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For independent studies comparing treatments $A$ and $B$ to common control $C$, a consistent log odds ratio comparison uses ${\hat{\delta }}_{AB}={\hat{\delta }}_{AC}-{\hat{\delta }}_{BC}$, with variance $v_{AC}+v_{BC}$. This follows from subtracting two estimates of additive effects relative to the same reference. With overlapping evidence, subtract twice their covariance. Causal comparability across the studies is supplied by transitivity in network meta-analysis, not by the variance calculation.

For two independent binomial distributions with event and nonevent counts $a,b$ and $c,d$, the odds ratio is $ad/(bc)$. The delta method applied to $\log (p/(1-p))$, whose derivative is $1/[p(1-p)]$, gives a group variance $1/[Np(1-p)]$. Substitution of the fitted proportion gives $1/a+1/b$ in the first group and $1/c+1/d$ in the second. Their independence gives the displayed sum. A normal approximation then gives a confidence interval on the log odds ratio scale; exponentiation produces an odds-ratio interval. Zero cells require another approximation or model.

For independent two-by-two tables with counts $a_i,b_i,c_i,d_i$ and total $n_i$, the Mantel–Haenszel pooled odds ratio is the displayed ratio of sums. When all $b_i{c}_i>0$, it is also the weighted arithmetic mean of study odds ratios with weights $b_i{c}_i/n_i$. These differ from inverse estimated variance weights for log odds ratios. Both constructions can represent a fixed-effect meta-analysis, but they should not be silently interchanged when interpreting a supplied plot.

Meta-regression relates study effect estimates to study-level explanatory variables. A common random-effects meta-analysis extension is $y_i=\alpha +\beta x_i+u_i+{\epsilon }_i$, where $\mathrm{Var}(u_i)={\tau }^2$ and $\mathrm{Var}({\epsilon }_i)=v_i$. For log odds ratios, $e^{\beta }$ is a ratio of underlying odds ratios per unit of $x$. Few studies, correlated design changes and exploratory variable selection limit interpretation; a study-level association does not identify an individual-level causal effect. The interpretation of study-level predictors is also discussed in the Cochrane methods handbook, section 10.11.

The prose names a second year that differs from the table label. The calculations use the four supplied counts, interpreting the first and second rows as the two displayed periods; no numerical conclusion depends on resolving that label discrepancy.

The first generalized linear model treats the four counts as independent Poisson random variables, with a log link and both row and column regression factors. The row-by-column interaction makes the log-linear model saturated. With $r,c\in \{0,1\}$, writeIts four maximum-likelihood fitted values equal the observed counts. Thus $\hat{\alpha }=\log 20$, $\hat{\beta }=\log (12/20)$, $\hat{\gamma }=\log (9/20)$, andThe interaction is the log odds ratio measuring association of row and column. Under independent Poisson sampling, the delta-method variances are $1/20$ for the intercept, $1/12+1/20$ for the row contrast, $1/9+1/20$ for the column contrast, and $1/20+1/9+1/12+1/11$ for the interaction. Their square roots reproduce the printed standard errors. Although the software labels the standardized ratios as t values, these are asymptotic normal Wald statistics, with known Poisson dispersion one, not exact finite-df Student tests.

The statistical saturated model has zero residual deviance and zero residual degrees of freedom; this is a perfect interpolation, not evidence of predictive adequacy. The null deviance $5.016056$ compares the four observed counts with an intercept-only model, whose four fitted means are $52/4=13$. Fisher scoring is the numerical score/information iteration used to fit the model; the iteration count is a convergence report, not a test statistic.

Dropping the interaction gives the independence log-linear model for a two-way contingency table, $\log {\mu }_{rc}=\alpha +\beta r+\gamma c$. Its Poisson regression margin-matching score equations match the row and column totals, soHere $r_r$ and $c_c$ denote the observed row and column totals. In particular the fitted row multiplier is $23/29$, the column multiplier is $20/32$, and the intercept is $\log (29\cdot 32/52)=2.881788$. These produce the second coefficient table. Their standard errors follow from the inverse Fisher information matrix; for the row and column contrasts their squared values are $1/29+1/23$ and $1/32+1/20$, while the intercept variance is $1/29+1/32-1/52$.

The Poisson deviance relative to the saturated fit isThe linear terms cancel because fitted and observed totals agree. The additive model has three coefficients, leaving one degree of freedom. Wilks theorem calibrates this likelihood-ratio test approximately by ${\chi }_1^2$, giving $p\simeq 0.2164$. The interaction Wald test instead gives $p\simeq 0.2192$; the two are only asymptotically equivalent.

Fisher's exact test conditions on both margins under independence. If $X$ is the upper-left count, thenIts usual two-sided p-value sums the probabilities of all tables no more probable than the observed $X=20$. That sum is $0.259724$, reproducing the output without an asymptotic approximation. It tests the same absence of row-column association, but conditioning and discreteness explain the numerical difference from the likelihood-ratio test.

In ordinary language, the male proportion among the cases is $20/29\simeq 69.0\%$ in the first row and $12/23\simeq 52.2\%$ in the second. The estimated case-sex odds ratio is $2.037$, but its approximate 95-percent confidence interval is wide, $(0.655,6.338)$. These small samples do not give convincing evidence that the sex composition of cases changed between the two periods. Failure to reject is not evidence that the compositions are exactly equal. Nor do these counts establish a male-to-female disease-incidence ratio or an incidence trend: population denominators and comparable ascertainment would be needed for those interpretations.

The population-averaged logistic regression describes $P(Y_{ij}=1\mid z_i,x_i)$ after averaging over unmeasured individual heterogeneity. Its intercept is a marginal log odds at reference baseline values and time zero, its treatment coefficient is a population conditional-on-baseline log odds ratio, and its time coefficient is the change in that marginal log odds per month. Time zero is an extrapolation here because the recorded visits begin later; centring time at the first visit would give a more interpretable intercept.

In the random-intercept logistic model, the intercept is instead the conditional log odds for a subject with $B_i=0$. The treatment and time coefficients apply at fixed $B_i$; the subject's actual intercept is ${\alpha }_C+B_i$. Even with a treatment-independent random intercept, the conditional and marginal coefficients generally differ through noncollapsibility of the odds ratio. Indeed, if $\eta ={\alpha }_C+{\varphi }_{C}z+{\beta }_C^{T}x+{\delta }_{C}t$, marginalization giveswhich is not generally a logistic curve with the same linear coefficients. To quantify random-intercept attenuation of marginal logistic slopes, let $p_B={\mathrm{logit}}^{-1}(\eta +B)$. ThenFor a nondegenerate finite random intercept this derivative lies strictly between zero and one and usually varies with $\eta$. Thus the marginal treatment contrast is an integrated, attenuated version of the conditional contrast, and the marginal time slope need not be constant. One should not equate the two sets of coefficients, even in a randomized controlled trial without treatment confounding.

For missing completely at random, an unweighted observed-response GEE with cluster-robust standard errors can consistently fit the marginal mean, and a correctly specified integrated GLMM likelihood is valid as well. Under missing at random depending on previous observed outcomes, ordinary unweighted GEE is generally biased because the remaining responses are selected by informative observed history. Correct inverse-observation-weighted estimating equations for longitudinal dropout recover the marginal target under the sequential MAR and positivity assumptions already stated. Covariate-only observation mechanisms are a simpler special case in which correctly conditioned unweighted mean equations can remain valid.

A correctly specified joint response GLMM, fitted by observed-data likelihood, is valid under ignorable missing at random with distinct parameters, including dependence of missingness on observed response history. This is a likelihood property, not a claim that every mixed model automatically fixes missingness. Under missing not at random, neither ordinary GEE, MAR-based weights nor a response-only GLMM is generally valid; the unseen-outcome dependence requires further modelling and sensitivity assumptions.

The command fits independent grouped binomial distributions $Y_j\sim \mathrm{Bin}(N_j,p_j)$, using the supplied totals as trial counts. The logistic regression has the additive linear predictorThe reference category is adult females. The age and gender coefficients are conditional log odds ratios: the fitted odds for a child are multiplied by $e^{-1.27079}\approx 0.281$, and those for a male by $e^{-0.37211}\approx 0.689$.

The coefficients are maximum-likelihood estimates, found by Newton's method or iteratively reweighted least squares. With design matrix $X$ and $W_{jj}=N_j{\hat{p}}_j(1-{\hat{p}}_j)$, the estimated covariance matrix is $(X^{T}WX{)}^{-1}$; the printed standard errors are square roots of its diagonal entries. Each Wald test compares a coefficient with zero using its estimate divided by its standard error, approximately a standard normal distribution under the null. The age and gender effects are strongly significant. The intercept test instead tests whether the reference probability is $1/2$.

The fitted probabilities are about $0.04476$ for adult females, $0.03128$ for adult males, $0.01298$ for female children, and $0.008981$ for male children. Adult females have the largest fitted probability, about 4.48%. The residual deviance is $0.06514$ on $4-3=1$ degree of freedom; comparison with a chi-squared distribution gives $p\approx 0.799$. There is no evidence here that an age–gender interaction is needed, under the assumed independent binomial distribution model.

Zero observed events do not mean that the underlying event probability cannot be estimated. For independent observations having a common Bernoulli distribution, the binomial likelihood with no events is $(1-p{)}^{1515}$, maximized at $\hat{p}=0$. A zero-event binomial upper confidence bound shows the remaining uncertainty: the one-sided 95% upper bound solves $(1-p_U{)}^{1515}=0.05$, givingThus the point estimate is zero, but a nonzero mortality risk is compatible with the observations. This bound is an illustration under independent equal-risk sampling; clustered data or unequal risks require their own uncertainty calculation.

The empirical unadjusted odds ratio is also defined:Its log odds ratio is $-\infty$, so the ordinary normal approximation for a log-odds confidence interval is unavailable. Exact likelihood-based limits can still describe uncertainty; a specified continuity correction would instead yield a finite, method-dependent point estimate.

In an unpenalized logistic regression with a separate coefficient for elective exposure, every exposed death outcome is zero. Decreasing that coefficient toward $-\infty$ increases the exposed observations' likelihood contributions and leaves the reference observations unchanged. This is quasi-complete separation: there is no finite ordinary maximum-likelihood estimate of the exposure coefficient, even after adding background predictors. It is reasonable to say that the usual finite adjusted coefficient could not be fitted; it is too strong to say that no risk estimate or statistical information is possible. A specified penalized likelihood or Bayesian inference with proper coefficient priors can provide a finite adjusted estimate, whose dependence on the regularization must be reported.

The two independent binomial distributions give fitted event probabilities $24/99$ and $12/100$. Their odds ratio and log odds ratio areUsing the log odds ratio variance from a two-by-two table,The stated normal approximation and quantile approximately two give the confidence intervalExponentiation gives the corresponding odds ratio interval $(1.0824,5.0876)$. Its lower endpoint exceeds one.

A fixed-effect meta-analysis models the studies as estimating a common underlying treatment effect. In a generic inverse-variance version, independent log odds ratios $y_i$ have approximate distributions $N(\delta ,v_i)$ and are pooled with weights $1/v_i$. The supplied forest plot instead has the Mantel–Haenszel pooled odds ratio weights: for event/nonevent cells $a_i,b_i,c_i,d_i$ and total $n_i$, the normalized weights are proportional to $b_i{c}_i/n_i$. They reproduce the printed percentages and yield a pooled odds ratio about $2.338$. This is still a common-effect analysis; the two weighting methods should not be silently identified.

A random-effects meta-analysis allows different true effects: $y_i\mid {\delta }_i\sim N({\delta }_i,v_i)$ with ${\delta }_i\sim N(\mu ,{\tau }^2)$. The between-study heterogeneity variance ${\tau }^2$ adds to sampling variance, giving marginal variance $v_i+{\tau }^2$. Its pooled estimate targets the mean of the study-effect distribution rather than one identical effect. Uncertainty about heterogeneity must also be considered.

The additional estimate points in the same direction as the pooled effect: its odds ratio is about $1.90$, compared with the original pooled value about $2.34$. Its standard error is relatively large, and the approximate two-standard-error interval for its log odds ratio is $0.64\pm 1.14=(-0.50,1.78)$, or about $(0.61,5.93)$ on the odds-ratio scale. This interval includes both no effect and the existing pooled effect. Thus it supplies compatible but imprecise evidence and would tend to pull the pooled estimate somewhat toward one while adding information. A change in the main conclusion is unlikely.

For scale, a conventional inverse-variance common-effect fit to the six rounded tabulated estimates can be updated by ${\hat{\delta }}_{\mathrm{new}}=(W\hat{\delta }+wy)/(W+w)$, with new weight $w=0.57^{-2}$. This gives the new study roughly eleven percent of the combined weight, a modest effect on the estimate. An exact update of the supplied Mantel–Haenszel pooled odds ratio needs the new two-by-two table, which the reported log ratio and standard error do not uniquely specify. The qualitative compatibility conclusion does not depend on pretending that the supplied plot used inverse-variance weights.

The log odds ratio for treatment relative to control is the difference of the two log odds:Thus the treatment-to-control odds ratio is $e^{\beta }$. Its direction refers to whatever event the binomial count records.

Use a proper normal distribution prior centered at zero on the log odds ratio. For example,puts approximately 95 percent of its mass between $-2$ and $2$, giving odds ratios roughly between $e^{-2}$ and $e^2$, close to $1/8$ and $8$. Centering at zero treats reciprocal odds ratios symmetrically. If the desired central 95 percent interval is exactly $(1/8,8)$, use standard deviation $\log 8/{\Phi }^{-1}(0.975)\simeq 1.0610$. A soft prior is appropriate for implausibility, whereas a bounded uniform prior would declare effects outside the limits impossible.

The overall mean $\mu$ and the between-study heterogeneity $\tau$ encode different information. A proper broad normal distribution prior such as $\mu \sim N(0,2^2)$ is one possible weak prior for the mean log odds ratio; information about the spread of trials alone does not determine its center.

A concrete prior calibration for normal random-effect range can make the factor-of-50 statement simultaneous across all six trials. Conditional on $\tau$, each pair difference has normal distribution ${\beta }_j-{\beta }_k\sim N(0,2{\tau }^2)$. Set $R=\log 50$, $m=\left(\genfrac{}{}{0ex}{}{6}{2}\right)=15$, $z={\Phi }^{-1}(1-0.05/(2m))$ andFor every $\tau \le A$, each pair exceeds $R$ in absolute value with probability at most $0.05/m$. The union bound therefore gives $\mathbb{P}({\max }_j{\beta }_j-{\min }_j{\beta }_j>R)\le 0.05$, and integrating over the uniform prior preserves that bound. This is one explicit interpretation of “very unlikely”; a different elicited probability would change the bound. A smoother proper scale prior could be calibrated similarly.

The proposed $1/\tau$ improper prior is unsuitable. The observed-data likelihood function approaches the positive common-effect likelihood as $\tau \downarrow 0$. After restricting $\mu$ and the intercepts to a compact interior region, it is bounded below there by a positive constant. HenceThis is an improper posterior from a log-uniform random-effect scale prior. Proper conditional sampling distributions do not repair the improper joint posterior, and an arbitrary tiny cutoff would make inference depend on that cutoff.

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.

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.

Write $y_{WP}$ and $y_{AP}$ for the estimated log odds ratios relative to the common control. Indirect treatment comparison uses the consistency relation ${\delta }_{WA}={\delta }_{WP}-{\delta }_{AP}$. For independent trials, subtraction adds their variances, givingThe substantive assumption is transitivity in network meta-analysis: the study populations, outcome definitions and follow-up are sufficiently comparable, particularly in their distributions of effect modifiers, that the two control comparisons estimate effects applicable to the same target population. The common control's name alone does not justify that assumption. This also assumes comparable marginal odds ratios; these need not equal covariate-adjusted odds ratios even without confounding.

Using the supplied rounded variances and the given standard normal quantile, the approximate confidence interval isOn the odds ratio scale this is $e^{-1.26}\simeq 0.284$, with approximate limits $(0.104,0.771)$. The estimated odds of recurrent stroke under treatment $W$ are about 28% of those under $A$, a reduction of about 72% in odds. The interval excludes equal odds, so this comparison provides evidence of lower odds under $W$ if the indirect treatment comparison assumptions hold. These are odds, not a 72% reduction in probability; a confidence interval describes the long-run coverage of the procedure, not a posterior probability for this particular interval.

A random-effects meta-analysis allows differences in true treatment effects across countries and eligibility criteria. For trial $i=1,\dots ,6$, let $y_i$ be its estimated log odds ratio and $v_i$ its estimated within-trial variance. A usual approximate statistical model isindependently across trials, with $b_i$ independent of sampling errors. Here $\mu$ is the mean true log odds ratio in the population of comparable trials, and ${\tau }^2\ge 0$ is between-trial heterogeneity, not additional sampling error. Therefore $y_i\sim N(\mu ,v_i+{\tau }^2)$, and for a fitted heterogeneity value,One can estimate ${\tau }^2$ by restricted maximum likelihood or another justified method; $({\sum }_i{w}_i{)}^{-1}$ is a plug-in conditional variance and does not fully account for estimating heterogeneity. With only six studies that uncertainty matters. Different eligibility rules motivate this random effect but do not by themselves establish exchangeability or remove bias; known effect modifiers may warrant stratification or regression.

Use leave-one-study-out influence analysis: fit all six trials, then omit Cohen 1989 and refit both $\mu$ and ${\tau }^2$. Report $\hat{\mu }-{\hat{\mu }}_{(-C)}$, the corresponding change in the odds ratio, and changes in the confidence interval and heterogeneity. As a diagnostic with heterogeneity held fixed, writing $W={\sum }_i{w}_i$ givesThe fitted Cohen weight would be $(0.15+{\hat{\tau }}^2{)}^{-1}$ and $y_C=-1.95$. A precise but discordant trial can have large influence; refitting assesses additional influence through heterogeneity. The other five trials' data are absent, so a numerical six-trial influence assessment is not identifiable from the displayed table.

For $J\ge 2$ conditionally independent effects ${\beta }_j\sim N(\mu ,{\tau }^2)$, each pair difference has normal distribution $N(0,2{\tau }^2)$. With $m=\left(\genfrac{}{}{0ex}{}{J}{2}\right)$, an upper bound $A=R/[\sqrt{2}{\Phi }^{-1}(1-\epsilon /(2m))]$ on $\tau$ ensures, by the union bound, that $\mathbb{P}({\max }_j{\beta }_j-{\min }_j{\beta }_j>R)\le \epsilon$. Any proper scale prior distribution supported on $(0,A)$ preserves this bound after averaging. This is conservative simultaneous calibration, rather than the weaker statement about one selected pair. For log odds ratios, a bound $R$ corresponds to a ratio-of-odds-ratios bound $e^R$.