Given a population $(X,Y)$ we hypothesize underlying population hasa regression line as follows. The conditional expectation is
$$\begin{aligned} & E(Y|x) = \beta_0 + \beta_1x & \text{PRF (1)} \end{aligned}$$
Including the error $\varepsilon$, the prediction of dependent variable would be
$$\begin{aligned} & Y = E(Y|x) + \varepsilon & \text{Prediction (2)} \end{aligned}$$
which is called simple linear regression model for population.
Random Variables: $X, Y|x, \varepsilon$
Parameters $\beta_0, \beta_1, \mu_X, \sigma_X^2, \ \ \mu_{Y|x}, \sigma_{Y|x}^2 = \sigma^2$
Questions:
- As per my understanding and as also asserted here, the predictor $X$ is fixed and known, not random. So the uncertainity is only on $Y$. So, in this case, shouldn't the equation notations be as follows?
$$\begin{aligned} & E(Y) = \beta_0 + \beta_1x & \text{PRF (3)} \end{aligned}$$
$$\begin{aligned} & Y = E(Y) + \varepsilon & \text{Prediction (4)} \end{aligned}$$
If my point 1 above is correct, then the characterization of variables and constants in play would be as follows?
- Random Variables: $Y, \varepsilon$
- Unknown Parameters: $\beta_0, \beta_1, \mu_{Y}, \sigma_{Y}^2 = \sigma^2$
- Known Parameters: $\mu_X, \sigma_X^2 $
Gung's take: As gung clarifies further here, assuming X as random complicates situation as below
$$\begin{aligned} & E(Y|x) = \beta_0 + \beta_1(x + \eta) & \text{PRF (5)} \end{aligned}$$
This implies, the standard assumption in our simple linear models (as they do not have that extra uncertainity added to slope), is that $X$ is fixed and known. So PRF is generally, $E(Y)$ not $E(Y|x)$. Is this conclusion right? How is taking $E(Y|x)$ but also $x$ as fixed, justified, as some books seem to follow that?
- My take: It depends on the experiment. Suppose below is an experiment where $X$ = width of palprebal fissure and $Y$ ocular surface area,OSA (just picked up first one I saw in Devore's).
\begin{array}{|c|c|} \hline i & 1 & 2 & 3 & 4 & 5 \\ \hline x_i & 0.40 & 0.42 & 0.48 & 0.51 & 0.57 \\ \hline y_i & 1.02 & 1.21 & 0.88 & 0.98 & 1.52 \\ \hline \end{array}
- If every time I repeat the experiment, if both $x_i$ and $y_i$ would vary, $X$ is also random and $E(Y|x)$ is more appropriate.
- If every time I repeat the experiment, if $x_i$ would be same and only $y_i$ would vary, $X$ is fixed and known and $E(Y)$ is more appropriate.
Is this conclusion right?
References (link leads to relevant page):
1. Gujarati's Basic Econometrics,
2. lecture
Update 1 - Ben's answer summary:
\begin{align}
& E(Y|x) = \beta_0 + \beta_1x & \scriptsize{X \text{ is fixed, not RV, in scope of regression} } & \checkmark \\
& E(Y) \neq \beta_0 + \beta_1x & \scriptsize{X \text{ is RV, marginal expectation, out of scope of regression} } \\
& E(Y|x) \neq \beta_0 + \beta_1x & \scriptsize{X \text{ is RV, conditional expectation, out of scope of regression? (awaiting his reply) } }\\
\end{align}