Dynamic Prostate Cancer Nomogram: Coefficients


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 = e
1 + e

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.

Model Type Model Variable Value
survival Preoperative BCR Intercept 6.15800897
Preoperative PSA -0.34852011
Preoperative PSA Spline 1 0.00251028
Preoperative PSA Spline 2 -0.00684538
Primary Biopsy Gleason Grade >3 -1.99233848
Secondary Biopsy Gleason Grade >3 -0.87483682
Clinical Stage 2A -0.47465258
Clinical Stage 2B -0.95565945
Clinical Stage 2C -0.74726801
Clinical Stage 3+ -1.26864741
C-index 0.79397522
Scaling Parameter 1.12572374
Model N 11552
survival Preoperative BCR (Cores) Intercept 5.31965275
Preoperative PSA -0.29814542
Preoperative PSA Spline 1 0.0020837
Preoperative PSA Spline 2 -0.0056708
Primary Biopsy Gleason Grade >3 -1.85423052
Secondary Biopsy Gleason Grade >3 -0.81497521
Clinical Stage 2A -0.28808935
Clinical Stage 2B -0.62486512
Clinical Stage 2C -0.28430869
Clinical Stage 3+ -0.75821157
No. of Negative Cores 0.07179485
No. of Positive Cores -0.03298566
C-index 0.80644193
Scaling Parameter 1.07654267
Model N 6216
survival Postoperative BCR Intercept 5.98892458
Preoperative PSA -0.23592057
Preoperative PSA Spline 1 0.00191662
Preoperative PSA Spline 2 -0.00527262
Primary Pathological Gleason Grade >3 -2.12300827
Secondary Pathological Gleason Grade >3 -0.81741524
Surgical Margin Status -0.95182943
Extracapsular Extension -0.53165034
Seminal Vesicle Invasion -0.64127396
Lymph Node Involvement -1.14839885
C-index 0.83725441
Scaling Parameter 0.99669679
Model N 12086
logistic Organ Confined Disease Intercept 0.71262161
Preoperative PSA -0.05226675
Primary Biopsy Gleason Grade >3 -0.77573582
Secondary Biopsy Gleason Grade >3 -0.51233547
Clinical Stage 2A -0.32201647
Clinical Stage 2B -0.65324803
Clinical Stage 2C -0.72206724
Clinical Stage 3+ -1.24134492
AUC 0.66812964
Model N 11552
logistic Organ Confined Disease (Cores) Intercept 0.49526516
Preoperative PSA -0.05095943
Primary Biopsy Gleason Grade >3 -0.73751745
Secondary Biopsy Gleason Grade >3 -0.44663154
Clinical Stage 2A -0.1168568
Clinical Stage 2B -0.34671504
Clinical Stage 2C -0.54514969
Clinical Stage 3+ -0.87575065
No. of Negative Cores 0.05807154
No. of Positive Cores -0.08284535
AUC 0.70106158
Model N 6216
logistic Extracapsular Extension Intercept -0.69680956
Preoperative PSA 0.04690053
Primary Biopsy Gleason Grade >3 0.69407012
Secondary Biopsy Gleason Grade >3 0.48588045
Clinical Stage 2A 0.32068552
Clinical Stage 2B 0.60090328
Clinical Stage 2C 0.65737146
Clinical Stage 3+ 1.19145673
AUC 0.65739489
Model N 11552
logistic Extracapsular Extension (Cores) Intercept -0.51392379
Preoperative PSA 0.04561469
Primary Biopsy Gleason Grade >3 0.65431256
Secondary Biopsy Gleason Grade >3 0.42847216
Clinical Stage 2A 0.13481419
Clinical Stage 2B 0.30853489
Clinical Stage 2C 0.4665139
Clinical Stage 3+ 0.76378608
No. of Negative Cores -0.05528222
No. of Positive Cores 0.08354868
AUC 0.69224805
Model N 6216
logistic Lymph Node Involvement Intercept -4.71127143
Preoperative PSA 0.04427068
Primary Biopsy Gleason Grade >3 1.90797075
Secondary Biopsy Gleason Grade >3 0.89152421
Clinical Stage 2A 0.54386908
Clinical Stage 2B 0.92753512
Clinical Stage 2C 1.24526645
Clinical Stage 3+ 1.63966106
AUC 0.8465941
Model N 11552
logistic Lymph Node Involvement (Cores) Intercept -3.92383186
Preoperative PSA 0.04127647
Primary Biopsy Gleason Grade >3 1.69555954
Secondary Biopsy Gleason Grade >3 0.73627978
Clinical Stage 2A 0.41819731
Clinical Stage 2B 0.56954741
Clinical Stage 2C 0.77240836
Clinical Stage 3+ 1.00598066
No. of Negative Cores -0.10567544
No. of Positive Cores 0.06398027
AUC 0.84591163
Model N 6216
logistic Seminal Vesicle Invasion Intercept -4.4812856
Preoperative PSA 0.05317635
Primary Biopsy Gleason Grade >3 1.57363228
Secondary Biopsy Gleason Grade >3 0.84143741
Clinical Stage 2A 0.4693118
Clinical Stage 2B 0.79378831
Clinical Stage 2C 1.20017536
Clinical Stage 3+ 1.43669426
AUC 0.8272523
Model N 11552
logistic Seminal Vesicle Invasion (Cores) Intercept -3.6100875
Preoperative PSA 0.04214384
Primary Biopsy Gleason Grade >3 1.56883605
Secondary Biopsy Gleason Grade >3 0.80784929
Clinical Stage 2A 0.17723063
Clinical Stage 2B 0.40811619
Clinical Stage 2C 0.6721717
Clinical Stage 3+ 0.74598682
No. of Negative Cores -0.15215667
No. of Positive Cores 0.07906369
AUC 0.85071111
Model N 6216
Knot Value
PSAPreopKnot1 0.2
PSAPreopKnot2 4.66
PSAPreopKnot3 7.12
PSAPreopKnot4 96.53

Last Updated: April 10, 2018