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Statistical methods: CUSUM, VLAD, risk adjustment, funnel plot. Which one to choose?

Graeme Hickey
October 02, 2016
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Statistical methods: CUSUM, VLAD, risk adjustment, funnel plot. Which one to choose?

Presented at the 30th Annual EACTS Meeting, Barcelona, Spain (1-5 October 2016)

Graeme Hickey

October 02, 2016
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  1. Monitoring Quality in Your Unit: State of the Art Graeme

    L. Hickey* Department of Biostatistics, University of Liverpool * No conflicts of interest
  2. None
  3. • Process monitoring control charts Source: Spiegelhalter DJ. Funnel plots

    for comparing institutional performance. Stat Med 2005;24:1185–202.
  4. Warning: Caution required with prolonged monitoring in cardiac surgery Source:

    Hickey GL et al. Dynamic trends in cardiac surgery: why the logistic EuroSCORE is no longer suitable for contemporary cardiac surgery and implications for future risk models. Eur J Cardio-Thoracic Surg 2013;43:1146–52. Logistic EuroSCORE
  5. 1190 D. J. SPIEGELHALTER NY Hospitals Volume of cases %

    mortality 0 500 1000 1500 2000 0 2 4 6 8 99.8 % limits 95 % limits NY Surgeons Volume of cases % mortality 0 200 400 600 800 1000 0 2 4 6 8 10 99.8 % limits 95 % limits Figure 3. Risk-adjusted 30-day mortality rates following coronary artery bypass grafts in New York state, 1997–1999, for 33 hospitals and 175 surgeons who conducted at least 25 operations in a hospital (separate results are given for the same surgeon op- erating in di erent hospitals, and some comprise an ‘all other’ category). The target is the overall rate of 2.2 per cent. 3.2. Change data A variety of measures of change in performance are possible. For example, changes in propor- Source: Spiegelhalter DJ. Funnel plots for comparing institutional performance. Stat Med 2005;24:1185–202.
  6. 1194 D. J. SPIEGELHALTER Emergency re-admission within 28 days of

    discharge % readmission 0 20000 40000 60000 80000 100000 120000 0 2 4 6 8 10 99.8 % limits 95 % limits Multiplicative over-dispersion % readmission 0 20000 40000 60000 80000 100000 120000 0 2 4 6 8 10 Over-dispersion factor 7.6 based on 10 % winsorised Additive over-dispersion Volume of cases % readmission 0 20000 40000 60000 80000 100000 120000 0 2 4 6 8 10 Random effects SD 0.0073 based on 10 % winsorised (a) (b) (c) Figure 6. Proportions of emergency re-admission within 28 days of discharge from 67 large acute or multi-service hospitals in England, 2000–2001. The target is the overall average rate of 5.9 per cent: (a) shows the over-dispersion around the unadjusted funnel plot; (b) is a multiplicative model in which the limits are expanded by an estimated over-disperion parameter , while (c) is an additive random-e ects model in which all the sample variances have an additional 2 term added. The current control limits can then be in ated by a factor ˆ around Â0 . For example, based on the approximate normal control limits, over-dispersed control limits can then be Source: Spiegelhalter DJ. Funnel plots for comparing institutional performance. Stat Med 2005;24:1185–202.
  7. None
  8. Available online http://ccforum.com/content/10/1/R28 and independently verified. All patients were followed

    up until death or hospital discharge. Model fit was assessed with a calibration curve, and model discrimination was measured by the area under the receiver operating characteristic curve, approximated by the trapezoi- dal method and estimation of 95% confidence intervals [17,18]. The cumulative E-O mortality chart uses patients indexed by order of admission to the ICU. A mathematical description is provided in Additional file 1. It has been described previously death) was recorded. The estimates of probability of death minus the observed outcomes were then accumulated for sequential admissions. The cumulative difference between the expected and observed number of deaths is displayed on the y-axis, for the sequence of patients. The x-axis displays sequential patient admissions, although the date of ICU admis- sion is used on the label for ease of interpretation. The risk-adjusted p chart [20] is a control chart plotting the observed mortality rate and expected mortality rate in groups of patients. It is presented in detail in Additional file 1. In this case we have chosen 2 units of the estimated SD above and Figure 1 Risk-adjusted p chart by blocks of 30 patients Risk-adjusted p chart by blocks of 30 patients. Probability of death estimated with UK APACHE, Royal Berkshire Hospital, 1 January 2003 to 30 June 2004. Source: Cockings JGL et al. Process monitoring in intensive care with the use of cumulative expected minus observed mortality and risk-adjusted P-charts. Crit Care 2006;10:R28.
  9. CUSUM curve ends at zero, indicating performance exactly as expected.

    patie vince surge these ness Virtu Now been comp theor can s lated desir apply the p fate deter her e “pseu pecte puter expec a p% on ch tion. serie Fig 1. Ten hypothetical patients with varying risks (Exp ϭ expected mortality), for a total risk of 2.0. The surgeon is operating exactly as expected, and 2 patients died (Obs ϭ observed mortality). Thus, the observed/expected (O/E) ratio is 1, and the cumulative sum (CUSUM) of the observed mortality minus expected mortality equals zero at the end. 362 THE STATISTICIAN’S PAGE GRUNKEMEIER ET AL CUMULATIVE SUM CURVES Source: Grunkemeier GL et al. Cumulative sum curves and their prediction limits. Ann Thorac Surg 2009;87:361–4.
  10. 4,920 valve operations from 1997 to 2004 [9]. Subse- quently,

    the PHS cardiac programs have performed 3,216 additional heart valve surgeries in which a predicted risk score could be obtained, allowing us to compare the observed mortality in this validation subset of subsequent patients to that expected from the PHS risk model. The CUSUM for these patients, constructed just as the exam- ple in Figure 1, using the PHS risk model for expected mortality, is shown in Figure 2. The horizontal axis is scaled by surgery number, and the operative years are given by vertical grid lines. (An alternative presentation would be to scale the horizontal axis by calendar time and depict the number of cases by vertical grid lines.) CUSUM Prediction Limits The O-E difference will almost never equal exactly zero, even when performance is as expected. Random varia- tion must be accounted for, before any clinical difference is attributed to performance, by constructing an interval estimate that contains the values that are consistent with the observed (point) estimate. A simple 95% confidence interval can be constructed using the normal (“bell- shaped”) approximation to the binomial distribution (Appendix). To produce prediction intervals for the CUSUM plot in Figure 2, we computed these 95% limits at each point (patient). As previously mentioned, it is not intuitively apparent why these prediction intervals should be expanding, or bullet-shaped, as the number of each point, we recruited 5,000 virtual surgeons, each of whom operated exactly as expected, with only random variability separating their results. The first 100 of these results are drawn in Figure 3 as light gray lines, and the quantiles for the middle 95% at each point (patient) for all 5,000 are drawn as thicker black lines. Note that they Fig 2. The cumulative sum of observed minus expected operative deaths for 3,217 heart valve surgeries, with 95% point-wise predic- tion limits. The horizontal axis is scaled by patient number, and the operative years are given by vertical grid lines. The odds ratio (OR), with 95% confidence interval (CI), gives an overall assessment of performance. Source: Grunkemeier GL et al. Cumulative sum curves and their prediction limits. Ann Thorac Surg 2009;87:361–4.
  11. Xt = max(0, Xt-1 + Wt ) Xt ≥ h

    ‘signalled’
  12. None
  13. Source: Steiner SH et al. Monitoring surgical performance using risk-adjusted

    cumulative sum charts. Biostatistics 2000;1:441–52. Unadjusted Adjusted To detect R1 = 2 To detect R1 = ½ To detect R1 = 2 To detect R1 = ½
  14. of ␣ and ␤) and the longer the sequence of

    operations needed graph is more intuitive because it is easier to identify changes in Figure 1. Cumulative failure charts for (a) surgical failure after off-pump CABG (OPCAB) and (b) 30-day mortality after orthotopic heart transplantation in adults. Expected failure rates (p 0 ) were set at overall failure rates for the programs as a whole: (a) 8.5% for 1 consultant and 4 residents and (b) 12% for 8 centers. Boundary lines were constructed to detect a 50% increase in failures (odds ratio, 1.5): (a) 3.7% (p 1 ؍ 12.2%) and (b) 5.0% (p 1 ؍ 17.0%). False-positive (␣) and false-negative (␤) error rates are 5% for both charts. Lines representing expected cumulative failures (— · · ) are shown in both charts, although these are not usually included. In (a), which depicts the consultant and 1 of the 4 residents, the consultant’s failure rate is similar to the overall failure rate (closely follows the — · · line), but is less than expected for the resident. The resident’s performance was confirmed as acceptable (or better) after 100 operations, when the lower boundary line was reached. In (b), which depicts 2 of the 8 transplant centers, performance at center A was consistently better than expected and was confirmed to be acceptable or better after 80 transplantations. Performance at center B was in line with overall mortality for the first 100 transplantations, but increased steadily thereafter. By transplantation 167, center B was close to the 5% upper boundary, having already crossed the 10% upper boundary (not shown). Rogers et al Statistics for the Rest of Us STATISTICS However, plotting boundary lines to detect deviations from accept- able performance is more intuitive with cumulative failures or cumulative log-likelihood ratio charts. Therefore, we consider the two types of chart to be complementary. A line with a gradient corresponding to the acceptable (ex- pected) failure rate could be added to cumulative failure charts, but VLAD or CRAM chart. The graph, which starts at 0, is incre- mented by 1 Ϫ p 0i for a failure and is decremented by p 0i for a success, where p 0i denotes the predicted probability of failure for operation i, derived from the appropriate risk model (Figure 4). The graph has a natural interpretation: it moves upward if the failure rate increases above that predicted by the risk model, moves Figure 2. Cumulative log-likelihood ratio test charts for (a) surgical failure after off-pump CABG (OPCAB) and (b) 30-day mortality after orthotopic heart transplantation in adults. Data and parameter settings for constructing boundary lines (p 0 , p 1 , ␣, and ␤) are the same as for Figure 1. Lines representing expected cumulative failures have not been included; note that such lines, if included, would not be horizontal through 0 but would slope downward from 0 toward the lower boundary, which denotes acceptance of H 0 . These figures provide an alternative representation of the data shown in Figure 1. Interpretation of the graphs in relation to the boundary lines is the same. The points at which the graphs for the resident and center A cross the lower acceptance boundary coincide with Figure 1. Statistics for the Rest of Us Rogers et al STATISTICS Cumulative failure charts Sequential probability ratio test chart tive data that will serve to improve the quality of health care. Although comparative performance of UK cardiac surgeons has been published in the public arena,15 operator specific data for percutaneous cor- onary intervention are not yet available. The task force of the American College of Cardio- logy and American Heart Association has recently published recommendations for standards to assess operator proficiency and institutional programme quality.16 We address these recommendations and provide a method to implement them in a UK setting. We used the north west quality improvement pro- gramme risk model and then used cumulative funnels and funnel plots to display the observed major adverse cardiovascular and cerebrovascular events against the predicted rate of these events. Comparative perfor- mance of UK cardiac surgeons has been disseminated using these plots.17 In cardiology, funnel plots have been used to interpret the dataset of the myocardial infarction national audit project (a UK cardiology dataset that provides specific performance tables).18 Weaimedtoshowthatoperatorspecificoutcomesafter percutaneous coronary intervention can be monitored successfully using funnel plots and cumulative funnel plots. METHODS A detailed database of clinical, procedural, and angiographic variables has been maintained on all patients undergoing percutaneous coronary inter- vention in our unit since 1994. The dataset is based on the British Cardiovascular Intervention Society national dataset,19 with several additional data ele- ments. The prospective acquisition of data is accom- plished by immediate input from the operators after enzyme levels but is not required in the national dataset. We considered Q wave myocardial infarction occurring in the context of angioplasty therapy for acute ST elevation myocardial infarction to be an outcome of the original coronary event and not a complication of percutaneous coronary intervention. Consecutive No of cases Major adverse cardiovascular and cerebrovascular events (%) 1 501 1001 1502 2002 2502 3002 3502 4002 4502 5002 r adverse cardiovascular rebrovascular events (%) 4 6 8 10 0 2 4 6 8 10 Predicted major adverse cardiovascular and cerebrovascular events Observed major adverse cardiovascular and cerebrovascular events Upper control limit Lower control limit Upper warning limit Predicted major adverse cardiovascular and cerebrovascular events Observed major adverse cardiovascular and cerebrovascular events Upper control limit Lower control limit Upper warning limit Cumulative funnel plot Sources: [1] Rogers C et al. Control chart methods for monitoring cardiac surgical performance and their interpretation. J Thorac Cardiovasc Surg 2004;128:811–9. [2] Kunadian B et al. Cumulative funnel plots for the early detection of interoperator variation: retrospective database analysis of observed versus predicted results of percutaneous coronary intervention. BMJ 2008;336:931–4. + many more
  15. always necessary model fit for purpose