```R
mean((training.data - predict(training.model))^2)
```

Code to evaluate LR using MSE in R

Implementing MSE in R

Mean Square Error (MSE) reflects the degree of difference between the estimated value and the actual value. If the predicted response value is close to the real response value, MSE will be very small. If the predicted response value is substantially different from the real response value, MSE will be very large.
$$ MSE = \frac{1}{m} \sum_{i=1}^{m} (y_m - \hat{y_m})^2 $$


University of Michigan - Ann Arbor

The extent to which the linear regression model fits the data, the quality of linear regression fit, can be achieved by:
1. $ R^2 $
2. Residual Standard Error
3. Goodnesss of Fit
4. F Statistics
5. Mean Square Error (MSE)

Assessing the accuracy of linear regression

The Residual Sum of Squares (RSS) is the sum of the squared residuals (differences between predicted and actual values) across all data points. It is defined by the formula:

$$RSS = e_1^2 + e_2^2 + \\dots + e_n^2$$

Alternatively, in simple linear regression, it can be written as:

$$RSS = \\sum_{i=1}^n (y_i - \\hat{\\beta}_0 - \\hat{\\beta}_1 x_i)^2$$

where $$\\hat{\\beta}_0$$ and $$\\hat{\\beta}_1$$ are the estimated model parameters. A smaller RSS indicates that the regression model fits the data more closely.

Residual Sum of Squares (RSS)

James, G., Witten, D., Hastie, T., & Tibshirani, R. (2013). An introduction to statistical learning (Vol. 112, pp. 3-7). New York: springer.

An Introduction to Statistical Learning with Applications in R

Residual Standard Error is an estimate of σ standard deviation. It assesses the quality of linear regression. It is the average deviation of a response from the true regression line.

RSE is considered a measure of lack of fit. So if it is large, the model is considered not to fit well whereas if RSE is small, the model may fit well.

$$ RSE = \sqrt{\dfrac{RSS}{n-2}}= \sqrt{\dfrac{\sum_{i=1}^{n} (y_i - \hat{y}_i)^2}{n-2}} $$
Where n − 2 is the degrees of freedom.

Residual Standard Error

Goodness of fit is a measure of the strength of the linear relationship between two variables. The regression line $$\hat{Y}_t = \hat{\alpha} + \hat{\beta} X_t$$ obtained by the least squares method is the best fitting line of all possible lines at least in the sense that the sum of squares of residuals is the smallest.

Goodness of Fit

R-Squared ($R^2$)—the proportion of variance explained—measures how well a prediction model for "regression" fits the given data. It is independent of the scale of $Y$.
$$0 \leq R^2 \leq 1$$
The larger the R-Squared value, the closer it is to 1, the better the regression model explains the variation of observations around its mean.
$$ R^2 = \frac{TSS-RSS}{TSS} = 1 - \frac{RSS}{TSS} = 1 - \frac{\sum_{i=1}^{n} (y_i - \hat{y}_i)^2}{\sum_{i=1}^{n} (y_i - \bar{y})^2} $$

R-Squared (Coefficient of Determination)

In simple linear regression, the F-statistic is used to test the null hypothesis $$H_0$$ that there is no relationship between the predictor and the response. If the null hypothesis is true, the F-statistic follows an F-distribution with 1 and n-2 degrees of freedom: $$ F = \frac{(TSS - RSS) / 1}{RSS / (n-2)} $$ where TSS is the total sum of squares and RSS is the residual sum of squares.

F-Statistic in Simple Linear Regression

For the simple linear regression equation $$\hat{y}=\hat{\beta}_0+\hat{\beta}_1x$$, the Residual Sum of Squares (RSS) is minimized when the coefficient estimates are $$\hat{\beta}_1=\dfrac{\sum_{i=1}^n (x_i-\bar{x})(y_i-\bar{y})}{\sum_{i=1}^n (x_i-\bar{x})^2}$$ and $$\hat{\beta}_0=\bar{y}-\hat{\beta}_1\bar{x}$$, where $$\bar{x}$$ and $$\bar{y}$$ are the sample means.

Least Squares Coefficient Estimates

Mallows' $$C_p$$ is a metric used to estimate the bias introduced into predicted responses when using an underspecified regression model. It is defined as:

$$C_p = \frac{RSS + 2d \hat{\sigma}^2}{n}$$

where $$RSS$$ is the Residual Sum of Squares, $$d$$ is the number of predictors in the model, $$n$$ is the number of observations, and $$\hat{\sigma}^2$$ is an estimate of the variance of the error associated with each response measurement (typically estimated using the full model containing all predictors). A smaller $$C_p$$ value indicates a better model with less bias.

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