11. Growth curves — is that error bar an SD or an SEM? · Dr. Seonggu Sohn's Data Visualization, First Steps

Fig. 4C follows tumour volume in mice over 21 days. Four conditions, 7 mice each. The legend says:

Data are presented as the mean ± SEM (n = 7)

That one line is half the figure. Without it the error bars are unreadable.

Same data, same means. Only the error bars change — and the further right you go, the more certain the difference looks.

Python Runs in your browser

import matplotlib.pyplot as plt
import viz, numpy as np, json

F = json.load(open("viz_facts.json"))
E = {d["g"]: d for d in F["err"]}
g = E["Cd47"]
wt, kd = np.array(g["wt"]), np.array(g["kd"])

def stats3(v):
    sd = v.std(ddof=1)
    return v.mean(), sd, sd/np.sqrt(len(v)), 2.0*sd/np.sqrt(len(v))

fig, ax = plt.subplots(1, 3, figsize=(10.5, 3.2), sharey=True)
for a, k, tag in zip(ax, [1, 2, 3], ["SD", "SEM", "95% CI (approx)"]):
    for i, v in enumerate([wt, kd]):
        m, sd, sem, ci = stats3(v)
        err = [sd, sem, ci][k-1]
        a.errorbar(i, m, yerr=err, fmt="o", ms=7, capsize=6, lw=2, color=viz.PAL[i])
        a.plot([i]*len(v) + np.random.default_rng(i).normal(0, .04, len(v)), v,
               ".", ms=5, color="#888", alpha=.8)
    a.set_xticks([0, 1]); a.set_xticklabels(["WT (n=4)", "KD (n=3)"], fontsize=8)
    a.set_title(tag, fontsize=9); viz.bare(a)
ax[0].set_ylabel("Cd47  log2(CPM+1)", fontsize=8)
fig.tight_layout()
print("SD %.3f / %.3f   SEM %.3f / %.3f" % (wt.std(ddof=1), kd.std(ddof=1),
      wt.std(ddof=1)/2, kd.std(ddof=1)/np.sqrt(3)))

Switch the growth-curve error bars between SD, SEM and CI, and turn on the per-animal lines.

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Growth curves hide one more thing that is easy to miss: the mice that died along the way.

If only the animals alive on day 21 enter the mean, then the later you look, the more only the robust survivors remain. When a mouse with a huge tumour is culled early, the group's mean tumour volume appears to fall — not because the treatment worked, but because the bad case left.

The dangerous way

join the mean of whoever is alive each day. n changes daily, and the chart never says so

What to do

draw the individual lines too, or print the number still alive under the axis. It is also why a survival curve is published next to it

The left is tidy and the right is messy. But whether one animal is an outlier is visible only on the right.

Python Runs in your browser

import matplotlib.pyplot as plt
import viz, numpy as np, json

F = json.load(open("viz_facts.json"))
G = F["growth"]
days = np.array(G["days"])

fig, ax = plt.subplots(1, 2, figsize=(10.5, 3.6), sharey=True)
for i, arm in enumerate(G["arms"]):
    R = np.array(arm["rows"])
    m = R.mean(axis=0); sem = R.std(axis=0, ddof=1)/np.sqrt(R.shape[0])
    ax[0].errorbar(days, m, yerr=sem, fmt="-o", ms=4, lw=1.6, capsize=3,
                   color=viz.PAL[i], label=arm["name"])
    for row in R:
        ax[1].plot(days, row, lw=.8, alpha=.55, color=viz.PAL[i])

ax[0].set_title("mean +- SEM", fontsize=9)
ax[1].set_title("every mouse", fontsize=9)
ax[0].legend(fontsize=6.5, frameon=False)
for a in ax:
    a.set_xlabel("day", fontsize=8); viz.bare(a)
ax[0].set_ylabel("tumour volume (mm3)", fontsize=8)
fig.tight_layout()
print("(fake data, for shape only)")

🤔 Check yourself

At n = 7, SEM is what fraction of SD?

🤔 Check yourself

What goes wrong if the legend does not say what the error bars are?

🤔 Check yourself

Some mice were culled part-way through a 21-day growth experiment. What does that do to the mean curve?