# Dynamic Prostate Cancer Nomogram: Coefficients

Share

#### Legend of Symbols:

β0 : Intercept term, denoted by “Intercept”
β(variable name) : Coefficient estimate, associated with a covariate/variable
γ : Scaling parameter in survival regression models, denoted by “Scaling Parameter”
: Model linear predictor, estimated by summation of the intercept term and all β values multiplied by that variable. For example, β0 + βvar1 * var1 + βvar2 * var2 + … βvarN * varN

### Linear Regression Models:

Linear regression models are used to predict continuous outcomes. The prediction is estimated by:

predicted result =

### Logistic Regression Models:

Logistic regression models are used to predict probabilities. The prediction is estimated by:

 predicted probability = eXβ 1 + eXβ

### Survival Regression Models:

Survival regression models are used to predict the probability that the patient will be free from an event (e.g. biochemical recurrence) at some time point (e.g. 5 years after surgery). They are also used to predict the probability that the patient will be free from an event at some time point, given that the patient has made it t* years without the event (denoted by P(t > 5 years | t > t*) ). (Note: For a baseline prediction of an event at 5 years, use t*= 0, and the numerator will simplify to 1.) The predictions are estimated by:

 P(t > 5 years |t > t*) = 1 + [(e-Xβ)t*]1 ⁄ γ 1 + [(e-Xβ)5]1 ⁄ γ

Note: A PSA test is used to determine biochemical recurrence. If a patient who was free from biochemical recurrence at his last PSA test 3 years after surgery comes in now at 4 years after surgery for a prediction, we can only provide him the probability of being free from biochemical recurrence at 5 years after surgery, given he was free from biochemical recurrence at 3 years after surgery.

### Restricted Cubic Spline Terms:

For some variables in our models, we are including restricted cubic spline terms, meaning that two additional terms will be added to each of the models. Because the knots for these terms are at the minimum, maximum, and tertiles, the knots can change on a daily basis. The knot values are included here. The formula to calculate the two spline terms from the knots is:

 sp1var = max(var - knot1, 0)3 - max(var - knot3, 0)3 knot4 - knot1 knot4 - knot3
 + max(var - knot4, 0)3 knot3 - knot1 knot4 - knot3

 sp2var = max(var - knot2, 0)3 - max(var - knot3,0)3 knot4 - knot2 knot4 - knot3
 + max(var - knot4,0)3 knot3 - knot2 knot4 - knot3

In this example, sp1var and sp2var are the additional spline terms, and var is the value of the variable being splined. The sp1var and sp2var values should be multiplied by their corresponding model coefficients in the model formula to obtain the necessary linear predictor.

### Prostate Cancer Death Models:

The models predicting probability of prostate cancer death have only one covariate, which is the survival probability at 5 years for the associated BCR model (preoperative BCR model with or without cores and postoperative BCR model with or without clinical stage and clinical grade). To calculate the predicted probability of prostate cancer death using any of the preoperative or postoperative models, first calculate the BCR-free probability at 5 years using the selected BCR model. Prostate cancer death predictions are then calculated using the corresponding prostate cancer death models in the same fashion using BCR-free survival probability at 5 years as the covariate.

Model Type Model Variable Value
survival Preoperative BCR (Cores) Intercept 6.59468768
Patient Age -0.00527992
Preoperative PSA -0.4707596
Preoperative PSA Spline 1 0.00407629
Preoperative PSA Spline 2 -0.01141213
Biopsy Gleason Grade Group 2 -1.0072232
Biopsy Gleason Grade Group 3 -2.02462439
Biopsy Gleason Grade Group 4 -2.49980646
Biopsy Gleason Grade Group 5 -2.80520499
Clinical Stage 2A -0.3285729
Clinical Stage 2B -0.65148102
Clinical Stage 2C -0.61816331
Clinical Stage 3+ -0.93665289
No. of Positive Cores -0.05378144
No. of Negative Cores 0.03456631
Scaling Parameter 1.06458144
C-index 0.78350156
Model N 11397
survival Preoperative BCR Intercept 6.64629541
Patient Age -0.00357632
Preoperative PSA -0.45242591
Preoperative PSA Spline 1 0.00381316
Preoperative PSA Spline 2 -0.0106668
Biopsy Gleason Grade Group 2 -1.13060939
Biopsy Gleason Grade Group 3 -2.20718758
Biopsy Gleason Grade Group 4 -2.64665119
Biopsy Gleason Grade Group 5 -3.00685127
Clinical Stage 2A -0.40121778
Clinical Stage 2B -0.8154684
Clinical Stage 2C -0.80353543
Clinical Stage 3+ -1.21438175
Scaling Parameter 1.10666154
C-index 0.77724865
Model N 15660
logistic Extracapsular Extension (Cores) Intercept -4.12316412
Patient Age 0.03002483
Preoperative PSA 0.24631397
Preoperative PSA Spline 1 -0.00173719
Preoperative PSA Spline 2 0.00483196
Biopsy Gleason Grade Group 2 0.63224646
Biopsy Gleason Grade Group 3 1.12707867
Biopsy Gleason Grade Group 4 1.16261692
Biopsy Gleason Grade Group 5 2.14168233
Clinical Stage 2A 0.23208997
Clinical Stage 2B 0.78324104
Clinical Stage 2C 0.61674852
Clinical Stage 3+ 1.4802089
No. of Positive Cores 0.08822019
No. of Negative Cores -0.06245231
AUC 0.7780047
Model N 11602
logistic Extracapsular Extension Intercept -4.30879948
Patient Age 0.02661133
Preoperative PSA 0.25524862
Preoperative PSA Spline 1 -0.00186027
Preoperative PSA Spline 2 0.00517635
Biopsy Gleason Grade Group 2 0.92552469
Biopsy Gleason Grade Group 3 1.43033276
Biopsy Gleason Grade Group 4 1.48595601
Biopsy Gleason Grade Group 5 2.55246777
Clinical Stage 2A 0.311154
Clinical Stage 2B 0.82486803
Clinical Stage 2C 0.88165982
Clinical Stage 3+ 1.73837598
AUC 0.75645662
Model N 15933
logistic Lymph Node Involvement (Cores) Intercept -5.55716924
Patient Age 0.01383913
Preoperative PSA 0.19989322
Preoperative PSA Spline 1 -0.00137288
Preoperative PSA Spline 2 0.00379338
Biopsy Gleason Grade Group 2 0.96607022
Biopsy Gleason Grade Group 3 1.98186216
Biopsy Gleason Grade Group 4 2.13543564
Biopsy Gleason Grade Group 5 2.74999239
Clinical Stage 2A 0.20425873
Clinical Stage 2B 0.56977406
Clinical Stage 2C 0.5132864
Clinical Stage 3+ 0.92021757
No. of Positive Cores 0.06832617
No. of Negative Cores -0.08827788
AUC 0.83693779
Model N 10452
logistic Lymph Node Involvement Intercept -5.95874353
Patient Age 0.00799722
Preoperative PSA 0.17541653
Preoperative PSA Spline 1 -0.00110186
Preoperative PSA Spline 2 0.00302836
Biopsy Gleason Grade Group 2 1.52555553
Biopsy Gleason Grade Group 3 2.57257188
Biopsy Gleason Grade Group 4 2.70542497
Biopsy Gleason Grade Group 5 3.42284006
Clinical Stage 2A 0.28950032
Clinical Stage 2B 0.62068918
Clinical Stage 2C 0.92853262
Clinical Stage 3+ 1.26893881
AUC 0.83009662
Model N 14372
logistic Organ Confined Disease (Cores) Intercept 4.04759847
Patient Age -0.02874416
Preoperative PSA -0.25142694
Preoperative PSA Spline 1 0.00170562
Preoperative PSA Spline 2 -0.0047362
Biopsy Gleason Grade Group 2 -0.66507525
Biopsy Gleason Grade Group 3 -1.24440478
Biopsy Gleason Grade Group 4 -1.24556939
Biopsy Gleason Grade Group 5 -2.29909232
Clinical Stage 2A -0.24874093
Clinical Stage 2B -0.77330003
Clinical Stage 2C -0.57018419
Clinical Stage 3+ -1.47216113
No. of Positive Cores -0.08590313
No. of Negative Cores 0.06308424
AUC 0.78299456
Model N 11591
logistic Organ Confined Disease Intercept 4.27287045
Patient Age -0.02603766
Preoperative PSA -0.25896629
Preoperative PSA Spline 1 0.0018342
Preoperative PSA Spline 2 -0.00509655
Biopsy Gleason Grade Group 2 -0.94949662
Biopsy Gleason Grade Group 3 -1.52910382
Biopsy Gleason Grade Group 4 -1.56125181
Biopsy Gleason Grade Group 5 -2.66333201
Clinical Stage 2A -0.32812056
Clinical Stage 2B -0.83319028
Clinical Stage 2C -0.85856085
Clinical Stage 3+ -1.73380764
AUC 0.76380641
Model N 15921
logistic Seminal Vesicle Invasion (Cores) Intercept -6.34416925
Patient Age 0.02294635
Preoperative PSA 0.27708214
Preoperative PSA Spline 1 -0.0022032
Preoperative PSA Spline 2 0.00615729
Biopsy Gleason Grade Group 2 1.01060597
Biopsy Gleason Grade Group 3 1.85561279
Biopsy Gleason Grade Group 4 1.99375384
Biopsy Gleason Grade Group 5 2.95409228
Clinical Stage 2A 0.13413385
Clinical Stage 2B 0.40933437
Clinical Stage 2C 0.76647585
Clinical Stage 3+ 0.80547871
No. of Positive Cores 0.07534199
No. of Negative Cores -0.12050282
AUC 0.85380288
Model N 11602
logistic Seminal Vesicle Invasion Intercept -6.22097691
Patient Age 0.01234823
Preoperative PSA 0.26493532
Preoperative PSA Spline 1 -0.00199037
Preoperative PSA Spline 2 0.00554875
Biopsy Gleason Grade Group 2 1.1152652
Biopsy Gleason Grade Group 3 1.94162375
Biopsy Gleason Grade Group 4 2.08991533
Biopsy Gleason Grade Group 5 3.04739929
Clinical Stage 2A 0.29693639
Clinical Stage 2B 0.58877349
Clinical Stage 2C 1.15792092
Clinical Stage 3+ 1.29125462
AUC 0.82526306
Model N 15929
survival Postoperative BCR (Clinical) Intercept 6.04319955
Patient Age 0.00485775
Preoperative PSA -0.31950732
Preoperative PSA Spline 1 0.00296384
Preoperative PSA Spline 2 -0.00832215
Biopsy Gleason Grade Group 2 -0.51949515
Biopsy Gleason Grade Group 3 -0.84474436
Biopsy Gleason Grade Group 4 -0.92636898
Biopsy Gleason Grade Group 5 -0.64625692
Pathologic Gleason Grade Group 2 -0.72052835
Pathologic Gleason Grade Group 3 -1.5812866
Pathologic Gleason Grade Group 4 -2.01427586
Pathologic Gleason Grade Group 5 -2.17877223
Clinical Stage 2A -0.23465389
Clinical Stage 2B -0.42748199
Clinical Stage 2C -0.31008802
Clinical Stage 3+ -0.36436117
Extracapsular Extension -0.60749453
Seminal Vesicle Invasion -0.44228091
Lymph Node Involvement -1.12576018
Surgical Margin Status -0.93808006
Scaling Parameter 0.95652618
C-index 0.83157639
Model N 13958
survival Postoperative BCR Intercept 5.96506387
Patient Age 0.00145939
Preoperative PSA -0.30135242
Preoperative PSA Spline 1 0.00272576
Preoperative PSA Spline 2 -0.007647
Pathologic Gleason Grade Group 2 -1.01847272
Pathologic Gleason Grade Group 3 -2.129518
Pathologic Gleason Grade Group 4 -2.75692185
Pathologic Gleason Grade Group 5 -2.74643755
Extracapsular Extension -0.68793978
Seminal Vesicle Invasion -0.4440814
Lymph Node Involvement -1.17787231
Surgical Margin Status -0.92386727
Scaling Parameter 0.96514265
C-index 0.8300361
Model N 14661
survival Preoperative Prostate Cancer Death (Cores) Intercept 2.49813214
Survival probability from full pre-operative BCR model with cores at 5 years (for prostate cancer death model) 2.24491531
Scaling Parameter 0.40929215
C-index 0.88551012
Model N 11594
survival Preoperative Prostate Cancer Death Intercept 2.51827179
Survival probability from full pre-operative BCR model at 5 years (for prostate cancer death model) 2.10140324
Scaling Parameter 0.40306513
C-index 0.87189325
Model N 14808
survival Postoperative Prostate Cancer Death (Clinical) Intercept 2.63126393
Survival probability from full post-operative BCR model with clinical stage and grade at 5 years (for prostate cancer death model) 2.16093221
Scaling Parameter 0.39229262
C-index 0.91242165
Model N 13142
survival Postoperative Prostate Cancer Death Intercept 2.62713618
Survival probability from full post-operative BCR model at 5 years (for prostate cancer death model) 2.1548633
Scaling Parameter 0.39613907
C-index 0.90508277
Model N 13763
Knot Value
PSAPreopKnot1 0.2
PSAPreopKnot2 4.8
PSAPreopKnot3 7.33
PSAPreopKnot4 307

Last Updated: December 15, 2023