# Peto Odds Ratio Meta-analysis

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Case-control studies of dichotomous outcomes (e.g. healed or not healed) can by represented by arranging the observed frequencies into fourfold (2 by 2) tables. The separation of data into different tables or strata represents a sub-grouping, e.g. into age bands. Stratification of this kind is sometimes used to reduce confounding.

Peto and colleagues presented an alternative method to the usual Mantel-Haenszel method for pooling odds ratios across the strata of fourfold tables (Yusuf et al. 1985). This method is not mathematically equal to the classical odds ratio but it has come to be known as the ’Peto odds ratio’. The Peto odds ratio can cause bias, especially when there is a substantial difference between the treatment and control group sizes, but it peforms well in many situations – seek the help of a statistician if you are considering using it (Greenland and Salvan, 1990; Fleiss, 1993).

Meta-analysis is used to investigate the combination or interaction of a group of independent studies, for example a series of fourfold tables from similar studies conducted at different centres.

This StatsDirect function examines the Peto odds ratio for each stratum (a single fourfold table) and for the group of studies as a whole.

For a single stratum odds ratio is estimated as follows:

EXPOSURE: | |||

Exposed | Non-Exposed | ||

OUTCOME: | Cases: | a | b |

Non-cases: | c | d |

- where psi hat is the Peto odds ratio, n = a+b+c+d, z_{p} is the asymptotically normal test statistic, CI is the 100(1-a)% confidence interval and z_{α/2} is a quantile from the standard normal distribution. V is both weighting factor and variance for the difference between observed and expected a, O-E.

For each table, the Peto odds ratio is displayed with an approximate confidence interval. The pooled Peto odds ratio for all strata is calculated under the assumption of a fixed effects model as follows:

- where the intermediate statistics are defined as above for individual strata.

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:

Observed frequencies may be entered in a workbook (see example in relative risk meta-analysis) or directly via the screen as multiple fourfold tables:

feature present | feature absent | |

outcome positive: | a | b |

outcome negative: | c | d |

__Example__

From Fleiss (1993).

The data represent deaths after heart attack in seven different studies where the effect of giving aspirin was investigated. The data are provided in the test workbook in columns marked "Exposed total", "Exposed cases", "Non-exposed total", "Non-exposed cases" and "Study".

For this example:

Peto odds ratio meta-analysis

Study | Table (xt, xc, nt, nc) | ||||

1 | 49 | 67 | 566 | 557 | MRC-1 |

2 | 44 | 64 | 714 | 707 | CDP |

3 | 102 | 126 | 730 | 724 | MRC-2 |

4 | 32 | 38 | 285 | 271 | GASP |

5 | 85 | 52 | 725 | 354 | PARIS |

6 | 246 | 219 | 2021 | 2038 | AMIS |

7 | 1570 | 1720 | 7017 | 6880 | ISIS-2 |

Study | O-E | Odds ratio | 95% CI | Peto weight (V) | ||

1 | -8.578692 | 0.721713 | 0.492493 | 1.057619 | 26.304751 | MRC-1 |

2 | -9.540876 | 0.68386 | 0.462479 | 1.011214 | 25.107478 | CDP |

3 | -10.780024 | 0.803583 | 0.607852 | 1.062341 | 49.29715 | MRC-2 |

4 | -3.447284 | 0.80134 | 0.487603 | 1.316943 | 15.565457 | GASP |

5 | -6.258224 | 0.793516 | 0.544402 | 1.156621 | 27.058869 | PARIS |

6 | 12.986074 | 1.132558 | 0.934809 | 1.372138 | 104.323777 | AMIS |

7 | -73.755746 | 0.895032 | 0.829531 | 0.965705 | 665.092282 | ISIS-2 |

Study | z | P(two sided) | |

1 | -1.672646 | P = 0.0944 | MRC-1 |

2 | -1.904087 | P = 0.0569 | CDP |

3 | -1.535355 | P = 0.1247 | MRC-2 |

4 | -0.873768 | P = 0.3822 | GASP |

5 | -1.203085 | P = 0.2289 | PARIS |

6 | 1.271412 | P = 0.2036 | AMIS |

7 | -2.859927 | P = 0.0042 | ISIS-2 |

Pooled odds ratio = 0.896843 (95% CI = 0.840508 to 0.956954)

Z (test of odds ratio differs from 1)= -3.289276 P = 0.001

Non-combinability of studies

Cochran Q = 9.967823 (df = 6) P = 0.126

I² (inconsistency) = 39.8% (95% CI = 0% to 73.3%)

Here we can say with 95% confidence that the true population odds of death for those who received aspirin after a heart attack from this set of studies is between 0.84 and 0.96 of the same odds for those not receiving aspirin.