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3

Great question - you've hit a great realization about PCA. Even though the important feature in the model is X2 because of the large coefficient, because X2 has such low variance, the PCA is essentially ignoring it by giving it a low weight. PCA is great for capturing the greatest variance among variables, but that is a different task than regression, ...

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Yes it makes sense, and all employees should be considered (up to the time they leave) I think you are slightly confusing the time to failure analysis: Probability of failing in each time period is independent of what happened before. However, a basic model with no time dependence fitted to eg up to 5 year employees will sooner or later churn, so your 20 ...

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You can just run as.data.frame() on the sempreds object to turn it into a data.frame. Note that this isn't actually prediction; you're estimating factor scores. This covered in any SEM textbook (e.g., Bollen 1989). Factor scores are imperfect estimates of the latent variable that can often be used in subsequent analyses or for descriptive purposes. That ...

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You can show this with a little bit of algebra. Because the statement, $y = \mu_y + (x - \mu_x)^\top\beta$ $\hspace{1cm}$(1) can be easily written in a way that gives $V_{xx}\beta = V_{xy}$. From (1), we have $E[(x-\mu_x)(y - \mu_y)] = E[(x - \mu_x)(x - \mu_x)^\top]\beta$ $\hspace{1cm}$(2) and the rest follows by construction. In other words, ...

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