Bias-Variance Tradeoff

Bias-Variance Tradeoff in Health Economic Models

Every estimate that feeds a health economic model, from a mean cost in one subgroup to the tail of an extrapolated survival curve, can miss the true value in two ways. It can be off target on average, or it can swing from one sample to the next. The bias-variance tradeoff describes how these two errors tend to move in opposite directions as a model gains flexibility, so the estimate with the smallest typical error is often slightly biased. Analysts meet this choice when they decide how many parameters a survival distribution should carry, how many predictors a risk equation should include and how far subgroup estimates should borrow from the whole trial. The mean squared error decomposition underlies each of these choices, from survival curves and prediction models to partial pooling and decision model structure, and in an illustrative subgroup cost example a deliberately biased estimate halves the typical error.

Splitting expected error into bias, variance and noise

The tradeoff rests on an identity for squared-error loss. It separates the expected error of a prediction into a part no model can remove, a part due to systematic error and a part due to sampling variability. For an outcome predicted at a given set of patient characteristics, the decomposition is as follows.

$$\mathrm{Err}(x_0) = E\left[(Y - \hat{f}(x_0))^2 \mid X = x_0\right] = \sigma^2_{\varepsilon} + \left[E\hat{f}(x_0) - f(x_0)\right]^2 + E\left[\hat{f}(x_0) - E\hat{f}(x_0)\right]^2$$

where $Y$ is the outcome, such as a patient's annual cost, $x_0$ is a particular set of patient characteristics, $f(x_0)$ is the true mean outcome for those characteristics, $\hat{f}(x_0)$ is the prediction from a model fitted to one sample, and $\sigma^2_{\varepsilon}$ is the variance of individual outcomes around their true mean. The second term is the squared bias, the gap between the average prediction over repeated samples and the truth; the third is the variance of the prediction across those samples. The same split applies to an estimator of a single model input, such as a mean cost or a hazard ratio, whose mean squared error equals squared bias plus variance.

Hastie, Tibshirani and Friedman summarise the usual pattern. The more complex the model, the lower the squared bias and the higher the variance, while error measured on the fitting data keeps falling and is therefore a poor guide to performance in new patients. The decomposition always holds; the opposing movement of its terms is a typical pattern rather than a law. Geman, Bienenstock and Doursat framed it as the bias/variance dilemma in their 1992 tutorial on nonparametric regression and neural networks. Unbiasedness is therefore one property among several, and an estimator that accepts a little bias can sit closer to the truth in a typical sample.

Survival curve choice and the price of extra parameters

Survival extrapolation is where health technology assessment meets the tradeoff most directly. In a parametric survival model the exponential distribution has one parameter, while the Weibull and Gompertz have two.

The NICE Decision Support Unit's Technical Support Document 14 (TSD 14) explains that the Akaike information criterion and the Bayesian information criterion weigh improved fit against the potentially inefficient use of additional parameters, with the BIC penalising them more heavily, and that such comparisons address fit to the observed data but not the plausibility of the curve beyond the end of follow-up. In bias-variance terms the penalty guards against variance, since a parameter that improves fit by chance in one trial does not improve predictions for the next cohort.

Technical Support Document 21 (TSD 21) extends the argument to flexible parametric models built from restricted cubic splines, which take increasingly complex shapes as knots are added. It notes that many spline parameters may be impractical when there are few events, that these models generally fit extremely well within the observed data without this making their extrapolations reliable, and that with many knots the extrapolation may rest on the trend near the end of follow-up, where events are few. Read through the decomposition, extra knots buy lower bias inside the trial at the cost of higher variance in the region that drives mean survival. TSD 21 recommends sensitivity analysis on the number of knots, and the remaining doubt about the right shape is structural uncertainty, which no within-sample criterion resolves.

Overfitting in clinical prediction models

Risk equations that predict events or costs from patient characteristics often supply baseline risks to economic models. A model developed on a modest data set with many candidate predictors can follow the development data closely (low bias), but its estimated effects vary widely from sample to sample (high variance), so its apparent performance is optimistic and its performance in new patients is usually lower. This is overfitting, which Steyerberg's textbook on clinical prediction models treats together with shrinkage as a remedy. Riley and colleagues proposed that the sample size for a model with a binary or time-to-event outcome should achieve a global shrinkage factor of at least 0.9, an absolute difference of no more than 0.05 between the apparent and adjusted Nagelkerke $R^2$, and precise estimation of the overall risk. A shrinkage factor of 0.9 corresponds to an expected shrinkage of 10%. The same paper describes the rule of thumb of at least 10 events per candidate predictor and notes that it has generated much debate.

Penalisation methods, including uniform shrinkage, ridge regression, the lasso and the elastic net, shrink predictor effects towards the null, trading a little bias for lower mean-square prediction error in new individuals. A later study by Riley and colleagues found that the tuning parameters controlling this shrinkage are themselves estimated with large uncertainty when the effective sample size is small, which can leave predictions miscalibrated; penalisation is least reliable when it is most needed. Internal validation by cross-validation, including k-fold cross-validation, or by bootstrapping estimates how much of the apparent fit is optimism, and TSD 14 lists these among useful ways to validate a fitted survival model internally.

Partial pooling and borrowing strength across subgroups and baskets

When an economic model needs separate inputs for several subgroups, the tradeoff appears in its plainest form. Estimating each subgroup on its own data is unbiased but noisy, while one pooled estimate is stable but biased for any subgroup that differs. A hierarchical model takes a middle course, and in the normal hierarchical model set out by Gelman and colleagues the estimate for subgroup $j$, given $\mu$ and $\tau$, is a precision-weighted average of the form below.

$$\hat{\theta}_j = \frac{\frac{1}{\sigma_j^2}\bar{y}_j + \frac{1}{\tau^2}\mu}{\frac{1}{\sigma_j^2} + \frac{1}{\tau^2}} = w_j \bar{y}_j + (1 - w_j)\mu$$

where $\hat{\theta}_j$ is the posterior mean for subgroup $j$ given $\mu$ and $\tau$, $\bar{y}_j$ is its observed mean, $\sigma_j^2$ is the sampling variance of that mean, $\mu$ is the overall mean, $\tau^2$ is the variance of true subgroup means around $\mu$, and $w_j = \tau^2 / (\tau^2 + \sigma_j^2)$ is the weight on the subgroup's own data. A $\tau^2$ of zero gives complete pooling, and a very large $\tau^2$ leaves each subgroup on its own data. The key assumption is exchangeability. When nothing beyond the data distinguishes the subgroups, their true means are treated as draws from a common distribution, and a characteristic known to separate them is better added to the model as a covariate than left to shrinkage. A random-effects model in meta-analysis has the same structure, with $\tau^2$ as the between-study variance.

Berry and colleagues describe how this shrinks each group's estimate towards the overall mean by an amount that depends on the relative precision of the estimates, and recall Stein's result that with at least three groups the unshrunk estimates are inadmissible, because shrinkage reduces the total mean squared error whatever the true group means. Their simulations of Phase II oncology designs show the bias side of the bargain. In their null scenario a group whose true response rate was 20%, against 5% to 10% in the other three groups, had a mean hierarchical estimate of 14%, and when only one group truly responded, power for that group fell to 65%. Basket trials, which test one targeted treatment across several tumour types, often use this kind of borrowing.

Complexity in decision-analytic model structure

The structure of a decision model raises a looser version of the same choice. In its best practice II-8, the joint ISPOR and SMDM task force on good modelling practice states that simplicity, or parsimony, is desirable for transparency, ease of analysis, validation and description, but that the model must be complex enough to represent faithfully the differences in value between the strategies compared. It adds that models that are too simple, the structural counterpart of underfitting, may lose face validity, while models that are too complex may be difficult to build, debug, analyse, understand and communicate.

Seen through the decomposition, omitted states or pathways produce structural bias, while each added state in a Markov model needs transition probabilities, costs and utilities estimated from often thin evidence, which widens parameter uncertainty. The parallel is an analogy rather than a formal result, because a decision model is rarely tuned against held-out data and the balance rests on judgement, external validation and sensitivity analysis.

Worked example of a subgroup cost estimate that borrows from the trial

The figures in this example are illustrative. A cost-effectiveness model needs the mean annual cost of care for patients aged 80 and over. The trial contains 25 such patients and 400 others, and individual annual costs have a standard deviation of £4,000 in both groups. To show the mechanics, the true mean for the older subgroup is assumed to be £400 higher than for the others; in practice this difference would be unknown.

1. Variance of each simple estimate. The subgroup's own mean has the following sampling variance.

$$V_s = \frac{4000^2}{25} = 640{,}000$$

where $V_s$ is in pounds squared. The mean of the other patients has the following sampling variance.

$$V_p = \frac{4000^2}{400} = 40{,}000$$

where $V_p$ is also in pounds squared.

2. Error of a weighted estimate. A compromise puts weight $w$ on the subgroup mean and $(1 - w)$ on the mean of the other patients, so its bias is $-(1 - w)b$ and its mean squared error is as follows.

$$\mathrm{MSE}(w) = (1 - w)^2 b^2 + w^2 V_s + (1 - w)^2 V_p$$

where $b$ is the true difference of £400 between the two means, the first term is the squared bias and the last two terms are the variance.

3. The best weight. Setting the derivative of the mean squared error to zero gives the following.

$$w^* = \frac{b^2 + V_p}{V_s + b^2 + V_p} = \frac{160000 + 40000}{640000 + 160000 + 40000} = \frac{200000}{840000} = 0.238$$

where $w^*$ is the weight with the smallest error. Its mean squared error is as follows.

$$\mathrm{MSE}(w^*) = \frac{V_s (b^2 + V_p)}{V_s + b^2 + V_p} = \frac{640000 \times 200000}{840000} = 152{,}381$$

where the terms are as defined above. The root mean squared error is about £390, and the bias at this weight is $(1 - 0.238) \times 400 = 305$, so the estimate understates the subgroup mean by about £305.

4. Interpretation. The best estimate understates the older subgroup's cost by about £305 on average, yet its typical error of about £390 is less than half the £800 of the subgroup-only estimate and below the £447 of full pooling. The weight has the same form as the hierarchical weight. Ignoring $V_p$ and replacing $b^2$ with $\tau^2$, a between-subgroup standard deviation of £400 gives $\frac{160000}{160000 + 640000} = 0.2$. Had the true difference been £2,000, the best weight would rise to $\frac{4000000 + 40000}{640000 + 4000000 + 40000} = 0.863$, giving a root mean squared error of about £743 against £800 for the subgroup-only estimate, while keeping the weight of 0.238 would give about £1,543 and full pooling about £2,010. The gain from borrowing shrinks as subgroups diverge, and a weight suited to similar subgroups does harm when they differ, which is why a hierarchical model estimates $\tau^2$ from the spread of several subgroups instead of fixing it.

Where the tradeoff is misread

Several errors recur when the tradeoff is applied to health economic evidence. The most common is to treat close fit to observed data as evidence of predictive accuracy, although TSD 14 and TSD 21 both stress that information criteria say little about extrapolation. Another is to assume that shrinkage fixes overfitting automatically, when its tuning is itself uncertain in small samples. Stein's result is also read as a promise for every subgroup, yet a group that truly differs is pulled towards the others, as the 20% response rate estimated at 14% shows, and a reimbursement decision for that group inherits the bias. Finally, mean squared error treats overestimates and underestimates of the same size as equally costly, whereas the consequences for an adoption decision depend on which side of the cost-effectiveness threshold the estimate falls.

Sources

• Berry SM, Broglio KR, Groshen S, Berry DA. Bayesian hierarchical modeling of patient subpopulations: efficient designs of Phase II oncology clinical trials. Clinical Trials. 2013;10(5):720-734. https://pmc.ncbi.nlm.nih.gov/articles/PMC4319656/
• Gelman A, Carlin JB, Stern HS, Dunson DB, Vehtari A, Rubin DB. Bayesian Data Analysis. 3rd ed. Boca Raton: CRC Press; 2013. Sections 5.4 and 5.6. http://www.stat.columbia.edu/~gelman/book/
• Geman S, Bienenstock E, Doursat R. Neural networks and the bias/variance dilemma. Neural Computation. 1992;4(1):1-58. https://doi.org/10.1162/neco.1992.4.1.1
• Hastie T, Tibshirani R, Friedman J. The Elements of Statistical Learning: Data Mining, Inference, and Prediction. 2nd ed. New York: Springer; 2009. Sections 7.2 and 7.3. https://hastie.su.domains/ElemStatLearn/
• Latimer N. NICE DSU Technical Support Document 14: Survival analysis for economic evaluations alongside clinical trials, extrapolation with patient-level data. Sheffield: Decision Support Unit, ScHARR, University of Sheffield; 2011, last updated March 2013. Sections 3 to 3.5. https://www.sheffield.ac.uk/nice-dsu/tsds/survival-analysis
• Riley RD, Snell KIE, Ensor J, Burke DL, Harrell FE, Moons KGM, Collins GS. Minimum sample size for developing a multivariable prediction model: Part II, binary and time-to-event outcomes. Statistics in Medicine. 2019;38(7):1276-1296. https://pmc.ncbi.nlm.nih.gov/articles/PMC6519266/
• Riley RD, Snell KIE, Martin GP, Whittle R, Archer L, Sperrin M, Collins GS. Penalization and shrinkage methods produced unreliable clinical prediction models especially when sample size was small. Journal of Clinical Epidemiology. 2021;132:88-96. https://pmc.ncbi.nlm.nih.gov/articles/PMC8026952/
• Roberts M, Russell LB, Paltiel AD, Chambers M, McEwan P, Krahn M. Conceptualizing a model: a report of the ISPOR-SMDM Modeling Good Research Practices Task Force-2. Value in Health. 2012;15(6):804-811. https://pmc.ncbi.nlm.nih.gov/articles/PMC4207095/
• Rutherford MJ, Lambert PC, Sweeting MJ, Pennington B, Crowther MJ, Abrams KR, Latimer NR. NICE DSU Technical Support Document 21: Flexible methods for survival analysis. Sheffield: Decision Support Unit, ScHARR, University of Sheffield; 2020. Section 3.2. https://doi.org/10.15131/shef.data.33849118
• Steyerberg EW. Clinical Prediction Models: A Practical Approach to Development, Validation, and Updating. 2nd ed. Cham: Springer; 2019. Chapters 5 and 13.