#!/usr/bin/env python3
# Copyright (c) Facebook, Inc. and its affiliates. All Rights Reserved.
from typing import List, Tuple, Union
import numpy as np
import pandas as pd
from ax.utils.common.logger import get_logger
logger = get_logger("Statstools")
num_mixed = Union[np.ndarray, List[float]]
[docs]def inverse_variance_weight(
means: np.ndarray, variances: np.ndarray, conflicting_noiseless: str = "warn"
) -> Tuple[float, float]:
"""Perform inverse variance weighting.
Args:
means: The means of the observations.
variances: The variances of the observations.
conflicting_noiseless: How to handle the case of
multiple observations with zero variance but different means.
Options are "warn" (default), "ignore" or "raise".
"""
if conflicting_noiseless not in {"warn", "ignore", "raise"}:
raise ValueError(
f"Unsupported option `{conflicting_noiseless}` for conflicting_noiseless."
)
if len(means) != len(variances):
raise ValueError("Means and variances must be of the same length.")
# new_mean = \sum_i 1/var_i mean_i / \sum_i (1/var_i), unless any var = 0,
# in which case we report the mean of all values with var = 0.
idx_zero = variances == 0
if idx_zero.any():
means_z = means[idx_zero]
if np.var(means_z) > 0:
message = "Multiple observations zero variance but different means."
if conflicting_noiseless == "warn":
logger.warning(message)
elif conflicting_noiseless == "raise":
raise ValueError(message)
return np.mean(means_z), 0
inv_vars = np.divide(1.0, variances)
sum_inv_vars = inv_vars.sum()
new_mean = np.inner(inv_vars, means) / sum_inv_vars
new_var = np.divide(1.0, sum_inv_vars)
return new_mean, new_var
[docs]def total_variance(
means: np.ndarray, variances: np.ndarray, sample_sizes: np.ndarray
) -> float:
"""Compute total variance."""
variances = variances * sample_sizes
weighted_variance_of_means = np.average(
(means - means.mean()) ** 2, weights=sample_sizes
)
weighted_mean_of_variance = np.average(variances, weights=sample_sizes)
return (weighted_variance_of_means + weighted_mean_of_variance) / sample_sizes.sum()
[docs]def positive_part_james_stein(
means: num_mixed, sems: num_mixed
) -> Tuple[np.ndarray, np.ndarray]:
"""Estimation method for Positive-part James-Stein estimator.
This method takes a vector of K means (`y_i`) and standard errors
(`sigma_i`) and calculates the positive-part James Stein estimator.
Resulting estimates are the shrunk means and standard errors. The positive
part James-Stein estimator shrinks each constituent average to the grand
average:
y_i - phi_i * y_i + phi_i * ybar
The variable phi_i determines the amount of shrinkage. For `phi_i = 1`,
`mu_hat` is equal to `ybar` (the mean of all `y_i`), while for `phi_i = 0`,
`mu_hat` is equal to `y_i`. It can be shown that restricting `phi_i <= 1`
dominates the unrestricted estimator, so this method restricts `phi_i` in
this manner. The amount of shrinkage, `phi_i`, is determined by:
(K - 3) * sigma2_i / s2
That is, less shrinkage is applied when individual means are estimated with
greater precision, and more shrinkage is applied when individual means are
very tightly clustered together. We also restrict `phi_i` to never be larger
than 1.
The variance of the mean estimator is:
(1 - phi_i) * sigma2_i
+ phi * sigma2_i / K
+ 2 * phi_i ** 2 * (y_i - ybar)^2 / (K - 3)
The first term is the variance component from `y_i`, the second term is the
contribution from the mean of all `y_i`, and the third term is the
contribution from the uncertainty in the sum of squared deviations of `y_i`
from the mean of all `y_i`.
For more information, see https://fburl.com/empirical_bayes.
Args:
means: Means of each arm
sems: Standard errors of each arm
Returns:
mu_hat_i: Empirical Bayes estimate of each arm's mean
sem_i: Empirical Bayes estimate of each arm's sem
"""
if np.min(sems) < 0:
raise ValueError("sems cannot be negative.")
y_i = np.array(means)
K = y_i.shape[0]
if K < 4:
raise ValueError(
"Less than 4 measurements passed to positive_part_james_stein. "
+ "Returning raw estimates."
)
sigma2_i = np.power(sems, 2)
ybar = np.mean(y_i)
s2 = np.var(y_i - ybar, ddof=3) # sample variance normalized by K-3
if s2 == 0:
phi_i = 1
else:
phi_i = np.minimum(1, sigma2_i / s2)
mu_hat_i = y_i + phi_i * (ybar - y_i)
sigma_hat_i = np.sqrt(
(1 - phi_i) * sigma2_i
+ phi_i * sigma2_i / K
+ 2 * phi_i ** 2 * (y_i - ybar) ** 2 / (K - 3)
)
return mu_hat_i, sigma_hat_i
[docs]def relativize(
means_t: Union[np.ndarray, List[float], float],
sems_t: Union[np.ndarray, List[float], float],
mean_c: float,
sem_c: float,
bias_correction: bool = True,
cov_means: Union[np.ndarray, List[float], float] = 0.0,
as_percent: bool = False,
) -> Tuple[np.ndarray, np.ndarray]:
"""Ratio estimator based on the delta method.
This uses the delta method (i.e. a Taylor series approximation) to estimate
the mean and standard deviation of the sampling distribution of the ratio
between test and control -- that is, the sampling distribution of an
estimator of the true population value under the assumption that the means
in test and control have a known covariance:
(mu_t / mu_c) - 1.
Under a second-order Taylor expansion, the sampling distribution of the
relative change in empirical means, which is `m_t / m_c - 1`, is
approximately normally distributed with mean
[(mu_t - mu_c) / mu_c] - [(sigma_c)^2 * mu_t] / (mu_c)^3
and variance
(sigma_t / mu_c)^2
- 2 * mu_t _ sigma_tc / mu_c^3
+ [(sigma_c * mu_t)^2 / (mu_c)^4]
as the higher terms are assumed to be close to zero in the full Taylor
series. To estimate these parameters, we plug in the empirical means and
standard errors. This gives us the estimators:
[(m_t - m_c) / m_c] - [(s_c)^2 * m_t] / (m_c)^3
and
(s_t / m_c)^2 - 2 * m_t * s_tc / m_c^3 + [(s_c * m_t)^2 / (m_c)^4]
Note that the delta method does NOT take as input the empirical standard
deviation of a metric, but rather the standard error of the mean of that
metric -- that is, the standard deviation of the metric after division by
the square root of the total number of observations.
Args:
means_t: Sample means (test)
sems_t: Sample standard errors of the means (test)
mean_c: Sample mean (control)
sem_c: Sample standard error of the mean (control)
cov_means: Sample covariance between test and control
as_percent: If true, return results in percent (* 100)
Returns:
rel_hat: Inferred means of the sampling distribution of
the relative change `(mean_t / mean_c) - 1`
sem_hat: Inferred standard deviation of the sampling
distribution of rel_hat -- i.e. the standard error.
"""
# if mean_c is too small, bail
epsilon = 1e-10
if np.any(np.abs(mean_c) < epsilon):
raise ValueError(
"mean_control ({0} +/- {1}) is smaller than 1 in 10 billion, "
"which is too small to reliably analyze ratios using the delta "
"method. This usually occurs because winsorization has truncated "
"all values down to zero. Try using a delta type that applies "
"no winsorization.".format(mean_c, sem_c)
)
m_t = np.array(means_t)
s_t = np.array(sems_t)
cov_t = np.array(cov_means)
c = m_t / mean_c
r_hat = (m_t - mean_c) / np.abs(mean_c)
if bias_correction:
r_hat = r_hat - m_t * sem_c ** 2 / np.abs(mean_c) ** 3
# If everything's the same, then set r_hat to zero
same = (m_t == mean_c) & (s_t == sem_c)
r_hat = ~same * r_hat
var = ((s_t ** 2) - 2 * c * cov_t + (c ** 2) * (sem_c ** 2)) / (mean_c ** 2)
if as_percent:
return (r_hat * 100, np.sqrt(var) * 100)
else:
return (r_hat, np.sqrt(var))
[docs]def agresti_coull_sem(
n_numer: Union[pd.Series, np.ndarray, int],
n_denom: Union[pd.Series, np.ndarray, int],
prior_successes: int = 2,
prior_failures: int = 2,
) -> Union[np.ndarray, float]:
"""Compute the Agresti-Coull style standard error for a binomial proportion.
Reference:
*Agresti, Alan, and Brent A. Coull. Approximate Is Better than 'Exact' for
Interval Estimation of Binomial Proportions." The American Statistician,
vol. 52, no. 2, 1998, pp. 119-126. JSTOR, www.jstor.org/stable/2685469.*
"""
n_numer = np.array(n_numer)
n_denom = np.array(n_denom)
p_for_sem = (n_numer + prior_successes) / (
n_denom + prior_successes + prior_failures
)
sem = np.sqrt(p_for_sem * (1 - p_for_sem) / n_denom)
return sem
[docs]def marginal_effects(df: pd.DataFrame) -> pd.DataFrame:
"""
This method calculates the relative (in %) change in the outcome achieved
by using any individual factor level versus randomizing across all factor
levels. It does this by estimating a baseline under the experiment by
marginalizing over all factors/levels. For each factor level, then,
it conditions on that level for the individual factor and then marginalizes
over all levels for all other factors.
Args:
df: Dataframe containing columns named mean and sem. All other columns
are assumed to be factors for which to calculate marginal effects.
Returns:
A dataframe containing columns "Name", "Level", "Beta" and "SE"
corresponding to the factor, level, effect and standard error.
Results are relativized as percentage changes.
"""
covariates = [col for col in df.columns if col not in ["mean", "sem"]]
formatted_vals = []
overall_mean, overall_sem = inverse_variance_weight(
df["mean"], np.power(df["sem"], 2)
)
for cov in covariates:
if len(df[cov].unique()) <= 1:
next
df_gb = df.groupby(cov)
for name, group_df in df_gb:
group_mean, group_var = inverse_variance_weight(
group_df["mean"], np.power(group_df["sem"], 2)
)
effect, effect_sem = relativize(
group_mean,
np.sqrt(group_var),
overall_mean,
overall_sem,
cov_means=0.0,
as_percent=True,
)
formatted_vals.append(
{"Name": cov, "Level": name, "Beta": effect, "SE": effect_sem}
)
return pd.DataFrame(formatted_vals)[["Name", "Level", "Beta", "SE"]]
