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: letsplot 4.10.1 | Python 3.13.13
Quality: 90/100 | Updated: 2026-06-10
"""
import os
import numpy as np
import pandas as pd
from lets_plot import *
LetsPlot.setup_html()
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_COLOR = "#D9D7D0" if THEME == "light" else "#3A3A36"
# Imprint palette — first series always #009E73
BRAND = "#009E73" # Inside funnel (expected, within confidence limits)
OUTLIER = "#AE3030" # Outside funnel — semantic anchor: potential bias / outlier
# Data: Meta-analysis of 15 RCTs comparing drug vs placebo
# Effect sizes are log odds ratios; null effect at 0
studies = [
{"study": "Adams 2015", "effect_size": 0.42, "std_error": 0.18, "n": 120},
{"study": "Baker 2016", "effect_size": 0.28, "std_error": 0.22, "n": 85},
{"study": "Chen 2016", "effect_size": 0.65, "std_error": 0.30, "n": 48},
{"study": "Davis 2017", "effect_size": -0.08, "std_error": 0.25, "n": 64},
{"study": "Evans 2017", "effect_size": 0.52, "std_error": 0.12, "n": 280},
{"study": "Foster 2018", "effect_size": 0.10, "std_error": 0.35, "n": 34},
{"study": "Garcia 2018", "effect_size": 0.38, "std_error": 0.15, "n": 180},
{"study": "Hughes 2019", "effect_size": 0.55, "std_error": 0.28, "n": 52},
{"study": "Ito 2019", "effect_size": 0.30, "std_error": 0.10, "n": 410},
{"study": "Jensen 2020", "effect_size": 1.05, "std_error": 0.32, "n": 40},
{"study": "Klein 2020", "effect_size": -0.15, "std_error": 0.14, "n": 205},
{"study": "Lee 2021", "effect_size": 0.48, "std_error": 0.20, "n": 100},
{"study": "Morgan 2021", "effect_size": 0.33, "std_error": 0.16, "n": 160},
{"study": "Nguyen 2022", "effect_size": 0.88, "std_error": 0.30, "n": 46},
{"study": "Olsen 2023", "effect_size": -0.05, "std_error": 0.38, "n": 28},
]
df = pd.DataFrame(studies)
# Inverse-variance weights and pooled effect estimate
weights = 1 / df["std_error"] ** 2
pooled_effect = (df["effect_size"] * weights).sum() / weights.sum()
df["iw"] = weights
# Classify studies: inside or outside the 95% funnel boundary
df["position"] = np.where(
(df["effect_size"] >= pooled_effect - 1.96 * df["std_error"])
& (df["effect_size"] <= pooled_effect + 1.96 * df["std_error"]),
"Inside funnel",
"Outside funnel",
)
# Pseudo 95% confidence funnel boundary
se_max = df["std_error"].max() + 0.05
se_range = np.linspace(0, se_max, 200)
funnel_upper = pooled_effect + 1.96 * se_range
funnel_lower = pooled_effect - 1.96 * se_range
funnel_df = pd.DataFrame(
{"x": np.concatenate([funnel_lower, funnel_upper[::-1]]), "y": np.concatenate([se_range, se_range[::-1]])}
)
funnel_lines_df = pd.DataFrame(
{
"x": np.concatenate([funnel_lower, funnel_upper]),
"y": np.concatenate([se_range, se_range]),
"side": ["lower"] * len(se_range) + ["upper"] * len(se_range),
}
)
# Pooled OR annotation (single-row data frame for geom_label)
annotation_df = pd.DataFrame(
{"x": [pooled_effect + 0.04], "y": [0.012], "label": [f"Pooled OR = {np.exp(pooled_effect):.2f}"]}
)
# Top 3 highest-weight inside-funnel studies for labeling
inside_top = df[df["position"] == "Inside funnel"].nlargest(3, "iw")
title = "funnel-meta-analysis · python · letsplot · anyplot.ai"
title_size = round(16 * min(1.0, 67 / len(title)))
plot = (
ggplot()
# Funnel confidence region (shaded polygon)
+ geom_polygon(aes(x="x", y="y"), data=funnel_df, fill=BRAND, alpha=0.07)
# Funnel boundary lines (95% CI dashed)
+ geom_line(
aes(x="x", y="y", group="side"), data=funnel_lines_df, color=BRAND, size=0.8, linetype="dashed", alpha=0.5
)
# Null effect reference line
+ geom_vline(xintercept=0, color=INK_MUTED, size=0.6, linetype="dashed", alpha=0.7)
# Pooled effect line
+ geom_vline(xintercept=pooled_effect, color=INK, size=1.1, alpha=0.85)
# Study points — sized by inverse-variance weight, colored by classification
+ geom_point(
aes(x="effect_size", y="std_error", color="position", size="iw"),
data=df,
shape=16,
alpha=0.88,
tooltips=layer_tooltips()
.title("@study")
.line("Effect (log OR)|@effect_size{.3f}")
.line("Std. error|@std_error{.3f}")
.line("Sample size|@n")
.line("Status|@position"),
)
+ scale_color_manual(values={"Inside funnel": BRAND, "Outside funnel": OUTLIER}, name="Classification")
+ scale_size(range=[2.0, 7.0], guide="none")
# Outlier labels with filled background — right-align Jensen 2020 to prevent canvas overflow
+ geom_label(
aes(x="effect_size", y="std_error", label="study"),
data=df[df["study"] == "Jensen 2020"],
color=OUTLIER,
fill=ELEVATED_BG,
size=3.2,
nudge_y=-0.022,
hjust=1.1,
fontface="bold",
show_legend=False,
)
+ geom_label(
aes(x="effect_size", y="std_error", label="study"),
data=df[df["study"] == "Klein 2020"],
color=OUTLIER,
fill=ELEVATED_BG,
size=3.2,
nudge_y=-0.022,
hjust=-0.15,
fontface="bold",
show_legend=False,
)
# Top-weight inside-funnel study labels
+ geom_text(
aes(x="effect_size", y="std_error", label="study"),
data=inside_top,
color=INK_SOFT,
size=2.8,
nudge_y=-0.018,
show_legend=False,
)
# Pooled OR annotation box
+ geom_label(
aes(x="x", y="y", label="label"),
data=annotation_df,
color=INK,
fill=ELEVATED_BG,
size=3.5,
fontface="bold",
hjust=0,
show_legend=False,
)
# Inverted y-axis: lower SE (higher precision) at the top
+ scale_y_reverse()
+ labs(x="Log Odds Ratio", y="Standard Error (precision ↑)", title=title)
+ ggsize(800, 450)
+ theme_minimal()
+ theme(
plot_title=element_text(size=title_size, face="bold", color=INK),
axis_title=element_text(size=12, color=INK),
axis_text=element_text(size=10, color=INK_SOFT),
panel_grid_major_x=element_blank(),
panel_grid_major_y=element_line(color=GRID_COLOR, size=0.3),
panel_grid_minor=element_blank(),
axis_line=element_line(color=INK_SOFT, size=0.4),
legend_position=[0.85, 0.82],
legend_title=element_text(size=10, face="bold", color=INK),
legend_text=element_text(size=10, color=INK_SOFT),
legend_background=element_rect(fill=ELEVATED_BG, color=INK_SOFT, size=0.5),
plot_background=element_rect(fill=PAGE_BG, color=PAGE_BG),
panel_background=element_rect(fill=PAGE_BG),
)
)
ggsave(plot, f"plot-{THEME}.png", scale=4, path=".")
ggsave(plot, f"plot-{THEME}.html", path=".")
Part of Meta-Analysis Funnel Plot for Publication Bias on anyplot.ai.