A funnel plot used in meta-analysis to assess publication bias by plotting individual study effect sizes against their precision (typically standard error). Studies scatter around a summary effect line, with pseudo 95% confidence limits forming an inverted funnel shape. In the absence of bias, studies distribute symmetrically around the summary effect; asymmetry suggests publication bias or systematic heterogeneity. This is a standard tool in systematic reviews and Cochrane-style meta-analyses.
""" anyplot.ai
funnel-meta-analysis: Meta-Analysis Funnel Plot for Publication Bias
Library: plotly 6.8.0 | Python 3.13.13
Quality: 88/100 | Updated: 2026-06-10
"""
import os
import numpy as np
import plotly.graph_objects as go
# Theme tokens — Imprint palette + adaptive chrome
THEME = os.getenv("ANYPLOT_THEME", "light")
PAGE_BG = "#FAF8F1" if THEME == "light" else "#1A1A17"
ELEVATED_BG = "#FFFDF6" if THEME == "light" else "#242420"
INK = "#1A1A17" if THEME == "light" else "#F0EFE8"
INK_SOFT = "#4A4A44" if THEME == "light" else "#B8B7B0"
INK_MUTED = "#6B6A63" if THEME == "light" else "#A8A79F"
GRID = "rgba(26,26,23,0.15)" if THEME == "light" else "rgba(240,239,232,0.15)"
# Imprint palette positions used
BRAND = "#009E73" # position 1 — inside-funnel studies
BLUE = "#4467A3" # position 3 — pooled effect line
RED = "#AE3030" # position 5 — semantic anchor for outliers (outside 95% CI)
# Funnel shading derived from BLUE
FUNNEL_FILL_95 = "rgba(68,103,163,0.10)" if THEME == "light" else "rgba(68,103,163,0.16)"
FUNNEL_LINE_95 = "rgba(68,103,163,0.45)" if THEME == "light" else "rgba(68,103,163,0.65)"
FUNNEL_FILL_99 = "rgba(68,103,163,0.05)" if THEME == "light" else "rgba(68,103,163,0.08)"
FUNNEL_LINE_99 = "rgba(68,103,163,0.18)" if THEME == "light" else "rgba(68,103,163,0.30)"
# Data: 15 RCTs comparing drug vs placebo — log odds ratios and standard errors
np.random.seed(42)
studies = [
"Adams et al. 2016",
"Baker et al. 2017",
"Chen et al. 2017",
"Davis & Park 2018",
"Evans et al. 2018",
"Fischer 2019",
"Gupta et al. 2019",
"Harris et al. 2020",
"Ibrahim et al. 2020",
"Jensen & Liu 2021",
"Kim et al. 2021",
"Lambert et al. 2022",
"Morales et al. 2022",
"Nielsen 2023",
"Olsen et al. 2023",
]
log_or = np.array(
[-0.52, -0.38, -0.71, -0.15, -0.45, -0.63, -0.29, -0.55, -0.42, -0.33, -0.80, -0.48, -0.36, -0.61, -0.10]
)
std_error = np.array([0.18, 0.25, 0.12, 0.30, 0.20, 0.15, 0.28, 0.17, 0.22, 0.26, 0.11, 0.19, 0.24, 0.14, 0.35])
# Inverse-variance weights and pooled effect
weights = 1.0 / std_error**2
pooled_effect = np.sum(weights * log_or) / np.sum(weights)
pct_weights = 100 * weights / weights.sum()
# Marker sizes proportional to study weight
w_norm = weights / weights.max()
marker_sizes = 14 + w_norm * 20 # 14–34 px range
# Classify studies inside vs outside the 95% funnel
outside_funnel = np.abs(log_or - pooled_effect) > 1.96 * std_error
inside_funnel = ~outside_funnel
# Funnel boundary lines
se_max = max(std_error) * 1.1
se_range = np.linspace(0, se_max, 200)
upper_95 = pooled_effect + 1.96 * se_range
lower_95 = pooled_effect - 1.96 * se_range
upper_99 = pooled_effect + 2.576 * se_range
lower_99 = pooled_effect - 2.576 * se_range
# Build figure
fig = go.Figure()
# 99% CI shaded region
fig.add_trace(
go.Scatter(
x=np.concatenate([lower_99, upper_99[::-1]]),
y=np.concatenate([se_range, se_range[::-1]]),
fill="toself",
fillcolor=FUNNEL_FILL_99,
line={"color": FUNNEL_LINE_99, "width": 1, "dash": "dot"},
showlegend=True,
name="99% CI region",
hoverinfo="skip",
)
)
# 95% CI shaded region
fig.add_trace(
go.Scatter(
x=np.concatenate([lower_95, upper_95[::-1]]),
y=np.concatenate([se_range, se_range[::-1]]),
fill="toself",
fillcolor=FUNNEL_FILL_95,
line={"color": FUNNEL_LINE_95, "width": 1.5},
showlegend=True,
name="95% CI region",
hoverinfo="skip",
)
)
# Null effect dashed reference line
fig.add_trace(
go.Scatter(
x=[0, 0],
y=[0, se_max],
mode="lines",
line={"color": INK_SOFT, "width": 1.5, "dash": "dash"},
showlegend=False,
hoverinfo="skip",
)
)
# Pooled effect line
fig.add_trace(
go.Scatter(
x=[pooled_effect, pooled_effect],
y=[0, se_max],
mode="lines",
line={"color": BLUE, "width": 2.5},
showlegend=True,
name=f"Pooled OR {np.exp(pooled_effect):.2f} (log {pooled_effect:.2f})",
hoverinfo="skip",
)
)
# Studies inside 95% CI funnel
if inside_funnel.any():
hover_inside = [
f"<b>{s}</b><br>Log OR: {e:.2f} (OR = {np.exp(e):.2f})<br>SE: {se:.3f}<br>Weight: {w:.1f}%"
for s, e, se, w in zip(
np.array(studies)[inside_funnel],
log_or[inside_funnel],
std_error[inside_funnel],
pct_weights[inside_funnel],
strict=False,
)
]
fig.add_trace(
go.Scatter(
x=log_or[inside_funnel],
y=std_error[inside_funnel],
mode="markers",
marker={
"size": marker_sizes[inside_funnel],
"symbol": "circle",
"color": BRAND,
"line": {"color": "#006B4E", "width": 1.5},
"opacity": 0.88,
},
text=hover_inside,
hovertemplate="%{text}<extra></extra>",
hoverlabel={"bgcolor": ELEVATED_BG, "bordercolor": BRAND, "font": {"size": 13, "color": INK}},
name="Within 95% CI",
showlegend=True,
)
)
# Studies outside 95% CI funnel (potential outliers)
if outside_funnel.any():
hover_outside = [
f"<b>{s}</b><br>Log OR: {e:.2f} (OR = {np.exp(e):.2f})<br>SE: {se:.3f}<br>Weight: {w:.1f}%<br>⚠ Outside 95% CI"
for s, e, se, w in zip(
np.array(studies)[outside_funnel],
log_or[outside_funnel],
std_error[outside_funnel],
pct_weights[outside_funnel],
strict=False,
)
]
fig.add_trace(
go.Scatter(
x=log_or[outside_funnel],
y=std_error[outside_funnel],
mode="markers",
marker={
"size": marker_sizes[outside_funnel],
"symbol": "diamond",
"color": RED,
"line": {"color": "#7A1F1F", "width": 1.5},
"opacity": 0.88,
},
text=hover_outside,
hovertemplate="%{text}<extra></extra>",
hoverlabel={"bgcolor": ELEVATED_BG, "bordercolor": RED, "font": {"size": 13, "color": INK}},
name="Outside 95% CI",
showlegend=True,
)
)
# Layout
title_text = "funnel-meta-analysis · python · plotly · anyplot.ai"
n = len(title_text)
title_size = round(16 * 67 / n) if n > 67 else 16
fig.update_layout(
autosize=False,
paper_bgcolor=PAGE_BG,
plot_bgcolor=PAGE_BG,
font={"color": INK, "family": "Arial, sans-serif"},
title={
"text": title_text,
"font": {"size": title_size, "color": INK},
"x": 0.5,
"xanchor": "center",
"y": 0.98,
"yanchor": "top",
},
xaxis={
"title": {"text": "Log Odds Ratio", "font": {"size": 12, "color": INK}, "standoff": 12},
"tickfont": {"size": 10, "color": INK_SOFT},
"gridcolor": GRID,
"linecolor": INK_SOFT,
"linewidth": 1,
"showline": True,
"zeroline": False,
"range": [-1.15, 0.55],
},
yaxis={
"title": {"text": "Standard Error", "font": {"size": 12, "color": INK}, "standoff": 10},
"tickfont": {"size": 10, "color": INK_SOFT},
"gridcolor": GRID,
"linecolor": INK_SOFT,
"linewidth": 1,
"showline": True,
"zeroline": False,
"autorange": "reversed",
"rangemode": "tozero",
},
legend={
"bgcolor": ELEVATED_BG,
"bordercolor": INK_SOFT,
"borderwidth": 1,
"font": {"size": 10, "color": INK_SOFT},
"x": 0.02,
"y": 0.98,
"xanchor": "left",
"yanchor": "top",
},
hovermode="closest",
margin={"l": 80, "r": 40, "t": 80, "b": 60},
)
# Direction annotations
fig.add_annotation(
x=0.04,
xref="paper",
y=se_max * 0.96,
text="← Favors Treatment",
showarrow=False,
font={"size": 11, "color": BRAND},
xanchor="left",
)
fig.add_annotation(
x=0.96,
xref="paper",
y=se_max * 0.96,
text="Favors Control →",
showarrow=False,
font={"size": 11, "color": INK_MUTED},
xanchor="right",
)
# Pooled and null labels near top of chart (where SE ≈ 0)
fig.add_annotation(
x=pooled_effect + 0.03,
y=0.005,
text=f"Pooled ({pooled_effect:.2f})",
showarrow=False,
font={"size": 10, "color": BLUE},
xanchor="left",
yanchor="top",
)
fig.add_annotation(
x=0.03,
y=0.005,
text="Null (0)",
showarrow=False,
font={"size": 10, "color": INK_MUTED},
xanchor="left",
yanchor="top",
)
# CI boundary labels — solid background for readability against funnel fill
ci_label_se = se_max * 0.58
fig.add_annotation(
x=pooled_effect + 1.96 * ci_label_se + 0.03,
y=ci_label_se,
text="95% CI",
showarrow=False,
font={"size": 10, "color": INK_SOFT},
bgcolor=PAGE_BG,
bordercolor="rgba(0,0,0,0)",
xanchor="left",
)
fig.add_annotation(
x=pooled_effect + 2.576 * ci_label_se + 0.03,
y=ci_label_se,
text="99% CI",
showarrow=False,
font={"size": 10, "color": INK_MUTED},
bgcolor=PAGE_BG,
bordercolor="rgba(0,0,0,0)",
xanchor="left",
)
# Subtitle
fig.add_annotation(
x=0.5,
xref="paper",
y=1.0,
yref="paper",
text="Asymmetry at low precision (wide SE) suggests possible publication bias • Red ◆ outside 95% CI, green ● inside",
showarrow=False,
font={"size": 10, "color": INK_MUTED},
xanchor="center",
yanchor="bottom",
)
# Save — 3200×1800 landscape (800×450 @ scale=4)
fig.write_image(f"plot-{THEME}.png", width=800, height=450, scale=4)
fig.write_html(f"plot-{THEME}.html", include_plotlyjs="cdn")
Part of Meta-Analysis Funnel Plot for Publication Bias on anyplot.ai.