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
OUTCOME:	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.

P values

confidence intervals