Nomograms

A nomogram is not a model

In the oncology and surgical literature the nomogram is close to a formatting requirement: fit the model, report the C-index, and finish with that horizontal ruler covered in tick marks. It appears so often that papers write sentences like “we developed a nomogram to predict five-year recurrence” — as though the nomogram were a kind of model.

It is not a model. It takes the linear predictor of a model that has already been fitted and rescales it onto a 0 to 100 point scale, then draws it. No new information enters, and nothing is estimated at the moment the picture is drawn. It is a hand-calculation device: it lets someone with a pen and a straight edge get the same number the software would give.

That makes “is this a good nomogram?” the wrong question. The question is whether the model behind it has been through internal validation, external validation and calibration. The second half of this page hands those three questions back with the answers this site has already measured on the same model.

What the ruler is drawn from

This page deliberately uses the same model as the calibration page: a Cox model with four predictors fitted on survival::rotterdam (2982 patients, 1713 events) at a 5-year horizon, externally validated on survival::gbsg (686 patients, 299 events).

Using the same model is what the argument needs. If a nomogram really is only a change of drawing, then it must not move a single number the calibration page measured.

Reading it takes three steps:

1. On each predictor axis find the patient’s value and read upwards to the points axis.
2. Add the four point scores and locate the sum on the total points axis.
3. Read downwards from there to the risk axis at the bottom.

Walking down the ruler once

Take a hypothetical patient: age 55, a 35 mm tumour, 4 positive nodes, grade 3. This patient sits deliberately near the middle of the development cohort, for the reason just given; the script records the same sentence as its selection_rule.

Now read the points off, one axis at a time. Not recomputed from the coefficients — interpolated on the axes of the picture itself, which is the same thing a reader with a ruler does:

Converting 42.80 points back gives a linear predictor of 0.280401, which through the baseline survival function gives a five-year risk of 52.2%.

The other route never touches the figure: multiply the four covariate values by their coefficients and add, giving a linear predictor of 0.280401 and, through the same baseline survival function, 52.2%.

Not just this patient: the whole cohort

One patient agreeing could be a patient chosen to agree. So the script does it for every patient in the development cohort: read each one’s points off the four axes, add them up, and compare against that patient’s own linear predictor.

Across 2982 patients the largest residual is 4.3e-14 and $R^2$ is 1.000000. Total points is an affine transformation of the linear predictor — a shift and a multiplication — and nothing else:

Here $\eta$ is the linear predictor, $\eta_0$ is the linear predictor at zero total points, and $k$ is how many points one unit of linear predictor is worth. For this model $k$ is 39.57 points and $\eta_0$ is -0.801. It is negative because rms reports a linear predictor centred on the mean covariate values, and the patient scoring zero total points sits at the low end of every axis, well below that mean.

Where the points scale comes from

A change of units needs an exchange rate. The rule rms::nomogram uses is to work out how far each predictor can move the linear predictor over the tick range of its own axis, give the predictor that moves it furthest a full 100-point axis, and scale every other axis by the same factor. The tick range is not the observed range: here the node axis is ticked out to 35 while the largest count in the cohort is 34, and the exchange rate is computed from the former.

In this model the widest axis is the positive node count: its coefficient multiplied by the span of its axis is the largest of the four, so it takes the full scale and the other three axes are shorter: grade 3 is the shortest at 13.14 points, with age just behind at 14.03 — and age is short enough that the script has to hand it an explicit set of ticks, or the defaults crowd into an unreadable band. That fixes the exchange rate at 39.57 points per unit of linear predictor.

There is also an implementation trap that has nothing to do with interpretation and everything to do with a page telling the truth: the package that draws the picture and the package that produced the reported coefficients must be the same fit. nomogram() only understands an rms::cph object, while most papers report coefficients from survival::coxph. The convergence tolerance cph uses by default is the looser of the two, and the coefficients then agree only to the fourth or fifth decimal — enough to make the claim that these are the same model not quite true.

The script behind this page tightens the tolerance and asserts the agreement with stopifnot():

The largest of the four differences is 4.6e-15, well inside the threshold the script enforces. If some later change ever pulls the two apart, the script stops instead of quietly shipping a figure of a model that is not the one in the text.

Not the same thing as a Fagan nomogram

The diagnostic accuracy page has a figure that is also called a nomogram. The two share the old idea of replacing arithmetic with geometry, and differ in everything else:

The most practical difference is physical: the Fagan figure asks you to draw a slanted line across three axes, while this one only ever asks for vertical alignment followed by addition. Anyone laying a ruler diagonally across a prediction-model nomogram has confused the two.

So what should be asked of the figure

Since the nomogram adds no information, judging it is the same as judging the model behind it. For this model, the site has measured three things:

Discrimination drops a little from development to external validation, which is expected. The row worth stopping on is the last one: the calibration slope is well below 1, and the whole of its 95% confidence interval lies below 1.

A slope below 1 means the predictions are too spread out: high-risk patients are pushed too high and low-risk patients pressed too low. Translated back onto this figure, that is the same sentence in pictorial form — both ends of the ruler are too long. A patient landing at the high end reads a risk above what patients like them actually experienced in survival::gbsg, and a patient at the low end reads one below.

Running it yourself

library(survival); library(rms)
data(cancer, package = "survival")

dev <- rotterdam
dev$rfs_time  <- pmin(dev$rtime, dev$dtime)
dev$rfs_event <- as.integer(dev$recur == 1 | dev$death == 1)
dev$size_mm   <- c("<=20" = 15, "20-50" = 35, ">50" = 60)[as.character(dev$size)]
dev$grade3    <- as.integer(dev$grade >= 3)

# Labels must be attached BEFORE cph(), or the picture keeps the bare
# variable names -- with no error anywhere.
label(dev$age)     <- "Age at diagnosis (years)"
label(dev$size_mm) <- "Tumour size (mm)"
label(dev$nodes)   <- "Positive nodes (count)"
label(dev$grade3)  <- "Grade 3 histology (1 = yes)"

dd <- datadist(dev); options(datadist = "dd")

# eps defaults to 1e-4, which leaves the coefficients agreeing with coxph
# only to about 1e-5.
fit <- cph(Surv(rfs_time, rfs_event) ~ age + size_mm + nodes + grade3,
           data = dev, surv = TRUE, x = TRUE, y = TRUE, eps = 1e-9, tol = 1e-12)
max(abs(coef(fit) - coef(coxph(Surv(rfs_time, rfs_event) ~
       age + size_mm + nodes + grade3, data = dev))))    # should be ~1e-15

H     <- 5 * 365.25
surv5 <- Survival(fit)
risk5 <- function(lp) 1 - surv5(H, lp)

nom <- nomogram(fit, fun = risk5, funlabel = "5-year risk")
plot(nom, xfrac = 0.34)

# The exchange rate lives in the object's own attributes
info <- attr(nom, "info")
info$sc          # points per unit of linear predictor
info$Intercept   # linear predictor at zero total points

# One patient, two routes
patient <- data.frame(age = 55, size_mm = 35, nodes = 4, grade3 = 1)
lp_direct <- as.numeric(predict(fit, newdata = patient, type = "lp"))

# Once labels are attached, nom's elements are named by LABEL; a lookup by
# variable name returns NULL, and NULL[["points"]] is NULL, not an error.
ax     <- nom[["Positive nodes (count)"]]
p_node <- approx(ax$nodes, ax$points, xout = patient$nodes)$y

# With all four axes read and summed:
# lp_from_points <- total_points / info$sc + info$Intercept
# risk5(lp_from_points) matches risk5(lp_direct) to floating-point precision

import numpy as np, pandas as pd
from lifelines import CoxPHFitter
# The same cohort harmonisation and model fit as B5-05, spelled out so
# this block runs on its own.

RD  = "https://vincentarelbundock.github.io/Rdatasets/csv/"
rot = pd.read_csv(RD + "survival/rotterdam.csv")

rot["rfs_time"]  = rot[["rtime", "dtime"]].min(axis=1)
rot["rfs_event"] = ((rot["recur"] == 1) | (rot["death"] == 1)).astype(int)
rot["size_mm"]   = rot["size"].map({"<=20": 15, "20-50": 35, ">50": 60})
rot["grade3"]    = (rot["grade"] >= 3).astype(int)

dev = rot

# No Python equivalent of rms::nomogram exists. The whole figure is this.
v    = ["age", "size_mm", "nodes", "grade3"]
fit  = CoxPHFitter().fit(dev[v + ["rfs_time", "rfs_event"]], "rfs_time", "rfs_event")
beta = fit.params_[v].to_numpy()

lo, hi = dev[v].min().to_numpy(), dev[v].max().to_numpy()
span   = np.abs(beta * (hi - lo))    # how far each axis moves the lp
scale  = 100 / span.max()            # the widest axis is worth the full scale

def points(x):                       # per-predictor points for one patient
    return beta * (np.asarray(x, float) - lo) * scale

p     = points([55, 35, 4, 1])
total = p.sum()
lp    = total / scale + float(beta @ lo)      # back to the linear predictor

# The direct route. Both are on the uncentred scale here, so they agree.
lp_direct = float(beta @ np.array([55, 35, 4, 1]))
print(p, total, lp, lp_direct)

# Drawing it: each axis is one line segment plus a set of ticks, so
# matplotlib's hlines and text are all the plotting this needs.

Four questions for a paper with a nomogram

1. Is there an external validation? A nomogram drawn only on the development cohort is an unvalidated model, nicely typeset.
2. Is there a calibration curve? The figure hands the reader an absolute risk, and whether an absolute risk is right is a calibration question, not a C-index question.
3. What are the points used for? If the paper cuts them into high, intermediate and low risk groups, where did the cut-points come from and has anyone reproduced them elsewhere? Points carry no meaning across studies.
4. What does it buy over the staging system already in use? That is a question for the decision curve or net reclassification, not for how the figure looks.

Common misuses

Reproducing every number on this page

/opt/homebrew/bin/Rscript figures/scripts/B5-09-nomogram.R

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