# Imputing Covariance for Meta-Analysis in the Presence of Interaction
## 2 Regression with Interaction  
Suppose we aim to conduct a meta-analysis to explore the interaction between falls and age and investigate whether the effect of falls on hip fractures varies across differentage levels. (Note: This example is for illustrative purposes only and is not intended to provide any evidence on the topic.) A Poisson regression model can be utilized to examine these effects. The Poisson model is specified as follows  
\[Y_{i}\sim Poisson(\mu_{i}) \tag{1}\]  
with the following log-linear relationship  
\[\log(\mu_{i}) =\hat{\beta}_{0}+\hat{\beta}_{1}\times fall+\hat{\beta}_{2}\times (fall\times age)+\hat{\beta}_{3}\times age\] \[=\hat{\beta}_{0}+(\hat{\beta}_{1}+\hat{\beta}_{2}\times age) \times fall+\hat{\beta}_{3}\times age\]  
The effect of falls can be calculated as  
\[ \begin{split}\beta&=\log(\mu_{fall=1})-\log(\mu_ {fall=0})\\ &=[\hat{\beta}_{0}+(\hat{\beta}_{1}+\hat{\beta}_{2}\times age) \times 1+\hat{\beta}_{3}\times age]-[\hat{\beta}_{0}+(\hat{\beta}_{1}+\hat{ \beta}_{2}\times age)\times 0+\hat{\beta}_{3}\times age]\\ &=(\hat{\beta}_{1}+\hat{\beta}_{2}\times age)\end{split}  \tag{2}\]  
If exponentiating \(\beta\) yields the rate ratio (RR) for the effect. The effect of falls depends on \((\hat{\beta}_{1}+\hat{\beta}_{2}\times age)\), and its \((1-\alpha)100\%\) confidence interval can be calculated as [5]  
\[(\hat{\beta}_{1}+\hat{\beta}_{2}\times age)\pm t_{n-p-1,\alpha/2}\sqrt{var( \hat{\beta}_{1})+age^{2}\times var(\hat{\beta}_{2})+2\times age\times cov( \hat{\beta}_{1},\hat{\beta}_{2})} \tag{3}\]  
In this model, \(n\) represents the sample size, \(p\) denotes the number of predictors, and for large samples, the \(t\) score can be approximated by the \(z\) score. The variable 'age' serves as an effect modifier. For a two-stage meta-analysis, the overall effects and variances of \(\hat{\beta}_{1}\) and \(\hat{\beta}_{2}\) can be calculated using the inverse-variance weighted average method [9; 10; 11; 12]. Moreover, when investigating interactions, it is crucial to combine these two regression coefficients to construct confidence intervals for the effect of falls across different age levels. We present a method for combining coefficients and their covariance in a two-stage meta-analysis when interactions are present. Suppose we have data from 10 cohort studies. For each cohort, we fit the specified Poisson regression model and obtain the following results. In Table 1, \(\hat{\beta}_{1i}\) represents the coefficient for falls (i.e., the main effect) in the \(i\)th cohort, while \(\hat{\beta}_{2i}\) corresponds to the interaction effect of fall \(\times\) age. \(\text{var}(\hat{\beta}_{1i})\) and \(\text{var}(\hat{\beta}_{2i})\) denote the variances of \(\hat{\beta}_{1i}\) and \(\hat{\beta}_{2i}\), respectively, and \(\text{cov}(\hat{\beta}_{1i},\hat{\beta}_{2i})\) is the covariance between these coefficients. Finally, \(n_{i}\) represents the sample size for each cohort. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Cohort** & \(\hat{\beta}_{1i}\) & \(\hat{\beta}_{2i}\) & \(\text{var}(\hat{\beta}_{1i})\) & \(\text{var}(\hat{\beta}_{2i})\) & \(\text{cov}(\hat{\beta}_{1i},\hat{\beta}_{2i})\) & \(n_{i}\) \\ \hline A & \(3.0014\) & \(-0.0240\) & \(2.9419\) & \(0.0005\) & \(-0.0333\) & \(5000\) \\ B & \(1.1488\) & \(-0.0677\) & \(14.6165\) & \(0.0029\) & \(-0.2000\) & \(30,000\) \\ C & \(1.5819\) & \(-0.0936\) & \(15.8097\) & \(0.0022\) & \(-0.1825\) & \(10,000\) \\ D & \(2.0349\) & \(-0.0139\) & \(3.6954\) & \(0.0318\) & \(-0.3230\) & \(3000\) \\ E & \(-4.1219\) & \(0.0225\) & \(5.2448\) & \(0.009\) & \(-0.0629\) & \(2000\) \\ F & \(1.2506\) & \(-0.0020\) & \(11.2628\) & \(0.002\) & \(-0.144\) & \(5000\) \\ G & \(2.3383\) & \(-0.0173\) & \(5.2458\) & \(0.0009\) & \(-0.0644\) & \(1000\) \\ H & \(-3.1343\) & \(0.0483\) & \(9.0698\) & \(0.0013\) & \(-0.103\) & \(2000\) \\ I & \(1.3066\) & \(-0.0005\) & \(13.7763\) & \(0.0031\) & \(-0.2000\) & \(800\) \\ J & \(3.7753\) & \(-0.021\) & \(15.0422\) & \(0.0019\) & \(-0.1635\) & \(3200\) \\ \hline \end{tabular}
\end{table}