Meta-Analysis | CI and PI in Meta-Analysis

1. Confidence Interval & Prediction Interval

A confidence interval (CI) is an interval which is expected to typically (like 95%) contain the parameter being estimated. Its counterpart is point estimation.
A prediction interval (PI) is an interval which is expected to typically (like 95%) contain a future observation, given what has already been observed.
A parameter is considered unknown but fixed. In contrast, a future observation is usually regarded as random.
In most cases, a CI is employed to estimate the mean of a random variable, whereas a PI is used to forecast the value of the next sample from that random variable.
For instance, in linear regression, a CI is constructed for $E(y_i|x_i)$ , while a prediction interval is established for $y_i|x_i$ . In this context, the PI is typically broader than the CI, assuming the same level of confidence.

In the REM case,

\hat{\theta} =(\sum_jw_j\hat{\theta_j}) / (\sum_j w_j)\\ se(\hat{\theta}) = \sqrt{1/\sum_j w_j}

CI for $\theta$ ,

\hat{\theta}\ \pm\ c\ \times\ se(\hat{\theta})

We use $\theta_k - \hat{\theta}$ to construct PI for $\theta_k$ ,

E(\theta_k - \hat{\theta}) = E(\theta+u_k) - E(\hat{\theta}) = E(u_k) = 0 \\ V(\theta_k - \hat{\theta}) = V(\theta_k) + V(\hat{\theta}) = \tau^2 + V(\hat{\theta})\\ \hat{V}(\theta_k - \hat{\theta}) = \hat{\tau}^2 + se(\hat{\theta})^2

Hence, PI for $\theta_k$ ,

\hat{\theta}\ \pm\ c\ \times\ \sqrt{\hat{\tau}^2 + se(\hat{\theta})^2}

We can see that PI for $\theta_k$ is broader than CI for $\theta$ .