Thomas Alexander Gerds

# Index of Prediction Accuracy (IPA)

## 1. Introduction

This vignette demonstrates how our software calculates the index of prediction accuracy1, 2. We distinguish three settings:

- uncensored binary outcome
- right censored survival outcome (no competing risks)
- right censored time to event outcome with competing risks

The Brier score is a loss type metric of prediction performance where lower values correspond to better prediction performance. The IPA formula for a model is very much the same as the formula for R2 in a standard linear regression model:

\begin{equation*} \operatorname{IPA} = 1-\frac{\text{BrierScore(Prediction model)}}{\text{BrierScore(Null model)}} \end{equation*}

## 2. Package version

    data.table:[1] ‘1.17.8’

    survival:[1] ‘3.8.3’

    riskRegression:[1] ‘2025.9.5’

    Publish:[1] ‘2025.7.24’

## 3. Data

For the purpose of illustrating our software we simulate data alike the data of an active surveillance prostate cancer study 3. Specifically, we generate a learning set (n=278) and a validation set (n=208). In both data sets we define a binary outcome variable for the progression status after one year. Note that smallest censored event time is larger than 1 year, and hence the event status after one year is uncensored.

    set.seed(18)
    astrain <- simActiveSurveillance(278)
    astest <- simActiveSurveillance(208)
    astrain[,Y1:=1*(event==1 & time<=1)]
    astest[,Y1:=1*(event==1 & time<=1)]

## 4. IPA for a binary outcome

To illustrate the binary outome setting we analyse the 1-year progression status. We have complete 1-year followup, i.e., no dropout or otherwise censored data before 1 year. We fit two logistic regression models, one including and one excluding the biomarker `erg.status`:

    lrfit.ex <- glm(Y1~age+lpsaden+ppb5+lmax+ct1+diaggs,data=astrain,family="binomial")
    lrfit.inc <- glm(Y1~age+lpsaden+ppb5+lmax+ct1+diaggs+erg.status,data=astrain,family="binomial")
    publish(lrfit.inc,org=TRUE)

Variable | Units | OddsRatio | CI.95 | p-value
--- | --- | --- | --- | ---
age |  | 0.98 | [0.90;1.06] | 0.6459
lpsaden |  | 0.95 | [0.66;1.36] | 0.7747
ppb5 |  | 1.09 | [0.92;1.28] | 0.3224
lmax |  | 1.08 | [0.83;1.41] | 0.5566
ct1 | cT1 | Ref |  |
| cT2 | 1.00 | [0.29;3.41] | 0.9994
diaggs | GNA | Ref |  |
| 3/3 | 0.60 | [0.27;1.34] | 0.2091
| 3/4 | 0.25 | [0.05;1.30] | 0.1006
erg.status | neg | Ref |  |
| pos | 3.66 | [1.90;7.02] | <0.0001

Based on these models we predict the risk of progression within one year in the validation set.

    astest[,risk.ex:=100*predictRisk(lrfit.ex,newdata=astest)]
    astest[,risk.inc:=100*predictRisk(lrfit.inc,newdata=astest)]
    publish(head(astest[,-c(8,9)]),digits=1,org=TRUE)

age | lpsaden | ppb5 | lmax | ct1 | diaggs | erg.status | Y1 | risk.ex | risk.inc
--- | --- | --- | --- | --- | --- | --- | --- | --- | ---
62.6 | -3.2 | 4.9 | 4.6 | cT1 | 3/3 | pos | 0.0 | 23.2 | 36.3
66.9 | -1.7 | 0.7 | 4.1 | cT1 | 3/3 | pos | 1.0 | 14.0 | 24.7
65.4 | -1.5 | 4.0 | 3.9 | cT1 | 3/3 | neg | 0.0 | 17.4 | 10.6
59.0 | -2.8 | 6.8 | 3.3 | cT2 | 3/4 | pos | 1.0 | 10.7 | 21.1
55.6 | -3.5 | 2.8 | 3.0 | cT1 | 3/3 | neg | 0.0 | 21.9 | 11.8
71.1 | -2.6 | 3.3 | 3.7 | cT1 | 3/3 | neg | 0.0 | 15.0 | 9.5

To calculate the Index of Prediction Accuracy (IPA) we call the `Score` function as follows on a list which includes the two logistic regression models.

    X1 <- Score(list("Exclusive ERG"=lrfit.ex,"Inclusive ERG"=lrfit.inc),data=astest,
                formula=Y1~1,summary="ipa",se.fit=1L,metrics="brier",contrasts=FALSE)
    X1

    Summary statistics IPA:

    Results by model:

    Key: <model>
               model    IPA  lower  upper
              <fctr> <char> <char> <char>
    1: Exclusive ERG    2.7   -1.6    6.9
    2: Inclusive ERG    7.3    0.4   14.2

    NOTE: Values are multiplied by 100 and given in %.
    NOTE: The higher IPA the better.

    Metric Brier:

    Results by model:

               model  Brier  lower  upper
              <fctr> <char> <char> <char>
    1:    Null model   15.2   11.9   18.6
    2: Exclusive ERG   14.8   11.5   18.2
    3: Inclusive ERG   14.1   11.1   17.2

    NOTE: Values are multiplied by 100 and given in %.
    NOTE: The lower Brier the better.

Both logistic regression models have a lower Brier score than the `Null model` which ignores all predictor variables. Hence, both models have a positive IPA. The logistic regression model which excludes the ERG biomarker scores IPA=2.68% and the logistic regression model which includes the ERG biomarer scores IPA = 7.29%. The difference in IPA between the two models is 4.62%. This means that when we omit `erg.status` from the model, then we loose 4.62% in IPA compared to the full model. It is sometimes interesting to compare the predictor variables according to how much they contribute to the prediction performance. Generally, this is a non-trivial task which depends on the order in which the variables are entered into the model, the functional form and also on the type of model. However, we can drop one variable at a time from the full model and for each variable compute the loss in IPA as the difference between IPA of the full model and IPA of the model where the variable is omitted.

    IPA(lrfit.inc,newdata=astest)

         Variable  Brier    IPA IPA.drop
           <fctr> <char> <char>   <char>
    1: Null model   15.2    0.0      7.3
    2: Full model   14.1    7.3      0.0
    3:        age   14.1    7.4     -0.1
    4:    lpsaden   14.1    7.6     -0.3
    5:       ppb5   14.2    6.9      0.4
    6:       lmax   14.1    7.2      0.1
    7:        ct1   14.1    7.3     -0.0
    8:     diaggs   14.6    4.4      2.9
    9: erg.status   14.8    2.7      4.6

    NOTE: Values are multiplied by 100 and given in %.
    NOTE: The higher IPA the better.
    NOTE: IPA.drop = IPA(Full model) - IPA. The higher the drop
    the more important is the variable for the full model.

## 5. IPA for right censored survival outcome

To illustrate the survival outome setting we analyse the 3-year progression-free survival probability. So, that the combined endpoint is progression or death. We fit two Cox regression models, one including and one excluding the biomarker `erg.status`:

    coxfit.ex <- coxph(Surv(time,event!=0)~age+lpsaden+ppb5+lmax+ct1+diaggs,data=astrain,x=TRUE)
    coxfit.inc <- coxph(Surv(time,event!=0)~age+lpsaden+ppb5+lmax+ct1+diaggs+erg.status,data=astrain,x=TRUE)
    publish(coxfit.inc,org=TRUE)

Variable | Units | HazardRatio | CI.95 | p-value
--- | --- | --- | --- | ---
age |  | 1.03 | [0.99;1.07] | 0.124
lpsaden |  | 1.10 | [0.94;1.29] | 0.230
ppb5 |  | 1.21 | [1.12;1.30] | <0.001
lmax |  | 1.06 | [0.94;1.19] | 0.359
ct1 | cT1 | Ref |  |
| cT2 | 0.97 | [0.57;1.66] | 0.916
diaggs | GNA | Ref |  |
| 3/3 | 0.53 | [0.37;0.76] | <0.001
| 3/4 | 0.32 | [0.18;0.58] | <0.001
erg.status | neg | Ref |  |
| pos | 1.80 | [1.35;2.38] | <0.001

Based on these models we predict the risk of progression or death within 3 years in the validation set.

    astest[,risk.ex:=100*predictRisk(coxfit.ex,newdata=astest,times=3)]
    astest[,risk.inc:=100*predictRisk(coxfit.inc,newdata=astest,times=3)]
    publish(head(astest[,-c(8,9)]),digits=1,org=TRUE)

age | lpsaden | ppb5 | lmax | ct1 | diaggs | erg.status | Y1 | risk.ex | risk.inc
--- | --- | --- | --- | --- | --- | --- | --- | --- | ---
62.6 | -3.2 | 4.9 | 4.6 | cT1 | 3/3 | pos | 0.0 | 67.5 | 80.7
66.9 | -1.7 | 0.7 | 4.1 | cT1 | 3/3 | pos | 1.0 | 48.5 | 60.3
65.4 | -1.5 | 4.0 | 3.9 | cT1 | 3/3 | neg | 0.0 | 67.4 | 60.8
59.0 | -2.8 | 6.8 | 3.3 | cT2 | 3/4 | pos | 1.0 | 51.1 | 70.1
55.6 | -3.5 | 2.8 | 3.0 | cT1 | 3/3 | neg | 0.0 | 41.5 | 35.5
71.1 | -2.6 | 3.3 | 3.7 | cT1 | 3/3 | neg | 0.0 | 65.5 | 57.5

To calculate the Index of Prediction Accuracy (IPA) we call the `Score` function as follows on a list which includes the two Cox regression models.

    X2 <- Score(list("Exclusive ERG"=coxfit.ex,"Inclusive ERG"=coxfit.inc),data=astest,
                formula=Surv(time,event!=0)~1,summary="ipa",se.fit=1L,metrics="brier",contrasts=FALSE,times=3)
    X2

    Summary statistics IPA:

    Results by model:

    Key: <model, times>
               model times    IPA  lower  upper
              <fctr> <num> <char> <char> <char>
    1: Exclusive ERG     3    6.4   -2.6   15.4
    2: Inclusive ERG     3   17.1    8.3   25.8

    NOTE: Values are multiplied by 100 and given in %.
    NOTE: The higher IPA the better.

    Metric Brier:

    Results by model:

               model times  Brier  lower  upper
              <fctr> <num> <char> <char> <char>
    1:    Null model     3   24.0   22.6   25.4
    2: Exclusive ERG     3   22.4   20.5   24.4
    3: Inclusive ERG     3   19.9   17.8   21.9

    NOTE: Values are multiplied by 100 and given in %.
    NOTE: The lower Brier the better.

It is sometimes interesting to compare the predictor variables according to how much they contribute to the prediction performance. Generally, this is a non-trivial task which depends on the order in which the variables are entered into the model, the functional form and also on the type of model. However, we can drop one variable at a time from the full model and for each variable compute the loss in IPA as the difference between IPA of the full model and IPA of the model where the variable is omitted.

    IPA(coxfit.inc,newdata=astest,times=3)

         Variable times  Brier    IPA IPA.drop
           <fctr> <num> <char> <char>   <char>
    1: Null model     3   24.0    0.0     17.1
    2: Full model     3   19.9   17.1      0.0
    3:        age     3   19.7   17.6     -0.6
    4:    lpsaden     3   20.1   16.2      0.8
    5:       ppb5     3   21.3   11.2      5.9
    6:       lmax     3   19.9   16.7      0.4
    7:        ct1     3   19.9   17.0      0.1
    8:     diaggs     3   20.8   13.0      4.1
    9: erg.status     3   22.4    6.4     10.7

    NOTE: Values are multiplied by 100 and given in %.
    NOTE: The higher IPA the better.
    NOTE: IPA.drop = IPA(Full model) - IPA. The higher the drop
    the more important is the variable for the full model.

## 6. IPA for right censored time to event outcome with competing risks

To illustrate the competing risk setting we analyse the 3-year risk of progression in presence of the competing risk of death without progression. We fit two sets of cause-specific Cox regression models 4, one including and one excluding the biomarker `erg.status`:

    cscfit.ex <- CSC(Hist(time,event)~age+lpsaden+ppb5+lmax+ct1+diaggs,data=astrain)
    cscfit.inc <- CSC(Hist(time,event)~age+lpsaden+ppb5+lmax+ct1+diaggs+erg.status,data=astrain)
    publish(cscfit.inc)

       Variable Units                1                2
            age       1.04 [1.00;1.09] 1.01 [0.95;1.07]
        lpsaden       1.13 [0.92;1.38] 1.09 [0.83;1.42]
           ppb5       1.14 [1.04;1.24] 1.39 [1.22;1.58]
           lmax       1.19 [1.03;1.39] 0.82 [0.67;1.00]
            ct1   cT1             Ref              Ref
                  cT2 1.31 [0.73;2.36] 0.31 [0.07;1.28]
         diaggs   GNA             Ref              Ref
                  3/3 0.54 [0.35;0.84] 0.56 [0.29;1.10]
                  3/4 0.44 [0.22;0.88] 0.19 [0.06;0.60]
     erg.status   neg             Ref              Ref
                  pos 2.20 [1.56;3.11] 1.20 [0.71;2.04]

Based on these models we predict the risk of progression in presence of the competing risk of death within 3 years in the validation set.

    astest[,risk.ex:=100*predictRisk(cscfit.ex,newdata=astest,times=3,cause=1)]
    astest[,risk.inc:=100*predictRisk(cscfit.inc,newdata=astest,times=3,cause=1)]
    publish(head(astest[,-c(8,9)]),digits=1,org=TRUE)

age | lpsaden | ppb5 | lmax | ct1 | diaggs | erg.status | Y1 | risk.ex | risk.inc
--- | --- | --- | --- | --- | --- | --- | --- | --- | ---
62.6 | -3.2 | 4.9 | 4.6 | cT1 | 3/3 | pos | 0.0 | 49.7 | 65.5
66.9 | -1.7 | 0.7 | 4.1 | cT1 | 3/3 | pos | 1.0 | 45.2 | 60.1
65.4 | -1.5 | 4.0 | 3.9 | cT1 | 3/3 | neg | 0.0 | 50.6 | 42.3
59.0 | -2.8 | 6.8 | 3.3 | cT2 | 3/4 | pos | 1.0 | 46.0 | 69.0
55.6 | -3.5 | 2.8 | 3.0 | cT1 | 3/3 | neg | 0.0 | 26.3 | 19.9
71.1 | -2.6 | 3.3 | 3.7 | cT1 | 3/3 | neg | 0.0 | 51.8 | 42.2

To calculate the Index of Prediction Accuracy (IPA) we call the `Score` function as follows on a list which includes the two sets of cause-specific Cox regression models.

    X3 <- Score(list("Exclusive ERG"=cscfit.ex,
                     "Inclusive ERG"=cscfit.inc),
                data=astest, formula=Hist(time,event)~1,
                summary="ipa",se.fit=1L,metrics="brier",
                contrasts=FALSE,times=3,cause=1)
    X3

    Summary statistics IPA:

    Results by model:

    Key: <model, times>
               model times    IPA  lower  upper
              <fctr> <num> <char> <char> <char>
    1: Exclusive ERG     3    5.0   -1.9   11.9
    2: Inclusive ERG     3   17.5   10.1   25.0

    NOTE: Values are multiplied by 100 and given in %.
    NOTE: The higher IPA the better.

    Metric Brier:

    Results by model:

               model times  Brier  lower  upper
              <fctr> <num> <char> <char> <char>
    1:    Null model     3   24.5   23.4   25.5
    2: Exclusive ERG     3   23.2   21.1   25.3
    3: Inclusive ERG     3   20.2   18.1   22.2

    NOTE: Values are multiplied by 100 and given in %.
    NOTE: The lower Brier the better.

It is sometimes interesting to compare the predictor variables according to how much they contribute to the prediction performance. Generally, this is a non-trivial task which depends on the order in which the variables are entered into the model, the functional form and also on the type of model. However, we can drop one variable at a time from the full model (here from both cause-specific Cox regression models) and for each variable compute the loss in IPA as the difference between IPA of the full model and IPA of the model where the variable is omitted.

    IPA(cscfit.inc,newdata=astest,times=3)

         Variable times  Brier    IPA IPA.drop
           <fctr> <num> <char> <char>   <char>
    1: Null model     3   24.5    0.0     17.5
    2: Full model     3   20.2   17.5      0.0
    3:        age     3   20.1   18.0     -0.5
    4:    lpsaden     3   20.4   16.8      0.8
    5:       ppb5     3   20.4   16.5      1.1
    6:       lmax     3   21.4   12.6      4.9
    7:        ct1     3   19.8   18.9     -1.4
    8:     diaggs     3   20.8   14.8      2.8
    9: erg.status     3   23.2    5.0     12.5

    NOTE: Values are multiplied by 100 and given in %.
    NOTE: The higher IPA the better.
    NOTE: IPA.drop = IPA(Full model) - IPA. The higher the drop
    the more important is the variable for the full model.

## Footnotes:

1

Michael W Kattan and Thomas A Gerds. The index of prediction accuracy: An intuitive measure useful for evaluating risk prediction models. Diagnostic and Prognostic Research, 2(1):7, 2018.

2

Thomas A Gerds and Michael W Kattan. Medical Risk Prediction: With Ties to Machine Learning. Chapman and Hall/CRC, 2021.

3

Berg KD, Vainer B, Thomsen FB, Roeder MA, Gerds TA, Toft BG, Brasso K, and Iversen P. Erg protein expression in diagnostic specimens is associated with increased risk of progression during active surveillance for prostate cancer. European urology, 66(5):851–860, 2014.

4

Brice Ozenne, Anne Lyngholm S{\o }rensen, Thomas Scheike, Christian Torp-Pedersen, and Thomas Alexander Gerds. riskregression: Predicting the risk of an event using Cox regression models. R Journal, 9(2):440–460, 2017.

Last update: 05 sep 2025 by Thomas Alexander Gerds.