Trapezoidal Rule

What the trapezoidal rule does

It is simple, transparent and especially useful when a function is available only at discrete time points. This page explains the single-interval and composite formulas, approximation error, irregular grids and applications to survival, costs and QALYs in health-economic models.

The method approximates the area implied by linear interpolation between observations. It does not recover unobserved curvature or correct biased, missing or poorly timed measurements.

One interval is approximated by one trapezoid

For a function (f(x)) observed at endpoints (a) and (b), the integral is:

$$I=\int_a^b f(x)\,dx$$

The trapezoidal approximation is:

$$T = \frac{b-a}{2} \left[f(a)+f(b)\right]$$

This is the interval width multiplied by the average of the two endpoint heights. Geometrically, it is the area under the straight line connecting ((a,f(a))) and ((b,f(b))).

The composite rule adds adjacent trapezoids

For an equally spaced grid (a=x_0<x_1<\cdots<x_n=b) with step size:

$$h=\frac{b-a}{n}$$

the composite trapezoidal rule is:

$$T_n =h\left[ \frac{1}{2}f(x_0) +\sum_{i=1}^{n-1}f(x_i) +\frac{1}{2}f(x_n) \right]$$

Interior points receive full weight because each belongs to two neighbouring trapezoids. The two outer endpoints receive half weight because each belongs to only one.

Unequal intervals require their actual widths

Health data are often observed at irregular times. In that case, a single common step size must not be assumed. The composite approximation is:

$$T = \sum_{i=0}^{n-1} \frac{x_{i+1}-x_i}{2} \left[f(x_i)+f(x_{i+1})\right]$$

Each interval contributes its own width multiplied by its average endpoint height. The observations must be sorted by the integration variable, and duplicated or reversed times must be resolved before calculation.

A worked example integrates a curved function

Consider:

$$I=\int_0^1 x^2\,dx$$

The exact integral is:

$$I=\left.\frac{x^3}{3}\right|_0^1=\frac{1}{3}\approx0.333333$$

Use four equal subintervals, so (h=0.25). The function values are:

The composite approximation is:

$$T_4 =0.25\left[ \frac{1}{2}(0) +0.0625+0.25+0.5625 +\frac{1}{2}(1) \right] =0.34375$$

The signed approximation error is:

$$T_4-I =0.34375-0.333333\ldots =0.0104167$$

Because (x^2) is convex, the secant lines lie above the curve and the trapezoidal rule overestimates the integral.

Curvature determines the direction and size of error

For one interval and a function with a continuous second derivative, there is some (\xi\in(a,b)) such that:

$$\int_a^b f(x)\,dx-T =- \frac{(b-a)^3}{12}f''(\xi)$$

For the composite rule on (n) equal intervals, there is some (\xi\in(a,b)) such that:

$$\int_a^b f(x)\,dx-T_n =- \frac{(b-a)}{12}h^2 f''(\xi)$$

When (f''(x)>0), the function is convex and (T_n) tends to exceed the true integral. When (f''(x)<0), the function is concave and (T_n) tends to fall below it. A linear function has zero second derivative and is integrated exactly.

Error falls quadratically as the grid is refined

For a sufficiently smooth function on a uniform grid, the global discretisation error is of order (h^2):

$$I-T_n=O(h^2)$$

Halving (h) therefore reduces the leading error by approximately a factor of four once the asymptotic regime applies. This convergence rate concerns numerical integration error, not sampling uncertainty or measurement error in the function values.

If (|f''(x)|\le M) throughout ([a,b]), an absolute error bound is:

$$|I-T_n| \le \frac{(b-a)}{12}h^2M$$

The bound may be conservative, and it cannot be evaluated without defensible information about curvature.

Richardson extrapolation estimates and reduces leading error

If (T_h) is the trapezoidal result with spacing (h) and (T_{h/2}) uses half that spacing, their leading errors differ by a factor of approximately four. An estimate of the finer-grid error is:

$$I-T_{h/2} \approx \frac{T_{h/2}-T_h}{3}$$

Combining the two results gives the Richardson-extrapolated estimate:

$$R =T_{h/2} +\frac{T_{h/2}-T_h}{3}$$

For a smooth function, this combination is the basis of Simpson-type improvement and cancels the leading (h^2) error term. The estimate is unreliable when the grids are not nested, the function is not sufficiently smooth or other errors dominate.

Simpson's rule and adaptive methods can improve accuracy

Simpson's rule replaces pairs of straight-line segments with a quadratic interpolant and can be much more accurate for smooth functions. Gaussian quadrature selects nodes and weights to integrate polynomial structure efficiently. Adaptive quadrature subdivides regions where an error estimate indicates greater difficulty.

More sophisticated integration does not improve accuracy when the limiting problem is sparse or biased observed data rather than numerical evaluation.

Area under a survival curve gives restricted mean survival

For a survival function (S(t)), restricted mean survival time through horizon (\tau) is:

$$\operatorname{RMST}(\tau) =\int_0^{\tau}S(t)\,dt$$

If survival probabilities are available at times (t_0,\ldots,t_n), a trapezoidal approximation is:

$$\widehat{\operatorname{RMST}}(\tau) \approx \sum_{i=0}^{n-1} \frac{t_{i+1}-t_i}{2} \left[\widehat{S}(t_i)+\widehat{S}(t_{i+1})\right]$$

Applying linear interpolation to a Kaplan–Meier step function is not identical to integrating the step function exactly. If the event-time steps are available, direct step-function integration preserves the estimator's form. Trapezoidal integration may instead be appropriate for a smooth fitted survival curve or for regularly reported survival values when linear interpolation is the intended approximation.

A survival example shows the time units

Suppose estimated survival is 1.00 at month 0, 0.90 at month 6 and 0.75 at month 12. The trapezoidal area through 12 months is:

$$\widehat{\operatorname{RMST}}(12) \approx \frac{6}{2}(1.00+0.90) +\frac{6}{2}(0.90+0.75)$$ $$=5.70+4.95=10.65\text{ months}$$

The output has units of time because a dimensionless survival probability is integrated over months. It should be divided by 12 if the downstream model requires years and the conversion convention is one year per 12 months.

QALYs can be approximated from utility over time

Quality-adjusted life-years are the integral of health-related utility over time. When utility (u(t)) is observed at discrete times in years, the trapezoidal estimate is:

$$\widehat{\operatorname{QALY}} = \sum_{i=0}^{n-1} \frac{t_{i+1}-t_i}{2} \left[u(t_i)+u(t_{i+1})\right]$$

This is often called area under the curve. The interpretation assumes linear utility change between observations. If health changes abruptly at a known event, a step or event-based model may better represent the trajectory.

Discounting also changes the integrand. With continuous discount rate (r), discounted QALYs are:

$$\operatorname{QALY}_{\text{disc}} = \int_0^T u(t)e^{-rt}\,dt$$

The trapezoidal rule should be applied to the discounted function values (u(t_i)e^{-rt_i}), or to another discretisation derived consistently from the chosen discount convention.

A QALY example requires consistent time units

Suppose utility is 0.80 at baseline, 0.70 at six months and 0.60 at twelve months. Converting time to years gives intervals of 0.5 years. The estimated QALYs during the year are:

$$\widehat{\operatorname{QALY}} = \frac{0.5}{2}(0.80+0.70) + \frac{0.5}{2}(0.70+0.60)$$ $$=0.375+0.325=0.700\text{ QALYs}$$

Using widths of six without converting months to years would produce quality-adjusted life-months rather than QALYs. Unit labelling is part of the calculation, not a cosmetic reporting step.

The trapezoidal rule is related to cycle correction but not identical

In a cohort state-transition model, rewards calculated only at cycle boundaries can misrepresent events occurring within cycles. The trapezoidal approximation averages occupancy or reward at the beginning and end of each cycle, which is closely related to a half-cycle correction under certain assumptions. The concepts should not be treated as universally interchangeable.

A trapezoidal state-reward approximation over cycle length (\Delta t) is:

$$R \approx \sum_{i=0}^{n-1} \frac{\Delta t}{2} \left[r(t_i)+r(t_{i+1})\right]$$

Its adequacy depends on how transitions and rewards occur within the cycle. More precise within-cycle correction may be needed when hazards change rapidly, events cluster near boundaries or rewards accrue discontinuously.

Missing observations are not solved by numerical integration

The trapezoidal rule requires values at the endpoints of every included interval. If an outcome is missing, silently joining the nearest available observations asserts a long linear trajectory across the gap. That is an imputation assumption, not merely an integration step.

Analysts should distinguish:

• Interpolation between observed points within a supported interval.
• Imputation of an unobserved measurement.
• Extrapolation beyond the last observation.
• Censoring caused by dropout or death.
• Structural rules, such as assigning utility after death.

Missing-data methods and sensitivity analyses should address why observations are absent and how that absence relates to outcomes.

Uncertainty in input values should be propagated

The trapezoidal rule itself returns a deterministic weighted sum for fixed inputs. When function values are estimated, the integrated area is also uncertain. Correlation among repeated measurements or adjacent survival estimates means that treating every point as independent can misstate variance.

For weights (w) and estimated values (y), a trapezoidal area can be written as:

$$T=w^{\mathsf T}y$$

If (\operatorname{Cov}(y)=\Sigma), then:

$$\operatorname{Var}(T) =w^{\mathsf T}\Sigma w$$

Bootstrap, multiple-imputation or model-based simulation can propagate uncertainty when the covariance is complex or the area is part of a larger analysis. Numerical integration error and statistical sampling uncertainty should be reported separately where both matter.

Spreadsheet implementation is transparent and auditable

A spreadsheet can calculate one row per interval with columns for left time, right time, width, left value, right value and trapezoid area. The interval formula is:

=(RightTime-LeftTime)*(LeftValue+RightValue)/2

The total integral is the sum of interval areas. A robust sheet should flag unsorted time points, duplicate times, missing endpoints, negative interval widths and inconsistent units. The first and last integration limits should be displayed explicitly rather than inferred from hidden rows.

Common errors change the area materially

The simplicity of the formula can hide assumptions about ordering, interpolation, units and endpoints. These assumptions should be visible whenever the result affects a health-economic decision.

• Applying the equal-spacing formula to irregular observations assigns the wrong interval weights.
• Sorting function values without sorting their corresponding times pairs the wrong points.
• Omitting or double-counting an endpoint changes the integral.
• Using months as if they were years changes QALYs or annual costs by a factor of 12.
• Integrating percentages as whole numbers rather than proportions changes results by a factor of 100.
• Extending the last observed value beyond follow-up is extrapolation, not trapezoidal integration.
• Integrating a Kaplan–Meier curve by straight lines differs from exact step-function integration.
• Rounding each interval before summation can accumulate avoidable error.
• Treating denser sampling as proof of unbiased measurement confuses discretisation with data quality.
• Reporting an area without its limits and units makes the result uninterpretable.

A defensible trapezoidal-rule workflow

A reliable calculation makes the integration target, grid, units and interpolation assumption explicit. Validation should compare the result with known functions or finer grids before the method is relied upon in a model.

1. Define the integral: State the function, lower and upper limits, and required output units.
2. Inspect the grid: Sort paired observations, check duplicates, confirm spacing and identify missing intervals.
3. Choose the interpolation: Confirm that straight-line change between points is scientifically reasonable.
4. Calculate interval areas: Use the actual width and average endpoint value for every interval.
5. Verify endpoints: Confirm that the first and last observations match the intended integration limits.
6. Check units: Convert time, percentages, currencies and outcome scales before summation.
7. Assess numerical error: Compare with a finer grid, exact benchmark, Richardson estimate or alternative quadrature where possible.
8. Address data uncertainty: Propagate sampling, censoring and missing-data uncertainty separately from discretisation error.
9. Test decision impact: Examine whether plausible interpolation or extrapolation alternatives change the economic conclusion.
10. Document assumptions: Record the data source, grid, method, horizon, discounting, missing-data handling and precision retained.

The trapezoidal rule is most defensible when the observed points are sufficiently dense for linear interpolation and the integration horizon is supported by the data. Its transparency is a strength only when the assumptions between those points are equally transparent.