Hi Multibilly,
You ask a great question. People get tripped up by this subject a lot. A key reason this happens – and I suspect it's what has happened in your case – is because this oft-quoted statement,
The overall risk of progression was 10% per year for the first 5 years, approximately 3% per year for the next 5 years, and 1% per year for the last 10 years.
is only roughly correct for a very specific purpose, and that purpose is very different from the one for which most people use the probabilities.
The statement is based on the graph you included in your posting, and actually appears in the paper.
If, however, you do probability calculations using the same graph, but in a way that is more relevant to smoldering myeloma patients, the correct probability sequence is NOT the 10% - 3% - 1% sequence quoted above. It's actually 13% - 7% - 4%, for the same time periods used in the more commonly quoted sequence, i.e., 0-5 years after diagnosis, 5-10 years after diagnosis, and 10-20 years after diagnosis.
More specifically, based on the numbers in the graph you posted,
- A smoldering myeloma patient has a 51% chance of progressing to either multiple myeloma or amyloidosis during the first 5 years after they are diagnosed. At any point during those first 5 years, if the patient is still smoldering, they have a 13.3% chance per year of progressing.
- If a smoldering myeloma patient is still smoldering 5 years after diagnosis, they have a 31% chance of progressing sometime in the 5 years after that. At any point during those next 5 years, if the patient is still smoldering, they have a 7.0% chance per year of progressing.
- Finally, if a smoldering myeloma patient is still smoldering 10 years after diagnosis, they have a 35% chance of progressing sometime in the 10 years after that. At any point during those 10 years, if the patient is still smoldering, they have a 4.3% chance per year of progressing.
Again, these probabilities are calculated from the same graph that was used by the graph's authors to come up with the 10% - 3% - 1% probability sequence. It's just that those probabilities, it turns out, apply to a situation very different than the one for which the sequence is usually used.
Instead, for most purposes, the 13% - 7% - 4% sequence is more appropriate.
I will explain more below how I came up with the 13-7-4 sequence and the other probabilities I mentioned above. For most people, the discussion below is probably more technical than they want to get. But, if statistics is your thing, feel free to read on!
Interpreting the Probability GraphAs the authors say in their paper, the graph you pointed to shows the
cumulative probability (their words) of someone progressing from smoldering myeloma to either multiple myeloma or amyloidosis at different points in time after their diagnosis.
What does that mean?
Well, the graph says that if you randomly selected 1000 newly diagnosed smoldering myeloma today, 510 of them (51%) will have progressed sometime within the next 5 years.
By 10 years from now, a total of 660 of the original 1000 (61%) will have progressed.
And, by 20 years from now, a total of 780 (78%) will have progressed.
What The Graph Means For ProbabilitiesNow, back to your question ... and using the numbers in the graph to calculate some probabilities.
You question can be rephrased as follows: For those smoldering myeloma patients who make it 5 years after diagnosis without progressing, what is their probability of progressing in the next 5 years (i.e., in the period between 5 and 10 years after diagnosis)?
Well, the graph shows that, 5 years from diagnosis, 490 (=1000-510) of an original sample of 1000 newly diagnosed smoldering myeloma patients will still have smoldering myeloma.
At 10 years from diagnosis, only 340 (=1000-660) of the original sample of 1000 will still have smoldering myeloma.
So, between the 5th and 10th years after diagnosis, 150 (490-340) patients out of the 490 who still had smoldering myeloma at the 5-year mark will progress, or about 31% of the group.
So, for someone with smoldering myeloma who still has not progressed 5 years from diagnosis, the probability of progressing sometime between 5 and 10 years after diagnosis is 31%.
If you do the math, the 31% probability means that, for someone who still has smoldering myeloma anytime between 5 and 10 years after diagnosis, their risk of progressing in a given year is 7.0%.
What about the probabilities for the period 10 to 20 years from diagnosis?
Well, 10 years from diagnosis, we already saw above that 340 patients of the original 1000 will still be smoldering, based on the estimates in the study authors' chart.
At 20 years from diagnosis, the chart says that only about 220 patients of the original 100 will still be smoldering.
So, for someone who has made it 10 years after diagnosis and is still smoldering, the chance they will progress during the next 10 years is 120 (340-220) divided by 340, or 35 percent.
This translates into a yearly risk of progression during that period, for someone who still has smoldering myeloma, of about 4.3% per year.
The same sort of calculations for the very first 5 years after diagnosis indicate that a smoldering myeloma patient has a 51 percent chance of progressing sometime during that time period.
And that implies a yearly risk of progression during the first 5 years after diagnosis, for someone who still is smoldering at any time during that time, of 13.3% per year.
Why These Numbers Differ From The Typically Cited NumbersThe yearly progression rates I just calculated differ from those that were in the original paper, and which you hear cited so often, for two reasons.
First, the authors calculated their rates of progression always on the basis of the entire original sample, not the number of patients who still had smoldering myeloma at the 5-year and 10-year milestones.
Second, they ignored the effect of compounding -- in this case, what you might call "inverse compounding". That isn't a big issue for short time periods, or when you're dealing with small yearly risks, but it matters for periods as long as 10 years, and it matters for risks on the order of 10% per year.
As an example, take the authors' rate of progression of 1% per year for years 10 to 20 after diagnosis.
They calculated this by saying that, if you have a sample of 1000 newly diagnosed smoldering patients, then from year 10 after diagnosis to year 20 after diagnosis, about 100, or 10%, will progress.
So, they said, that 10% over 10 years is about 1% per year.
But that's misleading. At 10 years from diagnosis, about 340 patients of an original sample of 1000 will still have smoldering myeloma. Of those, about 120 will progress in the next 10 years, for a probability of progression in those 10 years of 35%
for those patients who make it 10 years without progressing.
In other words, for a number that's really meaningful to most smoldering myeloma patients, you need to do the risk calculation based on
how many patients still have smoldering myeloma at 10 years after diagnosis, not how many there were at time of diagnosis.
Second, there's the issue of compounding. The issue is similar to what we all know about compound interest. If you make 10% interest per year on an investment, you will double your money in less than 10 years if you are reinvesting your interest.
The same sort of logic applies in this case when calculating yearly risk of progression. You can't just take the risk for an entire period of time and divide by the number of years. As it turns out, if you take that approach, you will underestimate the yearly risk of progression.
