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”
Xβ : 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:
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