Competing risks

David McAllister

2025-01-17

Learning outcomes

  • Understand relationship between rates and risks
  • Understand concept of competing risks
  • Get a feel for the impact of competing risks on outcomes and treatment effects

Overview

  • Spend some time on rates and risks

  • Proceed to competing risks

  • Discussion, equations and interactive plots (via a Shiny app)

Why there are some equations

  • Only if you find these helpful

  • Tried hard, but I am not a statistician - some errors possible, corrections are welcome

Scenarios where we might find competing risks

Cancer

  • Trial of a cancer treatment
    • Death from relapse
    • Death from non-relapse related causes

Cardiovascular

  • Trial of a treatment for myocardial infarction (heart attack)
    • Cardiovascular death
    • Bleeding death
    • Non-cardiovascular non bleeding death

Pregnancy

  • Long-term outcomes following chemotherapy - childbirth
    • Childbirth
    • Relapse

Exercise

Come up with some examples of competing risks

Rates and risks

Rates and risks

  • Rates - events per person-time
    • Single event per person - incidence rate
    • Otherwise - event rates
  • Risk - % or proportion of people experiencing an event

Synonyms

  • Rates - incidence density, force of mortality, force of morbidity, hazard rate
  • Risk - cumulative incidence

Uses

  • Rates - understanding causation, modelling
  • Risks - prediction, magnitude, public health/policy, clinical decision-making

Incidence rate and hazard rate

  • Hazard rate
    • mathematics/statistics
    • instantaneous risk
    • Conditional probability
  • Incidence rate
    • epidemiology
    • empirical examples
    • Number at risk

Equivalent concepts

In the absence of competing risks

1 to 1 relationship. Constant rates

\[ Risk = 1 - e^{-\lambda t}\]

\[\frac{1}{e^{\lambda t}} , \frac{1}{2.72^{\lambda t}} , \frac{1}{2.72^1} , \frac{1}{2.72^2} , \frac{1}{2.72^3} , \frac{1}{2.72^4}\]

\[\frac{1}{e^{\lambda t}}, \frac{1}{2.72^{\lambda t}}, \frac{1}{2.7}, \frac{1}{7.4}, \frac{1}{20.12}, \frac{1}{54.74}\]

\[\frac{1}{e^{\lambda t}}, \frac{1}{2.72^{\lambda t}}, 0.4, 0.14, 0.05, 0.02\]

100 events per 1,000 person years, 0.1 events per person-year; 10 years; \(\lambda t = 1\)

Risk and rates

Risk of event at different follow-up periods

rate 1 year(s) 2 year(s) 3 year(s) 4 year(s)
0.1 0.10 0.18 0.26 0.33
0.2 0.18 0.33 0.45 0.55
0.3 0.26 0.45 0.59 0.70
0.4 0.33 0.55 0.70 0.80

Exercise 1 Questions

  1. A machine has a constant failure rate of \(\lambda = 0.001\) failures per year. Calculate the risk of the machine failing within 10 years?
  2. A machine has a constant failure rate of \(\lambda = 0.010\) failures per year. Calculate the risk of the machine failing within 1 year?
  3. A machine has a constant failure rate of \(\lambda = 0.010\) failures per year. Calculate the risk of the machine failing within 10 years?
  4. A machine has a constant failure rate of \(\lambda = 0.10\) failures per year. Calculate the risk of the machine failing within 100 years?
  5. A machine has a constant failure rate of \(\lambda = 0.90\) failures per year. Calculate the risk of the machine failing within 10 years?
  6. Why are the answers to question 1 and question 2 the same?
  7. Why is it that for the first three we would have gotten a similar answer if we just multiplied the rate per year by the number of years, but this is not the case for the last two questions?
  8. Why is the answer to question 2 similar to, but not exactly, ten times the answer to question 2?
  9. What assumptions do all five questions make?

Exercise 1 Answers

  1. A machine has a constant failure rate of \(\lambda = 0.001\) failures per year. Calculate the risk of the machine failing within 10 years?
1 - exp(-0.001*10)
[1] 0.009950166
  1. A machine has a constant failure rate of \(\lambda = 0.010\) failures per year. Calculate the risk of the machine failing within 1 year?
1 - exp(-0.010*1)
[1] 0.009950166
  1. A machine has a constant failure rate of \(\lambda = 0.010\) failures per year. Calculate the risk of the machine failing within 10 years?
1 - exp(-0.010*10)
[1] 0.09516258
  1. A machine has a constant failure rate of \(\lambda = 0.10\) failures per year. Calculate the risk of the machine failing within 100 years?
1 - exp(-0.100*100)
[1] 0.9999546
  1. A machine has a constant failure rate of \(\lambda = 0.90\) failures per year. Calculate the risk of the machine failing within 10 years?
1 - exp(-0.9*10)
[1] 0.9998766

  1. Why are the answers to question 1 and question 2 the same? Because \(\lambda \times t\) is the same for both.

  2. Why is it that for the first three we would have gotten a similar answer if we just multiplied the rate per year by the number of years, but this is not the case for the last two questions? For the first three, the rate and follow-up time are short (ie \(\lambda \times t\) is small) whereas in the latter two questions it is large

  3. Why is the answer to question 2 similar to, but not exactly, ten times the answer to question 2? Because \(\lambda \times t\) is quite low for both examples and so the relationship between \(\lambda \times t\) and risk is close to linear in this range, but is not quite linear.

  4. What assumptions do all five questions make? There are no competing risks. The rates are constant over time

Time-varying rates

  • Rates varying continuously

\[ Risk = 1 - e^{-\int(\lambda_t dt)}\] - Rates constant within discrete time periods

  - 0-3 months 2.5 per 1000 person years
  - 4-12 months 1.0 per 1000 person years

\[ Risk = 1 - e^{- \sum(\lambda_t t_t)}\]

Area under the curve

Plot of varying rates and risk

Example using incidence rate at discrete time periods

Table for incidence rate at different time periods

time_period N_at_risk duration events person_time rate
1 1000 0.5 231 442.2 52.2
2 769 0.5 168 342.5 49.1
3 601 0.5 149 263.2 56.6
4 452 0.5 129 193.8 66.6
5 323 0.5 96 137.5 69.8
6 227 0.5 65 97.2 66.8
7 162 0.5 44 70.0 62.9
8 118 0.5 27 52.2 51.7
9 91 0.5 21 40.2 52.2
10 70 0.5 15 31.2 48.0

Table for incidence rate at different time periods, continued

time_period N_at_risk duration events person_time rate Risk
1 1000 0.5 231 442.2 52.2 23.1%
2 769 0.5 168 342.5 49.1 39.9%
3 601 0.5 149 263.2 56.6 54.8%
4 452 0.5 129 193.8 66.6 67.7%
5 323 0.5 96 137.5 69.8 77.3%
6 227 0.5 65 97.2 66.8 83.8%
7 162 0.5 44 70.0 62.9 88.2%
8 118 0.5 27 52.2 51.7 90.9%
9 91 0.5 21 40.2 52.2 93%
10 70 0.5 15 31.2 48.0 94.5%

Exercise 2

What happens to risks and effect estimates for different rates

https://ihwph-hehta.shinyapps.io/competing_risks/

Set “Rate of target event per 100 person-years:” to 10, and the “Rate ratio for effect of treatment on target event:” to 0.78. Leave the other settings as they are. Examine the effect of increasing the “Rate of target event per 100 person-years:”. What impact do these changes have on the:-

  1. Risk of target events

  2. Relative risk

  3. Absolute risk reduction

  4. What happens to the relationship between the rate ratio and risk ratio as you increase the target event rate?

  5. What implications does this have for interpreting rate ratios?

Exercise 2 - answers

What happens to risks and effect estimates for different rates

https://ihwph-hehta.shinyapps.io/competing_risks/

Set “Rate of target event per 100 person-years:” to 10, and the “Rate ratio for effect of treatment on target event:” to 0.78. Leave the other settings as they are. Examine the effect of increasing the “Rate of target event per 100 person-years:”. What impact do these changes have on the:-

  1. Risk of target events It increases, but by less with each increase.

  2. Relative risk It gets closer to one

  3. Absolute risk reduction Over the initial year it gets larger, but it is smaller over longer time periods.

  4. What happens to the relationship between the rate ratio and risk ratio as you increase the target event rate? It tends towards the null

  5. What implications does this have for interpreting rate ratios? Long follow-up times or large rates mean that rate ratios ratios cannot be interpreted as risk ratios

Censoring

Example of censoring

Calculating cumulative incidence (risk)

  • Statistical methods to cope with censoring

    • Kaplan-Meier

    \[ S_t = \prod_{i: t_i \le t}(1 -\frac{d_i}{n_i}) \]

    • Nelson-Aalen

\[{\tilde H}_{(t)}=\sum_{i: t_i \le t}(\frac{d_{i}}{n_{i}})\]

with \(d_{i}\) the number of events at \(t_{i}\) and \(n_{i}\) the total individuals at risk at \(t_{i}\)

  • Nelson Aalen - \(risk = 1 - e^{-H_{(t)}}\)

Worked example

see Excel spreadsheet “competing_risk_calculation.xlsx”

Summary: without competing risks

  • Rates lie between 0 and infinity
  • Risks lie between zero and 1
  • Risks can be estimated from rates (and vice versa)
  • doubling rate and doubling time have the same effect on the risk
  • The relationship between rates and risk is non-linear
  • The relationship between time and risk is non-linear
  • If the rate ratio is constant over time, the risk ratio will attenuate over time
  • Rate ratios are NOT risk ratios

Competing risks

What happens to risk of event if there is a competing event

https://ihwph-hehta.shinyapps.io/competing_risks/

Set “Rate of target event per 100 person-years:” to 10 and “Rate ratio for effect of treatment on target event:” to 0.78. Leaving the other settings as they are, gradually increase the event rate for competing events. NOTE THAT IN THIS SCENARIO THE TREATMENT IS NOT RELATED TO THE COMPETING EVENT. What impact do these changes have on the:-

  • Risk of target events
  • Odds ratio
  • Relative risk
  • Absolute risk reduction

Repeat the exercise varying the target event rate too.

Estimating risk if there are competing events

If no censoring

Relapse Death Either Relapse_Cum Death_Cum Either_Cum Relapse_Risk Death_Risk Either_Risk
17 10 27 17 10 27 0.017 0.010 0.027
19 7 26 36 17 53 0.036 0.017 0.053
24 8 32 60 25 85 0.060 0.025 0.085
11 10 21 71 35 106 0.071 0.035 0.106
20 11 31 91 46 137 0.091 0.046 0.137
18 10 28 109 56 165 0.109 0.056 0.165
21 11 32 130 67 197 0.130 0.067 0.197
27 7 34 157 74 231 0.157 0.074 0.231

What if there is censoring

Why

  • Dashed line shows KM estimate treating non-relapse death as censoring

  • Solid line shows KM estimate treating non-relapse death as censoring

  • Double counting censoring if sum these

Why is it different from censoring due to loss to follow-up

  • We assume that such censoring is non-informative

  • KM estimates the survival if those who died of a competing risk had had same underlying rates as those who did not die from a competing risk

Relationship between rates and risk

\[ Risk_1(t) = \int_0^t{S(t)\lambda_1(t) dt} \]

\[ S(t) = exp^{- \int_0^t{\lambda_1(t) + \lambda_2(t) dt}} \]

  • At any time, survival plus Risk1 plus Risk2 always equals 1.

  • Work through an example in excel

Area under the curve

Impact of this graphically

Cause-specific hazard ratios

  • Can also use Cox regression to estimate the cause-specific hazard

  • Same model, different interpretation

  • Cannot directly translate to risk

  • Instead combine the cause-specific hazards using the equation in previous slide to estimate the risk of each outcome

    • R packages such as mstate and msm allow combination of different models
    • Rely on simulation or bootstrapping to get 95 % confidence intervals

Modelling cumulative incidence directly

  • Can also estimate the cumulative incidence directly using Fine and Gray model

  • Produces regression coefficients for effect of a predictor on the sub-distributional hazard rate

  • Unlike the hazard rate from a Cox model this has no natural interpretation

Direction of effect on cause-specific hazard rates

Prognostic score Relapse Cox Death Cox
Very low 1 1
Low 1.01 (0.81-1.27) 1.57 (1.25-1.97)
Medium 1.28 (1.03-1.59) 2.01 (1.61-2.52)
High 1.57 (1.25-1.99) 2.68 (2.12-3.37)
Very high 2.67 (2.06-3.47) 3.98 (3.09-5.13)

Direction of effect on cause-specific hazard rates and risk

Prognostic score Relapse Cox Relapse Fine and Gray Death Cox Death Fine and Gray
Very low 1 1 1 1
Low 1.01 (0.81-1.27) 0.93 (0.75-1.16) 1.57 (1.25-1.97) 1.56 (1.24-1.96)
Medium 1.28 (1.03-1.59) 1.07 (0.87-1.33) 2.01 (1.61-2.52) 1.94 (1.55-2.42)
High 1.57 (1.25-1.99) 1.17 (0.93-1.48) 2.68 (2.12-3.37) 2.48 (1.96-3.12)
Very high 2.67 (2.06-3.47) 1.55 (1.19-2.02) 3.98 (3.09-5.13) 3.27 (2.5; .1.22)

What happens to risks and effect estimates if the treatment also affects the competing event

https://ihwph-hehta.shinyapps.io/competing_risks/

Set “Rate of target event per 100 person-years:” to 10, “Rate ratio for effect of treatment on target event:” to 0.78, “Rate of competing event per 100 person-years:” to 0.10 and “Rate ratio for effect of treatment on competing event:” to 1. Examine the effect of changing the “Rate ratio for effect of treatment on competing event:”. What impact do these changes have on the:-

  • Risk of target events
  • Odds ratio
  • Relative risk
  • Absolute risk reduction

Specifically, what happens to the effect of the treatment on the target event when the treatment also increases the competing risk. Do you think that this is generally a good thing?

Additional references

see https://docs.google.com/document/d/1eqVzqYM6ozlrt5I_4aqqe6OIR6OV6EJ8G6Tg1lBaQ4Q/edit?usp=sharing

You will also find the link to this on the front-page of https://github.com/dmcalli2/Advanced_epidemiology_course