Table 1 and standardised mean differences

What this table is answering

The first table in a paper is almost always a table of baseline characteristics, but it is not answering the same question in every design — this is the single most important thing on this page, and every other detail follows from it.

The difference is very practical. In an RCT, any difference between the arms can only come from luck, because allocation had nothing to do with patient characteristics; you already know the answer to “is there a difference”, and Table 1 only shows you how big it is. In an observational study the arms differ because they really are different — clinicians prescribe one drug to one kind of patient — so Table 1 is telling you the size of the confounding problem.

The example used here

survival::pbc is the Mayo Clinic trial in primary biliary cholangitis (PBC), 418 patients in total. It happens to contain the first two rows of the table above inside one file:

• 312 patients were randomised to D-penicillamine or placebo → that is the Table 1 of an RCT
• 106 patients declined randomisation but agreed to be followed in a registry → “randomised vs registry” is a genuine observational comparison

Same data, same seven or eight variables. Change how you split the rows and the meaning of the whole table changes with it.

How the standardised mean difference is computed

The standardised mean difference (SMD) converts “how far apart are these groups” into “how many standard deviations apart are they”. For a continuous variable:

The denominator is the root mean square of the two standard deviations, not the pooled SD. The two differ slightly when the groups are of unequal size, and Table 1 conventionally uses the former.

For a binary variable, treat the proportion as the mean of a Bernoulli variable, whose variance is fixed by the proportion itself:

A variable with three or more categories needs Austin’s multinomial extension: write the proportions of the $k$ categories as a $(k-1)$-dimensional vector $T$, and take the Mahalanobis distance using the average covariance matrix $S$ of the two groups:

What all three formulas share is that the denominator depends only on spread, never on sample size. That is exactly why the SMD replaces the p-value here, and the next section demonstrates it with an experiment.

Running it yourself

library(survival)
data(pbc, package = "survival")

pbc$arm <- factor(pbc$trt, levels = c(1, 2),
                  labels = c("D-penicillamine", "Placebo"))

# ── SMD by hand: continuous variable ────────────────────
smd_cont <- function(x, g) {
  ok <- !is.na(x) & !is.na(g); x <- x[ok]; g <- droplevels(factor(g[ok]))
  m <- tapply(x, g, mean); s <- tapply(x, g, sd)
  unname((m[1] - m[2]) / sqrt((s[1]^2 + s[2]^2) / 2))
}

# ── SMD by hand: binary variable ────────────────────────
smd_bin <- function(x, g) {           # x is coded 0/1
  ok <- !is.na(x) & !is.na(g); x <- as.numeric(x[ok]); g <- droplevels(factor(g[ok]))
  p <- tapply(x, g, mean)
  unname((p[1] - p[2]) / sqrt((p[1] * (1 - p[1]) + p[2] * (1 - p[2])) / 2))
}

smd_cont(pbc$age, pbc$arm)      # age
smd_bin(pbc$sex == "m", pbc$arm)  # sex

# ── The one-line version you would really use (tableone would not ──
# ── install on this machine, hence the manual computation above) ──
# library(tableone)
# CreateTableOne(vars = c("age", "sex", "bili", "albumin", "stage"),
#                strata = "arm", data = pbc) |> print(smd = TRUE, nonnormal = "bili")

import numpy as np
import statsmodels.api as sm

pbc = sm.datasets.get_rdataset("pbc", "survival").data
pbc["arm"] = pbc["trt"].map({1: "D-penicillamine", 2: "Placebo"})
rct = pbc.dropna(subset=["arm"])

def smd_cont(x, g):
    a, b = [x[g == lv].dropna() for lv in sorted(g.dropna().unique())]
    return (a.mean() - b.mean()) / np.sqrt((a.std(ddof=1) ** 2 + b.std(ddof=1) ** 2) / 2)

print(smd_cont(rct["age"], rct["arm"]))

# The one-line version:
# from tableone import TableOne
# TableOne(rct, columns=["age", "sex", "bili", "albumin", "stage"],
#          groupby="arm", nonnormal=["bili"], smd=True)

Two Table 1s from the same data

1. The RCT Table 1: D-penicillamine vs placebo

2. The observational Table 1: accepted randomisation vs declined it

Stacking the two sets of |SMD| values on one axis is the clearest way to see it. That plot is called a Love plot, and it is standard equipment in propensity score work:

The row worth stopping on: even the two randomised arms have 4 variables with |SMD| above 0.1, age being 0.270. This is not a failure of randomisation. It is what one draw of n = 312 routinely produces. On the other line, 5 variables exceed 0.1, and oedema reaches 0.393 — that is what “these two groups were never alike” looks like.

Why the SMD rather than a p-value

Here is an experiment using the data on this page. Take the 312 randomised patients and duplicate the whole file ten times, giving 3120 rows. Every group mean, every standard deviation and every distributional shape is unchanged; only the sample size is ten times larger.

The SMD barely moves — the 0.001 of drift comes only from the $n-1$ denominator inside the standard deviation — while the p-value drops by 11.6 orders of magnitude. A p-value encodes “how large is the difference” and “how many people are there” at the same time, and Table 1 only wants to ask the first. This is why a claims database study with tens of thousands of patients, such as one built on Taiwan’s National Health Insurance data, routinely reports p < 0.001 down the entire column, and why that column carries no information at all.

An RCT has a second and more fundamental reason. The null hypothesis — that both arms were drawn from the same population — is true by construction under randomisation. You are testing a hypothesis whose answer you already know, so any significant result is by definition a type I error. The RCT table on this page tested 8 variables; the smallest p-value is 0.018, with 1 below 0.05 — entirely as expected. If those tests were independent of one another, the probability of at least one “significant” result across 8 variables would be about 33.7%, and about 64.2% across the 20 variables a paper commonly reports. Baseline variables are usually correlated with one another — age with stage, bilirubin with albumin — so treat both figures as an order-of-magnitude guide rather than the exact probability for this table. The direction is unaffected: the more variables you test, the less a stray “significant” result deserves attention.

Where the |SMD| < 0.1 line came from

The threshold is quoted as if it were law, but its provenance is looser than that. It comes out of the propensity score literature — the usage in Normand and colleagues in 2001, generalised through Austin’s series of methodological papers — and the argument is that simulations show residual |SMD| below 0.1 usually leaves bias small enough to ignore after matching.

Three things to remember:

• It is a convention, not a test. There is no significant side and non-significant side of 0.1. There is no material difference between 0.09 and 0.11.
• For an important confounder even 0.05 may be too loose. Whether to tighten the threshold depends on how strongly that variable acts on the outcome, not on which row it happens to be.
• For an unimportant variable, exceeding 0.1 usually needs no action. An imbalance in a variable unrelated to the outcome cannot create confounding.

A looser threshold of 0.25 also circulates, from the earlier propensity score literature, so do not be surprised to see something other than 0.1 — but a paper must state which line it used.

Mean or median: look at the distribution, not at habit

The two tables above deliberately print mean ± SD and median (IQR) side by side, so you can see where the two part company. Serum bilirubin is the classic case:

The mean is 3.22 with an SD of 4.41, while the median is only 1.4 (IQR 0.8–3.4). The damning part is that mean − SD = -1.19, a negative number, when the smallest value actually observed is 0.3. Write it as 3.22 ± 4.41 and the symmetric interval that appears in the reader’s head simply does not exist in this dataset.

The rule is straightforward:

• Roughly symmetric → mean ± SD. It carries more information, and it is the quantity a t-test or a linear model will use later.
• Clearly skewed, outlier-prone, or bounded by nature (length of stay, hospital costs, bilirubin, CRP, follow-up time) → median (IQR).
• Reporting both in Table 1 is entirely acceptable. One extra column costs far less than misleading the reader.

Missing values: they deserve their own row, and they should get one

They do — and honesty is not the only reason. Look at the cholesterol row of the second table: every one of the 106 registry patients is missing that value, because the 106 patients who declined randomisation never had the trial’s blood panel drawn (chol, copper, alk.phos, ast, trig, ascites, hepato, spiders are all in the same position). This is not scattered missingness; it is an entire group never measured. There is no SMD to compute, so the cell can only be left blank and annotated.

If Table 1 prints percentages without missingness counts, this disappears from the page entirely and the reader assumes both groups were measured. A denominator shrinking quietly is the commonest source of misreading in a baseline table.

Three practical points:

• Print the count of missing values and the denominator. “n = 106, cholesterol missing in 106” reads very differently from “n = 106”.
• State which denominator the percentages use. Complete cases and the full group size give different numbers; both are in use, and either is fine as long as it is stated.
• Missingness can itself be the finding. In the example above, “not measured” and “declined randomisation” are two faces of the same fact. Treating missingness as a category of its own often exposes the selection mechanism.

How to read someone else’s Table 1

Four moves, from the top down:

1. Read the two column ns first, then the missingness in each row. Whether the denominator shrinks on the way down decides what every percentage below it means.
2. Check whether there is an SMD column. If there is not, estimate one in your head for the two or three variables you care about: the difference divided by roughly the standard deviation is a serviceable approximation.
3. Check which summary each continuous variable uses, and whether that summary fits the distribution. A mean ± SD on length of stay is a warning sign.
4. Look at who these people are. Median age, sex ratio, disease stage — do your patients fall inside this distribution? This step has nothing to do with balance, but it decides whether the paper is any use to you.

Common misuses

Reproducing every number on this page

/opt/homebrew/bin/Rscript figures/scripts/B1-01-table1.R

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