1. Diving into the Deep: Advanced Concepts Module 9

2. Violation of Distributional Assumptions What distribution do the following charts assume? –i-Chart –Xbar-Range –Xbar-Sigma –p Chart –np Chart –u Chart –c Chart

3. Violation of Distributional Assumptions What distribution do the following charts assume? –i-Chart—near normal –Xbar-Range—near normal –Xbar-Sigma—near normal –p Chart--binomial –np Chart--binomial –u Chart--Poisson –c Chart--Poisson

4. Making your distribution normal: Trial and Error Steps Plot original data in histogram and on a probability plot of some kind. If skewed, try lower power (i.e., take the square root of each value). Make probability plot on transformed data. –If looks better, take even lower powers. –If looks worse, take higher powers. Continue until probability plot looks reasonably straight.

5. Ladder of Powers Cubic Square Identity SQRT Log 1/SQRT 1/X 1/Square 1/Cubic This is your raw data Lower Powers Higher Powers Try these transformations until you get a distribution that looks like it comes from a normal distribution.

6. Average monthly number of preventable hospitalizations due to chronic disease in one IPC site, 2004-2005 of Months

7. Histograms of different power transformations.

8. Quantiles of preventable hospitalizations against quantiles of the normal distribution: Dots are right on the line for normally distributed data

9. Quantile Normal Probability Plots for Different Transformations Which transformation is the closest to normally distributed?

10. Quantile Normal Probability Plots for Different Transformations

11. Probability of the observed distribution under the hypothesis of a normal distribution ladder Hospitalizations Transformation chi2(2)P(chi2) cubic 16.360.000 square 12.380.002 raw 6.170.046 square-root 2.010.366 log 0.290.864 reciprocal root 8.040.018 reciprocal 18.250.000 reciprocal square 31.620.000 reciprocal cubic 36.560.000 The approximate probability that the original data are normal. The approximate probability that the log of each data point is normally distributed. Which one would you use? In STATA: Simply type in this command! Results

12. How do you use this transformation? Take the log of each value. Calculate centerline and control limits for the log. Exponentiate the centerline and control limits. Plot these values.

13. i-Chart assuming normal distribution of raw data.

14. i-Chart of the log of the number of hospitalizations.

15. i-Chart of No. of hospitalizations with control limits derived from the log of hospitalizations. Note: UCL for log of hospitalizations = 4.81, UCL for this graph=e 4.81 = 122.74

16. But what about the Central Limit Theorem?

17. How do I know if I can use the Poisson distribution? C-Chart and U-Chart? Practical Way….. If there is a large “n” and very small “p.” If the mean ≈ variance –Calculate variance using formula for estimation of population variance with a sample (sum of squared deviations divided by n-1) and compare to mean. Then you probably have a Poisson distributed variable.

18. How do I know if I can use the Poisson distribution? C-Chart and U-Chart? You can get technical if you need to. –It is theoretically possible to compare the distribution of units of area of opportunity (i.e., bed-days) over the range of possible events (0,1,2,3….50) to the distribution predicted by the mathematical formulae for the Poisson distribution. Save that for another course, please.

19. When can I assume the binomial distribution ? If you have a situation where each observation can be classified as yes or no, 0 or 1, etc. If average p is not close to 0 or 1. If n*p*(1-p) >/= 5 –THEN you can probably use the binomial, i.e., p-chart, np-chart.

20. What about sample size? Need 25 subgroups, more or less For p and np charts, n >/= 4/pBar For u and c charts, cBar >/= 4 See Tables by Benneyan.

21. Extra-binomial & Extra-Poisson Dispersion Can occur when subgroup sizes are too large. Your observed points may be spread out more widely than predicted by the binomial or Poisson distribution.

22. Is this data over-dispersed?

23. How do we solve this? Use smaller increments of time. Check with subject-matter experts to be sure that there are no special causes. Then use the p’ method available in CHARTrunner 3.6. –Adjusts the control limits by combining within- subgroup variation with between sub-group variance.

24. Notice the wider control limits with the p’ chart.

25. This method is available in CHARTrunner 3.6 p’ np’ u’ c’

26. Exercise E9 Enjoy

27. What did we learn from the exercise? Time intervals or subgroups can be “roped together”, i.e., not independent. This restricts their freedom to take on all possible values at any given time. This reduces the real sample size. Control limits are then too narrow. Question: Are monthly measures on the same 50 patients for percent screened for cancer over the last year roped together? Autocorrelated?

28. What can we do? Separate the time between subgroups, e.g., sample once every three months. Change what you are measuring. Use advanced charts to adjust for autocorrelation (limited). Switch to time-series analysis and model the autocorrelation. (Usually need help here.) Don’t worry unless autocorrelation is really high. Use a run chart without run chart rules.

29. Hazards of Pooling Unlike Streams of Data Can we combine –Males with females? –Data from the private sector with the public? –Hospital A with Hospital B? Confusing the effect of time with the effect of another variable is the hazard. Confusion=Confounding

30. Example 30 Day Mortality following CABG Hospital A Hospital B.5 1 n=500 n=5250 n=10000 N=5250

31. Example: Pooled Hospitals A and B 30 Day Mortality following CABG Hospitals A & B.5 1 n=10500 How is this possible?

32. Pooled Results Confused the effect of time with the effect of changing patient load of hospital A vs B.

33. Solutions Don’t combine unless there is no special cause between subgroups. Use indirect standardization. –See Hart and Hart, 2002, Appendix 2.

34. P-Chart comparing reporting sites on 1 executive order.

35. What is wrong with this comparison?

36. Needs Risk Adjustment When comparing outcomes sensitive to the severity of the patient’s condition, and When comparing across sites. Risk adjustment not as important when comparing the same patient population over time. Risk adjustment not as important for process measures that should be done regardless of severity.

37. Approaches Simple: Adjust for Age and Sex Complex: Age, Sex, Comorbidity, Other. –Requires many variables and proprietary or government software.

38. Read about risk-adjusted control charts.

39. Converted into a system with its own software.

40. Other Risk Adjustment Programs CMS JCAHCO AHRQ

41. Problems and Solutions We don’t collect enough variables centrally to risk adjust. Solutions: –Collect more data and apply software. –Stratify/Restrict your comparisons Males, aged 65-74, diagnosed with diabetes 5 years ago. –Compare over time (“percent improvement”) –Investigate case-mix as explanation for difference.