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 6.6811967
Preoperative PSA -0.44459221
Preoperative PSA Spline 1 0.00364527
Preoperative PSA Spline 2 -0.01000436
Primary Biopsy Gleason Grade >3 -2.1257242
Secondary Biopsy Gleason Grade >3 -0.90755871
Clinical Stage 2A -0.54790914
Clinical Stage 2B -0.9676701
Clinical Stage 2C -0.74788691
Clinical Stage 3+ -1.17721246
C-index 0.79922355
Scaling Parameter 1.1394952
Model N 10558
survival Preoperative BCR (Cores) Intercept 6.14381457
Preoperative PSA -0.45143969
Preoperative PSA Spline 1 0.00384609
Preoperative PSA Spline 2 -0.01059144
Primary Biopsy Gleason Grade >3 -2.05851781
Secondary Biopsy Gleason Grade >3 -0.82099126
Clinical Stage 2A -0.29035543
Clinical Stage 2B -0.59646663
Clinical Stage 2C -0.4519026
Clinical Stage 3+ -0.79070727
No. of Negative Cores 0.05975679
No. of Positive Cores -0.03313577
C-index 0.82117111
Scaling Parameter 1.0814495
Model N 5359
survival Postoperative BCR Intercept 6.35423706
Preoperative PSA -0.30452901
Preoperative PSA Spline 1 0.00283125
Preoperative PSA Spline 2 -0.0078509
Primary Pathological Gleason Grade >3 -2.24310058
Secondary Pathological Gleason Grade >3 -0.85939601
Surgical Margin Status -0.88850077
Extracapsular Extension -0.53120801
Seminal Vesicle Invasion -0.59314997
Lymph Node Involvement -1.18652025
C-index 0.84056082
Scaling Parameter 1.00207822
Model N 10940
logistic Organ Confined Disease Intercept 0.72159898
Preoperative PSA -0.05572401
Primary Biopsy Gleason Grade >3 -0.8339879
Secondary Biopsy Gleason Grade >3 -0.54290901
Clinical Stage 2A -0.31781681
Clinical Stage 2B -0.66088364
Clinical Stage 2C -0.62862475
Clinical Stage 3+ -1.2687455
AUC 0.67289197
Model N 10558
logistic Organ Confined Disease (Cores) Intercept 0.48559304
Preoperative PSA -0.06135733
Primary Biopsy Gleason Grade >3 -0.75917868
Secondary Biopsy Gleason Grade >3 -0.42046589
Clinical Stage 2A -0.11674405
Clinical Stage 2B -0.3569897
Clinical Stage 2C -0.37818337
Clinical Stage 3+ -0.98212899
No. of Negative Cores 0.06281136
No. of Positive Cores -0.0917364
AUC 0.71253031
Model N 5359
logistic Extracapsular Extension Intercept -0.69762059
Preoperative PSA 0.0489202
Primary Biopsy Gleason Grade >3 0.73584198
Secondary Biopsy Gleason Grade >3 0.51859291
Clinical Stage 2A 0.30909082
Clinical Stage 2B 0.6029168
Clinical Stage 2C 0.56517529
Clinical Stage 3+ 1.21028544
AUC 0.66071185
Model N 10558
logistic Extracapsular Extension (Cores) Intercept -0.47046815
Preoperative PSA 0.0532227
Primary Biopsy Gleason Grade >3 0.6593498
Secondary Biopsy Gleason Grade >3 0.40266113
Clinical Stage 2A 0.13022831
Clinical Stage 2B 0.29508755
Clinical Stage 2C 0.28976819
Clinical Stage 3+ 0.8531101
No. of Negative Cores -0.06051063
No. of Positive Cores 0.08833464
AUC 0.70085374
Model N 5359
logistic Lymph Node Involvement Intercept -4.78603008
Preoperative PSA 0.04647603
Primary Biopsy Gleason Grade >3 2.01834524
Secondary Biopsy Gleason Grade >3 0.91667795
Clinical Stage 2A 0.54169879
Clinical Stage 2B 0.97420613
Clinical Stage 2C 1.09028922
Clinical Stage 3+ 1.59202656
AUC 0.85670802
Model N 10558
logistic Lymph Node Involvement (Cores) Intercept -3.90664548
Preoperative PSA 0.04648719
Primary Biopsy Gleason Grade >3 1.85080788
Secondary Biopsy Gleason Grade >3 0.77070173
Clinical Stage 2A 0.37888753
Clinical Stage 2B 0.57940132
Clinical Stage 2C 0.57618944
Clinical Stage 3+ 0.9105159
No. of Negative Cores -0.11228264
No. of Positive Cores 0.05721468
AUC 0.85580481
Model N 5359
logistic Seminal Vesicle Invasion Intercept -4.47585322
Preoperative PSA 0.05520405
Primary Biopsy Gleason Grade >3 1.60366838
Secondary Biopsy Gleason Grade >3 0.85036906
Clinical Stage 2A 0.51591678
Clinical Stage 2B 0.7556456
Clinical Stage 2C 1.10535272
Clinical Stage 3+ 1.35042904
AUC 0.82966998
Model N 10558
logistic Seminal Vesicle Invasion (Cores) Intercept -3.64087032
Preoperative PSA 0.04444051
Primary Biopsy Gleason Grade >3 1.61342422
Secondary Biopsy Gleason Grade >3 0.79980864
Clinical Stage 2A 0.24802096
Clinical Stage 2B 0.32708694
Clinical Stage 2C 0.61429198
Clinical Stage 3+ 0.57498157
No. of Negative Cores -0.13962727
No. of Positive Cores 0.07543931
AUC 0.85040492
Model N 5359
Knot Value
PSAPreopKnot1 0.21
PSAPreopKnot2 4.6
PSAPreopKnot3 7.03
PSAPreopKnot4 96.53

Last Updated: May 31, 2016