# Dynamic Prostate Cancer Nomogram: Coefficients

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#### 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 6.693875
Patient Age -0.00749195
Preoperative PSA -0.45344992
Preoperative PSA Spline 1 0.00386074
Preoperative PSA Spline 2 -0.01079389
Biopsy Gleason Grade Group 2 -1.0529841
Biopsy Gleason Grade Group 3 -2.0850453
Biopsy Gleason Grade Group 4 -2.51372173
Biopsy Gleason Grade Group 5 -2.8320576
Clinical Stage 2A -0.35824667
Clinical Stage 2B -0.64440576
Clinical Stage 2C -0.61990254
Clinical Stage 3+ -1.02150628
No. of Positive Cores -0.05206349
No. of Negative Cores 0.04163068
Scaling Parameter 1.08815849
C-index 0.78997145
Model N 13537
survival Preoperative BCR Intercept 7.00157453
Patient Age -0.00642905
Preoperative PSA -0.48533206
Preoperative PSA Spline 1 0.00414155
Preoperative PSA Spline 2 -0.01158187
Biopsy Gleason Grade Group 2 -1.12333324
Biopsy Gleason Grade Group 3 -2.1860852
Biopsy Gleason Grade Group 4 -2.64539245
Biopsy Gleason Grade Group 5 -3.00742926
Clinical Stage 2A -0.45329289
Clinical Stage 2B -0.84437351
Clinical Stage 2C -0.91932975
Clinical Stage 3+ -1.33843876
Scaling Parameter 1.11543002
C-index 0.78514932
Model N 14974
logistic Extracapsular Extension (Cores) Intercept -4.41310776
Patient Age 0.03329037
Preoperative PSA 0.19544533
Preoperative PSA Spline 1 -0.00118811
Preoperative PSA Spline 2 0.00327714
Biopsy Gleason Grade Group 2 0.68772997
Biopsy Gleason Grade Group 3 1.13269736
Biopsy Gleason Grade Group 4 1.23487789
Biopsy Gleason Grade Group 5 2.10105262
Clinical Stage 2A 0.21471067
Clinical Stage 2B 0.71371831
Clinical Stage 2C 0.60575591
Clinical Stage 3+ 1.44391394
No. of Positive Cores 0.11123539
No. of Negative Cores -0.05181734
AUC 0.77987197
Model N 13440
logistic Extracapsular Extension Intercept -4.36644071
Patient Age 0.02824699
Preoperative PSA 0.22926508
Preoperative PSA Spline 1 -0.00153902
Preoperative PSA Spline 2 0.00426701
Biopsy Gleason Grade Group 2 0.92649489
Biopsy Gleason Grade Group 3 1.38288589
Biopsy Gleason Grade Group 4 1.50616362
Biopsy Gleason Grade Group 5 2.43913196
Clinical Stage 2A 0.28084029
Clinical Stage 2B 0.86575467
Clinical Stage 2C 0.92857596
Clinical Stage 3+ 1.74854159
AUC 0.75896068
Model N 14851
logistic Lymph Node Involvement (Cores) Intercept -6.35881026
Patient Age 0.01586012
Preoperative PSA 0.28237058
Preoperative PSA Spline 1 -0.00221804
Preoperative PSA Spline 2 0.00616788
Biopsy Gleason Grade Group 2 1.09614179
Biopsy Gleason Grade Group 3 2.0302702
Biopsy Gleason Grade Group 4 2.2618141
Biopsy Gleason Grade Group 5 2.95080621
Clinical Stage 2A 0.3216301
Clinical Stage 2B 0.53980149
Clinical Stage 2C 0.79540086
Clinical Stage 3+ 0.86576301
No. of Positive Cores 0.08002588
No. of Negative Cores -0.07377657
AUC 0.84510344
Model N 12309
logistic Lymph Node Involvement Intercept -6.60038112
Patient Age 0.01120972
Preoperative PSA 0.29793074
Preoperative PSA Spline 1 -0.00237714
Preoperative PSA Spline 2 0.0066205
Biopsy Gleason Grade Group 2 1.43597898
Biopsy Gleason Grade Group 3 2.42410932
Biopsy Gleason Grade Group 4 2.62563632
Biopsy Gleason Grade Group 5 3.41432454
Clinical Stage 2A 0.37735952
Clinical Stage 2B 0.68802559
Clinical Stage 2C 1.16380623
Clinical Stage 3+ 1.23807322
AUC 0.83633414
Model N 13606
logistic Organ Confined Disease (Cores) Intercept 4.35221885
Patient Age -0.0323717
Preoperative PSA -0.20028874
Preoperative PSA Spline 1 0.00114922
Preoperative PSA Spline 2 -0.00315903
Biopsy Gleason Grade Group 2 -0.72634412
Biopsy Gleason Grade Group 3 -1.25699952
Biopsy Gleason Grade Group 4 -1.29752232
Biopsy Gleason Grade Group 5 -2.25788378
Clinical Stage 2A -0.23363868
Clinical Stage 2B -0.71391258
Clinical Stage 2C -0.57122334
Clinical Stage 3+ -1.45903072
No. of Positive Cores -0.108532
No. of Negative Cores 0.05351084
AUC 0.78551481
Model N 13427
logistic Organ Confined Disease Intercept 4.32614934
Patient Age -0.02750214
Preoperative PSA -0.23329264
Preoperative PSA Spline 1 0.00150617
Preoperative PSA Spline 2 -0.00416692
Biopsy Gleason Grade Group 2 -0.95627213
Biopsy Gleason Grade Group 3 -1.48980407
Biopsy Gleason Grade Group 4 -1.56000262
Biopsy Gleason Grade Group 5 -2.55847357
Clinical Stage 2A -0.3016105
Clinical Stage 2B -0.88032957
Clinical Stage 2C -0.90212421
Clinical Stage 3+ -1.76093271
AUC 0.76634167
Model N 14838
logistic Seminal Vesicle Invasion (Cores) Intercept -6.65111311
Patient Age 0.02664219
Preoperative PSA 0.26810237
Preoperative PSA Spline 1 -0.0021296
Preoperative PSA Spline 2 0.00594383
Biopsy Gleason Grade Group 2 1.17361557
Biopsy Gleason Grade Group 3 2.00220714
Biopsy Gleason Grade Group 4 2.1598454
Biopsy Gleason Grade Group 5 3.0589873
Clinical Stage 2A 0.16415575
Clinical Stage 2B 0.37132908
Clinical Stage 2C 0.73116561
Clinical Stage 3+ 0.6451354
No. of Positive Cores 0.08094435
No. of Negative Cores -0.12517844
AUC 0.85344015
Model N 13434
logistic Seminal Vesicle Invasion Intercept -6.97424875
Patient Age 0.01975243
Preoperative PSA 0.29093038
Preoperative PSA Spline 1 -0.00228885
Preoperative PSA Spline 2 0.00639119
Biopsy Gleason Grade Group 2 1.30718733
Biopsy Gleason Grade Group 3 2.15458057
Biopsy Gleason Grade Group 4 2.27405845
Biopsy Gleason Grade Group 5 3.28350819
Clinical Stage 2A 0.26469091
Clinical Stage 2B 0.61733764
Clinical Stage 2C 1.26943224
Clinical Stage 3+ 1.15217769
AUC 0.83119521
Model N 14845
survival Postoperative BCR (Clinical) Intercept 6.30010374
Patient Age 0.00317242
Preoperative PSA -0.35464602
Preoperative PSA Spline 1 0.00332495
Preoperative PSA Spline 2 -0.00933372
Biopsy Gleason Grade Group 2 -0.48663298
Biopsy Gleason Grade Group 3 -0.85113182
Biopsy Gleason Grade Group 4 -0.88576073
Biopsy Gleason Grade Group 5 -0.56472928
Pathologic Gleason Grade Group 2 -0.73273629
Pathologic Gleason Grade Group 3 -1.59634431
Pathologic Gleason Grade Group 4 -2.04296963
Pathologic Gleason Grade Group 5 -2.23724523
Clinical Stage 2A -0.22330695
Clinical Stage 2B -0.42809521
Clinical Stage 2C -0.32579901
Clinical Stage 3+ -0.41639404
Extracapsular Extension -0.58873074
Seminal Vesicle Invasion -0.48024863
Lymph Node Involvement -1.15349432
Surgical Margin Status -0.95852634
Scaling Parameter 0.96315633
C-index 0.83929893
Model N 13470
survival Postoperative BCR Intercept 6.23257641
Patient Age 2.5361e-4
Preoperative PSA -0.34431919
Preoperative PSA Spline 1 0.00318132
Preoperative PSA Spline 2 -0.00892723
Pathologic Gleason Grade Group 2 -1.03042113
Pathologic Gleason Grade Group 3 -2.16779384
Pathologic Gleason Grade Group 4 -2.78081863
Pathologic Gleason Grade Group 5 -2.79342781
Extracapsular Extension -0.65710185
Seminal Vesicle Invasion -0.47268471
Lymph Node Involvement -1.20414399
Surgical Margin Status -0.94994848
Scaling Parameter 0.97611662
C-index 0.83594157
Model N 14118
survival Preoperative Prostate Cancer Death (Cores) Intercept 2.4413028
Survival probability from full pre-operative BCR model with cores at 5 years (for prostate cancer death model) 2.13380007
Scaling Parameter 0.39857269
C-index 0.88283975
Model N 13216
survival Preoperative Prostate Cancer Death Intercept 2.45605961
Survival probability from full pre-operative BCR model at 5 years (for prostate cancer death model) 2.11147846
Scaling Parameter 0.40270535
C-index 0.87773153
Model N 13802
survival Postoperative Prostate Cancer Death (Clinical) Intercept 2.57292327
Survival probability from full post-operative BCR model with clinical stage and grade at 5 years (for prostate cancer death model) 2.12976519
Scaling Parameter 0.38922156
C-index 0.9144047
Model N 12369
survival Postoperative Prostate Cancer Death Intercept 2.56187213
Survival probability from full post-operative BCR model at 5 years (for prostate cancer death model) 2.11178069
Scaling Parameter 0.38973665
C-index 0.90584489
Model N 12911
Knot Value
PSAPreopKnot1 0.2
PSAPreopKnot2 4.78
PSAPreopKnot3 7.3
PSAPreopKnot4 307

Last Updated: June 2, 2022