# SENSITIVITY ANALYSIS AND INFERENCE. Lecture 12

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2 SENSITIVITY ANALYSIS AND INFERENCE Lecture 12

3 Points of Discussion 1. Univariate Sensitivity Analyses 2. Tornado Diagrams (many univariate) 3. Bivariate/multivariate sensitivity analyses 4. How to think about confidence intervals 5. How to calculate/interpret confidence intervals with primary data (bootstrapping) 6. How to calculate/interpret confidence intervals with simulations (probabilistic SA) 7. How to combine 5 and 6 8. Cost-effectiveness acceptability curves 9. How to transform back to something that looks more like a net benefit measure

4 Sensitivity analysis Simple definition Vary a critical component of the calculation by a meaningful amount Objective To determine whether results are insensitive to substantial but plausible variation in a parameter To judge the robustness of conclusions To indicate where more detailed studies to obtain better data to pin down the ICER are needed

5 What are critical components? Prices and quantities Sensitivity, specificity, and prevalence Functional relationships Are risks always the same? Multiple exposures Different personal characteristics Health related quality of life measures Discount rate Measures of costs and effectiveness may be differentially sensitive to changes

6 Univariate sensitivity analysis Vary one parameter at a time over a range Best estimate, low value, high value 95% confidence interval Extreme values in plausible range What to do if it is difficult to find a value Perform threshold analyses Assess (usually reasoning-based) whether value is likely to be above or below threshold of interest Can suggest a need for further research to determine range of parameter

7 Threshold analysis Solve equation for what value of a given parameter would make the incremental cost effectiveness ratio equal to a chosen value \$50,000/QALY \$100,000/QALY The same ICER as was found for another intervention in a previous analysis Can be relatively simple algebra in some cases May be more intricate if looking at one parameter in a complex decision tree Search over multiple values

8 Threshold Sensitivity Analysis Reproduced with permission from Frick KD, et al. Modeled cost-effectiveness of the experience corps Baltimore based on a pilot randomized trial. Journal of Urban Health 2004;81: All rights reserved.

9 Pros and cons of univariate sensitivity analyses Pros Can indicate whether anything alters the conclusion about the ICER s relationship with the acceptable range Cons In the real world more than one thing varies at a time No sense of the likelihood of being at different places in the range Ignoring correlation between variation in multiple parameters can overstate uncertainty

10 What can we do with univariate sensitivity analyses? Single univariate sensitivity analysis Not much Multiple univariate sensitivity analyses Tornado diagram Provides a graphical way to show the varying effects of different parameters

11 Tornado Diagram Methods 1. Calculate the point estimate of the ICER 2. Calculate the most favorable ICERs (lowest \$/QALY) when varying each parameter of interest 3. Calculate the least favorable ICERs (highest \$/QALY) when varying each parameter of interest 4. Calculate the range for each parameter 5. Order parameters from widest range to most narrow range 6. Draw graph centered at point estimate

12 Tornado Diagram Example Prevalece Sensitivity Specificity Cost of Diagnosis \$1,000s/QALY

13 Reporting on a Tornado Diagram Give ranges for parameters Give ranges for ICERs Tell which parameters were tested Sometimes do not show all on tornado diagram Interpret the variable for which the uncertainty about the value has the biggest effect on the ICER Suggests need for more information

14 Why might we need more than univariate sensitivity analysis? The ICER depends on multiple parameters Ratio of two numbers Numbers are functions of other parameters Parameters may be be correlated Interaction of certain factors may imply that total effect is larger or smaller than the sum of the parts

15 What to do with multiple sources of uncertainty? Bivariate or more generally multivariate sensitivity analysis How to choose which variables and how to choose how to have them vary together? Combination of extreme values Only useful if demonstrates insensitivity to changes Unlikely that all extreme parameters will occur together Look for set of values that provides best and worst case All most of least favorable rather than all highest or lowest Simulation Return to this one later

16 Confidence intervals for ICERs Confidence interval of ratio is not simple because Even if absolute and incremental are normally distributed Even if absolute and incremental effectiveness are normally distributed The ratio of two normally distributed variables is not normally distributed

17 Complication #1 Zero Denominator If the 95% effectiveness confidence interval includes zero, cannot calculate a 95% confidence interval for the ICER ICER goes to infinity as effect goes to zero If the CI for the effect includes zero, the CI for the ICER is discontinuous Sign of ICER changes when cross zero effectiveness Both negative and positive infinity

18 Early Efforts at CIs Confidence Box Concept Calculate separate confidence intervals for the numerator and denominator of the incremental cost-effectiveness ratio Draw the box formed by the minimum and maximum values in the confidence intervals of each measure Use the ratios implied by the northwest and southeast corners

19 Confidence Box - Graphically \$ Works best if both the NE and SW corners of the box are in the first quadrant 95% CI Effect

20 Confidence Box - Implications Assumes costs and effects are independent Each confidence interval is calculated with no consideration of the confidence interval for the other measure Potential uncertainty about Zero effectiveness in the denominator Dominance

21 Confidence Box What is the Real Interval? Recall that CIs reflect Type I error Proportion of times that the true parameter lies outside the estimated interval in repeated sampling Chance that the true parameter is within the interval is (1-α) Based on probability, the chance of two events simultaneously is (1- α) 2 The confidence interval formed by the confidence box is approximately a 90% confidence interval (0.95*0.95) So, what (if anything) do we know about the wedge formed by the rays from the origin through the corners?

22 More Confidence Box Problems - Graphically \$ a A a b B b A & B are areas with simultaneous extreme values but that is not all that is left out of confidence interval Effect From Briggs & Fenn. Health Economics. 7:723.

23 Confidence Ellipse - Concept Consider the fact that the costs and effects may not be independent More effective treatment may cost more Individuals who live longer have an opportunity to accrue greater costs over time We can use an ellipse to capture the confidence interval rather than a box

24 Confidence Ellipse - Implication If costs and effectiveness are positively correlated the CI will be smaller In contrast, the CI will be wider if costs and effectiveness are negatively correlated Thus, correlation can substantially influence the interpretation of the results

25 Confidence Ellipse Independent Costs and \$ Effects Note that the circle contains a greater proportion of the probability density function than the square, but that the relationship between the wedges is ambiguous Effect

26 Confidence Ellipse - Positively Correlated Costs and Effects \$ Effect

27 Confidence Ellipse Negatively Correlated Costs \$ and Effects Effect

28 Confidence Ellipse Comparison with Confidence Boxes To calculate ellipse use the joint probability density Relationship between ellipses and boxes The ellipse contains more of the probability density function than the box But, the wedges formed by the rays tangential to the box and ellipse contains more of the probability density than either shape alone

29 Confidence Interval Objective Find a way to determine the wedge that contains 95% of the probability density function Issue remains that the confidence interval may still include zero effectiveness Alternatives Mathematical expression Bootstrapping Cost effectiveness acceptability curves

30 Fieller - Concept Distribution of ratio Assume that the costs and effects follow a joint normal distribution R 2 var R E C ( E) + var( C) 2R cov( E, C) ( 0,1) Calculate exact CI such that the wedge in a graph on the CE plane will be exactly 95% Results are conditional on correctness of parametric assumptions ~ N

31 Fieller - Graphical \$ Box Fieller Box Fieller Relationship between Fieller wedge and box wedge is ambiguous. Primarily know that Fieller wedge is exact CI. Effect

32 Fieller -Mathematically Find roots of quadratic equation R 2 [ ( )] E z var 2 cov(, ) α / 2 E R E C zα / 2 E C [ var( )] C z C = 0 α / 2 [ ] Roots = [-b ± sqrt(b 2 4ac)] / 2a

33 Bootstrapping - Concept Empirical sample distribution Multiple draws from a data set Calculate statistics based on each draw Draw conclusions based on those results

34 Bootstrapping Basic Method Draw sample of same size as original sample with replacement from data Obtain confidence limits by taking observations at 2.5 percentile and 97.5 percentile Distinct from probabilistic sensitivity analysis because drawing observations from data set rather than random points in a distribution

35 Bootstrapping-type Example Reproduced with permission from Frick KD, et al. Modeled cost-effectiveness of the experience corps Baltimore based on a pilot randomized trial. Journal of Urban Health 2004;81: All rights reserved.

36 Bootstrapping Interpretation Note that cannot necessarily interpret CI directly May be better to ask questions like: What proportion of draws lead to ICERs below \$50,000 per QALY? What proportion of draws lead to ICERs below \$100,000 per QALY What proportion of draws indicate that the intervention is dominated?

37 Probabilistic sensitivity analyses through simulation Perhaps sensitivity analyses are not useful Not based on the likelihood of values occurring Probabilistic sensitivity analyses Use information on priors about each parameter A somewhat Bayesian approach How likely is it that the ICER is in an acceptable range Can indicate usefulness of additional

38 What does a probabilistic sensitivity analysis really do? Define a distribution of a parameter Draw a random value from the distribution Calculate the ICER (usually just two alternatives) with the value drawn Repeat the process a number of times Report the mean & standard deviation of the distribution, or percentiles of the distribution, or what proportion of the results are below a threshold value for the ICER

39 Cost-Effectiveness Acceptability Curves Crossover between CEA and CBA CEAC s can be used for two purposes Summarize (for every possible value of health effect) the evidence in favor of the intervention leading to a positive net benefit Construct a confidence interval

40 CEAC - Approaches Parametric Use joint normality assumption from Fieller s theorem and looks at the proportion of the joint density that is costeffective Non-parametric Use bootstrapping or probabilistic sensitivity analysis and determine what proportion of ICER estimates lie below any value of health effect

41 CEAC Graphical Example (1) Q \$

42 CEAC Graphical Example (2) Probability that net benefit is greater than zero Dollar value of a QALY

43 CEAC Graphical Example (2) Probability that net benefit is greater than zero Dollar value of a QALY

44 Lack of upper end of CEAC Frick et al. Journal of Urban Health example from several slides back is perfect example The distribution is so unfavorable for the intervention in question that there is never a 97.5% chance that the intervention has a positive net benefit Alternatively, may have so many cost saving alternatives that never have only 2.5% with a positive net benefit

45 Compare & Contrast Techniques Generally If parametric and non-parametric results are similar, use parametric results Replicability Generally more powerful The acceptability curve tends toward a Bayesian approach, i.e. the probability that the intervention is cost-effective Some authors have compared theoretical implications of techniques Some authors have compared empirical results with techniques

46 Back to Net Benefits While some had problems with CBA because of the need to place a dollar value on life, it is easier to work with confidence intervals NMB = λ E C λ = dollar value of a QALY var(nmb) = λ 2 var( E) + var( C) 2λ cov( E, C)

47 Briggs AH, et al. Thinking outside the box: recent advances in the analysis and presentation of uncertainty in costeffectiveness studies. Annu Rev Public Health. 2002;23: Reprinted, with permission, from the Annual Review of Public Health, Volume by Annual Reviews

48 Polsky et al. s Monte Carlo Experiment Repeated samples from data Box method using a different assumption about how wide to make the individual cost/effect confidence intervals Taylor series method An alternative mathematical approximation Fieller Non-parametric bootstrap

49 Polsky et al. s objectives Determine which method came closest to the target miscoverage level of 5% Determine whether or not the miscoverage was symmetric 2.5% of time entire CI is greater than true value and 2.5% of time entire CI is less than true value

50 Polsky et al. Results Fieller was most consistently near 5% Bootstrap was also fairly consistent Box and Taylor methods were less consistent as the correlation between costs and effectiveness increased in the positive range Taylor series method had asymmetric missed coverage

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