1. Introduction
A/B testing (also known as split testing or bucket testing) is a method of comparing two versions of a webpage or app against each other to determine which one performs better. It is essentially a special case of multiple variants of a page which are shown to users at random, and statistical analysis is used to determine which variation performs better for a given key performance indicator (KPI). Sometimes, the results from A/B testing are used as a baseline to be compared with other digital marketing services, e.g. personalization. Thus, it is important that A/B testing or multivariate testing results are truly valid so that we make the correct inference. One important aspect is to get rid of any confounding. In other words, we want to make sure that the randomly assigned groups have the same profile distributions. In this way any difference in the KPI actually results from the testing objects, instead of from some characteristics of the users such as age or gender. Despite the large volume of literature on A/B testing design, to our knowledge, there has not been any papers which aims to validate the balance of population distribution of different groups. We want to point out that our distribution testing methods are nonparametric by design and can have much more general applications beyond A/B testing, such as in randomized medical clinical trials, in design of experiments in manufacturing fault detection, and in bandits problem in personalized medicine, basically anywhere when uniformly random data splitting is required in a multidimensional space.
The rest of this paper has the following structure: Section 2 discusses past literature on theory and application of randomized controlled trials (RCT) and observational studies. Section 3 introduces some statistical tools, i.e., distancecovariance analysis, propensity score method, randomized chisquare test, and a resampling technique, to test equivalence of multidimensional categorical distribution. Section 4 compares and analyzes the above methods on simulated data. Section 5 applies our methods on real traffic data of online marketing from Adobe Experience Cloud. Section 6 discusses further generalization and implementations of our methodology.
2. Past Work
Randomized controlled trials are the simplest method to yield unbiased estimates of the treatment effects under a minimal set of assumptions. Researchers have been developing new methodology that boosts the efficiency of randomized trials when sample size is small. Sample size may not be a problem in some ecommerce settings as the number of online visitors can easily build up to hundreds of thousands. However, in other settings such as clinical trials, the number of samples (patients) is often small
(bhat2017, ). (bhat2017, ) formulates online/offline AB testing as a (computationally challenging) dynamic optimization problem and develops approximation and exact algorithms. In particular, the paper assumes that response is linear in the treatment and covariates: , where or according to whether subject is assigned to treatment or control group, and denotes covariates of subject. The objective is to maximize overall precision, which is the inverse of the standard error of
. In online setting, is measurable. It can also incorporates other goals such as controlled selection bias and endogenous stopping. Despite clean theoretical guarantees and relatively easy implementation of this algorithm, the linear model is not applicable in many real cases, for example, if is 0 or 1 denoting whether a user convert or not. In this case, a generalized linear model makes more sense but we are unclear about any theoretical results in this scenario. Also the theory requires to have elliptical distribution, which is not the case in many real applications, though one real experiment with categorical data in the paper still performs well. More importantly, the number of covariates may not be clear at the beginning of A/B testing as online user profile may not be available at the time of experiments. Since the current paper is concerned with experiments in ecommerce, the sample size does not pause a big issue.There are numerous works on the implementation, validity, and efficiency of A/B testing. (kohavi2009, )
gives a detailed summary of the things data scientists need to pay attention to when carrying out A/B testing, and in general multivariate testing, such as determining the right sample size before testing and stopping the experiments when buggy or unintentionally poor results come up. When there are more than one variant to test, such as the choice of the picture or the color of the text on a website, online experimenters can either do multiple pairwise comparison of each individual variant, and apply methods of multiple testing adjustments, such as Bonferroni adjustment, if the number of variants is small, or do a multivariate test altogether. This paper mentions some possibilities that A/B testing split might not be truly random. One of the most common in practice is the effect of robots. Robots can introduce significant skew into estimates, enough to render the split invalid, and cause many metrics to be significant when they should not have been. For some websites robots are thought to provide up to half the pageviews on the site
(kohavi2004, ). However, it is difficult to clearly delineate between the human users and robots (tan2002, ). Despite mentioning possible confounding factors in online experiments, this paper does not go into details on how to get rid of them, and how to test whether the splitting of traffic is truly random.When randomized trials are not possible, techniques in observational studies are needed to get valid inference. (gordon2016, )
focuses on the evidence of advertising measurements from big field experiments in Facebook. It uses the advertising effects measured from RCTs as benchmark, and compares the causal effects measured from observational studies to the benchmark. Two methods proposed by the paper to get rid of confounding in non randomized experiments are matching and regression adjustments. Exact matching and propensity matching are two common ways in practice for the former. Inverseprobabilityweighed regression adjustment (IPWRA)
(wooldridge2007, ) incorporates propensity information into regression adjustments to provide more robust results. This paper also mentions checking the equivalence of distribution in A/B split so as to create valid benchmark. Yet it simply checks each covariate independently without adjusting for multiplicity. This can incur false positives when the number of covariates is large and cannot detect difference in covariance structure. A/B testing is a design experiment so we do not need to use the methodology of observational studies in our paper.There are also papers on the continuous monitoring of A/B testing and how to effectively run a large number of A/B testing. If we plot online experiments results continuously and declare significance the first time pvalue is smaller than a predefined threshold, we will find that the actual type I error can be much larger than the threshold.
(johari2015, ) introduces methods to continuously monitoring A/B while still controlling type I error or the false discovery rate. (tang2010, ) describes overlapping experiments infrastructure that helps run experiments faster and produce better decisions. (zhang2017, ) describes an adaptive sample size modification for cold start of A/B testing. Network effect can also come into play, especially for experiments conducted in companies like Facebook or LinkedIn. (gui2015, ) proposes an A/B testing effect analysis by taking into account network effect into a linear model.All these works contribute to the validity or efficiency of online experiments. We emphasize that our ”randomness check” has a very general application, regardless of how traffic is split, the distribution of the covariates, or whether the experiments are done offline or monitored continuously.
3. Methodology
To formulate the problem mathematically, assume the distributor has assigned incoming users to
groups. Each user is represented by a row vector of length
, with each entry representing certain profile information, such as age, area, gender, or number of times of past visit. Hence, we have data sets, each of dimension . Our goal is to infer whether the split achieves desired randomness by testing whether the dimensional data have the same distribution. Throughout our analysis, we assume each column is categorical and each row (user) is independent. These are reasonable assumptions because many important profile informations, such as gender, age, region, are categorical. We assume the number of user in each group is large, and the network effect is negligible compared to the scale of user numbers.In the following subsections, we first state the method that is used currently by online experimenters to do the A/B split validity check (gordon2016, ) as baseline. Then we describe a method proposed by (rizzo2010, )
, called distance components (DISCO), of measuring the total dispersion of the samples, which admits a partition of the total dispersion into components analogous to the variance components in ANOVA. We will also apply propensity method and introduce a randomized chisquare test. Each of these distribution test method has its own advantage as we show in Section
4. Finally, we propose a resampling technique that controls familywise error rate (FWER) under any condition while still maintaining high power compared to some other multiplicity adjustment methods, as again shown in Section 4. Our contribution is to apply and modify existing testing methods to the ubiquitous A/Bmulti testing validity check.3.1. Baseline
(gordon2016, ) and some other papers apply Ftest ^{1}^{1}1see https://en.wikipedia.org/wiki/Ftest for an introduction of Ftest. to each of the column (covariate) or some key columns of the split data sets, and claim the split is truly random if none of the pvalue is smaller than 0.05. This is perhaps the most straightforward and simple way of approaching the randomization check problem, but with some potential issues. First, Ftest can only test difference in mean, but cannot detect other distributional differences, such as variance (we will see in the next subsection DISCO provides a solution to this). Second, even if each column has the same distribution among the data sets, the interaction among the columns can be different for each data set, which also contribution to multidimensional distribution heterogeneity. Third, multiplicity adjustment is needed if the dimension is large, which is often the case for online marketing data. Otherwise, even an A/A test can have very high false positive rate.
To address the third issue, we apply a resampling technique (introduced later) to adjust for multiplicity. It has the advantage of controlling FWER under any dependence structure of pvalues, while maintaining high power compared to other pvalue adjustment methods, such as Bonferroni or Holmes method.
3.2. DISCO onestep test
(rizzo2010, ) propose a new method, called distance components (DISCO), of measuring the total dispersion of the samples in multidimensions, which admits a partition of the total dispersion into components analogous to the variance components in ANOVA. They introduce a measure of dispersion based on Euclidean distances between all pairs of sample elements, for any power , which is called the index, of distances such that . The method is based on the following key definitions and theorem.
Suppose that and are independent and identically distributed (i.i.d.), and and are i.i.d., independent of . If is a constant such that and , define the distance (energy distance) between the distributions of and as
(1) 
Then we have the following theorem.
Theorem 3.1 ().
Suppose that are i.i.d. with distribution , are i.i.d. with distribution , and is independent of . If is a constant such tha and , then the
following statements hold:
(i)
(ii) If , then if and only if .
(iii) If , then if and only if .
The proof of Theorem 3.1 can be read in (rizzo2010, ). Based on this theorem, DISCO statistics, which can be thought of as a variation of the ANOVA statistics, can be developed. Define the empirical distance between distributions as follows. For two dimensional samples and , the distance between and is defined as
(2) 
where
(3) 
Note that is withinsample dispersion and is betweensample dispersion. Similar to ANOVA analysis, we can also write total dispersion as summation of betweensample and withinsample dispersion here. Let be dimensional samples with sizes . The sample distance statistic that takes the role of ANOVA sum of squares for treatments is the weighted sum of dispersion statistics:
(4) 
Similarly, the total dispersion of the observed response is
(5) 
where is the pooled sample, and the withinsample dispersion is
(6) 
Note that we have , and when , the decomposition is exactly the ANOVA decomposition of the total squared error: . Hence, ANOVA is a special case of DISCO method.
Based on the decomposition, the final statistics for testing equality of distribution is
(7) 
with (if , the above statistics can only test equality of mean). The distribution of is complicated so (rizzo2010, ) uses permutation, which is a simplified version of our resampling technique, to obtain rejection thresholds. As we will see in Section 4
, DISCO onestep test, though simple to implement, does not provide information about which dimensions are problematic when the null hypothesis of equal distribution is rejected.
3.3. Propensity score method
Propensity score comparison ^{2}^{2}2see https://en.wikipedia.org/wiki/Propensity_score_matching tests whether the dimensional data sets have the same distribution by fitting a model to the data to obtain the likelihood of a data point being assigned to a particular data set. To be specific, combine the data sets to be one big data set , which is also the dimensional covariate space. The response is a length vector with each entry being the class label (1,2,…,or
) of each data point. We can then fit logistic regression, treebased model, support vector machine, or any class prediction model to the data to obtain a predicted label for each data point. Here we use multiple logistic regression. If the
original data sets truly have the same distribution, the predicted labels should have about uniform distribution on
. We use chisquare test ^{3}^{3}3see https://en.wikipedia.org/wiki/Chisquared_test on a contingency table to test the uniformity of distribution. However, we can show, for logistic regression, that only when the data sets are exactly the same is the distribution of the predicted labels truly uniform. Otherwise, the predicted label will tend to be the same as the actual label due to overfitting. To resolve this, we can randomly choose a proportion of rows in each data set to be training data, and the rest are test data. But this reduces the effective test sample size in the stage of chisquare test. For example, if we choose , then only 1/5 of the data are used to chisquare test. This can shrink power when the total number of data points are not too large. Here, like in DISCO onestep test introduced in the previous subsection, we use permutation method to get the rejection threshold for the pvalue obtained from the chisquare test.The propensity score method, in some sense, takes into account both marginal distributions and interactions because all the covariates appear together on the right hand side of the link function. Interaction terms can also be added, but it is unclear how many of the interaction terms are enough since adding all the interactions are quite unfeasible when the number of covariates is large. However, as we will show in the simulation section, the propensity score method is not very sensitive to nonmean difference (such as variance difference) in distribution. Next, we will give a more direct approach to testing marginal and interaction homogeneity.
3.4. Randomized chisquare test
Since the data are categorical, the distribution is determined by the multinomial distribution parameters , where each denotes the probability of seeing a particular combination of categories from the columns, with , where is the total number of categories. With columns in total, the total number of categories can be huge ( is most likely a lower bound). Theoretically, we can create a contingency table and apply chisquare test to test whether the multinomial distribution is independent of data sets (equivalent to the data sets having the same distribution). Yet this is not feasible under computation time constraints. Nor can we reduce the dimensions of the table without losing information. For example, two distributions can have the same twoway interaction but different threeway interaction. However, if we have further information about the number of categories of each column, we can reduce accordingly. A simple example is that all columns are binary (two categories). The distribution in this case is determined by marginal distribution and twoway interaction. Without further information, we compromise by introducing randomization: choosing a small subset of columns each time, say columns, to do chisquare test, and then repeat the process times. The argument is that even if the nonnull columns are sparse, by repeatedly choosing times for relatively large , the nonnull columns still have large probability of being picked at least once. Assume there are columns in total, the probability that one single column is not picked even once is , which is approximately when is small. Thus, and have equal impacts on the selection probability. In practice, increasing costs less computation than increasing .
Note that randomized chisquare only applies to categorical data among the three methods we introduce. In practice, we can “categorize” continuous data into buckets, which sometimes gives us higher heterogeneous distribution detection power than applying the other methods directly on the continuous data.
3.5. A resampling technique
Resampling ^{4}^{4}4see https://en.wikipedia.org/wiki/Resampling_(statistics)
is often used when the distribution of the test statistics is not known or hard to derive. In our randomized chisquare test, for example, the distribution of Pvalues depends on the unknown distribution of the original data sets. Thus, we can use resampling to set the threshold of rejection. If we are in single hypothesis testing scenario (onestep DISCO and propensity score methods), our proposed resampling technique is equivalent to permutation test (see Algorithm 1)
^{5}^{5}5In all the algorithms here, without loss of generality we assume the computed statistics are Pvalues..It is easy to see that Algorithm 1 controls FWER exactly, because all permutations have equal probability under the null. Under multiple hypotheses testing scenario, we modify Algorithm 1 a little (see Algorithm 2).
We claim that Algorithm 2 controls FWER under any hypotheses configuration.
Theorem 3.2 ().
Assume Pvalues are marginally Uniform(0,1). Under any covariance structure, Algorithm 2 controls FWER, so it controls FDR as well.
Proof.
The proof is conceptually simple. Assume there are nonnull and null hypotheses among the hypotheses. Then the probability of making at least one false rejection is the probability that the minimum of a subset of null Pvalues is less than the th percentile of the distribution of the minimum of total null Pvalues, which is less than . ∎
In practice, resampling can also be applied after hypotheses selection (see Algorithm 3).
Empirical results show that Algorithm 3 also controls FWER, thus FDR, when the selection rule satisfies certain properties. That is, when nonnull hypotheses are more likely to be selected than null hypotheses. Intuitively, this controls FWER because the minimum of a subset of a total null Pvalues is stochastically larger than the minimum of all the null Pvalues.
We will compare this resampling technique to other multiplicity adjustment methods such as Holm’s procedure ^{6}^{6}6See https://en.wikipedia.org/wiki/Holm0Bonferroni_method for Holm’s procedure., and the BenjaminiYekutieli (BY) procedure proposed in (benjamini2001, ) in the next section.
4. Simulations and analysis
Since real data sets are huge and messy, we first use simulations to compare our methods introduced in the previous section. Heterogeneity in multidimensional distributions has two compositions: (a) heterogeneity in marginal distribution of one (or more) dimension(s); (b) heterogeneity in the covariance (interaction) structure among dimensions. In the following simulations, the data is generated with either (a) or (b), or both.
For randomized chisquare method, deciding the choices of the maximum number of columns to sample each time () and the number of times to sample () is tricky. When dimensions become large, we hope to increase to capture more complicated interaction structure. Yet the number in each cell of the table will decrease, which will diminish the power of chisquare test. Empirically, when the number of columns is not too large, and if the sum of each column of the table is greater than or equal to 5, we pick between and , where is the dimension of the data sets.
We do not do any feature selection to reduce dimension in our simulations, because in reality online marketing data has a great range of variety depending on the company, and each company has its own way of trimming data sets (either with general feature selection technique such as LASSO, or with domain knowledge). Thus, we do not apply Algorithm 3 in the previous section in this paper.
4.1. Detection of heterogeneity in marginal distribution or interaction structure
In this first simulation, we consider detection of marginal distribution difference and interaction structure difference separately. We simulate four data sets, each with 10 columns (dimensions) and 100 rows. Three of them have iid entry with multinomial distribution with probability vector (0.25,0.25,0.25,0.25) (so the number of categories of each column is also four).
In Scenario 1, the last data set has different marginal distribution than the other three data sets: for weak signal, the probability vector is (0.3,0.25,0.25,0.2); for medium signal, it is (0.4,0.25,0.2,0.15); for strong signal, it is (0.5,0.2,0.2,0.1). For each heterogeneity we also vary the number of columns (from 1 to 10) that have heterogeneous marginal distribution to test the sensitivity of each method.
In Scenario 2, the last data set has different interaction structure among columns keeping the marginal distribution the same as the other three sets (the other three have independent columns): first sort each column, and then for strong signal, each column is rotated by 10 (turning an array to . Here
denotes concatenation) from the previous column, so the columns are positively correlated; for medium signal, after rotation 40% of the data in each column is permuted to mitigate the correlation; for weak signal, after rotation 80% of each column is permuted randomly. Like in Scenario 1, we also vary the number of columns (from 2 to 10) with heterogeneous interaction structure. We compare four methods: baseline ttest, propensity score method with resampling, DISCO method, and randomized chisquare test with resampling.
From Figure 1 and 2 we see that baseline ttest has high power for detecting marginal heterogeneity, but not for detecting interaction heterogeneity. This agrees with our conjecture. Thus, in the following simulations, we will not test this method anymore. DISCO and propensity score methods have relatively higher power in detecting marginal difference, while randomized chisquare method has significant advantage in detecting interaction heterogeneity. Furthermore, while DISCO or propensity score methods can only tell whether the multidimensional distributions are the same, randomized chisquare test can also flag individual problematic columns when the distributions are imbalanced. The flagged columns may not be exhaustive due to randomness in column selection, but it definitely provides a starting point for the followup detail diagnosis procedure.
We also see from the power plots that interaction heterogeneity is in general harder to detect than marginal distribution heterogeneity since the former has convex power line (power only increase significantly when the number of heterogeneous columns is large) while the latter has concave power line.
4.2. Varying dimension
We next vary the dimension of data sets from 10 to 50 while holding the number of rows to be 100. We compare the three methods, propensity methods, DISCO, and randomized chisquare test, on weak heterogeneity (weak marginal heterogeneity + weak interaction heterogeneity as defined in the previous simulation), medium heterogeneity (weak marginal heterogeneity + medium interaction heterogeneity), and strong heterogeneity (medium marginal heterogeneity + medium interaction heterogeneity). In each of the three signal level the number of heterogeneous columns is 1/5 of the dimensions.
From Figure 3 we see randomized chisquare behaves almost the same, or a little better, when the difference in distribution is not too large, while the other two methods, especially DISCO, behave significantly better when the dimension of data sets is relatively large. In other words, randomized chisquare test has higher power in detecting many small effects.
We also notice that compared to the other two methods, increasing the dimension of the data has little positive effect in power for randomized chisquare test. The reason is that since the portion of heterogeneous columns is unchanged as dimension increases, the probability of picking the ”problematic columns” is also relatively unchanged as dimension increases. For the other two methods, the power has an increasing trend as dimension increases.
We use resamling procedure to get threshold of rejection in all the four methods, so here we also compare the power of resampling procedure, and one popular multiplicity adjustment procedure, the Holm’s procedure ^{7}^{7}7See https://en.wikipedia.org/wiki/Holm%E2%80%93Bonferroni_method for an introduction of Holm’s procedure. Both resampling and Holm’s procedure can control type I error under any Pvalue structure.
Figure 4 plots the power comparison of applying randomized chisquare test + resampling and applying randomized chisquare test + Holm’s procedure. We see that for any heterogeneity level, resampling has higher power than Holm’s procedure, which justifies the use of resampling procedure in multiple hypotheses testing problems.
4.3. Simulated realworld scenario
Next, we simulate a Californialocal real online marketing A/B testing scenario. Assume the company is an online retailer for fashion, skin care, and cosmetics. The data is collected from a 24hour window from 12AM February 21st to 12AM February 22nd. There are 8 columns representing area of living, browser type, gender, age, employment status, income, accumulated number of visits (in the past year), and converted or not before respectively. Table 1 below summarizes the encoding details of these 8 features.
Column names  Encoding 

Area of living  0: south Cal; 1: north Cal; 2: mid Cal 
Browser type  0: Chrome; 1: Safari; 2: Firefox; 
3: Internet Explorer; 4: others  
Gender  0: male; 1: female 
Age  0: ¡20; 1: 2030; 2: 3040; 
3: 4050; 4: ¿50  
Employment status  0: student; 1: employed; 2: unemployed 
Income  0: ¡50,000; 1: 50,000100,000; 
2: 100,000200,000; 3: ¿200,000  
Accumulated  0: ¡3; 1: 310; 2: 1020; 3: ¿20 
number of visits  
Converted before  0: no; 1: yes 
The number of categories for each column ranges from 2 to 5. The real interaction structure among these 8 columns can be complicated. We simplify this relationship and summarize it in Figure 5 below. Area of living and Browser type are independent variables, while the other 6 variables have complicated correlations. It is by no means accurate. For example conditioning on employment status, age and gender may not be independent in reality. We consider the scenario where the data traffic is not randomly split for a period of time. For example, almost all traffic is assigned to set A for two hours in the morning, which results in set A has more data than set B. To balance the number of data points in the two sets, the experimenter assigns most of the traffic in the evening to set B. A direct result is that employment status has different distributions in set A and B — unemployed people take up much higher proportion in the morning time when employed people and students are busy working than they do in the evening time.
Figure 5 also shows a way to generate our data. All marginal distributions are multinomial. Area of living and Browser type are independent which can be easily generated. Then we generate the Employment status with certain probability vector. Conditioning on Employment status, we generate age (the edge with 1), gender (the edge with 2), and income (the edge with 3). Then conditioning on income, gender, age, we generate Accumulated number of visits (the edges with 4). Finally we generate Converted or not conditioning on income and Accumulated number of visits (edges with 5). A detailed data generation procedure is summarized in the Appendix. In particular, the probability vector for Employment status is (0.3,0.3,0.4) in set A (more people in the morning) and (0.4,0.4,0.3) in set B (more people in the evening), which results in difference in marginal distribution and correlation structure in other dimensions. We control the number of data points to be 800 in either data sets.
The data is simulated 100 times. The power for the propensity method, DISCO method, and randomized chisquare test is 0.93, 0.91, and 0.87 respectively. All three methods can detect the multidimensional distribution heterogeneity with high power. Again there is a tradeoff: randomized chisquare method has comparably lower power but can point out the most imbalanced columns.
Column names  Number of times being rejected 

Area of living  28 
Browser type  23 
Employment status  163 
gender  43 
age  85 
income  44 
Accumulated number of visits  50 
Converted or not  43 
Table 2 displays the number of time each column gets rejected in 200 simulations with (sampling columns 10 times per simulation) and (sampling maximum 3 columns each time). We see Area of living and Browser type have significantly fewer times of being rejected because they are balanced and independent. Employment status has the largest number of sampling times since it is directly affected by the timing of the day, with the rest 5 columns having milder imbalance depending on its correlation with Employment status.
5. Anonymous real data
We also try our tests on some autopersonalization datasets provided by Adobe Digital Marketing Cloud. Autopersonalization is personalized recommendation to individual customers learned from past customer behavior in the database. Usually a random A/B splitting is needed to serve as baseline to evaluate any recommendation methodology. We obtain such a pair of randomlysplit data sets, and test the propensity method, DISCO method, and randomized chisquare test on a data set from the online marketing section of a company. The data is split into set A and B, each of dimension , with 2 to 50 categories per column. For privacy concern, we do not get to know the encodings of the columns and the detailed data collecting procedure. However, propensity score with resampling, DISCO method, and randomized chisquare tests all reject the hypothesis that the two distributions are the same. Randomized chisquare also provides some combinations of columns that are the most imbalanced, one of which is columns 196, 57, 248, 260, 271, 342, 374, and 239. These 8 columns have 50 different combinations (for example if one column can take on 2 values, another 3 values, then the two columns together can take on at most values). Figure 6 shows a bar plot of the 15 combinations that have the most counts from both data sets, and a ratio of counts in set B to those in set A. We see the two categorical distributions do differ a great deal, with counts ratio being as high as 3. If we get more information on the these two sets we can do further analysis on these selected columns to identify the source of heterogeneity.
6. Discussion
In summary, A/B testing has a wide range of application in online marketing and other areas, so making sure the test indeed provides valid results is crucial. This paper formulates the validity of A/B testing results as a hypothesis testing problem in multidimensional space. We propose three ways to test whether two or more data sets come from the same distribution and each has its own advantage. The propensity score and DISCO methods can both be generalized to continuous data, but randomized chisquare test is only applicable to categorical data. Of course, the simplest way is to categorize the continuous data. In fact, even when the data is onedimensional normal with the same mean but difference variance, both propensity and DISCO methods have very low power, but categorize each data point to be the floor of and apply randomized chisquare test yields high power.
Besides testing equivalence of distribution after the experiments have been carried out, we also hope to make sure the online experiments are correctly carried out from the beginning. This is a meaningful and promising area of research that involves multivariate analysis, multiple testing, feature selection, observational data and sequential analysis. There has been active research going on in some parts of this big area, including the papers we mentioned in Section 2. We also need to work closely with data engineers to effectively implement these methodology.
Appendix
.1. Realworld simulation data generation process

Generate Area of living Multinomial ((0.33,0.33,0.34)).

Generate Browser type Multinomial ((0.3,0.3,0.2,0.15,0.05)).

For data set A, generate Employment status Multinomial ((0.3,0.3,0.4)); for data set B, generate Employment status Multinomial ((0.4,0.3,0.3)).

gender Employment status Multinomial ((0.5,0.5)); gender Employment status Multinomial ((0.6,0.4)); gender Employment status Multinomial ((0.3,0.7)).

age Employment status Multinomial ((0.7,0.2,0.05,0.03,0.02)); age Employment status Multinomial ((0.1,0.2,0.3,0.3,0.1)); age Employment status Multinomial ((0.1,0.02,0.04,0.04,0.8)).

income Employment status Multinomial ((0.7,0.2,0.1,0)); income Employment status Multinomial ((0.2,0.3,0.3,0.2)); income Employment status Multinomial ((0.8,0.15,0.05,0)).

ANOV (Accumulated number of visits) (income & age ) Multinomial ((0.7,0.15,0.1,0.05)); ANOV (income & gender & age ) Multinomial ((0.5,0.2,0.2,0.1)); ANOV (income & gender & age ) Multinomial ((0.15,0.2,0.25,0.4)); ANOV other combinations of age, gender and income Multinomial ((0.25,0.25,0.25,0.25)).

Conv (converted or not) (income & ANOV ) Multinomial ((0.9,0.1)); Conv (income & ANOV ) Multinomial ((0.85,0.15)); Conv (income & ANOV ) Multinomial ((0.65,0.35)); Conv (income & ANOV ) Multinomial ((0.8,0.2)).
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