# Absolute risks of benefits and harms from PSA testing for use in the decision aid entitled "PSA testing for prostate cancer: it's your choice"

## Table of Contents

Author: Dr Mark Clements, Associate Professor, Department of Medical Epidemiology and Biostatistics, Karolinska Institutet, Stockholm, Sweden

## Statements of absolute risks of benefits and harms from PSA testing

The following draft statements use results from the European Randomised Study of Prostate Cancer (ERSPC) and Australian data on PSA testing, prostate cancer incidence, prostate cancer mortality and prostate cancer survival to estimate cumulative risks of benefit and harm over 20 years of testing from ages 50 to 69 years. The estimates have been based on the best evidence available and will need to be updated as further evidence becomes available.

### Statement 1

*Out of 1000 men who have prostate cancer testing every 2 years from 50 to 69 years of age, 1 man avoids dying from prostate cancer because of testing, 1 man avoids being diagnosed with advanced prostate cancer, and 4 men still die from prostate cancer.* (Estimate of number of men who avoid dying from prostate cancer is based on 13 years of follow-up of men in the ERSPC Study and Australian PSA testing, prostate cancer incidence, prostate cancer mortality data and all-cause mortality data. Estimates of number of men who avoid being diagnosed with advanced prostate cancer is based on 13 years of follow-up in the ERSPC and New South Wales (NSW) prostate cancer incidence and survival data. NSW data were used because NSW is the only state that has survival specific to the different extents of disease at diagnosis.)

### Statement 2

*Out of 1000 men who have prostate cancer testing every 2 years from 50 to 69 years of age, 72 men are diagnosed with prostate cancer. Of these, 25 men experience over-diagnosis – that is they are diagnosed and treated for a cancer that would not have caused any trouble within 20 years – and 47 men are diagnosed with prostate cancer that is not over-diagnosed.* (Estimate of number of men who experience over-diagnosis of prostate cancer is based on 13 years of follow-up of men in the ERSPC Study and 20 years of follow-up for PSA testing and prostate cancer prevalence in Australia.)

### Statement 3

*Out of 1000 men who have prostate cancer testing every 2 years from 50 to 69 years of age, 93 men will experience one or more false positive test results– that is they will have an abnormal PSA test and a following biopsy of the prostate that does not find any cancer, and 907 men will not experience a false positive result.* (Estimate of number of men who experience a false positive result is based on 13 years of follow-up of men in the ERSPC Study and Australian prostate cancer prevalence.)

## Methods

In outline, we describe the mathematical derivations for these estimates, list the data used in the calculations, and provide R code used for the calculations.

### Mathematical derivations

#### Prostate cancer mortality

To estimate the twenty-year risk of prostate cancer death, we combined (i) prostate cancer mortality rates for 1980–1984 (Australian Institute of Health and Welfare; \(\mu_1\)), (ii) other causes mortality rates for 2010-2014 (Australian Institute of Health and Welfare; \(\mu_0\)), and (iii) the participation-adjusted mortality rate ratio from ERSPC (Schröder et al 2014; HR=0.73). Adjusted for competing risks, the risk equations are:

\begin{align*} \mathrm{Risk}_\mathrm{unscreened} & = \int_{50}^{70} \exp\left( - \int_{50}^u (\mu_0(w)+\mu_1(w))dw\right) \mu_1(u) du \\ \mathrm{Risk}_\mathrm{screened} & = \int_{50}^{70} \exp\left( - \int_{50}^u (\mu_0(w)+\mathrm{HR}\mu_1(w))dw\right) \textrm{HR}\mu_1(u) du \end{align*}#### Advanced prostate cancer incidence

To estimate the twenty-year probability of being diagnosed with advanced prostate cancer and being alive at twenty years, we combined (i) the incidence rate ratio for advanced prostate cancer from ERSPC (Schröder et al 2012; HR=0.70), (ii) the ratio of initial and subsequent diagnoses to initial diagnoses of advanced prostate cancer in the unscreened arm of ERSPC (Schröder et al 2012; \(\kappa=1.46\)), (iii) prostate cancer incidence rates for distant extent from New South Wales for 1980–1982 (unpublished data from the NSW Central Cancer Registry; \(\lambda_0\)), (iv) other causes mortality rates for 2010-2014 (Australian Institute of Health and Welfare; \(\mu_0\)), and (v) relative survival for distant extent prostate cancer from NSW (NSW Cancer Institute, 2012; Table 54; \(\mu_2\) as a function of time since diagnosis). Then

\begin{align*} \mathrm{Risk}_\mathrm{unscreened\ and\ alive} & = \int_{50}^{70} \exp\left( - \int_{50}^u (\kappa\lambda_0(w)+\mu_0(w))dw\right) \kappa\lambda_0(u)\times\\ &\qquad\quad \exp\left( - \int_u^{70} (\mu_0(w)+\mu_2(w-u)) dw\right) du \\ \mathrm{Risk}_\mathrm{screened\ and\ alive} & = \int_{50}^{70} \exp\left( - \int_{50}^u (\mathrm{HR}\kappa\lambda_0(w)+\mu_0(w))dw\right) \textrm{HR}\kappa\lambda_0(u)\times\\ &\qquad\quad \exp\left( - \int_u^{70} (\mu_0(w)+\mu_2(w-u)) dw\right) du \end{align*}#### Prostate cancer diagnosis - "Over-diagnosis"

To estimate the twenty-year prevalence of prostate cancer by screening status, we combined published estimates of (i) prostate cancer prevalence for Australia for men aged 70 years in 2007 (Australian Institute of Health and Welfare 2012; average of estimates for ages 60–69 years and 70–79 years, \(R=0.061\)), (ii) prevalence of ever having had a prostate cancer [PSA] test for Australian men aged 70 years in 2011–2012 (Australian Bureau of Statistics 2013; based on an estimate for ages 65–74 years, \(p=0.554\)), and (iii) the prostate cancer incidence hazard ratio from the European Randomised Study of Prostate Cancer (ERSPC; Schröder et al 2014; \(\phi=1.57\); this HR was unadjusted for participation since no participation-adjusted incidence HR has been published). The equation was assumed to be:

\[ 1-R = p(1-R_0)^\phi+(1-p)(1-R_0) \]

Solving this equation for the prevalence of prostate cancer diagnosis in the unscreened, \(R_0\), we calculated the prevalence of prostate cancer in the screened \((=1-(1-R_0)^\phi)\).

#### Negative biopsies – “False positives”

To estimate the twenty-year prevalence of negative biopsy, we combined estimates for (i) the twenty-year prevalence of prostate cancer for men screened (Australian Institute of Health and Welfare 2012), (ii) the number of biopsies from the ERSPC screening arm (=20921), and (iii) the number of men diagnosed with prostate cancer in the ERSPC screening arm (=9163). The twenty-year prevalence of prostate cancer for those screened was multiplied by (20921-9163)/9163 = 1.28 to calculate the number of negative biopsies for those screened.

### Data sources

ageGroup | pcMortRate19801984 | pcMortRate20102014 | allMortRate20102014 | pcIncRate19821986 | otherMortRate20102014 |
---|---|---|---|---|---|

50 | 2.777466e-05 | 1.942580e-05 | 0.0033619511 | 0.0001262 | 0.0033425253 |

55 | 1.056985e-04 | 7.435364e-05 | 0.0051199883 | 0.0004696 | 0.0050456347 |

60 | 2.986520e-04 | 2.163039e-04 | 0.0078476402 | 0.0012188 | 0.0076313363 |

65 | 6.941749e-04 | 4.760841e-04 | 0.0124648228 | 0.0026592 | 0.0119887387 |

ageGroup | advancedInc19801982 |
---|---|

50 | 2.571E-05 |

55 | 9.647E-05 |

60 | 2.194E-04 |

65 | 3.997E-04 |

tm | advancedSurv |
---|---|

0 | 0.6348783 |

1 | 0.4737844 |

2 | 0.3184537 |

3 | 0.2876821 |

4 | 0.2513144 |

The other parameters for men are given in Table 4.

Name | Value | Description | Reference |
---|---|---|---|

PrevPCa | 0.061 | Prevalence of prostate cancer, Australian males aged 70 years, 2007 (\(R\)) | Australian Institute of Health and Welfare |

EverPCaTesting | 0.554 | Prevalence of ever prostate cancer testing, 2011-2012 (\(p\)) | Australian Bureau of Statistics 2013 |

HRDx | 1.57 | Prostate cancer incidence hazard ratio, ERSPC (\(\phi\)) | Schröder et al 2014 |

HRAdvDx | 0.70 | Advanced prostate cancer incidence hazard ratio, ERSPC | Schröder et al 2012 |

RatioAdvDx | 1.46 | Initial and subsequent versus initial diagnoses of advanced prostate cancer, ERSPC | Schröder et al 2012 |

HRMort | 0.73 | Prostate cancer mortality hazard ratio adjusted for compliance, ERSPC | Schröder et al 2014 |

NegBxPerPCaDx | 1.28 | Negative biopsy per prostate cancer diagnosis for screened men, ERSPC | Schröder et al 2014 |

The prevalence of prostate cancer for Australian males aged 70 years (PrevPCa, \(R\) in the mathematical formulation) was calculated from published estimates from AIHW. The value for age 70 years was calculated from the average of men aged 60–69 years and 70–79 years. The published values were for 26 year prevalence, however incidence is comparatively uncommon before age 50 years and PSA testing was negligible before 1988. Similarly, we have used an estimate of ever having had a PSA test from the Australian Health Survey at age 70 years; this is an approximation of twenty-year prevalence, where we have assumed that almost all men who have a PSA test prior to age 70 years have at least one PSA test between age 50 and 70 years.

R code for the competing risks calculation is given below.

competing <- function(start,stop,length=2001,HR=1, mu0=rates$otherMortRate20102014, mu1=rates$pcMortRate19801984, age=rates$ageGroup) { tm <- seq(start,stop,length=length) index <- findInterval(tm,age) wt <- tm*0+1; wt[1] <- wt[length(wt)] <- 0.5 dt <- tm[2]-tm[1] surv <- exp(-dt*cumsum(mu0[index]+HR*mu1[index])) sum(surv*wt*mu1[index]*dt*HR) } illnessDeath <- function(start,stop,length=2001,HR=1, kappa, mu0=rates$otherMortRate20102014, lambda=nswRates$advancedInc19801982, age=rates$ageGroup, mue=nswSurv$advancedSurv, tme=nswSurv$tm) { tm <- seq(start,stop,length=length) index <- findInterval(tm,age) tm2 <- tm-start index2 <- findInterval(tm2,tme) wt <- tm*0+1; wt[1] <- wt[length(wt)] <- 0.5 dt <- tm[2]-tm[1] surv0 <- exp(-dt*sum(mu0[index])) # baseline survival across the interval surv1 <- exp(-dt*cumsum(HR*kappa*lambda[index])) # before dx surv2 <- exp(-dt*rev(cumsum(rev(mue[index])))) # after dx sum(surv0*surv1*wt*kappa*lambda[index]*dt*HR*surv2) }

## Results

#### Summary

Parameter | Study | Screen | Unscreened |
---|---|---|---|

Prostate cancer mortality | ERSPC (13 years) | 6 per 1000 | 8 per 1000 |

Australia (50–69 years) | 4 per 1000 | 5 per 1000 | |

Advanced prostate cancer incidence and alive | ERSPC (13 years) | Not available | Not available |

Australia (50–69 years) | 1 per 1000 | 2 per 1000 | |

Prostate cancer diagnosis ("over-diagnosis") | ERSPC (13 years) | 81 per 1000 | 60 per 1000 |

Australia (50–69 years) | 72 per 1000 | 47 per 1000 | |

Negative biopsies ("false positives") | ERSPC (13 years) | 104 per 10000 | Not available |

Australia (50–69 years) | 93 per 1000 | Not available |

### Calculations

The value for `NegBxPerPCaDx`

was calculated from the screening arm of the ERSPC, where the difference in the number of biopsies and the number of men diagnosed with prostate cancer was divided by the number of men diagnosed with prostate cancer (=(20921-9163)/9163).

We model for not having a prevalent prostate cancer as a mixture of those screened and unscreened. Let \(S\) be the probability of not being diagnosed with prostate cancer in the Australian population (= 1 - `PrevPCa`

), \(S_0\) be not being diagnosed with prostate cancer under no prostate cancer testing, \(HR\) be the hazard ratio for prostate cancer incidence from the ERSPC, and \(p\) be the proportion of men ever having a prostate cancer test from the Australian Health Survey. Then \(S=p S_0^{HR}+(1-p)S_0\).

The twenty-year risks are:

value <- function(name) values$Value[values$Name == deparse(substitute(name))] S0 <- uniroot(function(S0) (1-value(PrevPCa)) - (value(EverPCaTesting)*S0^value(HRDx) + (1-value(EverPCaTesting))*S0), c(0,0.99))$root cat(sprintf("MortRiskUnscreened=%3.1f\tMortRiskScreened=%3.1f\n", 1000*competing(50,70), 1000*competing(50,70,HR=value(HRMort)))) cat(sprintf("AdvPCaAliveUnscreened=%4.2f\tAdvPCaAliveScreened=%4.2f\n", 1000*illnessDeath(50,70,kappa=value(RatioAdvDx)), 1000*illnessDeath(50,70,HR=value(HRAdvDx),kappa=value(RatioAdvDx)))) cat(sprintf("PrevPCa=%4.1f\t\tPrevPCaUnscreened=%4.1f\t\tPrevPCaScreened=%4.1f\n", 1000*value(PrevPCa), 1000*(1-S0), 1000*(1-S0^value(HRDx)))) cat(sprintf("PrevNegBx=%4.1f\t\ttPrevNegBxUnscreened=%4.1f\tPrevNegBxScreened=%4.1f\n", 1000*value(PrevPCa)*value(NegBxPerPCaDx), 1000*(1-S0)*value(NegBxPerPCaDx), 1000*(1-S0^value(HRDx))*value(NegBxPerPCaDx)))

MortRiskUnscreened=5.2 MortRiskScreened=3.8 AdvPCaAliveUnscreened=1.51 AdvPCaAliveScreened=1.06 PrevPCa=61.0 PrevPCaUnscreened=46.8 PrevPCaScreened=72.5 PrevNegBx=78.1 tPrevNegBxUnscreened=59.9 PrevNegBxScreened=92.7

## Commentary

The estimated incidence and mortality risks will be higher for older men and for longer follow-up. Moreover, the PSA screening hazard ratios for incidence and mortality are expected to vary by age, length of follow, intensity of testing, biopsy compliance and choice of prostate cancer test. Mortality will further vary by choice of treatment modality. The baseline mortality rates assume that the rates for the period 1980–1984 are representative, with no secular trend.

Estimates of false positive results, as measured by negative biopsies, assume that the ratios of negative biopsies to prostate cancer diagnoses are similar between the ERSPC and Australia.

There is statistical uncertainty and potential bias in these estimates. They are estimates of population averages and may not accurately represent the risk for any given individual.

Created: 2017-08-29 Tue 12:42