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Ann Thorac Surg 2003;76:663-667
© 2003 The Society of Thoracic Surgeons

Statistician's page

Cumulative sum techniques for assessing surgical results

Gary L. Grunkemeier, PhDa*, Ying Xing Wu, MDa, Anthony P. Furnary, MDa

a Providence Health System, Portland, Oregon, USA

* Address reprint requests to Dr Grunkemeier, 9205 SW Barnes, #33, Portland, OR 97225, USA.
e-mail: gary.grunkemeier{at}providence.org

The report by Novick and colleagues [1], in this issue of The Annals of Thoracic Surgery uses a cumulative sum (CUSUM) technique to assess the learning curve in telerobotic surgery. Novick and associates [2] have previously used CUSUMs to describe the learning curve of an academic surgeon, the change from on-pump to off-pump coronary bypass surgery [3], and the learning curve for off-pump surgery [4]. From these studies, they found that CUSUM provided " ... a more sensitive indicator of a cluster of surgical failures than standard statistical techniques" [1].

Background

CUSUM analysis was introduced 50 years ago in the United Kingdom (UK) using the terminology of industrial quality control [5], and was first used to monitor surgical performance 10 years ago [6]. Since then, several authors from the UK have extended the theory to accommodate the varying risk of cardiac surgery mortality [7–13]

Constant risk of failure
The original idea, as used by Novick and coworkers [1], is to plot the cumulative sum of "adjusted" failures by patient number, where the adjustment consists of subtracting a fraction of a failure for each patient, representing the expected or acceptable failure rate. The units on the vertical axis are then "excess failures." If the "process" is performing as expected, the resulting cumulative sum will be a jagged line hovering around the horizontal axis. For example, if the expected failure rate is 10%, then 0.1 (10% of a failure) is subtracted from the cumulative sum for each patient. When a patient fails, 1.0 (100% of a failure) is added, resulting in a net rise of 0.9 (1.0 to 0.1) at that point. Nothing is added to the expected risk if the patient does not fail; the drop stays at 0.1. Thus, if exactly 1 of the first 10 patients fails (as expected), then the cumulative sum at the tenth patient will be zero (1.0 to 10 x 0.1 = 0.0). If the process is experiencing more failures than expected the CUSUM curve will rise above the horizontal axis. This is what happened in Fig 1 of Novick and associates [1], but most of the rise occurred in the first 20 patients. After that the curve remained relatively flat, indicating a period of acceptable performance. If the process is performing better than expected, the CUSUM will drop below the horizontal axis, indicating fewer failures than expected.



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  		Fig 1. Output of a logistic regression risk model for operative mortality after coronary bypass surgery in four hospitals. Each surgery is represented by a single dot, with its horizontal position indicating the date of surgery. The vertical position indicates the logit (log odds) of death; the higher the dot the higher the risk of dying. The larger black dots indicate patients who died, and the smaller gray dots represent patients who lived. Four typical patients are indicated by arrows. In decreasing order of desirability they are: a patient with a high risk who lived (in Hospital A), a patient with a low risk who lived (in Hospital B), a patient with a high risk who died (in Hospital C), and a patient with a low risk who died (in Hospital D).

 
Variable risk of failure
The standard CUSUM technique "expects" a constant risk of failure for each patient. This is appropriate for a new procedure, such as telerobotics, which has no individualized risk models yet available. But for more well studied areas of cardiac surgery, the expected mortality or failure rate for each patient may not be constant, but may be known from one of the many risk models available, such as those generated from international [14], national [15], regional [16], or local [17] databases. Using such a risk model a modification of the CUSUM plot can be constructed by subtracting not a fixed amount for each patient, but varying amounts based on individual preoperative risks as given by the model. The report of de Laval and coworkers [6] gave examples using both constant and individualized expected risks. Lovegrove and colleagues [7] plotted cumulative expected minus observed deaths versus patient number, calling the result a variable life-adjusted display (VLAD). Poloniecki and associates [8] produced essentially the same display, but called it cumulative risk-adjusted mortality, later abbreviated CRAM [18, 19]. The term "CUSUM" refers to methods associated with the classic, constant risk technique. We will hence use "Cusum" to refer to these risk-adjusted cumulative sum methods.

Example: coronary artery bypass graft operative mortality
We illustrate the risk-adjusted Cusum using cardiac surgery mortality data from ten Providence Health System (PHS) hospitals in four western states that have contributed to a collaborative cardiac database. A logistic regression risk model for operative death after coronary bypass surgery was developed from 12 risk factors using 12,641 patients operated on from January 1997 through June 2002.

Logistic regression produces a score, called a logit (the logarithm of the odds of death), for each patient; the higher the logit, the higher the risk of death [20]. Figure 1 illustrates the results of the PHS risk model applied to four hospitals from 1998 to 2000. The darker symbols indicate patients who died. It is difficult to determine from these raw results how the hospitals are performing. Hospital C has the most deaths, but it also has the most patients. Hospital D had only one death in 1998, but not many patients. These raw results are transformed into a Cusum plot in two steps.

First, for more direct interpretation, the logits are transformed into probabilities [20]. The top halves of the panels in Fig 2 demonstrate the risks of death for each survivor on the probability scale, between 0 (0% risk of dying) and 1 (100% risk). For the each patient who dies, one is subtracted from the expected probability of death, so these points fall into the lower half of each panel (between -1 and 0). Thus Fig 2 is a plot of the expected (E) minus observed (O) outcome for each patient, where E is the probability of death from the risk model and O is 0 for survivors and 1 for deaths. If a patient had a 48% risk of death and did not die his point would lie at + 0.48 (arrow in hospital A); conversely, if a patient had a 44% risk of death and did die his point on this graph would lie at -0.56 (arrow in hospital C).



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  		Fig 2. Transformed output for the four hospitals in Fig 1, with the same 4 typical patients are indicated (arrows). The vertical position depicts the acceptability of the outcome, computed as expected (probability) minus observed (0 for survivors, 1 for deaths) mortality. The top halves of each of the four panels represent patients who lived (positive outcomes), with the vertical axis indicating their expected probability of death. The higher patients in this half (e.g., arrow in hospital A) had the highest risk, so their survival is the best possible outcome; the lower patients (arrow in hospital B) had lower estimated risk, and still had a good, but not as unexpected, outcome. The bottom halves of the panels represent patients who died (negative outcomes) scaled according to their estimated probability of death, minus one. The lowest patients in this half (arrow in hospital D) had the lowest risk, and hence the worst possible outcome; the higher patients (arrow in hospital C) had higher estimated risk and still a bad, but not as unexpected, outcome.

 
It is still impossible to assess performance from the E–O points plotted in Fig 2, but the cumulative summation of the values of these points across time, as in Fig 3, renders the information quite amenable to visual interpretation. If the deaths are occurring as expected from the risk model, the plot will be relatively flat (Fig 3, hospital D). If the overall observed mortality is less than that which was expected, cumulative lives are saved and the plot rises above the horizontal axis (Fig 3, hospital A). And more deaths than expected results in a fall below the horizontal axis (Fig 3, hospital C). Novick and coworkers [1], and others, have plotted O–E instead of E–O (see Comment section on labeling the vertical axis).



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  		Fig 3. For each hospital the jagged line is the risk-adjusted cumulative sum over time of the values plotted in Fig 2. The height of the line at each date equals the sum of the contributions for each patient who had surgery on or before that date. Each patient contributes a value between -1 (worst case) to + 1 (best case), according to the value on the vertical axis in Fig 2. Thus, excursions above the dashed horizontal line indicate lives saved, compared with expected, and below the horizontal line excess deaths. An upward slope indicates improvement, a downward slope worsening. Such trends are easy to appreciate in this portrayal, and virtually impossible to detect before cumulating (Fig 2). The 95% pointwise two-sided prediction limits are plotted as aids for interpretation, but do not provide exact values for hypothesis testing (see text).

 
Prediction limits
When the individual risk is taken to be constant, classis CUSUM theory provides for control limits to signal an "alert" or "alarm" when the failure rate has increased [5, 6]. These limits incorporate the implied continuous testing; that is, they account for the "multiple comparison" problem of generating many unplanned p values at essentially every point on the horizontal axis. For the risk-adjusted Cusum, the situation becomes more complex, and some authors do not include control limits [6, 7], or include limits that do not account for the implied multiple comparisons [8]. Sherlaw-Johnson and colleagues [9] computed a prediction interval for the point at the end of the curve that does not account for multiple testing, but does provided an "indication of the inherent variability" to assist "the process of interpretation." We extended this method to add prediction intervals at each point on the curve (Fig 3). Note these pointwise limits do not account for the multiple testing; nevertheless, they do seem to provide a useful way of assessing the variability, and they seem to agree qualitatively with a more complex method that does produce accurate probabilities (see Appendix).

Comment

Cumulative sum techniques are an informative, visually helpful tool for presenting data and studying trends. CUSUM is a 50-year-old method that has been used on cardiac surgery mortality for the past 10 years. Treasure and associates [21]22 recently wrote an overview that covers much of the material in this study, and Grigg and colleagues [13] provide a thorough review on a more technical level.

Labeling the vertical axis
Plotting cumulative expected minus observed (E-O) mortality on the vertical axis (Fig 3) means that if the line goes up, the actual deaths are fewer than expected; the vertical axis could be labeled "lives saved." This was used in several reports [7–9] and we used it for the ease of explaining the transition from the individual (Fig 2) to the cumulative values (Fig 3). Plotting cumulative observed minus expected (O-E) on the vertical axis, as Novick and colleagues [1] and others [5, 6, 12, 13] have done, means that if the line goes up, the actual deaths are more than expected; the vertical axis could be labeled "excess deaths." Both styles of labeling are correct; which is used is a matter of taste, or emphasis.

Labeling the horizontal axis
Most reports used operation number on the horizontal axis, but some used date of surgery. The latter is advantageous when comparing trends within or between providers because the unit of reporting for quality assessment is usually calendar time (year or quarter of year). We used date of surgery (Fig 3), but similar curves would result from using number of cases if the caseload is constant across time. That this was the case can be inferred by the smoothness of the prediction curves in Fig 3. They increase as the numbers of patients increase; that they do so rather smoothly indicates that the number of patients is fairly constant over time.

Prediction limits
Prediction limits aid in deciding when a deviation from the horizontal axis is more than would be expected due to random variation. In Fig 3, when the Cusum curve goes outside of the prediction bands, it gives a suggestion of a statistically significant difference. There is a caveat with this, however, because it does not account for the implied multiple comparisons. A recent study, discussed in the Appendix, provides cumulative sum curves (Fig 4) that do provide correct significance tests, and that "naturally complement intuitively attractive plots of cumulative observed-expected mortality" [12] (Fig 3).



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  		Fig 4. The jagged lines in this figure depict a method for assessing statistical significance based on formal hypotheses tests (see Appendix). The thicker lines are for testing the hypothesis of an observed to expected odds ratio of 2 (alternative hypothesis) versus 1 (null hypothesis); that is, testing for a poor result versus performance as expected. The thinner lines are for testing the hypothesis of an observed to expected odds ratio 1/2 (alternative) versus 1 (null); that is, testing for a superior result versus performance as expected. Horizontal lines are given for the rejecting the null hypotheses (above the horizontal axis) and for rejecting the alternative hypotheses (below the horizontal axis) with probabilities of type I error (alpha) of 0.05 and type II error (beta) of 0.50.

 
The precise assignment of statistical significance may not be an essential requirement for cumulative sum curves, anyway. It can be overshadowed by the inherent tradeoff between examining long versus short runs of data. If the Cusum is reset to 0 at periodic intervals (yearly, quarterly), then a significant trend may never be discovered. And if it is never reset, statistical significance will likely be achieved eventually. This effect is pervasive in statistical work: small studies will not find clinically important differences to be statistically significant, and large studies will find clinically unimportant differences to be significant. Realization of this has effect has moved current opinion away from hypothesis testing (Fig 4) and towards interval estimation (Fig 3).

Acknowledgments

The authors are grateful to Jeanne Zerr for guidance on content, to Ling Zhang for clerical support and to the following PHS hospitals for sharing their coronary bypass surgery data: Alaska: Providence Anchorage Medical Center; Washington: Providence Everett Medical Center, Providence Campus, Swedish Medical Center (Seattle), Providence St. Peter Hospital (Olympia), Providence Yakima Medical Center (Yakima); Oregon: Providence Portland Medical Center, Providence St. Vincent Medical Center (Portland); and California: Providence St. Joseph Medical Center (Burbank), Providence Holy Cross Medical Center (Mission Hills), Little Company of Mary Hospital (Torrance).

Appendix

Prediction and control limits
The risk-adjusted Cusum at time t is the cumulative sum of the expected (0 < E < 1) minus the observed (O = 0 or 1) mortality for all patients operated from the start of the analysis period up to time t (instead of E-O, many authors use O-E; see Comments section). If a process is operating at the expected risk, the Cusum will hover around the horizontal axis. Some departures from this line are to be expected due to random variation. It is important to have an indication of where the limits of random variation stop and divergent performance, good or bad, begins. For the prediction limits in Fig 3, we used the recommendation of Sherlaw-Johnson and coworkers [9], but computed it for each point along the horizontal axis, rather than at just the end of the curve, as they did. The standard error (SE) of the risk-adjusted Cusum at time t is the square root of the cumulative sum of E(1-E) for all patients operated from the start of the analysis period up to time t. Then 90%, 95%, or 99% two-sided prediction limits are obtained by multiplying the SE by 1.64, 1.96, or 2.58, respectively (usual quantiles of the standard normal distribution). This same formula has been used as pointwise confidence limits for the Cusum.

For an ongoing process, confidence or prediction intervals constructed at each point on the horizontal axis do not maintain their nominal size because of the implied multiple comparisons. For correct significance tests they should be augmented by a method based on formal hypothesis testing. This method uses a cumulative sum, not of intuitive units like "lives saved" or "excess deaths," but rather on units of "logarithm of the likelihood ratio" of the alternative to the null hypothesis [10, 11]. The larger this is, the more the evidence favors the alternative hypothesis over the null hypothesis; the resulting cumulative sums can be used to provide correct p values. Spiegelhalter and colleagues [12] recently used this approach to establish threshold values indicating rejection of the null or alternative hypothesis (Fig 4). Following others [10–12], we chose as the null hypothesis that the odds ratio (OR) equals 1, that is that the observed mortality is as expected, and two different alternative hypotheses: OR = 2 to test for worse results than expected, and OR = 1/2 to test for better results than expected. The control limits are determined by a function of alpha and beta, the Type I and II error rates of the hypothesis test, respectively. We used alpha = 0.05, and for comparison with confidence intervals, beta = 0.50 [22]. The results in Fig 4 agree qualitatively with the simpler and more intuitive confidence interval method depicted in Fig 3, but appear more conservative, as expected. Significance derived from these two figures should not be expected to compare directly because the curves in Fig 4 are based on specific alternative hypotheses.

References

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