spark-abtest

Analyzing A/B testing with massive data can be tedious: you need to understand the details of the formulas for various statistical tests, and get the required sufficient statistics on a distributed computing cluster, then pull the sufficient statistics locally and manually calculate the final results. This project provides a series of concise, consistent and high-performance functions to make it easier for data scientists/data analysts to analyze A/B testing in Apache Spark.

CI:

setup

spark-abtest is a SparkSessionExtensions, you can use it in most spark clients like spark-sql/spark-shell/spark-submit/pyspark/spark-connect with some extra configurations.

--conf spark.jars=target/scala-2.12/spark-abtest-0.1.0-3.5.0_2.12.jar \ 
--conf spark.sql.extensions=org.apache.spark.sql.extra.AbtestExtensions

you can download the corresponding spark and scala version jar from releases.

for example, in spark-shell

$SPARK_HOME/bin/spark-shell \
  --conf spark.jars=target/scala-2.12/spark-abtest-0.1.0-3.5.0_2.12.jar \
  --conf spark.sql.extensions=org.apache.spark.sql.extra.AbtestExtensions

example

init

from pyspark.sql import SparkSession

spark = (
    SparkSession.builder
        .config("spark.jars", "spark-abtest-0.1.0-3.5.0_2.12.jar")
        .config("spark.sql.extensions", "org.apache.spark.sql.extra.AbtestExtensions")
        .getOrCreate()
)

N = 100000
theta = 0.4
sdf = spark.sql(f"""
select user_id, t, mt, y, x
       ,x * {theta} + randn() as y2 -- continuous response variable, and correlation with x
  from (
       select id as user_id
              ,if(rand() < 0.5, 1.0, 0.0) as t -- treat/control group = 0.5:0.5
              ,case when rand() < 0.4 then 't0'
                    when rand() < 0.7 then 't1' 
                    else 't2' end as mt -- multi treat: base/t1/t2 = 0.4:0.3:0.3
              ,if(rand() < 0.1, 1.0, 0.0) as y --0/1 response variable, like is_visit, is_click...
              ,randn() as x --control variable
         from range({N})      
       ) t1     
""")
sdf.createOrReplaceTempView("sdf")
pdf = sdf.toPandas() # local pandas dataframe
pdf.head(5)

	user_id	t	mt	x	y2
0	0	1.0	t0	-0.64626599	1.54772415
1	1	0.0	t0	-0.85735783	1.62497210
2	2	0.0	t0	0.63050955	-0.73287976
3	3	1.0	t0	0.98298468	0.44892532
4	4	0.0	t1	-0.83978164	-1.26275208

0/1 response: prop test/chisq test

two arm

spark.sql("select prop_test(y, t) as ret from sdf").select("ret.*").toPandas() # select("ret.*") to expand all struct fields

	estimate	pvalue	lower	upper	chisq
0	[0.09937538912654897, 0.10111732956242904]	0.36465656	-0.00200086	0.00548474	0.82179266

import pandas as pd
from scipy.stats import chi2_contingency
observed = pd.crosstab(pdf['y'], pdf['t'])
chi2, p, dof, expected = chi2_contingency(observed)
(p, chi2)

(0.3646565557469007, 0.8217926576217617)

in addition to p-values and chisq, spark-abtest, also provides confidence intervals (for more detail, refer to prop.test or chisq.test function in R) and mean response ratio for each treat.

multi arm

spark.sql("select prop_test(y, factor(mt, array('t0', 't1', 't2'))) as ret from sdf").select("ret.*").toPandas() # select("ret.*") to expand all struct fields

	estimate	pvalue	lower	upper	chisq
0	[0.10143841774048547, 0.10020523100420008, 0.0...	0.38456362	NaN	NaN	1.91129209

observed = pd.crosstab(pdf['y'], pdf['mt'])
chi2, p, dof, expected = chi2_contingency(observed)
(p, chi2)

(0.38456361940111916, 1.9112920872850279)

continuous response: ttest

two arms and unequal variance, aka Welch’s t-test

spark.sql("select ttest(y2, t) as ret from sdf").select("ret.*").toPandas() # select("ret.*") to expand all struct fields

	delta	stderr	tvalue	pvalue	lower	upper	y0	y1
0	-0.00240728	0.00683655	-0.35211824	0.72475033	-0.01580684	0.01099229	-0.00010504	-0.00251232

from scipy import stats
x = pdf[pdf['t'] == 1.0]['y2']
y = pdf[pdf['t'] == 0.0]['y2']
ret = stats.ttest_ind(x, y, equal_var=False)
ci = ret.confidence_interval(0.95)
f"tvalue: {ret.statistic:.6f}, pvalue: {ret.pvalue:.6f}, lower: {ci.low:.6f}, upper: {ci.high:.6f}"

'tvalue: -0.352118, pvalue: 0.724750, lower: -0.015807, upper: 0.010992'

two arms and equal variance

spark-abtest

spark.sql("select ttest(y2, t, 0.95, 'two-sided', true) as ret from sdf").select("ret.*").toPandas() # select("ret.*") to expand all struct fields

	delta	stderr	tvalue	pvalue	lower	upper	y0	y1
0	-0.00240728	0.00683657	-0.35211763	0.72475079	-0.01580686	0.01099231	-0.00010504	-0.00251232

this actually equal to anova or ols

spark.sql("select anova(y2, t) as ret from sdf").select("ret.*").toPandas() # inline expand array of struct

	delta	stderr	tvalue	pvalue	lower	upper	my0	my
0	-0.00240728	0.00683657	-0.35211763	0.72475079	-0.01580686	0.01099231	-0.00010504	-0.00251232

spark.sql("select inline(ols(y2, array(1.0, t))) from sdf").toPandas()

	delta	stderr	tvalue	pvalue	lower	upper
0	-0.00010504	0.00484427	-0.02168431	0.98269982	-0.00959976	0.00938967
1	-0.00240728	0.00683657	-0.35211763	0.72475079	-0.01580686	0.01099231

python scipy.stat

from scipy import stats
x = pdf[pdf['t'] == 1.0]['y2']
y = pdf[pdf['t'] == 0.0]['y2']
ret = stats.ttest_ind(x, y)
ci = ret.confidence_interval(0.95)
f"tvalue: {ret.statistic:.6f}, pvalue: {ret.pvalue:.6f}, lower: {ci.low:.6f}, upper: {ci.high:.6f}"

'tvalue: -0.352118, pvalue: 0.724751, lower: -0.015807, upper: 0.010992'

mannwhitneyua test

from scipy.stats import mannwhitneyu
_, p = mannwhitneyu(x, y)
print(p)

spark.sql("""
select ttest(y2_rank, t).pvalue as mannwhitneyu_pvalue
  from (
       select avg_rank() over (order by y2) as y2_rank
              ,t
         from sdf
       ) t1
""").toPandas()

0.6348526173470306

	mannwhitneyu_pvalue
0	0.63485152

multi arm: anova

ttest is not suitable for more than two treatment groups, in spark-abtest, use anova instead

spark.sql("select inline(anova(y2, factor(mt, array('t0', 't1', 't2')))) from sdf").toPandas() # inline expand array of struct

	delta	stderr	tvalue	pvalue	lower	upper	my
0	-0.00320884	0.00540173	-0.59403883	0.55248749	-0.01379615	0.00737848	-0.00320884
1	0.00752212	0.00755395	0.99578717	0.31935597	-0.00728352	0.02232777	0.00431329
2	-0.00696294	0.00969043	-0.71853771	0.47242748	-0.02595606	0.01203018	-0.01017177

import statsmodels.api as sm
from statsmodels.formula.api import ols

anova = ols('y2 ~ C(mt)', data=pdf).fit().summary()
anova.tables[1]

Variance Reduction: cuped/ancova1/ancova2

refer deep-dive-into-variance-reduction or alexdeng book chapter 10

import pandas as pd
pd.set_option('display.float_format', lambda x: f'{x:,.8f}')
spark.sql("select ttest(y2, t) as ret from sdf").select("ret.*").toPandas() # select("ret.*") to expand all struct fields

	delta	stderr	tvalue	pvalue	lower	upper	y0	y1
0	-0.00240728	0.00683655	-0.35211824	0.72475033	-0.01580684	0.01099229	-0.00010504	-0.00251232

spark.sql("select cuped(y2, t, x) as ret from sdf").select("ret.*").toPandas() # select("ret.*") to expand all struct fields

	delta	stderr	tvalue	pvalue	lower	upper	y0	y1	x0	x1	theta
0	-0.00250543	0.00633984	-0.39518820	0.69270484	-0.01493143	0.00992058	-0.00010504	-0.00251232	0.00021016	0.00045288	0.40439612

spark.sql("select ancova1(y2, t, x) as ret from sdf").select("ret.*").toPandas() # select("ret.*") to expand all struct fields

	delta	stderr	tvalue	pvalue	lower	upper	my0	my	mx0	mx
0	-0.00250543	0.00633990	-0.39518449	0.69270758	-0.01493155	0.00992069	-0.00010504	-0.00251232	0.00021016	0.00045288

spark.sql("select ancova2(y2, t, x) as ret from sdf").select("ret.*").toPandas() # select("ret.*") to expand all struct fields

	delta	stderr	tvalue	pvalue	lower	upper	my0	my	mx0	mx
0	-0.00250543	0.00633993	-0.39518271	0.69270890	-0.01493161	0.00992075	-0.00010504	-0.00251232	0.00021016	0.00045288

randomization unit > analysis unit: delta method or cluster-robust standard error

when randomization unit is larger than analysis unit, usually we need to use delta method or cluster-robust variance estimator. functions prefixed with `cluster`` can handle these problems.

sample data

randomization unit is user_id and analysis unit is user_id + visit_id

import pandas as pd
import numpy as np

num_user = 1000
lambda_ = 10
pcdf_ = pd.DataFrame({
    "user_id": range(num_user),  # each user may have multi visit
    # treat/control group = 0.5:0.5
    "num_visit": np.random.poisson(lambda_, size=num_user),
    "t": np.random.choice([0.0, 1.0], p=[0.5, 0.5], size=num_user),
    "mt": np.random.choice(
        ['t0', 't1', 't2'],
        p=[0.4, 0.3, 0.3],
        size=num_user
    ),  # multi treat: base/t1/t2 = 0.4:0.3:0.3,
    # 0/1 respone variable, like is_visti, is_click...
    "mp": np.random.uniform(size=num_user),
    "mx": np.random.randn(num_user),  # control variable
})


def visit_df(idf_):
    idf = idf_.iloc[0]
    num_visit = idf['num_visit']
    y = np.random.choice(
        [0.0, 1.0], p=[1.0 - idf['mp'], idf['mp']], size=num_visit)
    x = np.random.normal(idf['mx'], size=num_visit)
    my2 = idf['mx'] * theta + np.random.randn()
    y2 = np.random.normal(my2, size=num_visit) 
    ret = pd.DataFrame({
        "user_id": idf["user_id"],
        "visit_id": range(num_visit),
        "t": idf["t"],
        "mt": idf["mt"],
        "y": y,
        "x": x,
        "y2": y2
    })
    return ret


pcdf = pcdf_.groupby("user_id", group_keys=False).apply(visit_df)
scdf = spark.createDataFrame(pcdf)
scdf.createOrReplaceTempView("scdf")
pcdf.head(5)

	visit_id	mt	y	x	y2
0	0	t0	1.00000000	-2.00802713	-1.70662868
1	1	t0	1.00000000	-2.74050928	-0.55966320
2	2	t0	1.00000000	-2.34139859	-1.58488664
3	3	t0	1.00000000	-1.97814922	-1.20993685
4	4	t0	1.00000000	-2.32212300	-1.17054778

0/1 or continuous response: cluster_ttest

spark.sql("select cluster_ttest(y2, t, user_id) as ret from scdf").select("ret.*").toPandas()

	delta	stderr	tvalue	pvalue	lower	upper	my0	my
0	0.01441253	0.07627543	0.18895374	0.85013291	-0.13510268	0.16392773	-0.03379021	-0.01937769

delta method with variance reduction

spark.sql("select cluster_ancova1(y2, t, x, user_id) as ret from scdf").select("ret.*").toPandas()

	delta	stderr	tvalue	pvalue	lower	upper	my0	my	mx0	mx
0	0.00309024	0.07224579	0.04277391	0.96588261	-0.13852608	0.14470655	-0.03379021	-0.01937769	-0.06156514	-0.01019905

spark.sql("select cluster_ancova2(y2, t, x, user_id) as ret from scdf").select("ret.*").toPandas()

	delta	stderr	tvalue	pvalue	lower	upper	my0	my	mx0	mx
0	0.00312417	0.07226738	0.04323066	0.96551853	-0.13853445	0.14478279	-0.03379021	-0.01937769	-0.06156514	-0.01019905

ols/cluster_ols

spark.sql("select inline(ols(y2, array(1.0, t, x, t * x))) from scdf").toPandas()

	delta	stderr	tvalue	pvalue	lower	upper
0	-0.01991035	0.02028927	-0.98132440	0.32645656	-0.05968141	0.01986070
1	0.00272057	0.02928153	0.09291066	0.92597640	-0.05467713	0.06011826
2	0.22544998	0.01383548	16.29505891	0.00000000	0.19832965	0.25257031
3	-0.01093015	0.02040205	-0.53573755	0.59215188	-0.05092228	0.02906199

import statsmodels.api as sm
import statsmodels.formula.api as smf

ret = smf.ols(formula='y2 ~ t * x', data=pcdf).fit().summary().tables[1]
pd.DataFrame(ret)

	0	1	2	3	4	5	6
0		coef	std err	t	P>\|t\|	[0.025	0.975]
1	Intercept	-0.0199	0.020	-0.981	0.326	-0.060	0.020
2	t	0.0027	0.029	0.093	0.926	-0.055	0.060
3	x	0.2254	0.014	16.295	0.000	0.198	0.253
4	t:x	-0.0109	0.020	-0.536	0.592	-0.051	0.029

spark.sql("select inline(cluster_ols(y2, array(1.0, t, x, t * x), user_id)) from scdf").toPandas()

	delta	stderr	tvalue	pvalue	lower	upper
0	-0.01991035	0.05397288	-0.36889552	0.71221342	-0.12570808	0.08588737
1	0.00272057	0.07213233	0.03771632	0.96991462	-0.13867333	0.14411447
2	0.22544998	0.03126138	7.21177413	0.00000000	0.16417138	0.28672857
3	-0.01093015	0.04018747	-0.27197896	0.78564382	-0.08970569	0.06784539

import statsmodels.api as sm
import statsmodels.formula.api as smf

fit = smf.ols(formula='y2 ~ t * x', data=pcdf).fit()
ret = fit.get_robustcov_results(cov_type='cluster', groups=pcdf['user_id']).summary().tables[1]
pd.DataFrame(ret)

	0	1	2	3	4	5	6
0		coef	std err	t	P>\|t\|	[0.025	0.975]
1	Intercept	-0.0199	0.054	-0.369	0.712	-0.126	0.086
2	t	0.0027	0.072	0.038	0.970	-0.139	0.144
3	x	0.2254	0.031	7.212	0.000	0.164	0.287
4	t:x	-0.0109	0.040	-0.272	0.786	-0.090	0.068

all functions

prop_test(y, t, confLevel = 0.95, alternative = ‘two-sided’, varEqual = false, removeNaN = true, correct = true)
ttest(y, t, confLevel = 0.95, alternative = ‘two-sided’, varEqual = false, removeNaN = true)
cuped(y, t, x, confLevel = 0.95, alternative = ‘two-sided’, varEqual = false, removeNaN = true)
ols(y, X, confLevel = 0.95, stdErrorType = ‘const’, removeNaN = true)
anova(y, t, confLevel = 0.95, stdErrorType = ‘const’, removeNaN = true)
ancova1(y, t, confLevel = 0.95, stdErrorType = ‘const’, removeNaN = true)
ancova2(y, t, confLevel = 0.95, stdErrorType = ‘const’, removeNaN = true)
cluster_ols(y, X, cid, confLevel = 0.95, stdErrorType = ‘HC1’, removeNaN = true)
cluster_anova(y, t, x, cid, confLevel = 0.95, stdErrorType = ‘HC1’, removeNaN = true)
cluster_ancova1(y, t, x, cid, confLevel = 0.95, stdErrorType = ‘HC1’, removeNaN = true)
cluster_ancova2(y, t, x, cid, confLevel = 0.95, stdErrorType = ‘HC1’, removeNaN = true)
avg_rank(): like rank(), but return average rank rather than min rank when ties

except avg_rank(window rank), all functions are aggregate functions.

input

y: metrics variable, double type, prop_test also accept boolean metrics col
t: treat variable
- two arms (A/B test): numeric type of 0/1, or boolean type
- multi arm(A/B/C… test):
  - wrapped by factor(t, array('t0', 't1', 't2')), the array is different values of t
  - the first is base group
  - ttest does’t accept multi treatment
x: control variable, double type
confLevel: confidence level of the interval, default value is 0.95
alternative: a character string specifying the alternative hypothesis, must be one of “two.sided” (default), “greater” or “less”. You can specify just the initial letter.
varEqual: is used to estimate the variance otherwise the Welch (or Satterthwaite) approximation to the degrees of freedom is used
removeNaN: if true, any of y/t/x contains nan, the corresponding row will be removed, like na.action = ‘omit’
stdErrorType: a character string specifying the estimation type, for ols/anova/ancova1/ancova2, default is const, for cluster_ols/anova/ancova1/ancova2, default is HC1

output

two arms (A/B test): return a struct with some fields:
- delta: the estimated difference in means of y
- stderr: the standard error of the mean (difference), used as denominator in the t-statistic formula
- tvalue: the value of the t-statistic, equal to delta / tvalue
- pvalue: the p-value for the test
- lower: lower confidence limits for delta
- upper upper confidence limits for delta
- y0: mean of y in the treat group (t = 0)
- y1: mean of y in the control group (t = 1)
- x0: mean of x in the treat groups (t = 0)
- x1: mean of x in the treat group (t = 0)
- theta: for cuped only
- my: mean of y in the corresponding treat group
- mx: mean of x in the corresponding treat group
multi arm (A/B/C… test): return an array of struct for each treat

Name		Name	Last commit message	Last commit date
Latest commit History 10 Commits
.github/workflows		.github/workflows
.vscode		.vscode
project		project
src		src
.gitignore		.gitignore
LICENSE		LICENSE
README.md		README.md
build.sbt		build.sbt
scalastyle-config.xml		scalastyle-config.xml

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spark-abtest

setup

example

init

0/1 response: prop test/chisq test

two arm

multi arm

continuous response: ttest

two arms and unequal variance, aka Welch’s t-test

two arms and equal variance

spark-abtest

python scipy.stat

mannwhitneyua test

multi arm: anova

Variance Reduction: cuped/ancova1/ancova2

randomization unit > analysis unit: delta method or cluster-robust standard error

sample data

0/1 or continuous response: cluster_ttest

delta method with variance reduction

ols/cluster_ols

all functions

input

output

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spark-abtest

setup

example

init

0/1 response: prop test/chisq test

two arm

multi arm

continuous response: ttest

two arms and unequal variance, aka Welch’s t-test

two arms and equal variance

spark-abtest

python scipy.stat

mannwhitneyua test

multi arm: anova

Variance Reduction: cuped/ancova1/ancova2

randomization unit > analysis unit: delta method or cluster-robust standard error

sample data

0/1 or continuous response: cluster_ttest

delta method with variance reduction

ols/cluster_ols

all functions

input

output

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