations.  
As shown in Equation (4), there could be multiple predictor variables in the model that explain the response variable. Some of these are insignificant, while others are not. The existence of insignificant or irrelevant predictors in the model decreases the precision and conceals the genuine predictor. Most regression analyses repeatedly carry out the process of removing those irrelevant predictors from the model until only significant predictors are left in the model. Any variable with a \(p\)-value less than the significance level 0.05 is significant.