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Example
Section* *

Consider the following example for the difference in sample means between an inhaled steroid and montelukast in asthmatic children. The outcome variable is FEV 1 (L) from four clinical trials. Note that only the first study yields a statistically significant result (*p*-value < 0.05).

k |
\(Y_k\) | \(S_k\) | \(w_k\) | p-value |

1 |
0.070 | 0.032 | 977 | 0.028 |

2 |
0.043 | 0.049 | 416 | 0.375 |

3 |
0.058 | 0.052 | 370 | 0.260 |

4 |
0.075 | 0.041 | 595 | 0.067 |

The overall treatment effect is *Y* =

\( \dfrac{(0.070\times977)+(0.043\times416)+(0.058\times370)+(0.075\times595)}{977+416+370+595} \)

Overall effect \(Y = \dfrac{152.363}{2358} = 0.065\)

Overall estimated variance \(S^2 = \dfrac{1}{2358} = 0.0004 \left(S = 0.0206\right)\)

\(\dfrac{|Y|}{S} = \dfrac{|0.065|}{0.0206} = 3.155\), and the *p*-value = 0.002

The magnitude of the effect, or the 95% confidence interval is [0.025, 0.105]

It appears that there is a degree of homogeniety between these studies...

The statistic for testing homogeneity is *Q* = 0.303 which does not exceed 7.81, the 95^{th} percentile from the \(\chi_3^2 \) distribution. Therefore, we have further evidence that the studies are homogenous, although the small number of studies involved in this overview does not give this result very much power.

Based on the evidence presented above, we can conclude that the inhaled steroid is significantly better than montelukast in improving lung function in children with asthma.

The weighted analysis for the fixed-effects approach described previously corresponds to the following linear model for the \(k^{th}\) study, \(k = 1, 2, \dots , K\):

\(Y_k = \theta + e_k\)

where \(Y_k\) is the observed effect in the \(k^{th}\) study, \(\theta\) is the pooled population parameter of interest (difference in population treatment means, natural logarithm of the population odds ratio, etc.) and \(e_k\) is the random error term for the \(k^{th}\) study.

It is assumed that \(e_1 , e_2 , \dots , e_k\) are independent random variables with \(e_k\) having a N (0 , \(\sigma_k^2\)) distribution. The variance term \(\sigma_k^2\) then reflects intra-study variability and its estimate is \(S_k^2\). Usually, \(Y_k\) and \(S_k\) are provided as descriptive statistics in the \(k^{th}\) study report.