### FITTED REGRESSION LINE AND RANDOM ERROR

_{0}+β

_{1}X+ε. The regression model is divided into two parts, the fitted regression line, ‘β

_{0}+β

_{1}X’ and random error, ‘ε.’ The need of error term is justified by the gap between the line and observed dots because regression line does not go through all the observed values as appeared in Figure 1A.

_{1}to χ

_{5}in Figure 1B. Please find the conceptual distribution of Y, which corresponds with subgroup of χ

_{1}, lying on upper vertical direction. The distribution is displayed as a bell-shaped normal distribution with a mean, μ

_{y|χ1}, at the center. The symbol, μ

_{y|χ1}means expected population mean of Y when X variable has the value χ

_{1}. In accordance with the previous sections on regression, the expected mean of Y can be symbolized as Ŷ, which is equal to the fitted line, β

_{0}+β

_{1}X’.

_{y|χ1}to μ

_{y|χ5}as X changes from χ

_{1}to χ

_{5}. The conceptual model suggests that the expected mean of Y can be depicted as the straight line ‘β

_{0}+β

_{1}X’ by connecting the expected mean of Y matched with subgroups of X. We call the straight line as ‘mean function’ because the expected mean of Y is expressed as the function of ‘β

_{0}+β

_{1}X’. Please clearly understand that there are numerous means by numerous subgroups of continuous X, and they are linearly connected to make the linear mean function, ‘β

_{0}+β

_{1}X’. Therefore, we should be able to reasonably assume that the mean function of Y has the form of fitted regression line when we apply the SLR model.

*e.g.*, μ

_{y|χ1},..., μ

_{y|χ5}. What would remain after removing the conditional means? Like Figure 1C, after removing means, the mean of difference in all the subgroups will be changed to zero while the shape of distribution will remain unchanged. There will be an identical distribution of random errors with mean of zeros for all the values of X.

^{2}). The equation means that the distribution of errors follows the normal distribution with mean of zero and variance of a constant, σ

^{2}. What does the constant variance tell us? The constant variance shows that all the error distribution for all the subgroups have the same variance. In other words, the shape of the error terms is the same for all the subgroups as shown in Figure 1C.

_{0}+β

_{1}X+ε, the distribution of Y for each subgroup is Y~N(β

_{0}+β

_{1}X, σ

^{2}) for a given X. According to the definition, the distribution of subgroup for the given value of χ

_{1}is a normal distribution with the mean of β

_{0}+β

_{1}χ

_{1}and with the constant variance of σ

^{2}.

*) which locates on the line, and the observed value Y*

_{i}*for the*

_{i}*i*

^{th}observation in Figure 1A. An observed residual, e

*, is represented as Y*

_{i}*−Ŷ*

_{i}*. If the linear regression model is correctly applied to the observed data, the observed errors from the actual data should be in accordance with the assumption on the distribution of random error. In addition, if the fitted line correctly represents the mean response, the means of residuals for all the subgroups should be near zero and the shapes of residual distributions by subgroups follow the assumed normal distribution, similar to the conceptual error distribution in Figure 1C.*

_{i}### FOUR BASIC ASSUMPTIONS AND RESIDUALS

### 1. Linearity

*etc*., which means that residual and X variable are unrelated. In other words, the residuals appear randomly scattered in relation to X. Also, most observations should lie near the regression line, while observations far away from the line are less frequent, according to the characteristic of assumed normal distribution.

### 2. Independence

### 3. Normality

### 4. Equality of variance

^{2}, as shown in Figure 1C. In SLR, we can plot observed residuals against X, because the fitted value and X are linearly related, as shown in Figure 1C. However, in multiple linear regression with more than one X, we need to consider a scatter plot of residuals and fitted values, Ŷ, which is a linear combination of Xs.