# Linear regression model, SCE,SCT,SCM and model's error

Could you please check if what I've done is correct? and how could I improve some of them? Thank you in advance.

Suppose I have the following data (the original data its like 20 data with decimal numbers):

$$\begin{array}{c|lcr} y &2&4&3&70&9 \\ \hline x & 0.24&21&33&11&10 \end{array}$$

1. Draw a dispersion diagram of the data.

2. Adjust a simple linear regression model.

3. Find the SSE,SCModel, SCTotal.

4. Find the model's errors.

Solution.

1. Using the instruction plot(x,y,xlab="These are x's",ylab="These are y's").

2. I think we want to find $$\beta_0$$ and $$\beta_1$$ in our segment line, that is we need to calculate $$\hat \beta_0=\frac{(\sum y_i)(\sum x_i^2)-(\sum x_iy_i)(\sum x_i)}{n\sum x_i^2-(\sum x_i)^2}$$ and $$\hat \beta_1=\frac{n(\sum x_iy_i)-(\sum x_i)(\sum y_i)}{n\sum x_i^2-(\sum x_i)^2}$$ with $$n=5$$ and substitute them in the model $$y=\beta_0+\beta_1x$$

Am I correct?

1. I think this is calculated with $$SSE=y'(I-M)y,\ where \ y=(2,4,3,70,9)',M=X(X'X)^{-1}X'$$ and X=$$\left( \begin{matrix} 1 & .24 \\ 1 & 21 \\ 1 & 33 \\ 1 & 11 \\ 1 & 10 \end{matrix}\right)$$

$$SCTotal=SSE+SSModel$$ and $$SCTotal=y'y$$ so $$SSModel$$ can be calculated with $$SCTotal-SSE$$

Am I correct?

1. Here am I asked to find $$error=y-\hat y$$ right?

How can I do that?

1. It is called scatter plot generally.
2. I would replace "Adjust" by "Fit"

Get the $$\hat\beta_1$$ first, then $$\hat\beta_0 = \bar y - \hat\beta_1 \bar x$$. Your $$\hat\beta_0$$ is complicated and I did not verify it.

1. Suppose you mixed up SC and SS. All of them should be SS. SSTotal = $$(y-\bar y)'(y-\bar y)$$

2. In the simple linear regression model $$y=\beta_0 + \beta_1 x + \epsilon$$ $$\epsilon$$ is called error or error term. Its estimate is $$y-\hat y$$ and is called residual.

The question "Find the model's error" maybe means to find something that violates the assumptions of simple linear regression. For example the linearity between $$y$$ and $$x$$.

• Hi, I was observing and I wonder why $\hat\beta_0 = \bar y - \hat\beta_1 \bar x$ ? Why the equality holds? Why did you take $\overline y$ and $\overline x$? Oct 30, 2018 at 1:13
• By least square method, you get that $\hat \beta_0$. See en.wikipedia.org/wiki/Simple_linear_regression. Which equality holds? Taking $bar y$ and $\bar x$ are easy for calculation? Oct 30, 2018 at 1:20
• Ah I see, I did not know the alternative way to calculate $\hat\beta_0: (\hat\beta_0 = \bar y - \hat\beta_1 \bar x)$. I was referring to that same equality $(\hat\beta_0 = \bar y - \hat\beta_1 \bar x)$. I think yes, $\overline x,\overline y$ are easy to calculate using R or other software. Oct 30, 2018 at 1:28
• Any other questions? My answers are understandable? Oct 30, 2018 at 1:33
• supposing $\hat β_0=5,\hat β_1=10$, then $ϵ_1=2−5−10*0.24 = -5.4$, $ϵ_2=4−5−10*21 = -211$, ... Oct 30, 2018 at 2:43