Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 35 / 2

Three distinct benefits of a meta-analysis are greater precision, an explicit synthesis of the effect, and assessment of variation across studies.

Combining compatible independent estimates can reduce the standard error, improve statistical power and provide more informative confidence intervals than individual small trials. A prespecified pooled estimate summarises the direction and magnitude of the evidence quantitatively, instead of informally counting statistically significant results. Finally, between-study heterogeneity and subgroup comparisons can reveal whether effects vary with populations or interventions, helping assess generalisability. These benefits require comparable estimands and an appraisal of study validity; pooling cannot repair systematic within-study bias.

The event and nonevent cell counts are $(a,b)=(30,60)$ and $(c,d)=(20,40)$. Thus the estimated odds ratio and log odds ratio areFor independent binomial groups, the delta method givesUsing the permitted normal approximation quantile of 2, the approximate 95% interval isThe observed odds of death are equal between groups. The confidence interval includes zero and allows appreciable effects in either direction; it does not demonstrate equivalence. Exponentiating gives an odds ratio interval approximately $(0.493,2.028)$, so the data are compatible with roughly half to twice the death odds for the first treatment relative to the second. The tabulated standard error is rounded.

Use $w_i=1/v_i$ for the inverse within-study variances, and $m=21$. For Cochran's Q statistic, the DerSimonian–Laird estimator of the between-study variance isConsequently the estimated heterogeneity variance and standard deviation areThese describe dispersion of underlying study log odds ratios, not ordinary sampling error of a single trial. The between-study heterogeneity is additional to the within-study variances.

The I-squared statistic isIt estimates the share of variation beyond that expected from sampling error in this collection of studies. Approximately half the variation is attributed to heterogeneity on that scale; this is neither the percentage of trials with different effects nor a percentage change in mortality. The PDF omits the symbol naming its heterogeneity estimate, so both ${\hat{\tau }}^2$ and $\hat{\tau }$ are reported explicitly.

A fixed-effect meta-analysis assigns inverse-variance weights. Using the unrounded within-study variance from part (b), this trial has $w=8$, so its percentage weight isUsing the printed rounded standard error of $0.35$ instead gives approximately $2.04\%$; the raw cell calculation yields the more precise answer.

For a random-effects meta-analysis, the corresponding weight and percentage weight areWith ${\hat{\tau }}^2=0.05$, its unnormalised weight is $40/7\approx 5.714$. The supplied sums of fixed-effect weights do not determine the denominator of the random-effects percentage; the individual $v_i$ are needed. Positive heterogeneity makes the weight distribution more even. Percentage weight measures the trial's contribution to a pooled mean with fixed weights; total influence can also involve re-estimation of heterogeneity.

Within-study bias is a systematic displacement of a study's effect estimate from its intended true effect because of its design, conduct, analysis or reporting. Examples include faulty allocation concealment, differential outcome assessment without suitable blinding, informative loss to follow-up, deviations from the intended treatment analysis, and selective reporting of outcomes or analyses. This differs from chance sampling error; the bias need not diminish as a study becomes larger.

It can shift a pooled meta-analysis effect and can create or obscure between-study heterogeneity. A random-effects meta-analysis accommodates dispersion, not systematic invalidity. Publication or non-inclusion of whole studies is a separate selection problem at the synthesis level.

Two useful approaches are risk-of-bias sensitivity analyses and explicit bias adjustment. First, appraise the relevant bias domains and compare the full synthesis with a prespecified synthesis restricted to studies with more credible methods, or stratify by those domains; discuss the resulting loss of precision and possible confounding of study characteristics. Second, if substantive information supports plausible bias magnitudes, use a bias-adjusted meta-analysis with uncertainty about those adjustments, and assess the effect across plausible values. Simply assigning a generic quality score or downweighting a study's sampling variance does not by itself remove its bias.

For an independent subgroup contrast in meta-analysis, the difference of disjoint independent subgroup estimates has estimated variance equal to the sum of the two variances. The subgroup contrast and standard error areThe Wald test of equal subgroup mean log odds ratios therefore givesUnder the null, $Z$ is approximately standard normal and $Z^2$ has a chi-squared distribution with one degree of freedom. A quantile-2 confidence interval for the difference is $(0.29-0.8,0.29+0.8)=(-0.51,1.09)$. There is no statistically persuasive evidence of a subgroup difference. The result does not establish equal effects. Nor does a between-trial subgroup comparison, by itself, identify a causal advantage of one technique: other study characteristics may differ.