The RMSE represents the variance of the model errors and is an absolute measure of fit which has units identical to the response variable. The goodness of fit can be quantified using the root mean squared error (RMSE) and R-squared metrics. If the data fit well to the line, then the relationship is likely to be a real effect. This is done by investigating the variance of the data points about the fitted line. The next step is to determine whether the relationship is statistically significantand not just some random occurrence. Ggplot(aes(x = sqrt(disp), y = sqrt(mpg))) + The mpg and disp relationship is already linear but it can be strengthened using a square root transformation. Common transformations include natural and base ten logarithmic, square root, cube root and inverse transformations. If the relationship is non-linear, a common approach in linear regression modelling is to transformthe response and predictor variable in order to coerce the relationship to one that is more linear. It just indicates whether a mutual relationship, causal or not, exists between variables. Note that correlation does not imply causation. The strength of the relationship can be quantified using the Pearson correlation coefficient.

Upon visual inspection, the relationship appears to be linear, has a negative direction, and looks to be moderately strong. The slope and intercept can also be calculated from five summary statistics: the standard deviations of x and y, the means of x and y, and the Pearson correlation coefficientbetween x and y variables. The line of best fit is calculated in R using the lm() function which outputs the slope and intercept coefficients. The modelling application of OLS linear regression allows one to predict the value of the response variable for varying inputs of the predictor variable given the slope and intercept coefficients of the line of best fit. Where y is the response (dependent) variable, m is the gradient (slope), x is the predictor (independent) variable, and c is the intercept. The linear equation (or equation for a straight line) for a bivariate regression takes the following form: y = mx + c If the relationship between two variables appears to be linear, then a straight line can be fit to the data in order to model the relationship. Ordinary Least Squares (OLS) linear regression is a statistical technique used for the analysis and modelling of linear relationships between a response variable and one or more predictor variables.

Ols regression excel