Why is Listwise Deletion Standard Error Too Small?

Question

I am going over Stef Van Burren's "Flexible Imputation of Missing Data" and don't understand why Table 1.1 from section 1.3 below gives the standard error for listwise deletion as too large.

In more detail, section 1.3.1 states "Under MCAR, listwise deletion produces standard errors and significance levels that are correct for the reduced subset of data, but that are often larger relative to all available data"

Why does listwise deletion produce standard errors that are correct for the reduced subset off data, but often larger relative to all available data? I don't see why the standard error wouldn't be the same under MCAR, as the missingness distribution is the same for the missing and observed data. Thank you!

Answer 1

Suppose we have $n$ observations on some normal random variable $X_1$ which is fully observed, and we randomly make 50% of it MCAR to form a new variable called $X_2$. We want to estimate the mean and find the standard error of the estimate. How will the standard error differ in these two variables?

You correctly note that under MCAR the distribution remains the same, so $\mu_{X_1} = \mu_{X_2}$ and $\sigma^2_{X_1}=\sigma^2_{X_2}$. Therefore $E[\bar{x}_1] = E[\bar{x}_2]$ and $E[s^2_1]=E[s^2_2]$

But, when we calculate the standard error we need to divide by the square root of the sample size. For $X_1$ that is dividing by $\sqrt{n}$. For $X_2$ we instead divide by $\sqrt{n/2}$, as the sample size has been reduced by 50% because of missingness. Therefore (in this example) the standard error of $\bar{x}_2$ will be $\sqrt{2}$ times as large as the standard error of $\bar{x}_1$.

In this example with only 1 variable there isn't anything more that can be done and we'd have to accept this limitation. But often in practice we'll have other data on the individuals, say $Z$, which is related to $X$ somehow. By using that other available data, for example by informing a multiple imputation procedure, we can improve the estimation efficiency and achieve smaller standard errors than the naive listwise deletion approach.