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] ‘1.14.2’

    survival:
    [1] ‘3.2.13’

    riskRegression:
    [1] ‘2022.3.8’

    Publish:
    [1] ‘2020.12.23’

## 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=0L,metrics="brier",contrasts=FALSE)
    X1

    Metric Brier:

    Results by model:

               model Brier IPA
    1:    Null model  15.2 0.0
    2: Exclusive ERG  14.8 2.7
    3: Inclusive ERG  14.1 7.3

    NOTE: Values are multiplied by 100 and given in %.
    NOTE: The lower Brier the better, the higher IPA 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
    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=0L,metrics="brier",contrasts=FALSE,times=3)
    X2

    Metric Brier:

    Results by model:

               model times Brier  IPA
    1:    Null model     3  24.0  0.0
    2: Exclusive ERG     3  22.4  6.4
    3: Inclusive ERG     3  19.9 17.1

    NOTE: Values are multiplied by 100 and given in %.
    NOTE: The lower Brier the better, the higher IPA 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
    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.
    Warning message:
    In `[.data.table`(r2, , `:=`(IPA.drop, IPA[Variable == "Full model"] -  :
      Invalid .internal.selfref detected and fixed by taking a (shallow) copy of the data.table so that := can add this new column by reference. At an earlier point, this data.table has been copied by R (or was created manually using structure() or similar). Avoid names<- and attr<- which in R currently (and oddly) may copy the whole data.table. Use set* syntax instead to avoid copying: ?set, ?setnames and ?setattr. If this message doesn't help, please report your use case to the data.table issue tracker so the root cause can be fixed or this message improved.

## 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=0L,metrics="brier",
                contrasts=FALSE,times=3,cause=1)
    X3

    Metric Brier:

    Results by model:

               model times Brier  IPA
    1:    Null model     3  24.5  0.0
    2: Exclusive ERG     3  23.2  5.0
    3: Inclusive ERG     3  20.2 17.5

    NOTE: Values are multiplied by 100 and given in %.
    NOTE: The lower Brier the better, the higher IPA 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
    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: 09 Mar 2022 by Thomas Alexander Gerds.