1. Cumulative sum techniques for assessing surgical results
Gary L Grunkemeier, PhD, Ying Xing Wu, MD, Anthony P Furnary, MD 
The Annals of Thoracic Surgery 
Volume 76, Issue 3, Pages (September 2003)
DOI: /S (03)

2. 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).
The Annals of Thoracic Surgery  , DOI: ( /S (03) )

3. 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.
The Annals of Thoracic Surgery  , DOI: ( /S (03) )

4. 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).
The Annals of Thoracic Surgery  , DOI: ( /S (03) )

5. 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 Annals of Thoracic Surgery  , DOI: ( /S (03) )