Treatment effect at a chosen covariate value from a centred meta-regression

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theta_z = d + b * (z - x_bar)

Function

Precision-weighted regression of study effect estimates on study characteristics

Computational function

• Computational function for a baseline-risk adjusted odds ratio, events avoided and ICER from a centred meta-regression

  Takes the inputs a decision modeller holds, a baseline risk for the model population, the fitted coefficients of a centred meta-regression on baseline log-odds and the cost and QALY consequences of an event, and returns the adjusted odds ratio, the absolute effect and the incremental cost-effectiveness ratio. It converts the risk to log-odds, applies HE-FM-MREG-005, exponentiates the result and applies the odds ratio to the baseline odds, so each step uses the result of the one before. The inputs therefore differ from the formula's variables, which need the covariate already on the log-odds scale.

  Inputs and outputs: p_0 is the baseline risk of the event over the model period in the decision population, required, above 0 and below 1, a probability; d is the log odds ratio at the trials' mean baseline log-odds; b is the interaction per unit of baseline log-odds; x_bar is the mean baseline log-odds used for centring; c_tx is the cost of treatment per person in GBP; s_ev is the cost saved per event avoided in GBP; q_ev is the QALY gain per event avoided; the outputs are z, the baseline log-odds, theta_z, the adjusted log odds ratio, OR_z, the adjusted odds ratio, p_1, the treated risk, N_av, events avoided per 1,000 treated, C_net, the net cost per 1,000 treated in GBP, Q_gain, the QALYs gained per 1,000 treated, and ICER, in GBP per QALY.

  Assumption: The meta-regression was fitted on the logit scale with the true baseline log-odds as covariate, centred at x_bar, and the decision population's risk lies inside the range of the trials. Costs and QALYs accrue within one period with no discounting, as in the article's illustrative example, and C_net and Q_gain are positive.

  Worked example (Article's 5% baseline risk population): With d of -0.40, b of 0.30 and x_bar of -1.70, a 5% baseline risk gives an odds ratio of about 0.462 and 26.3 events avoided per 1,000. At GBP 300 per person, GBP 2,000 saved and 0.5 QALYs gained per event avoided, the ICER is about GBP 18,800 per QALY, as in the article. p_0 = 0.05; d = -0.40; b = 0.30; x_bar = -1.70; c_tx = 300; s_ev = 2000; q_ev = 0.5; z = -2.9444; theta_z = -0.7733; OR_z = 0.4615; p_1 = 0.02371; N_av = 26.29; C_net = 247424; Q_gain = 13.144; ICER = 18824

  Worked example (Zero interaction returns the unadjusted result): With b set to 0 the odds ratio stays at about 0.670 whatever the baseline risk, giving 15.9 events avoided per 1,000 and an ICER of about GBP 33,700 per QALY, the article's result with the unadjusted effect. p_0 = 0.05; d = -0.40; b = 0; x_bar = -1.70; c_tx = 300; s_ev = 2000; q_ev = 0.5; z = -2.9444; theta_z = -0.40; OR_z = 0.6703; p_1 = 0.03408; N_av = 15.92; C_net = 268155; Q_gain = 7.961; ICER = 33683

  Excel: =LN(B1/(1-B1)) in B9, =B2+B3*(B9-B4) in B10, =EXP(B10) in B11, =B1/(1-B1)*B11/(1+B1/(1-B1)*B11) in B12, =1000*(B1-B12) in B13, =1000*B5-B13*B6 in B14, =B13*B7 in B15 and =B14/B15 in B16. With p_0, d, b, x_bar, c_tx, s_ev and q_ev in cells B1 to B7, cells B9 to B16 return z, theta_z, OR_z, p_1, N_av, C_net, Q_gain and the ICER.

  R: mr_adjust <- function(p0, d, b, xbar, c_tx, s_ev, q_ev) { z <- log(p0/(1-p0)); th <- d+b*(z-xbar); or_z <- exp(th); o0 <- p0/(1-p0); p1 <- o0*or_z/(1+o0*or_z); n_av <- 1000*(p0-p1); c_net <- 1000*c_tx-n_av*s_ev; q_gain <- n_av*q_ev; c(z = z, theta_z = th, OR_z = or_z, p_1 = p1, N_av = n_av, C_net = c_net, Q_gain = q_gain, ICER = c_net/q_gain) } Vectorised over p0, so a vector of baseline risks returns the adjusted results for each subgroup; mr_adjust(0.05, -0.40, 0.30, -1.70, 300, 2000, 0.5) reproduces the article's example.

  Python: def mr_adjust(p0, d, b, xbar, c_tx, s_ev, q_ev): z = math.log(p0/(1-p0)); th = d+b*(z-xbar); or_z = math.exp(th); o0 = p0/(1-p0); p1 = o0*or_z/(1+o0*or_z); n_av = 1000*(p0-p1); c_net = 1000*c_tx-n_av*s_ev; q_gain = n_av*q_ev; return z, th, or_z, p1, n_av, c_net, q_gain, c_net/q_gain Uses the math module, where math.log is the natural logarithm, and returns the eight outputs in the order listed above.

  Test (Effect at the trials' mean risk equals d): Setting the baseline risk to the risk whose log-odds is x_bar returns an adjusted log odds ratio equal to d. Expected result is TRUE. Excel check =ABS(B2+B3*(LN((1/(1+EXP(-B4)))/(1-1/(1+EXP(-B4))))-B4)-B2)<1E-9

  Test (Protective adjusted effect lowers the treated risk): When the adjusted log odds ratio is negative, the treated risk is below the baseline risk and the events avoided are positive. Expected result is TRUE. Excel check =IF(B10<0,AND(B12<B1,B13>0),TRUE)

  Common error (Multiplying the baseline risk by the odds ratio): Treating the odds ratio as a risk ratio gives a treated risk of p_0 times OR_z. At a 5% risk the error is small, 26.9 instead of 26.3 events avoided per 1,000, but at a 35% risk it gives 25.5 instead of 17.0, half as many again. The odds ratio has to be applied to the baseline odds.

  Source: Dias S, Sutton AJ, Welton NJ, Ades AE. NICE DSU Technical Support Document 3, Heterogeneity, subgroups, meta-regression, bias and bias-adjustment. Decision Support Unit; 2011, last updated April 2012. Section 4.3 on un-centring the effect to any covariate value z and section 4.4 on meta-regression on baseline risk with the covariate on the logit scale.

  z = log(p_0 / (1 - p_0)); theta_z = d + b * (z - x_bar); OR_z = exp(theta_z); p_1 = p_0 / (1 - p_0) * OR_z / (1 + p_0 / (1 - p_0) * OR_z); N_av = 1000 * (p_0 - p_1); C_net = 1000 * c_tx - N_av * s_ev; Q_gain = N_av * q_ev; ICER = C_net / Q_gain

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Implementations

• Excel

  Adjusted relative effect from a centred meta-regression in one cell

  With the effect at the mean in a cell named EffectAtMean, the interaction in Interaction, the mean covariate in CovMean and the population's covariate value in CovValue, the first formula returns theta_z. The second takes a baseline risk in a cell named Risk and converts it to log-odds inside the formula.

  =EffectAtMean+Interaction*(CovValue-CovMean); =EffectAtMean+Interaction*(LN(Risk/(1-Risk))-CovMean)

Assumptions

• Covariate centred on the scale of the linear predictor

  The covariate is centred at its mean on the scale used in the model. For meta-regression on baseline risk with a logit link, TSD 3 centres on the mean baseline log-odds, so z must also be a log-odds and x_bar a mean of log-odds.

• Decision population inside the covariate range of the trials

  The line is used within the range of covariate values the trials cover, 4% to 35% control-group risk in the article's illustrative example, where a 5% target lies inside the range.

• One interaction term common to all treatments in a network meta-regression

  In a network meta-analysis TSD 3 favours a single interaction term common to all treatments relative to the reference, since separate terms could lead a strict cost-effectiveness analysis to recommend different treatments for different subgroups. One b then adjusts every comparison with the reference.

Worked examples

• Adjusted log odds ratio for a 5% baseline risk population

  With d of -0.40 at the trials' mean log-odds of -1.70, an interaction of 0.30 and a decision population with a 5% baseline risk, a log-odds of about -2.944, the adjusted log odds ratio is about -0.773, an odds ratio of about 0.462, as in the article.

  d = -0.40; b = 0.30; z = -2.9444; x_bar = -1.70; theta_z = -0.7733

• Centred meta-regression effect at the mean covariate value

  At z equal to the mean of -1.70, a baseline risk of about 15.4%, the adjustment term is zero and the effect is d.

  d = -0.40; b = 0.30; z = -1.70; x_bar = -1.70; theta_z = -0.40

• Centred meta-regression effect at a 35% baseline risk

  At the top of the illustrative range, a 35% baseline risk or a log-odds of about -0.619, the adjusted log odds ratio is about -0.0757, an odds ratio of about 0.927, so little benefit remains (computed here for illustration).

  d = -0.40; b = 0.30; z = -0.6190; x_bar = -1.70; theta_z = -0.0757

Common errors

• Entering the baseline risk instead of its log-odds

  Using z as 0.05 rather than the log-odds of -2.944 gives an adjusted log odds ratio of 0.125, an odds ratio of about 1.13, which reverses the direction of the effect. The covariate value has to be on the scale on which the model was fitted.

• Using the centred effect with an uncentred covariate value

  Adding b times z to d without subtracting x_bar gives a log odds ratio of about -1.283 for a 5% risk, an odds ratio of about 0.277 instead of 0.462, which overstates the benefit in the decision model.

• Adopting an adjusted effect as the base case without a pre-specified interaction

  With few trials and one covariate, an effect adjusted by meta-regression would normally be a scenario unless the interaction was pre-specified, plausible and precise. NICE asks whether the relative effect is constant over baseline risks when the model's comparator event rate differs greatly from those in the meta-analysis.

Sources

• Un-centring a meta-regression and baseline risk adjustment in NICE DSU TSD 3

  Dias S, Sutton AJ, Welton NJ, Ades AE. NICE DSU Technical Support Document 3, Heterogeneity, subgroups, meta-regression, bias and bias-adjustment. Decision Support Unit; 2011, last updated April 2012. Section 4.3 (centred covariate values, effects estimated at the mean covariate and un-centred to any value z), section 4.4 (covariate on the scale of the linear predictor, centred on the mean log-odds) and section 4.2 (single common interaction term for decision making).

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• Relative effect over different baseline risks in the NICE manual

  National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance, the manual (PMG36). 2022, last updated 13 October 2026. Section 3.4.10, on exploring known clinical heterogeneity with subgroup analyses and meta-regression and on assessing whether relative effectiveness is constant over different baseline risks when it is used in an economic model.

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