Creating a Monthly "running" or "moving-average" SMR
with a Confidence Interval
Standardized Mortality Ratios (SMRs), like all mortalities, are subject to considerable variation over time. So a monthly SMR would not be very useful. What would be more useful, is an SMR for, say, the last 18 (or 12 or 24) months. The larger the group you're calculating the SMR for, the less relative variation you would expect to see month-to-month, so for a large group of patients (say, 100) you may be able to use a 12-month SMR; for a smaller group you'd want to move more toward 24 months or even longer. With the method described here, your SMR is always updated for the most recent time period.
The advantage of standardizing your mortalities, is that you are pre-adjusting for at least some variables that affect mortality significantly, such as age, race, sex, and primary renal diagnosis. It's less surprising if several 90-year-old patients die, than several 60-year-olds. Note that you can calculate SMRs for any group of patients. You could calculate an SMR for all female patients, for instance, or an SMR for all patients seen by a particular nephrologist.
Calculating a "running" or "moving-average" SMR is pretty easy. Here's how you'd go about it.
Step 1. For each patient, look up the patient's expected annual mortality rate ("xamr") in a table of national mortalities, according to the patient's age, race, sex, primary renal diagnosis ("PRD"), or according to whatever patient characteristics are in the table, as some national tables are different from others. You could use the US Renal Data System's ("USRDS") tables, which you can find at http://www.usrds.org/reference.htm (click on "section H"), or the University of Michigan Kidney Epidemiological and Cost Center's tables, which you can find at http://www.med.umich.edu/kidney/usr/ratetab.txt, or any other tables you can find. You don't need to use national tables, you can use a table for any group larger than the group you're calculating the SMR for, or any group you'd like to standardize your mortality to.
Keep the xamr for each patient in your QI records, and update them annually. These numbers will be between 0 and 1. For instance, using the USRDS tables for all modalities, a 58-year-old white female with a PRD of diabetes has an xamr of 0.245. If you can get it, also jot down the standard error ("se") of the rate. The USRDS tables have these, the KECC tables don't. For our 58-year-old diabetic white female, the se is 0.006.
If you have an se, you can calculate a 95% confidence interval ("95% ci") - the range within which 95% of all patients would fall. The 95% ci is 2 standard errors on either side of the expected rate. So the lower 95% ci for our hypothetical patient would be (0.245-2x0.006) or 0.233, and the upper 95% ci would be (0.245+2x0.006) or 0.257, meaning that 95% of patients would fall between 0.233 and 0.257.
To be most accurate, you'd want to update your xamr on the patient's birthday, but you can update everyone all at once, once a year, for convenience. Most expected mortality tables are in 5-year increments, so an easy way to be most accurate, would be to update everyone's expected rates once a year, when the USRDS or KECC (or whoever issues the table you're using) updates their table. Then, whenever a patient has a birthday that makes their age divisible by 5 (65, 70, 75, etc), look up a new xamr for them.
Step 2. For each patient each month, determine the number of days they were in the group for which you're calculating the SMR. For instance, if you're calculating an SMR for all of the patients seen by a particular nephrologist, count up the number of days that the patient was under that nephrologist's care during that month. If you're using the modality-specific USRDS rates, you'll need to keep track of when patients change modality, and credit the appropriate number of days to each modality.
Step 3. Divide the result of step 2 (number of days) by 365 (the number of days in a year) to get patient-years (or fraction thereof). If a patient was there all month, just mark down 0.083 (1/12) for them. This gives you the fraction of a year that the patient was assigned to the group. We have to get this number into annual terms, because our expected mortality rates are in annual terms.
Step 4. Multiply the number you got in step 3 (patient-years), by the numbers you got in step 1 (xamr and the 95% ci) to get the patient’s expected mortality ("pxm") and it’s 95% ci. If our hypothetical patient was there all month, then her pxm would be (0.245x0.083) or 0.020, her lower 95% ci would be (0.233x0.083) or 0.019, and her upper 95% ci would be (0.257x0.083), or 0.021. Write these numbers in the patient's QI records: "xamr 0.245 for 0.083 patient-years, pxm 0.020 for the month, 95% ci 0.019 to 0.021."
Step 5. Now add up these numbers for all patients in the group, for each month. This gives you the total expected mortality ("txm") in your group for the month. You know the total actual mortality ("tam") - it's just the number of people who died. Your SMR is simply tam divided by txm.
But as we said at the beginning, a one-month SMR isn't usually much good, because deaths don't occur evenly in time. However, having these numbers for each month, you can easily calculate your "running" SMR for any number of months. To do so for any series of months, simply...
Step 6. Add up the txm's for each month in the series, add up the tam's for each month in the series, and then divide the total tam by the total txm. Also add each patient's lower 95% ci to get a lower group 95% ci, and add their upper 95% ci, to get an upper group 95% ci., and divide tam by each of those numbers.
Let's work through a highly simplified example. Suppose you had 10 patients in your group, each of which had the same xamr as the hypothetical patient mentioned above (0.245). Then the txm would be 0.20 deaths (see step 4: 0.245x0.083x10), the lower 95% ci would be 0.19 (0.233x0.083x10), and the upper 95% ci 0.21 deaths (0.257x0.083x10). If this condition applied to 12 months running, then the txm for the 12 months would be 2.40 (0.20x12), with a lower 95% ci of 2.28 (0.19x12) and an upper 95% ci of 2.52 (0.21x12).
Of course, in real life, each patient would have their own unique xamr, and many patients would be in the group for less than the entire month, so the txm for each month would be different, and we'd have to add up the pxm for each patient, and add up the txm for each month, to get the txm for the 12 months, rather than just multiplying by 10 patients and 12 months.
In our oversimplified example, though, we expect 2.40 deaths (2.28 to 2.52) within these 12 months. If we find 2 actual deaths, then our SMR, which is actual deaths divided by expected deaths, is 2 / 2.40 or 0.83, with a lower 95% ci of 0.79 (2 / 2.52) and an upper 95% ci of 0.88 (2 / 2.28). Had we found 3 actual deaths, our SMR would have been 3 / 2.40 or 1.25, with a lower 95% ci of 1.19 (3 / 2.52) and an upper 95% ci of 1.32 (3 / 2.28). Notice that we divide actual mortality by the lower 95% ci of expected mortality to get the upper 95% ci of the SMR, and divide by our upper 95% ci of expected mortality to get the lower 95% ci of the SMR.
In the following month, we would drop the oldest month from our calculation, and add the latest month. Let's say, for instance, that in the following month, when we added up the pxm for each patient, we got 0.18, with a lower 95% ci of 0.16 and an upper 95% ci of 0.20. The previous 11 months had txm of 0.20 each with a 95% ci of 0.19 to 0.21. Our 12-month txm would then be 2.38 (11x0.20 + 0.18) with a lower 95% ci of 2.25 (11x0.19 + 0.16) and an upper 95% ci of 2.51 (11x0.21 + 0.20). If we had 2 actual deaths within this latest 12 months (don't forget to exclude actual deaths in the month that we dropped), then our 12-month running SMR would be 2 / 2.38 or 0.84, with a 95% ci of 0.80 (2 / 2.51) to 0.89 (2 / 2.25).
The SMR is a ratio, so a number above 1.0 indicates more mortality than expected, and a number below 1.0 indicates less mortality than expected. You can easily convert the ratio to a percentage, just multiply by 100. An SMR of 1.00 is 100% of expected mortality, an SMR of 1.25 is 125% of expected mortality (25% more than expected), and an SMR of 0.75 is 75% of expected mortality (25% less than expected).
Let’s summarize...
For each patient in each month in each group, you’d keep track of their expected annual mortality rate (which you’d look up in a table) and their patient-years (the number of days they were in the group during the month, divided by 365), and the product of these two fractions, which is the patient’s expected mortality for that month in that group. If you wanted to, and your table included standard errors, you could also keep track of their 95% confidence intervals.
For each month, you’d keep track of the total expected mortality (or expected deaths) for the group (or for each group, if you’re monitoring several different groups), by simply adding the expected mortalities for each patient. You could also add each patient’s 95% confidence interval to get the group 95% confidence interval, if you had the data. You’d also keep track of the number of actual deaths.
Each month you would add up the total expected mortalities for the previous y (say, 12 or 24) months, add up the actual deaths which occurred during those y months, and divide the actual by the expected, to get your running y-month SMR. If you wanted you could also calculate the running 95% confidence interval on the SMR.