Interpolation

What interpolation does

It fills an internal gap in a defined domain; estimating beyond the available range is extrapolation and usually requires stronger assumptions.

The method is common in health-economic models when outcomes, costs, utilities, hazards, or calibration targets are available only at selected times or parameter values. The choice of method matters because different interpolants can agree at the known points while producing materially different values between them.

The points that anchor an interpolation

Known locations $x_0,x_1,\ldots,x_n$ are called nodes, with corresponding values $y_i=f(x_i)$. An interpolating function $p(x)$ is constructed so that it passes through those observations.

$$p(x_i)=y_i,\qquad i=0,1,\ldots,n$$

Interpolation does not create new empirical information. It distributes information between nodes according to the method's assumptions about shape, smoothness, monotonicity, and scale.

Linear interpolation between two points

Linear interpolation joins two adjacent nodes with a straight line. It is transparent and stable, but it assumes a constant rate of change over the interval.

For $x_0\leq x\leq x_1$:

$$p(x)=y_0+\frac{x-x_0}{x_1-x_0}(y_1-y_0)$$

The weight on the upper node is $w=(x-x_0)/(x_1-x_0)$, so the same formula can be written as:

$$p(x)=(1-w)y_0+wy_1,\qquad 0\leq w\leq1$$

Because the weights are non-negative and sum to one, the interpolated value lies between $y_0$ and $y_1$. This range-preserving property can be useful for bounded outcomes, although it does not guarantee that the straight-line path reflects the real process.

A worked health-utility example

Suppose mean health-state utility is $0.80$ at month 2 and $0.65$ at month 5. If a model needs the value at month 3 and assumes linear change between observations, the interpolation weight is $(3-2)/(5-2)=1/3$.

$$p(3)=0.80+\frac{3-2}{5-2}(0.65-0.80) =0.80+\frac{1}{3}(-0.15)=0.75$$

The interpolated utility at month 3 is $0.75$. This calculation estimates the value at a time point; estimating QALYs over the interval requires integration or a specified within-interval approximation and should not be confused with the interpolation itself.

Piecewise linear interpolation

With several nodes, piecewise linear interpolation applies the two-point formula within each adjacent interval. The resulting curve is continuous at the nodes but usually has a change in slope, or kink, at each internal node.

Piecewise linear interpolation is often appropriate when the available evidence does not justify a smoother shape. It can nevertheless misrepresent processes with pronounced curvature, abrupt changes between measurement times, or known mechanistic constraints.

Polynomial interpolation

A single polynomial of degree at most $n$ can pass through $n+1$ distinct nodes. The Lagrange form expresses that polynomial as a weighted sum of the observed values.

$$p_n(x)=\sum_{i=0}^{n}y_iL_i(x)$$

where:

$$L_i(x)=\prod_{\substack{j=0\\j\neq i}}^{n} \frac{x-x_j}{x_i-x_j}$$

High-degree global polynomials can oscillate severely between evenly spaced nodes, especially near the boundaries. Adding or moving one node can also change the curve across the entire domain, so passing exactly through every point is not evidence that the interpolation is plausible.

Spline interpolation

Splines use low-degree polynomials in separate intervals and impose continuity conditions where the pieces meet. Cubic splines typically provide a smooth curve while avoiding many of the oscillations associated with one high-degree polynomial.

A cubic spline can enforce continuity of the fitted value, first derivative, and second derivative at internal knots. Boundary conditions—such as natural, clamped, or not-a-knot rules—affect the curve near the ends and must be stated when they can influence results.

Standard cubic splines may overshoot observed ranges or violate monotonicity. Shape-preserving methods, such as monotone piecewise cubic interpolation, are preferable when the quantity must remain ordered, non-negative, or within a clinically meaningful bound.

Interpolating on a transformed scale

The scale of interpolation encodes an assumption about how the quantity changes. Linear interpolation on the original scale is not equivalent to linear interpolation after a logarithmic, logit, or complementary log-log transformation.

For two positive values, log-linear interpolation is:

$$\log p(x)=(1-w)\log y_0+w\log y_1$$

which gives:

$$p(x)=y_0^{1-w}y_1^w$$

Log interpolation preserves positivity and represents proportional rather than additive change. For probabilities, interpolation on the logit scale can preserve values within $(0,1)$, but values exactly equal to zero or one require a justified treatment rather than an arbitrary numerical adjustment.

Interpolation in two dimensions

Models sometimes require a value between grid points defined by two inputs, such as age and time or dose and baseline risk. Bilinear interpolation performs linear interpolation along each dimension within one rectangular grid cell.

Let $u=(x-x_0)/(x_1-x_0)$ and $v=(y-y_0)/(y_1-y_0)$. For corner values $f_{00}$, $f_{10}$, $f_{01}$, and $f_{11}$:

$$f(x,y)=(1-u)(1-v)f_{00} +u(1-v)f_{10} +(1-u)vf_{01} +uvf_{11}$$

The four weights are non-negative and sum to one when the target lies inside the grid cell. Bilinear interpolation is continuous across grid cells but generally does not produce continuous first derivatives at their boundaries.

How interpolation error behaves

Interpolation error depends on node spacing, the curvature or smoothness of the underlying function, data error, and the selected method. Dense nodes do not guarantee accuracy when observations are biased, noisy, or poorly located relative to rapid changes.

For linear interpolation of a twice-differentiable function, there is some $\xi$ between the nodes such that:

$$f(x)-p(x)=\frac{f''(\xi)}{2}(x-x_0)(x-x_1)$$

The expression shows that error is zero at the nodes and increases with curvature and interval width. In practice, $f''(\xi)$ is usually unknown, so empirical validation, alternative methods, and sensitivity analysis are often more informative than a theoretical bound alone.

Interpolation is not smoothing

An interpolating curve passes through the supplied data values, whereas a smoothing method is allowed to miss individual observations to estimate an underlying trend. This distinction matters when the data include sampling error or measurement noise.

Exact interpolation of noisy observations can reproduce noise as though it were signal. Regression splines, penalised splines, Gaussian processes, or other smoothing models may be more appropriate when the objective is estimation rather than exact passage through every point.

Interpolation is not extrapolation

Interpolation stays inside the range supported by surrounding nodes, while extrapolation projects outside that range. The same formula may be computable outside the interval, but its evidential meaning changes and the error can grow rapidly.

Spreadsheet formulas should explicitly flag values below the minimum or above the maximum node rather than silently extending the end segment. When extrapolation is required, it should use a separately justified model and be labelled as extrapolation.

Time interpolation in economic models

Clinical outcomes may be observed at visits that do not align with a model's cycle boundaries. Interpolation can map those observations to required time points, but the mapping must respect the meaning and timing of each measure.

• Health utilities may be interpolated between assessments when a within-interval trajectory is clinically plausible.
• Cumulative incidence or survival should be interpolated on a scale consistent with the event process and must preserve monotonicity.
• Hazards and rates should not be treated as probabilities; the relevant time unit and transformation must be retained.
• Costs may accrue discretely at events rather than smoothly, making interpolation inappropriate for some resource items.
• Treatment effects may wane, begin after a delay, or change abruptly, requiring a clinically justified time function rather than automatic straight lines.

Interpolating a survival probability linearly can differ from interpolating cumulative hazard and transforming back. The method should match the model's structural assumptions and be tested for its effect on life-years, QALYs, and costs.

Preserving constraints

Many health-economic quantities have mathematical or clinical constraints that a generic interpolant can violate. A method should be selected and validated against those constraints, not only against visual smoothness.

Clipping invalid values after interpolation can conceal a poor method and create flat regions or biased totals. It is better to use an interpolant that respects the constraint by construction and to document any remaining boundary treatment.

Uncertainty in interpolated values

An interpolated point is often treated as deterministic even though the node values and assumed curve are uncertain. Ignoring this uncertainty can make downstream model results appear more precise than the evidence supports.

Probabilistic analysis should preserve dependence among node values and apply the interpolation consistently within each simulation draw. Structural sensitivity analysis can compare linear, transformed-linear, spline, and clinically specified shapes where the choice is genuinely uncertain.

If interpolation parameters are fitted from data, their estimation uncertainty should be propagated using an appropriate covariance structure, bootstrap, posterior distribution, or other method consistent with the model. Independently sampling neighbouring node values can create implausible zigzags when the source estimates are correlated.

Spreadsheet implementation

A transparent spreadsheet should separate node data, interval selection, interpolation weights, generated values, and validation checks. Lookup logic must identify the two correct surrounding nodes and must not silently use approximate matches on unsorted data.

For linear interpolation, an auditable calculation records $x_0$, $x_1$, $y_0$, $y_1$, the weight $w$, and the resulting $p(x)$. Required checks include duplicate node locations, missing node values, division by zero when $x_0=x_1$, targets outside the supported range, discontinuities, boundary violations, and exact recovery of every input node.

Spreadsheet chart trendlines are not a substitute for a documented interpolation algorithm. Displayed curves may use rendering rules that are difficult to audit and may not match the values used in the economic model.

A practical workflow

A defensible interpolation begins by identifying why an internal estimate is needed and what behaviour is plausible between observations. The following workflow makes the mathematical and health-economic assumptions traceable.

1. Define the target variable, units, domain, time scale, and reason interpolation is required.
2. Inspect node order, spacing, precision, missingness, uncertainty, outliers, and measurement timing.
3. State required bounds, monotonicity, continuity, smoothness, and mechanistic constraints.
4. Choose the original or transformed scale on which change is assumed to be simple.
5. Select the simplest interpolation method that respects the evidence and constraints.
6. Calculate values with full precision and preserve the exact input nodes.
7. Validate boundary behaviour, continuity, constraints, special cases, and independently calculated examples.
8. Compare plausible alternative methods where curve choice could change a decision-relevant result.
9. Propagate node and structural uncertainty through the downstream model.
10. Document nodes, method, scale, software, assumptions, checks, and the treatment of out-of-range requests.

Interpreting and reporting interpolated results

An interpolated value is conditional on the observed nodes and the assumed behaviour between them. Extra decimal places reflect computation, not necessarily additional empirical precision.

Reporting should identify the node source, interpolation method, transformation, constraints, validation tests, uncertainty treatment, and whether any requested value required extrapolation. Common errors include confusing interpolation with extrapolation, using a high-degree polynomial because it fits every node, ignoring non-linearity or bounds, interpolating incompatible quantities, and presenting generated values as new observations.