# Correlation Meta-analysis

Menu location: **Analysis_Meta-Analysis_Correlation**.

This function enables you to calculate an overall correlation coefficient (r) from a set of correlations.

Two methods are used: The Hedges-Olkin method is based on a conventional summary meta-analysis with a Fisher Z transformation of the correlation coefficient (Hedges and Olkin, 1985). The Hunter-Schmidt method is effectively a weighted mean of the raw correlation coefficient (Schmidt and Hunter, 1990).

Neither of these methods is completely suitable for either a small number of studies (less than 30) or a heterogeneous set of studies (Field, 2001). The Hedges-Olkin method tends to over estimate the pooled effect whereas the Schmidt-Hunter method under-estimates it a little when the correlation is greater than 0.5. The least biased estimate of the true population correlation is provided by the Schmidt-Hunter method. The Hedges-Olkin method reduces the risk of type I error when compared with the Schmidt-Hunter method, but only when the studies are homogeneous. For heterogeneous studies you should consider consulting with a Statistician about an alternative multilevel modelling approach.

The inconsistency of results across studies is summarised in the I² statistic, which is the percentage of variation across studies that is due to heterogeneity rather than chance – see the heterogeneity section for more information.

DATA INPUT:

You enter the correlation coefficient and the sample size for each study. You may also enter a title for each study.

__Example__

The following data represent the relationship between drug misuse and delinquency. A small homogeneous set of studies is used for convenience here, and because the results are widely cited on-line across a variety of public code bases that implement the algorithms. Note that correlation studies are usually heterogeneous, in which case at least 30 studies should be used:

Correlation | Sample size |

0.51 | 131 |

0.48 | 129 |

0.30 | 155 |

0.21 | 121 |

0.60 | 111 |

0.46 | 119 |

0.22 | 112 |

0.25 | 145 |

To analyse these data in StatsDirect first prepare them in three workbook columns and label these columns appropriately. Alternatively, open the test workbook using the file open function of the file menu. Then select correlation from the meta-analysis section of the analysis menu, and then select the columns 'Correlation' and 'Sample size' as prompted.

For this example:

Study | Size | Correlation | Approximate 95% CI | % Weight (fixed, random, size) | |

1 | 131 | 0.51 | 0.370922 | 0.626703 | 12.812813, 12.609861, 12.805474 |

2 | 129 | 0.48 | 0.334935 | 0.602837 | 12.612613, 12.559901, 12.609971 |

3 | 155 | 0.30 | 0.149418 | 0.436981 | 15.215215, 13.129365, 15.151515 |

4 | 121 | 0.21 | 0.03273 | 0.37446 | 11.811812, 12.347632, 11.827957 |

5 | 111 | 0.60 | 0.465688 | 0.707292 | 10.810811, 12.050626, 10.85044 |

6 | 119 | 0.46 | 0.305281 | 0.591057 | 11.611612, 12.291224, 11.632454 |

7 | 112 | 0.22 | 0.03591 | 0.389649 | 10.910911, 12.082088, 10.948192 |

8 | 145 | 0.25 | 0.090686 | 0.396837 | 14.214214, 12.929303, 14.173998 |

__Hedges-Olkin fixed effects__

Pooled correlation = 0.383132 (95% CI = 0.329008 to 0.434748)

Z (test correlation differs from 0) = 12.760536 P < 0.0001

__Non-combinability of studies__

Cochran Q = 27.382264 (df = 7) P = 0.0003

Moment-based estimate of between studies variance = 0.023363

I² (inconsistency) = 74.4% (95% CI = 37% to 85.7%)

__Hedges-Olkin random effects__

Pooled correlation = 0.387032 (95% CI = 0.277911 to 0.486292)

Z (test Correlation) = 6.512137 P < 0.0001

__Bias indicators__

Begg-Mazumdar: Kendall's tau = 0.071429 P = 0.9049 (low power)

Egger: bias = 10.348335 (95% CI = -20.031214 to 40.727885) P = 0.4365

__Schmidt-Hunter__

Weighted mean correlation (95% CI): 0.374262 (0.278809 to 0.469714)

Test of association for weighted mean correlation: 7.684871 P < 0.0001

Observed variance across studies: 0.018974

Variance due to sampling error: 0.005828

Variance in the population correlations: 0.013146

95% Credibility interval for weighted mean correlation: 0.14954 to 0.598984

Indicators of homogeneity/heterogeneity:

- 1. Residual standard deviation (should be smaller than 1/4 WMC: 0.093565): 0.114656
- 2. Percent of observed variance accounted for by sampling error (should be at least 75%): 30.717078
- 3. Chi-square test of heterogeneity: 26.044144 P = 0.0005

Here we cannot make a confident statement about the pooled correlation coefficient because the set of studies is clearly not homogeneous and the number of studies to too small for standard correlation meta-analysis methods to be reliable.