Difference between revisions of "2015 FutureUserEngagementPredictiona"

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revenue, the number of visits, etc.), also known as an over-
 
revenue, the number of visits, etc.), also known as an over-
 
all evaluation criterion [18]. The ability of the metric to
 
all evaluation criterion [18]. The ability of the metric to
detect the statistically signi�cant difference when the treat-
+
detect the statistically significant difference when the treat-
 
ment effect exists is referred to as the sensitivity of the test.
 
ment effect exists is referred to as the sensitivity of the test.
 
Improvement of the sensitivity is quite challenging, because
 
Improvement of the sensitivity is quite challenging, because
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in the treatment group.
 
in the treatment group.
  
and challenging [4], that is why a modi�cation of an existing
+
and challenging [4], that is why a modification of an existing
 
standardized metric is preferred.
 
standardized metric is preferred.
  
The variance reduction techniques (the strati�cation and
+
The variance reduction techniques (the stratification and
 
control variates), used in Monte Carlo simulations, were also
 
control variates), used in Monte Carlo simulations, were also
 
applied [7] in the A/B testing. Though, they do not have
 
applied [7] in the A/B testing. Though, they do not have
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Finally, our approach could be used simultaneously with
 
Finally, our approach could be used simultaneously with
other sensitivity improvement methods (i.e., with strati�ca-
+
other sensitivity improvement methods (i.e., with stratifica-
 
tion, control variates [7], transformation of the metric, �l-
 
tion, control variates [7], transformation of the metric, �l-
 
tration of the treatment group [18], etc.) and, hence, is not
 
tration of the treatment group [18], etc.) and, hence, is not
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cides with the emerging Internet companies' needs: to run
 
cides with the emerging Internet companies' needs: to run
 
more online controlled experiments and to get their results
 
more online controlled experiments and to get their results
on a signi�cant level faster. Specifically, the major contri-
+
on a significant level faster. Specifically, the major contri-
 
butions of our work include:
 
butions of our work include:
  
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real online experiments run at Yandex, first, demon-
 
real online experiments run at Yandex, first, demon-
 
strating the growth of the number of statistically sig-
 
strating the growth of the number of statistically sig-
ni�cant A/B experiments and, second, detecting the
+
nificant A/B experiments and, second, detecting the
 
treatment effect of an A/B test faster (up to 50% of
 
treatment effect of an A/B test faster (up to 50% of
 
saved time) with the same level of statistical signi�-
 
saved time) with the same level of statistical signi�-
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who were not a�ected by the service change in the treat-
 
who were not a�ected by the service change in the treat-
 
ment group [27]. More sophisticated variation reduction
 
ment group [27]. More sophisticated variation reduction
techniques were proposed in [7]: the strati�cation and the
+
techniques were proposed in [7]: the stratification and the
 
usage of control variates based on pre-experiment data. The
 
usage of control variates based on pre-experiment data. The
 
authors of [16] noted that the reduction of skewness of the
 
authors of [16] noted that the reduction of skewness of the
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of the evaluated treatment variant of the service. The dif-
 
of the evaluated treatment variant of the service. The dif-
 
ference between the evaluation metrics over groups should
 
ference between the evaluation metrics over groups should
be controlled by a statistical signi�cance test. In our study,
+
be controlled by a statistical significance test. In our study,
 
we utilize widely applicable two-sample t-test (as in [7, 27,
 
we utilize widely applicable two-sample t-test (as in [7, 27,
 
6]) to decide weather the metric of the treatment user group
 
6]) to decide weather the metric of the treatment user group
B is signi�cantly larger or smaller than that of the control
+
B is significantly larger or smaller than that of the control
 
one. This test is based on t-statistic:
 
one. This test is based on t-statistic:
 
�T =
 
�T =
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than the threshold pval � 0:05 (commonly used [18, 14, 7,
 
than the threshold pval � 0:05 (commonly used [18, 14, 7,
 
27, 16]), then the test rejects the null hypothesis, and the
 
27, 16]), then the test rejects the null hypothesis, and the
difference �T is assumed to be statistically signi�cant. The
+
difference �T is assumed to be statistically significant. The
 
additional details of the A/B testing framework could be
 
additional details of the A/B testing framework could be
 
found in the survey and practical guide [18].
 
found in the survey and practical guide [18].
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served during the A/B experiment (i.e., on the metric values
 
served during the A/B experiment (i.e., on the metric values
 
MX([0; t)) for some t � T) [14]. This method does not pro-
 
MX([0; t)) for some t � T) [14]. This method does not pro-
vide the signi�cance level of the forecast difference due to
+
vide the significance level of the forecast difference due to
 
the absence of data per experimental unit (i.e., per user in
 
the absence of data per experimental unit (i.e., per user in
 
our case), and, thus, could not be used to improve sensitivity
 
our case), and, thus, could not be used to improve sensitivity
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late into the growth of the metric sensitivity. For example,
 
late into the growth of the metric sensitivity. For example,
 
we may expect that a 10% growth of prediction quality over
 
we may expect that a 10% growth of prediction quality over
a baseline may lead to a signi�cant sensitivity improvement,
+
a baseline may lead to a significant sensitivity improvement,
 
while further growth of quality does not improve the ob-
 
while further growth of quality does not improve the ob-
 
tained metric.
 
tained metric.
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users) in order to, first, reduce the data size, which should
 
users) in order to, first, reduce the data size, which should
 
comply with computational constrains, and, second, apply
 
comply with computational constrains, and, second, apply
the paired two-sample t-test to measure the signi�cance level
+
the paired two-sample t-test to measure the significance level
 
of the obtained results.
 
of the obtained results.
 
The performance measure. Let us consider the naive
 
The performance measure. Let us consider the naive
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riodicity, and entropy features are derivable from the source
 
riodicity, and entropy features are derivable from the source
 
time series. Therefore, we expect that not all these fea-
 
time series. Therefore, we expect that not all these fea-
tures could provide a signi�cant pro�t in prediction quality
+
tures could provide a significant pro�t in prediction quality
 
w.r.t. the source time series. Besides, utilization of the large
 
w.r.t. the source time series. Besides, utilization of the large
 
number of features may lead to overfitting of the model15
 
number of features may lead to overfitting of the model15
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We see that both the periodicity and entropy features im-
 
We see that both the periodicity and entropy features im-
 
prove prediction quality, contrariwise, none of the deriva-
 
prove prediction quality, contrariwise, none of the deriva-
tive features shows any signi�cant improvement with p-value
+
tive features shows any significant improvement with p-value
 
� 0:05. We selected the features that showed noticeable
 
� 0:05. We selected the features that showed noticeable
and signi�cant improvement in several prediction tasks (i.e.,
+
and significant improvement in several prediction tasks (i.e.,
 
targets and models) or for several user engagement mea-
 
targets and models) or for several user engagement mea-
 
15In our work, we tried to train models based on all features.
 
15In our work, we tried to train models based on all features.
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scalar features). The nRMSE values for these 5 feature sets,
 
scalar features). The nRMSE values for these 5 feature sets,
 
for 2 models, and for 6 different targets are presented in
 
for 2 models, and for 6 different targets are presented in
Table 3, all differences are signi�cant with p-value � 0:05.
+
Table 3, all differences are significant with p-value � 0:05.
 
First, we see that our feature set Best outperforms all
 
First, we see that our feature set Best outperforms all
 
baselines. Second, we see that the feature set Best provides
 
baselines. Second, we see that the feature set Best provides
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have the worst prediction quality w.r.t. the naive average
 
have the worst prediction quality w.r.t. the naive average
 
baseline. Moreover, the prediction quality for these two tar-
 
baseline. Moreover, the prediction quality for these two tar-
gets is very di�cult to improve in comparison with the oth-
+
gets is very dificult to improve in comparison with the oth-
 
ers. These observations coincide with our findings on the
 
ers. These observations coincide with our findings on the
 
measures' persistence across time (compare the correlations
 
measures' persistence across time (compare the correlations
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CpQ) does not have a noticeable effect on the quality, while
 
CpQ) does not have a noticeable effect on the quality, while
 
the user loyalty measures S and ATpS have a noticeable and
 
the user loyalty measures S and ATpS have a noticeable and
signi�cant in
+
significant in
 
uence on the prediction of any studied target
 
uence on the prediction of any studied target
 
(e.g., the ablation of all S-features decreases the nRMSE
 
(e.g., the ablation of all S-features decreases the nRMSE
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lation them into features, and on the sizes of the observed
 
lation them into features, and on the sizes of the observed
 
and the forecast time periods. We conclude that the use
 
and the forecast time periods. We conclude that the use
of almost all studied features could signi�cantly improve the
+
of almost all studied features could significantly improve the
 
quality of user engagement prediction, and the dependance
 
quality of user engagement prediction, and the dependance
 
of this quality on the parameters of the prediction task agrees
 
of this quality on the parameters of the prediction task agrees
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to check the correctness of the experimentation [18, 4]. We
 
to check the correctness of the experimentation [18, 4]. We
 
considered a commonly used threshold pval = 0:05 for the
 
considered a commonly used threshold pval = 0:05 for the
p-value of the statistical signi�cance test (i.e., of two-sample
+
p-value of the statistical significance test (i.e., of two-sample
 
t-test [7, 27, 6], see Sec. 3 for details).
 
t-test [7, 27, 6], see Sec. 3 for details).
 
In this section, the per-user metrics based on six user en-
 
In this section, the per-user metrics based on six user en-
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that have been trained in advance17. We consider forecast
 
that have been trained in advance17. We consider forecast
 
time periods up to the 28-th day since the experiment start,
 
time periods up to the 28-th day since the experiment start,
and, hence, we get several novel modi�cations for each met-
+
and, hence, we get several novel modifications for each met-
 
ric M 2 M. In this section, we refer to them as FUBPAX!Y ,
 
ric M 2 M. In this section, we refer to them as FUBPAX!Y ,
 
where X is the considered duration of the experiment in
 
where X is the considered duration of the experiment in
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as if it was conducted for 21 days. Additionally, we predicted
 
as if it was conducted for 21 days. Additionally, we predicted
 
the metric value at the post-experiment periods. Such met-
 
the metric value at the post-experiment periods. Such met-
ric modi�cations, which are referred to as FUBPApost
+
ric modifications, which are referred to as FUBPApost
 
X!Y , pre-
 
X!Y , pre-
 
dict the value of M over (Y 􀀀X) days after the end of X-day
 
dict the value of M over (Y 􀀀X) days after the end of X-day
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the time for the p-value threshold 0:05 [18, 4], which is used
 
the time for the p-value threshold 0:05 [18, 4], which is used
 
in our work. Each of the metrics C, PT, CpQ, each of all
 
in our work. Each of the metrics C, PT, CpQ, each of all
their FUBPA modi�cations, and each of the post-experiment
+
their FUBPA modifications, and each of the post-experiment
modi�cations (i.e., FUBPApost
+
modifications (i.e., FUBPApost
 
X!Y ) of ATpS failed one A/A test.
 
X!Y ) of ATpS failed one A/A test.
All other metrics and all other modi�cations did not fail any
+
All other metrics and all other modifications did not fail any
A/A test. Hence, all our metrics and all our modi�cations
+
A/A test. Hence, all our metrics and all our modifications
 
have an acceptable rate of A/A fails.
 
have an acceptable rate of A/A fails.
 
Example. We start our study from consideration of an
 
Example. We start our study from consideration of an
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sults for this experiment are presented in Fig. 4. We see
 
sults for this experiment are presented in Fig. 4. We see
 
that the metric does not detect the treatment effect during
 
that the metric does not detect the treatment effect during
the first 7 days of the experiment. However, its modi�cation
+
the first 7 days of the experiment. However, its modification
FUBPA4!17 shows a statistically signi�cant difference 􀀀0:4%
+
FUBPA4!17 shows a statistically significant difference 􀀀0:4%
 
17They have been trained on the data obtained during the
 
17They have been trained on the data obtained during the
 
time period of February and March, 2013, before all our
 
time period of February and March, 2013, before all our
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for this example). Note that the signs of the actual and
 
for this example). Note that the signs of the actual and
 
the predicted differences Di� are the same. Moreover, it is
 
the predicted differences Di� are the same. Moreover, it is
observed for all FUBPA modi�cations. The magnitudes of
+
observed for all FUBPA modifications. The magnitudes of
 
the predicted Di� for FUBPAX!Y are also of the same order.
 
the predicted Di� for FUBPAX!Y are also of the same order.
 
However, their exact values are sometimes far from the ac-
 
However, their exact values are sometimes far from the ac-
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ble 4 summarizes the number of A/B experiments, whose
 
ble 4 summarizes the number of A/B experiments, whose
 
treatment effect is detected (i.e., pval � 0:05) by each of the
 
treatment effect is detected (i.e., pval � 0:05) by each of the
actual key metrics and each of its FUBPA modi�cations19.
+
actual key metrics and each of its FUBPA modifications19.
 
The best result in each column is underlined. Additionally,
 
The best result in each column is underlined. Additionally,
 
the number of A/B experiments, whose treatment effect is
 
the number of A/B experiments, whose treatment effect is
detected by a considered modi�cation and is not detected
+
detected by a considered modification and is not detected
 
by the corresponding baseline metric, is indicated in brack-
 
by the corresponding baseline metric, is indicated in brack-
 
ets. First, we see that the metrics based on the number
 
ets. First, we see that the metrics based on the number
 
of sessions measure S (i.e., the actual per-user metric and
 
of sessions measure S (i.e., the actual per-user metric and
its FUBPA modi�cations) detect the treatment effects in a
+
its FUBPA modifications) detect the treatment effects in a
 
fewer number of A/B tests than the ones based on the ab-
 
fewer number of A/B tests than the ones based on the ab-
 
sence time measure ATpS, which is assumed to be a novel
 
sence time measure ATpS, which is assumed to be a novel
Line 1,019: Line 1,019:
 
Third, one could note that, for instance, the actual met-
 
Third, one could note that, for instance, the actual met-
 
ric C detects the treatment effects in 8 A/B tests, while its
 
ric C detects the treatment effects in 8 A/B tests, while its
modi�cation FUBPApost
+
modification FUBPApost
 
14!21 detects the ones in 8 A/B tests
 
14!21 detects the ones in 8 A/B tests
 
also, but one of them is new (i.e., it is not detected by C).
 
also, but one of them is new (i.e., it is not detected by C).
It means that a FUBPA modi�cation does not always de-
+
It means that a FUBPA modification does not always de-
 
tect the treatment effect in a test, where the baseline metric
 
tect the treatment effect in a test, where the baseline metric
 
18The absence time should decrease for an improvement [9].
 
18The absence time should decrease for an improvement [9].
Line 1,029: Line 1,029:
 
Table 4: The number of A/B tests whose treatment
 
Table 4: The number of A/B tests whose treatment
 
effect is detected by each UE metric and its FUBPA
 
effect is detected by each UE metric and its FUBPA
modi�cations.
+
modifications.
 
Metric S Q C PT CpQ ATpS
 
Metric S Q C PT CpQ ATpS
 
Actual@14 day 2 1 8 4 15 4
 
Actual@14 day 2 1 8 4 15 4
Line 1,048: Line 1,048:
 
further in this section). Nonetheless, in all those treatment
 
further in this section). Nonetheless, in all those treatment
 
effects that were detected both by a baseline metric and its
 
effects that were detected both by a baseline metric and its
FUBPA modi�cations, the sign and the magnitude of the
+
FUBPA modifications, the sign and the magnitude of the
 
relative difference value Di� are the same. This means that
 
relative difference value Di� are the same. This means that
our modi�cations at least do not harm the decision taken af-
+
our modifications at least do not harm the decision taken af-
 
ter a successful experiment: to accept the evaluated change
 
ter a successful experiment: to accept the evaluated change
 
of the web service, or to reject it. In total, the baseline met-
 
of the web service, or to reject it. In total, the baseline met-
Line 1,071: Line 1,071:
 
ther, we consider only those A/B tests that have no detected
 
ther, we consider only those A/B tests that have no detected
 
treatment effect w.r.t. the baseline metric over 7 days. For
 
treatment effect w.r.t. the baseline metric over 7 days. For
each FUBPA modi�cation and for each A/B test, we look
+
each FUBPA modification and for each A/B test, we look
at the FUBPA modi�cation and check if it expects appear-
+
at the FUBPA modification and check if it expects appear-
 
ing of the treatment effect in the future measurement of the
 
ing of the treatment effect in the future measurement of the
 
baseline metric (i.e., in the future 7 days, 14 days, etc., de-
 
baseline metric (i.e., in the future 7 days, 14 days, etc., de-
Line 1,080: Line 1,080:
 
nally, Table 5 summarizes this information over considered
 
nally, Table 5 summarizes this information over considered
 
A/B tests and the baseline metric CpQ obtaining the statis-
 
A/B tests and the baseline metric CpQ obtaining the statis-
tics for each type of the FUBPA modi�cation. Thus, we are
+
tics for each type of the FUBPA modification. Thus, we are
 
interested in how good our approach is at prediction of the
 
interested in how good our approach is at prediction of the
 
detection of the treatment effect in the future (i.e., at the
 
detection of the treatment effect in the future (i.e., at the
14-th day of an A/B test) in the case of the non-signi�cant
+
14-th day of an A/B test) in the case of the non-significant
 
difference observed at the current moment (i.e., at the 7-th
 
difference observed at the current moment (i.e., at the 7-th
 
day of an A/B test). From Table 5, one sees that almost
 
day of an A/B test). From Table 5, one sees that almost
all FUBPA modi�cations have very good positive and nega-
+
all FUBPA modifications have very good positive and nega-
 
tive predictive values (i.e., Col. 1 and Col. 4). So, one could
 
tive predictive values (i.e., Col. 1 and Col. 4). So, one could
 
conclude that our FUBPA technique can be used in prac-
 
conclude that our FUBPA technique can be used in prac-
Line 1,117: Line 1,117:
 
of an A/B test, then we are interested to know if this effect
 
of an A/B test, then we are interested to know if this effect
 
remains at the 14-th day or disappears. This case is very
 
remains at the 14-th day or disappears. This case is very
important in practice, because the statistically signi�cance
+
important in practice, because the statistically significance
 
difference in the first days of an A/B experiment may be
 
difference in the first days of an A/B experiment may be
 
caused by the Primacy and Novelty effects [18, 14, 16], and,
 
caused by the Primacy and Novelty effects [18, 14, 16], and,
Line 1,148: Line 1,148:
 
could be applied in many cases and could be combined with
 
could be applied in many cases and could be combined with
 
other sensitivity improvement techniques. For example, we
 
other sensitivity improvement techniques. For example, we
could combine the FUBPA with the strati�cation technique,
+
could combine the FUBPA with the stratification technique,
 
which is proposed in [7]. On the one hand, we could directly
 
which is proposed in [7]. On the one hand, we could directly
apply strati�cation of users (e.g., w.r.t. the used browser)
+
apply stratification of users (e.g., w.r.t. the used browser)
to the FUBPA modi�cation of the key metric. On the other
+
to the FUBPA modification of the key metric. On the other
 
hand, we could stratify users in the training set of the user
 
hand, we could stratify users in the training set of the user
 
behavior predictor and obtain an individual predictor for
 
behavior predictor and obtain an individual predictor for
Line 1,157: Line 1,157:
 
a particular browser). After that, during an A/B test, we
 
a particular browser). After that, during an A/B test, we
 
would apply the proper predictor for each stratum.
 
would apply the proper predictor for each stratum.
Besides, a FUBPA modi�cation of a metric could be re-
+
Besides, a FUBPA modification of a metric could be re-
 
garded as a self-su�cient evaluation metric, because it is
 
garded as a self-su�cient evaluation metric, because it is
 
Table 6: The persistence of the treatment effect of
 
Table 6: The persistence of the treatment effect of
Line 1,181: Line 1,181:
 
way by means of the predictor. Therefore, we can apply
 
way by means of the predictor. Therefore, we can apply
 
any previously proposed approaches of sensitivity improve-
 
any previously proposed approaches of sensitivity improve-
ment, treating a FUBPA modi�cation as a key metric. For
+
ment, treating a FUBPA modification as a key metric. For
 
instance, one can (a) remove those users from the treatment
 
instance, one can (a) remove those users from the treatment
 
group, who were not a�ected by the evaluated change [18, 5];
 
group, who were not a�ected by the evaluated change [18, 5];
Line 1,214: Line 1,214:
 
ag to make the decision during an A/B test to
 
ag to make the decision during an A/B test to
 
continue or to stop it based on the probabilities of obtain-
 
continue or to stop it based on the probabilities of obtain-
ing the signi�cant treatment effect. Hence, the results of
+
ing the significant treatment effect. Hence, the results of
 
our study coincide with the emerging needs of modern web
 
our study coincide with the emerging needs of modern web
 
companies to run more and fast controlled experiments on
 
companies to run more and fast controlled experiments on

Latest revision as of 19:56, 15 January 2020

Subject Headings: User Engagement Measure.

Notes

Cited By

Quotes

Abstract

Modern Internet companies improve their services by means of data-driven decisions that are based on online controlled experiments (also known as A/B tests). To run more online controlled experiments and to get statistically significant results faster are the emerging needs for these companies. The main way to achieve these goals is to improve the sensitivity of A/B experiments. We propose a novel approach to improve the sensitivity of user engagement metrics (that are widely used in A/B tests) by utilizing prediction of the future behavior of an individual user. This problem of prediction of the exact value of a user engagement metric is also novel and is studied in our work. We demonstrate the effectiveness of our sensitivity improvement approach on several real online experiments run at Yandex. Especially, we show how it can be used to detect the treatment effect of an A/B test faster with the same level of statistical significance.

1. INTRODUCTION

Online controlled experiments, or A/B testing, have be- come the state-of-the-art technique to improve web services based on data-driven decisions [16]. It is utilized by many web companies, e.g., web search engines such as Bing [6, 16] and Google [28], social networks like Facebook [1], etc. The largest ones have designed special experimental plat- forms that allow them to run A/B tests at large scale (e.g., hundreds of concurrent experiments per day) [28, 15]. An A/B test compares two variants of a service1 at a time by exposing them to two user groups and by measuring the 1e.g., a current version of the service (the control variant) and a new one (the treatment variant)

difference between them in terms of a key metric (e.g., the revenue, the number of visits, etc.), also known as an over- all evaluation criterion [18]. The ability of the metric to detect the statistically significant difference when the treat- ment effect exists is referred to as the sensitivity of the test. Improvement of the sensitivity is quite challenging, because the better it is, the smaller changes of the service could be detected by an experiment.

The straightforward way to improve the sensitivity, or the power, of a controlled experiment is to increase the amount of the observed statistical data that could be done either by increasing the population of users, participating in the experiment, or by conducting the experiment for a su�cient period of time. Both approaches have disadvantages. First, the amount of data is upper-bounded by the available web service tra�c, and the growth of tra�c per experiment re- duces the number of experiments conducted per year [28, 7]. Second, increasing the size of a treatment group in the treat- ment group is always not desirable, as any treatment may have a negative effect2. Third, the sooner an experiment �nishes, the sooner the treatment variant of the system is shipped or is returned for rework. So, the described ways of improving the sensitivity of an A/B test are not preferred, because they are contrary to the main purposes of an exper- imentation platform: to run many experiments and to get their results fast [28].

The power of a controlled experiment depends on the vari- ance of the key metric used in the test [18], hence, a variance reduction could serve as another way to improve the sensi- tivity. The basic examples of such approaches are discussed in [18]: the use of a key metric with lower variance and the elimination of users who were not a�ected by the service change in the treatment group3. The applicability of these approaches is limited and strongly depends on the case of a conducted experiment. Namely, an experimental platform usually has a standardized success criteria, which utilizes a small set of key metrics to make the final decision on the treatment variant of the service [15]. These metrics are usu- ally selected with respect to some business-related criteria of a considered service and are aligned with its long-term goals (like the number of sessions per user for a search engine [14]). Hence, finding an alternative for them is non-trivial 2Actually, experiments with negative result form a notice- able fraction among all experiments [13].

3For instance, if an update of the ranking algorithm of a search engine is made only for the small amount of queries, then only the users submitted these queries should be kept in the treatment group.

and challenging [4], that is why a modification of an existing standardized metric is preferred.

The variance reduction techniques (the stratification and control variates), used in Monte Carlo simulations, were also applied [7] in the A/B testing. Though, they do not have the above-mentioned drawbacks and lead to a noticeable variance reduction, these techniques are based on the uti- lization of pre-experiment data and, hence, are limited in their applicability as well.

In our paper, we propose a novel method to improve sen- sitivity of online controlled experiments that exploits the prediction of the future behavior of an individual user based solely on the user's behavior observed during the experi- ment. The intuition of this method is the following. We know (as mentioned above) that the longer an A/B experi- ment is conducted, the higher its sensitivity. Therefore, we can try to peek into the future behavior of each user partic- ipated in the experiment, and, based on the predicted user behaviors, calculate the evaluation metric for the experiment as if it was conducted beyond its actual time period. Hence, we expect improvement of the sensitivity as if the experi- ment was extended, while the actual experiment duration was not increased.

While our approach could be applied to any online metric of system performance (which is calculated over user data) we investigate it for the case of user engagement metrics. User engagement re

ects how often the user solves her needs (e.g., to search something) by means of the considered ser- vice (e.g., a search engine). On the one hand, these metrics are measurable in the short-term experiment period, and, on the other hand, they are predictive of the long-term success of the company [14, 15, 16, 25]. That is why engagement metrics are often considered to be most appropriate for on- line evaluation. In this study, we pay special attention to the metrics that re ect the loyalty aspect of engagement: the state-of-the-art number of sessions per user metric [14, 27] (which is accepted as the \North-star" for A/B testing evaluation in major search engine companies like Bing [15, 16]) and the recently proposed absence time metric [9]. Our approach raises the problem of prediction of the future in- dividual user engagement. To the best of our knowledge, no existing study on user engagement investigated such a prediction problem4.

Finally, our approach could be used simultaneously with other sensitivity improvement methods (i.e., with stratifica- tion, control variates [7], transformation of the metric, �l- tration of the treatment group [18], etc.) and, hence, is not an a substitute, but rather a complement to them. To sum up, our study considers the problem, which coin- cides with the emerging Internet companies' needs: to run more online controlled experiments and to get their results on a significant level faster. Specifically, the major contri- butions of our work include:

� The solution to the problem of predicting the exact future values of engagement metrics of individual user behavior.

� Utilization of this predictor for the sensitivity improve- ment of online controlled experiments. 4The only study [27] was devoted to the binary prediction of user engagement increase/decrease in the future week. In our study, we focus on predicting the exact values of the user engagement metrics.

� Validation of our sensitivity improvement approach on real online experiments run at Yandex, first, demon- strating the growth of the number of statistically sig- nificant A/B experiments and, second, detecting the treatment effect of an A/B test faster (up to 50% of saved time) with the same level of statistical signi�- cance.

The rest of the paper is organized as follows. In Section 2, the related work on A/B testing and user engagement is dis- cussed. In Section 3, our sensitivity improvement approach is presented. In Section 4, the six studied user engagement measures are introduced and a brief analysis of them is pre- sented. In Section 5, our user engagement prediction prob- lem is stated and its solution is evaluated by a large series of experiments. In Section 6, we show the results of applying our sensitivity improvement approach to several A/B exper- iments and discuss its possible extensions. In Section 7, the study's conclusions and our plans for the future work are presented.

2. RELATED WORK

We compare our research with other studies in two as- pects. The first one relates to online controlled experiments and, specifically, to the sensitivity improvement. The second one concerns user engagement with web services. Online controlled experiment studies. The theoreti- cal details of the online controlled experiment methodology were widely described in the existing works [22, 17, 18], and we conduct our experiments in accordance with them. A rich practical experience of using this methodology for dif- ferent evaluation cases in many web companies was studied recently [13, 19, 28, 15]. These studies concern, inter alia, experiments with different components of a web service (the user interface [13, 8], ranking algorithms [27, 8], etc.), large- scale experimental infrastructure [28, 15], different aspects of user behavior and interaction with a web service (clicks [14, 16], speed [19, 16], abandonment [16], periodicity [8], etc.), and so on. Some of existing works focused on the study of the trustworthiness of the results of an A/B test. Various pitfalls and puzzling results of online controlled ex- perimentation were shared in [4, 14] and several \rules of thumb" were discussed in [16]. In our work, we were aware of all these experiences and pitfalls during our A/B experi- mentation.

Basic sensitivity improvement techniques for an online controlled experiment were concerned in [18]: increasing of the experiment duration or of the user population, partic- ipated in the experiment [5]; the use of another evaluation metric with lower variance [8]; and the elimination of users who were not a�ected by the service change in the treat- ment group [27]. More sophisticated variation reduction techniques were proposed in [7]: the stratification and the usage of control variates based on pre-experiment data. The authors of [16] noted that the reduction of skewness of the evaluation metric used in an A/B experiment could also im- prove the sensitivity (e.g., transformation of the metric or capping its values [16]). The sensitivity improvement for two-stage online controlled experiments was proposed in [6]. All these methods have their own disadvantages and limita- tions in their applicability (as discussed in Section 1). We propose an alternative approach to improve the sensitivity, which is based on predicting future user behavior and does not possess the limitations of the ones described above: no use of pre- or post-experiment data; no dependance on the particular case of the service change (or comparison) under evaluation; and applicability to a wide range of metrics that could be predicted for an individual user.

User engagement with web services. Existing stud- ies of user engagement with a web service and, particularly, a search engine are three-fold. First, several studies focused on analysis of user behavior. Some of them discovered the relationship between search success and search engine reuse [30, 11]. Some others concerned behavioral patterns of users (e.g., simultaneous usage of several search engines [30] or pe- riodicity of user engagement with a search engine usage [8]) and models of web sites with respect to user behavior (e.g., w.r.t. multitasking user behavior [20] or w.r.t. popularity, ac- tivity, and loyalty among users [21]). In our work, we present a brief analysis of the correlations between several user en- gagement metrics including the state-of-the-art \number of sessions" and the recently proposed \absence time". Second, some studies focused on the prediction of future changes in user engagement. Prediction of how a user switches (no switch, persistent switch, or oscillating behavior) be- tween search engines during 26 weeks was studied in [30]. The authors of [27] predict user engagement increase/decrease in the future week (they did not attempt to put this into any practical use, e.g., for sensitivity improvement). Among the features of the classiffer they built, there were both engage- ment measures (the numbers of sessions, queries, clicks) and some non-engagement ones (query types, user satisfaction, etc.) observed during either one or three preceding weeks. The output of such a classiffer could not be straightforwardly integrated in the A/B testing methodology to improve the sensitivity of evaluation metric due to the lack of the inter- pretable quantity of that output (see Sec. 3 for details). The prediction of the exact values of a user engagement metric provides an approximation of the original evaluation met- ric, which thus can serve as an interpretable metric itself, but it was not investigated in existing studies to the best of our knowledge. In our work, we introduce such prediction methodology for different user engagement metrics includ- ing the commonly used numbers of sessions, queries, clicks, and the recently proposed absence time.

Third, there are papers devoted to user engagement as an evaluation metric in online controlled experiments. In [14, 15, 16] the number of sessions per user was stated to be one of the key metrics used in Bing's experimentation platform. The authors of [27] evaluated different changes in search relevance of a popular search engine by means of A/B testing with respect to this engagement metric and several non-engagement measures re ecting query types and user satisfaction. The absence time (the time between two user visits) on a par with other engagement metrics was applied to compare different ranking algorithms used at Yahoo! An- swers [9] and to evaluate a web search engine changes in ranking algorithm and user interface [3]. The novel period- icity engagement metrics of user behavior (resulted from the Discrete Fourier transform of 4 state-of-the-art engagement measures) were applied in [8] to evaluate, by means of A/B experiments, different changes of the search engine rank- ing algorithm, changes of the user interface, and changes of the engine's e�ciency. In our work, we study the sensitiv- ity improvement of controlled experiments with several user engagement metrics that are widely used in the described literature (see details in Sec. 4).

3. FRAMEWORK

Background. In A/B testing, users, participated in an experiment, are randomly exposed to one of two variants of the service, the control (A) and the treatment (B) variants (e.g., the current production version of the service and its update), in order to compare them [18, 14, 16]. The com- parison is based on the evaluation metric (also known as the online service quality metric, the overall evaluation criterion, etc.). In this paper, we consider \per user metrics" [4], which are calculated for each individual user and, then, are aver- aged over all users in each experiment variant5. Namely, let us consider a measure M(u; T) of user interaction with the service (e.g., the number of clicks) during the experiment period T. For each user group UA and UB, we calculate the average values6 MX(T) = avgu2UX M(u; T);X = A;B. After that, their absolute and relative differences are calculated7: �T = MB(T) 􀀀 MA(T) and Di�T = � � �T=MA(T): Nonetheless, the quantities �T and Di�T could not serve themselves as indicators of positive or negative consequences of the evaluated treatment variant of the service. The dif- ference between the evaluation metrics over groups should be controlled by a statistical significance test. In our study, we utilize widely applicable two-sample t-test (as in [7, 27, 6]) to decide weather the metric of the treatment user group B is significantly larger or smaller than that of the control one. This test is based on t-statistic: �T = q �2A (M(T)) � jUAj􀀀1 + �2B (M(T)) � jUBj􀀀1; (1) where �2X (M(T)) is the standard deviation of the measure M(�; T) over users UX;X = A;B. The larger the absolute value of the t-statistic, the lower the probability (also known as p-value) of the null hypothesis, which assumes that the observed difference is caused by random uctuations, and the variants are not actually different. If the p-value is lower than the threshold pval � 0:05 (commonly used [18, 14, 7, 27, 16]), then the test rejects the null hypothesis, and the difference �T is assumed to be statistically significant. The additional details of the A/B testing framework could be found in the survey and practical guide [18]. The future user behavior prediction approach. As it was shown above, any evaluation metric is calculated based on some time period T = [0; T), i.e., on the exper- iment period. One way to improve the sensitivity of an A/B test is to conduct it longer [18], i.e., to increase the length of the period T. The main idea of this paper is to make it virtually by peeking into the future user behavior. Suppose that, for a considered measure M and for each user u 2 U, we are able to forecast the value of the measure M, calculated over some forecast Tf -day period based on some user behavior data FTp (u) observed during the first Tp days of the target period. We denote this predicted value of the true measure by PM;Tf (FTp (u)). If the length T of an A/B experiment equals to Tp, then we are able to predict the user 5This type of metrics (e.g., the number of sessions per user) is very popular in the online controlled experimentation [14]. However, there are also frequently used non-per user met- rics [4] like the annual revenue [16].

6In order to distinguish the method of calculating the value M(u; T) for a user, from the value MX(T), calculated for a user group X, the first one is referred to as a measure and the second one is referred to as a metric in our work. 7The factor � is randomly chosen once in our study in order to hide real values for con�dentiality reasons.

measure M(u; �) over the post-experiment period Tf = [0; Tf ) and calculate the evaluation metric MX(�);X = A;B; (based on the predicted values) as of the A/B test was conducted beyond its actual time period. We will use the notations eM (u; Tf ) = PM;Tf (FT (u)) � M(u; Tf ) and eM X(Tf ) � MX(Tf ) for the predicted user measure and its average values over the user groups X = A;B respectively. Thereby, we forecast the metric M(�; Tf ), which would be calculated, of the A/B test period has been [0; Tf ) instead of [0; Tp). We refer to this method of improving an evaluation metric as the future user behavior prediction approach (FUBPA). First, our approach differs from the naive one, where pre- diction of the metric value MX(Tf ) for each group X = A;B; (or of their difference �T) is based on their trends, ob- served during the A/B experiment (i.e., on the metric values MX([0; t)) for some t � T) [14]. This method does not pro- vide the significance level of the forecast difference due to the absence of data per experimental unit (i.e., per user in our case), and, thus, could not be used to improve sensitivity of A/B tests. On the contrary, our approach provides data for each user and, therefore, allows to calculate t-statistic. Second, our approach differs from the utilization of a bi- nary classiffer of increase/decrease of a considered user mea- sure in a post-experiment period (as it is proposed in [27] for the number of sessions). A fundamental shortcoming of such a classiffer is that it could not be straightforwardly inte- grated in the A/B testing methodology. In fact, its outputs should be transformed to an evaluation metric first. One clear solution is to take the fraction of users whose engage- ment is predicted to grow. Another solution could be the averaged probability of growth estimated by the classiffer. In any case, there is no reason why the obtained metric will correlate with the original user engagement metric accepted in the company. Hence, its difference Di�T could not pro- vide an insight on the difference of the original metric, which may be fatal both for understanding the A/B experiment's result and for the taken decision. For instance, a drop in the fraction of users with increased number of clicks on ads could be observed simultaneously with a growth in the num- ber of clicks per user. The latter evaluation metric has the direct connection with the annual revenue of a considered company. On the contrary, in our approach, the approxi- mation eM X(Tf ) is trained to be as close as possible to the baseline metric M. Finally, the approximation is a surrogate of different sim- ple measures FTp (u) (which are used as features in the pre- dictor). Therefore, the FUBPA could be also treated as a method to combine a series of evaluation metrics ([4]), tar- geting the combined metric to be similar to a desirable eval- uation metric in future. The rest of this section is devoted to the theoretical analysis of the FUBPA approach. Relation between prediction quality and sensitiv- ity. Denote m(u) = M(u; Tf ) the metric over users u 2 U. Denote em(u) = eM (u; Tf ) its predictor. W.l.o.g., we assume that the mean value E(m) equals 0. We also assume that our prediction is unbiased, i.e., its mean value E( em) is also equal to zero. In this case, the standard deviation fie of the prediction error e := m􀀀 em is, by definition, the Root Mean Square Error (RMSE) of the predictor em. We have �2m = �2 e + �2 em + 2Cov(e; em); (2) where �2 � denotes the standard deviation of a variable �. Note that, for any predictor m0, the optimal predictor in the class fcm0 j c 2 Rg is cm0 := (Cov(m;m0)=�2 m0 )m0, since it provides the minimum of the mean squared error �2 m􀀀cm0 = c2�2 m0􀀀2cCov(m;m0)+�2m . Therefore, if the class of prediction models is closed under scalar multiplication8, for the optimal predictor em, we have Cov(m; em) = �2 em, and thus Cov(e; em) = Cov(m 􀀀 e m; em) = Cov(m; em) 􀀀 �2 em = 0: Therefore, the identity (2) implies that, in the case of an optimal predictor, the RMSE and the standard deviations of the metric �m and its predictor � em are connected by the following identity �2 em = �2m 􀀀 RMSE2(m; em): (3) This implies the following clear finding. The better the pre- diction quality in terms of RMSE, the greater the variance of the obtained surrogate measure em. On the one hand, increasing of the prediction quality provides a better ap- proximation of the original measure m(u) and may lead to a better online metric M(Tf). On the other hand, better pre- diction implies greater variance, which may a�ect the sensi- tivity of the metric (see t-statistics in Eq. (1)). Hence, we conclude that the prediction quality does not directly trans- late into the growth of the metric sensitivity. For example, we may expect that a 10% growth of prediction quality over a baseline may lead to a significant sensitivity improvement, while further growth of quality does not improve the ob- tained metric.

4. USER ENGAGEMENT

Engagement measures.

We use the logs of Yandex[1], one of the most popular global search engines, in order to study user engagement. For each user10, we study 6 popular engagement measures, which represent both loyalty and activity aspects of user engagement. For a considered period of time (a day, a week, a month, etc.), we study the following engagement measures calculated over this time period:

8 the absence time per session (ATpS).

Following common practice [12, 14, 9, 27, 3, 8], we define a session as a sequence of user actions whose dwell times are less than 30 minutes. The presence time PT is measured as the sum of all session lengths (in seconds) observed during the considered time period, while the total absence time is measured as the length of the considered time period minus the presence time. Note that the measures S, Q, C, and PT are additive with respect to the time period. The measure ATpS is calculated as the total absence time divided by the number of user sessions S11. The number of clicks per query CpQ could be regarded as the CTR of the search engine result pages as well. The measures S and ATpS represent the user loyalty [25, 14, 9, 27, 8], whereas the measures Q, C, PT, and CpQ represent the user activity [18, 25, 9, 7, 8] aspects of user engagement [25]. The set of all measures is denoted by M = fS; Q; C; PT; CpQ; ATpSg. Before proceeding to the main problems studied in this paper, we present a brief analysis of these measures. We investigate the relationships between them and their persistence across time in order to have a better interpretation of the prediction quality obtained in the next section.

8 This condition is satisfied by a wide range of prediction models including linear model and the state-of-the-art Friedman's gradient boosting decision tree model [10], which are applied in Section 5.

10 We use cookie IDs [18] to identify users as done in other studies on user engagement [9, 20, 27, 8].

Figure 1: The joint distributions of users U w.r.t. each pair of the studied measures M calculated over a 2-week period.

Correlation between measures. We calculate the cor- relations between the studied measures in the following way. Let one consider two measures M1; M2 2 M, a certain set of users U, and a certain time period T. Then, these measures Mi(u; T); i = 1; 2; are calculated for each user u 2 U over the whole time period T (e.g., the total number of sessions dur- ing the entire period). We consider a user as a random event and calculate the Pearson's correlation coe�cient CorrU over users U. In the analysis of this section, we use a 2-week period12 from March, 2013 as the time period T, and we consider all active users of the search engine during the period T as the set U (its size jUj � 107). Table 1 reports the val- ues of the correlations CorrU between all measures from M. We highlighted in boldface the ones with the high- est absolute value in each column, while the smallest ones are underlined. First, one sees that all additive measures S, Q, C, and PT are noticeably well correlated (CorrU � 0:81). Second, the measure CpQ has low correlation with other mea- sures (jCorrUj � 0:083) except with the number of clicks C (CorrU = 0:175). Third, the absence time ATpS is mostly negatively correlated with the other measures. This obser- vation coincides with the intuition that the more frequently a user utilizes the service (higher the number of sessions, queries, etc.), the lower the absence time. Next, we plot the joint distributions13 of users U w.r.t. each pair of the studied measures M calculated over the 11In our study, we also considered another definition of this measure: the sum of times between consecutive user sessions divided by the number of absences (i.e., by S 􀀀 1), and, if only one session is observed, then it is equal to 0. However, such variant of measure demonstrated lower persistence over time, noticeably lower predictability, and the OEC based on it failed unacceptable number of A/A tests. 12It is a popular length of A/B experiments [14, 27, 16, 8]. 13We hide all absolute values for con�dentiality reasons. Table 1: Correlations CorrU between all engagement measures M calculated over a 2-week period. CorrU S Q C PT CpQ ATpS S | 0:843 0:810 0:831 0:028 􀀀0:595 Q 0:843 | 0:888 0:910 􀀀0:001 􀀀0:483 C 0:810 0:888 | 0:904 0:175 􀀀0:478 PT 0:831 0:910 0:904 | 0:081 􀀀0:461 CpQ 0:028 􀀀0:001 0:175 0:081 | 􀀀0:083 ATpS 􀀀0:595 􀀀0:483 􀀀0:478 􀀀0:461 􀀀0:083 | 2-week period T (i.e., 15 heat maps in total) in Fugure 1. First, the previously observed close relationships between all additive measures S, Q, C, and PT are also seen in their joint distributions: the user population is concentrated near the main diagonal in the heat maps in Fig. 1 (1, 2, 3, 6, 7, and 10). The most consistent pattern is observed for the distri- bution of the number of queries Q and the number of clicks C (Fig. 1 (6)), which reveals a clear linear dependence between their logarithms. The maps Fig. 1 (4, 8, 11, 13, and 15) ex- plain the very small correlation of CpQ with other measures reported in Table 1. Second, the negative correlation of the absence time ATpS with the other measures is also seen in Fig. 1 (5, 9, 12, 14, and 15), where the user population is concentrated along the secondary diagonal of the heat maps. Long-term measure persistence. We also identify the relationship between the values of the same measure calcu- lated over two different time periods. Namely, the Pearson's correlation coe�cient over users U is calculated for the val- ues M(�; T1) and M(�; T2) of each measure M 2 M, where T1 and T2 are two consecutive 2-week periods14. The correlation are presented at the bottom of Fig. 2. The highest value is highlighted in boldface, and the lowest one is underlined. The joint distributions of users U w.r.t. each engagement measure from M calculated over the 2-week periods are also captured in Fig. 2. Almost all measures (except for CpQ) have a high persistence over time (CorrU � 0:61). Hence, we expect that the values of these measures over the future may be better predicted by their values over the observed pe- riod than the ones of CpQ. In this section, we presented the main user engagement measures under our study and brie y analyzed them in dif- ferent ways. These observations will help us clearly under- stand the findings in the next section devoted to the predic- tion of user engagement. 5. USER ENGAGEMENT PREDICTION In our work, we study the user engagement prediction problem in the following setting. We suppose that one has user data observed during a period of time Tp (the past, or the observed time period), and, based on these data, one needs to predict the exact value of a target engagement mea- sure calculated over an increased period of time Tf � Tp (the forecast time period) for each individual user. We con- sider the 6 engagement measures M presented in the previ- ous section as our targets. The models. We utilized two models to predict the exact values of the targets. The first one is a state-of-the-art Fried- man's gradient boosting decision tree model [10]. We used a proprietary implementation of the machine learning algo- 14In our study, we consider all users U that used the search engine during the period T1 (and may use it during the fore- cast period), while users who started using it in the period T2 are not included in the set U. 260 Figure 2: Correlations and the joint distributions of users U w.r.t. to each engagement measure from the set M between two consecutive 2-week periods. rithm with 1000 iterations and 1000 trees, which appeared to be the best settings during validation. The second model is a linear regression model with L2 regularization. The learning rate of the decision tree model and the regulariza- tion parameter of the linear model were adjusted by means of cross-validation technique during optimization. The op- timal settings were found with respect to the mean squared error used as the loss function. The data. In order to conduct our experiments, we used the logs of user search activity on the popular search engine Yandex. We collected user data from two non-intersecting 34-day periods of 2013 year, the earlier period was used to form the training data set and the later one was used to collect the test data set. Since the day of the week may have a considerable impact on the user behavior, we would not like the data set to be biased to some particular day of the week the observed period Tp begins. Therefore, for each of the 34-day periods, each user is represented by 7 feature vectors in the data set (treated as 7 different examples) that are calculated based on the raw user behavior data truncated at 0; 1; : : : ; 6 first days, so that the observed period begins on the 1st, 2nd, . . . , 7th day of the 34-day period. Further, we randomly sampled these data sets 20 times, obtaining smaller training and test data sets of equal size (of > 105 users) in order to, first, reduce the data size, which should comply with computational constrains, and, second, apply the paired two-sample t-test to measure the significance level of the obtained results. The performance measure. Let us consider the naive average prediction model (avgU), which, for each user from the test data set, predicts the target value as the average of the values of that target over all users in the training data set. One could treat it as a model, which utilizes zero features. Denote the RMSE value calculated over the test set for a given observed and forecast periods (Tp and Tf ), for a given target (tg), for a given prediction model (m), and for a given feature set (�), by RMSE(Tp; Tf ; tg; m; �). In the results reported below, we use the normalized RMSE (nRMSE) defined as nRMSE(Tp; Tf ; tg; m; �) = RMSE(Tp; Tf ; tg; m; �) RMSE(Tp; Tf ; tg; avgU;?)

This performance measure allow us, first, to hide the RMSE values due to con�dential reasons, second, to compare the prediction quality between different targets and forecast pe- riods, and, finally, to improve our understanding of weather a studied prediction model is better or not than a naive baseline model (the average prediction) by comparing the nRMSE w.r.t. 1. The observed and target periods. In our experi- mentation, we compare different feature sets and prediction models for 14-day observed periods Tp and for 28-day fore- cast period Tf . We remind that these are popular lengths of A/B experiments [14, 27, 16, 8]. However, the dependence Table 2: The top-20 features w.r.t. the improvement over the baseline feature set in terms of RMSE for the measure S with jTpj = 14 and jTf j = 28. Decision Trees Linear Regression M Feature Impr. M Feature Impr. S lgavg 􀀀0:43% S DFTA[1] 􀀀0:28% ATpS lgTSc[14] 􀀀0:40% S lgDFTA[1] 􀀀0:25% ATpS EntPerm[5] 􀀀0:20% C GrPos1 􀀀0:24% S EntAp[2] 􀀀0:20% PT GrPos1 􀀀0:23% ATpS lgTSc[13] 􀀀0:19% PT GrPos2 􀀀0:23% S sum2 􀀀0:18% S lgmax 􀀀0:22% ATpS lgEntPerm[5] 􀀀0:18% S std 􀀀0:21% Q EntSmpl[6] 􀀀0:18% Q GrPos1 􀀀0:21% ATpS EntPerm[6] 􀀀0:17% S GrPos1 􀀀0:21% ATpS lgavg 􀀀0:17% Q lgGrPos1 􀀀0:21% S DFTA[1] 􀀀0:17% S lgGrPos1 􀀀0:21% Q EntAp[2] 􀀀0:16% PT EntSh 􀀀0:20% ATpS EntPerm[8] 􀀀0:16% ATpS EntSh 􀀀0:20% ATpS EntPerm[8] 􀀀0:15% S lgstd 􀀀0:20% S lgEntAp[2] 􀀀0:14% ATpS lgEntSh 􀀀0:19% S lgsum2 􀀀0:14% S EntAp[2] 􀀀0:18% ATpS EntSort[8] 􀀀0:13% ATpS DFTA[1] 􀀀0:18% Q EntAp[6] 􀀀0:13% S lgmax 􀀀0:18% S GrPos4 􀀀0:13% PT lgEntSh 􀀀0:17% ATpS EntPerm[7] 􀀀0:13% ATpS EntPerm[2] 􀀀0:17% of the prediction quality on the period lengths is analyzed at the end of this section. Features. Now we discuss the features we utilize in the user engagement prediction. Each engagement measure M 2 M can be translated into several scalar features in different ways. Specifically, we consider the following calculation and transformation methods. The total feature. First, we calculate the measure M over the observed time period Tp (e.g., for M = S, it is the total number of sessions over the time period Tp). This is the same as the predicted target, but over the observed time period Tp instead of the forecast one Tf . We denote this feature by Total. The time series. Second, we calculate the measure M over each day of the time period Tp and obtain a daily time series of length jTpj. We denote such feature vector by TSd. Then, for each day t 2 Tp, we calculate the measure M over the time period that starts on the first day of Tp and finishes on the day t. Thus, we obtain a cumulative time series of length jTpj. This feature vector is denoted by TSc. The statistics features. Next, we consider the daily time series TSd = fxtgjTpj t=1 and get basic statistics over it: the minimal (min), maximal (max), average (avg) values, the standard deviation (std), the median (med), the sum (sum), the sum of squares (sum2), and the sum of cubes (sum3) over days. Additionally, we calculate the variation of the time series (i.e., var : = PjTpj􀀀1 t=1 jxt+1 􀀀 xtj) and the num- ber (GrPos� ) of the values of the series not less than the threshold � , which is chosen from among 1; 2; 4; 10; 20; 40; and 100. Thus, we consider here 16 scalar features in total. The derivative features.We calculate the finite differences of the first order FD1 := fxt+1􀀀xtgjTpj􀀀1 t=1 and of the second order FD2 := fxt+2 􀀀2xt+1 +xtgjTpj􀀀2 t=1 over the daily time series TSd. These 2jTpj􀀀3 scalar features are analogs of the �rst and second derivatives in the discrete case. Similar first order finite differences over weekly time series were used in the classiffer [27] and had one of the highest weights in the prediction settings. We also sort in descending order the differences FD1 (FD2) and replace each difference by its index in the original series, obtaining the integer sequence 261 FDrank 1 (FDrank 2 ), which describes the day of the highest difference, the day of the lowest difference, etc. The periodicity features. We apply the discrete Fourier transform (DFT) [29] to the daily time series TSd: Xk = PjTpj t=1 xte􀀀i!k(t􀀀1); !k = 2�k jTpj ; k 2 ZjTpj: Thus, we ob- tain jTpj complex numbers (DFT), then we get their real parts (amplitudes DFTA) and their imaginary parts (phases DFTPh). These 2jTpj scalar features encode periodicity in user engagement and were studied in the paper [8], where their long-term persistence is found to be better than the one of the daily time series. We also sort in descending order the amplitudes DFTA and replace each amplitude by its index in the original series, obtaining the integer se- quences DFTrank A . This sequence describes which frequen- cies !k; k 2 ZjTpj; dominate among others, which one has the lowest amplitude, etc. The entropy features. In order to get the average amount of information contained in daily user engagement series, we calculate entropy for our series TSd in the following ways: (a) the Shannon entropy EntSh as in [26]; (b) the Permuta- tion entropies EntPerm as in [2], and the Sorting entropies EntSort as in [2] of orders n = 2; ::; jTpj 􀀀 1; (c) the Ap- proximate entropies EntAp as in [23, 24], and the Sample entropies EntSmpl as in [24] for m = 2; ::; jTpj 􀀀 2. Thus, we consider 14jTpj + 2 scalar features for each of 6 engagement measures M (i.e., 1188 in total). All features described above are considered both in normal and in loga- rithmic (where applicable) scales in order to better catch the differences between values of different magnitudes. Besides, we consider the day-of-the-week of the first day of the ob- served period Tp as an additional categorical feature DoW in order to better catch the in uence of the position of the period w.r.t. the week cycle. Feature selection. Note that statistics, derivative, pe- riodicity, and entropy features are derivable from the source time series. Therefore, we expect that not all these fea- tures could provide a significant pro�t in prediction quality w.r.t. the source time series. Besides, utilization of the large number of features may lead to overfitting of the model15 and unreasonable consumption of computational resources. Hence, we conduct feature selection by studying the pro�t in prediction quality of each scalar feature w.r.t. a baseline feature set. We consider the set fTotal;TSdg calculated for all user engagement measures M as the baseline feature set (it consists of 6jTpj + 6 = 90 scalar features) in our fea- ture selection procedure. Then, we train our models under the setup described at the beginning of this section for the baseline set and for this set extended by one of the other scalar features. We present the top-20 features w.r.t. the improvement of the baseline prediction quality in terms of RMSE for the number of sessions measure S in Table 216. We see that both the periodicity and entropy features im- prove prediction quality, contrariwise, none of the deriva- tive features shows any significant improvement with p-value � 0:05. We selected the features that showed noticeable and significant improvement in several prediction tasks (i.e., targets and models) or for several user engagement mea- 15In our work, we tried to train models based on all features. However, these models were outperformed by the models based on the time series features solely. 16We removed the identical features, such as avg; ReDFT[0], DFTA[0], etc. from the lists. Table 3: Comparison of feature sets in terms of nRMSE (relative improvement w.r.t. the first col- umn) of prediction of each of the measures M for jTpj = 14 and jTf j = 28. Measures: Target measure All (i.e., M) Features: Total Total;TSd Total Total;TSd Best Target & Model 1 feat. 15 feat. 6 feat. 90 feat. 307 feat. S LR 0:334 􀀀4:92% 􀀀0:12% 􀀀5:11% 􀀀5:58% DT 0:34 􀀀6:38% 􀀀0:32% 􀀀6:48% 􀀀7:59% Q LR 0:37 􀀀2:65% 􀀀0:64% 􀀀3:59% 􀀀5:04% DT 0:411 􀀀4:49% 􀀀1:17% 􀀀4:28% 􀀀5:62% C LR 0:375 􀀀2:52% 􀀀1:12% 􀀀4:21% 􀀀5:4% DT 0:394 􀀀5:31% 􀀀1:9% 􀀀5:23% 􀀀6:49% PT LR 0:372 􀀀2:49% 􀀀1:35% 􀀀4:47% 􀀀4:94% DT 0:388 􀀀4:82% 􀀀1:86% 􀀀5:41% 􀀀6:57% CpQ LR 0:423 0% 􀀀0:12% 􀀀0:08% 􀀀0:46% DT 0:44 􀀀0:03% 􀀀1:18% 􀀀1:02% 􀀀2:5% ATpS LR 0:458 􀀀0:3% 􀀀0:2% 􀀀0:64% 􀀀1:78% DT 0:456 􀀀1:9% 􀀀0:67% 􀀀2:63% 􀀀2:96% sures. These features are: lgTSd[13 􀀀 14], TSc[12 􀀀 13], lgTSc[12 􀀀 14], GrPos� for � = 1; 2; 4; 10; 20; 40; and 100, min, max, lgmax, avg, lgavg, std, med, sum2, lgsum2, sum3, var, ReDFT[1], ImDFT[1], DFTA[1], DFTrank A [1], EntSh, EntPerm[3 􀀀 6], EntSort[2; 3; 4 􀀀 8], EntAp[2􀀀9], EntSmpl[3􀀀6], and DoW. All these features together with the baseline ones (Total and TSd) calculated over all user engagement measures are denoted by Best (307 scalar features in total). Comparison of feature sets and prediction mod- els. We evaluate the feature set Best for both the linear regression model and the decision tree model with respect to the following 4 baselines. The first baseline set consists of one scalar feature Total calculated for the target measure we predict. The second baseline includes 6 scalar features Total calculated for each user engagement measure fromM. The last two baseline sets contain both Total and the daily time series TSd calculated either for the target measure (15 scalar features in total), or for each measure from M (90 scalar features). The nRMSE values for these 5 feature sets, for 2 models, and for 6 different targets are presented in Table 3, all differences are significant with p-value � 0:05. First, we see that our feature set Best outperforms all baselines. Second, we see that the feature set Best provides a higher improvement over the fourth baseline set than the addition of each scalar feature individually, what we did in our feature selection procedure (compare the top of Table 2 and two first rows in Table 3). Therefore, we conclude that the new features carry different information for prediction and are not interchangeable. Third, the decision trees and the linear models show sim- ilar results. On the one hand, the linear regression model sometimes outperforms the former one for several targets, especially for a low number of features. On the other hand, we see that the decision tree model demonstrates a higher rate of improvement with the growth of the size of the fea- ture set than the linear one. In the next experiments we mostly report the results only for the decision trees model, because it demonstrates the best results for loyalty measures (S and ATpS) and it is the state-of-the-art model. Fourth, we see that the number of sessions S is the most predictable target measure in terms of nRMSE for any used feature set (e.g., the use of only one feature Total reduces 262 Figure 3: The dependence of prediction perfor- mance on (a) different Tp = 1; ::; 14 for Tf = 28 and (b) different Tf = 15; ::; 28 for Tp = 14 in terms of nRMSE. the RMSE of the naive average baseline by 66.6%). The other additive target measures Q, C, and PT have almost the same nRMSE values and they are a little bit worse than the ones of S. The ratio target measures CpQ and ATpS have the worst prediction quality w.r.t. the naive average baseline. Moreover, the prediction quality for these two tar- gets is very dificult to improve in comparison with the oth- ers. These observations coincide with our findings on the measures' persistence across time (compare the correlations CorrU;Tp=f in Fig. 2 and the first column of Table 3). Fifth, the most noticeable quality improvement is observed, when we add the daily time series TSd into the feature set (see col. 2 and 4 in the table) for all additive engage- ment measures (S, Q, C, and PT). This effect is expected, because all our novel features are derived from these time series (even the cumulative time series due to additivity of these measures). Note also that the quality for CpQ is im- proved noticeably when the feature set is extended by the other measures or the novel features, while the quality for ATpS also increases with the addition of the daily time series TSd in the feature set. At last, in order to understand the dependence of the pre- diction quality on the UE measures M used in the set of features, we conduct the experiments to evaluate the drop in the performance caused by removal of each measure M 2 M. For all targets, we observe the highest performance drop, when we remove the features calculated for the target mea- sure (e.g., ablation of all S-features decreases the nRMSE for prediction of the number of session S by 0.94%). In the other cases, the ablation of one of the measures connected with the activity aspect of user engagement (Q, C, PT, and CpQ) does not have a noticeable effect on the quality, while the user loyalty measures S and ATpS have a noticeable and significant in uence on the prediction of any studied target (e.g., the ablation of all S-features decreases the nRMSE for prediction of the presence time PT by 0.24%). Finally, Fig. 3 demonstrates the nRMSE values for our best predic- tors learned for different combinations of the length Tp of the observed time period and the length Tf of the forecast one. We see that the closer these lengths are, the better the prediction quality. To sum up, in this section, we studied the problem of pre- diction of the exact value of one of the six user engagement measures. We investigated the dependance of the prediction quality on the utilization of different user engagement mea- sures, on the calculation/transformation techniques of trans- lation them into features, and on the sizes of the observed and the forecast time periods. We conclude that the use of almost all studied features could significantly improve the quality of user engagement prediction, and the dependance of this quality on the parameters of the prediction task agrees with the user engagement analysis provided in Section 4. 6. A/B EXPERIMENTS Experimental setup. In our paper, we consider 32 A/B experiments conducted on real users of the search engine (Yandex) in order to validate our approach of improving the sensitivity of key metrics (FUBPA, see Section 3 for de- tails). Each experiment ran for at least two weeks. The user samples used in the A/B tests are all uniformly randomly selected, and the control and the treatment groups are of ap- proximately the same size (at least, hundreds of thousands of users). Each experiment evaluates a change in one of the main components of the search engine: a change in the rank- ing algorithm, the engine response time, or a change in the user interface. During our A/B experimentation, 23 control experiments (so-called A/A tests) were conducted in order to check the correctness of the experimentation [18, 4]. We considered a commonly used threshold pval = 0:05 for the p-value of the statistical significance test (i.e., of two-sample t-test [7, 27, 6], see Sec. 3 for details). In this section, the per-user metrics based on six user en- gagement measures from M (introduced and analyzed in Sec. 4) are considered as our baseline evaluation criteria. Then, we study the modiffed variants of these metrics ob- tained by means of the future user behavior prediction ap- proach (FUBPA) described in Sec. 3. Namely, for each user participated in an A/B test, we predict her future engage- ment based only on the data observed during the A/B test's period by means of the best decision tree model from Sec. 5, that have been trained in advance17. We consider forecast time periods up to the 28-th day since the experiment start, and, hence, we get several novel modifications for each met- ric M 2 M. In this section, we refer to them as FUBPAX!Y , where X is the considered duration of the experiment in days and Y is the length of the forecast period in days. For instance, FUBPA7!21 is the evaluation metric, which predicts (based on the first 7 days of the experiment) the value of the considered metric M over 21 days since the experiment start, as if it was conducted for 21 days. Additionally, we predicted the metric value at the post-experiment periods. Such met- ric modifications, which are referred to as FUBPApost X!Y , pre- dict the value of M over (Y 􀀀X) days after the end of X-day experiment period. A/A tests. First of all, we check our metrics against 23 A/A tests. An A/A test should be failed about 5% of the time for the p-value threshold 0:05 [18, 4], which is used in our work. Each of the metrics C, PT, CpQ, each of all their FUBPA modifications, and each of the post-experiment modifications (i.e., FUBPApost X!Y ) of ATpS failed one A/A test. All other metrics and all other modifications did not fail any A/A test. Hence, all our metrics and all our modifications have an acceptable rate of A/A fails. Example. We start our study from consideration of an example A/B experiment. It evaluates a treatment, which is an improvement of the ranking algorithm of the search engine. We look at the absence time metric ATpS. The re- sults for this experiment are presented in Fig. 4. We see that the metric does not detect the treatment effect during the first 7 days of the experiment. However, its modification FUBPA4!17 shows a statistically significant difference 􀀀0:4% 17They have been trained on the data obtained during the time period of February and March, 2013, before all our A/B experiments that were conducted from April, 2013 to September, 2014. 263 Figure 4: The Di� and pval of ATpS observed during an example A/B test and of the estimations of ATpS by the FUBPAX!Y with different values of X and Y . (pval � 0:048). Thus, the treatment effect is correctly18 de- tected at the 4-th day since the experiment start by means of FUBPA, while the baseline metric ATpS does not detect the effect till the 8-th day (i.e., one saves 50% of the time for this example). Note that the signs of the actual and the predicted differences Di� are the same. Moreover, it is observed for all FUBPA modifications. The magnitudes of the predicted Di� for FUBPAX!Y are also of the same order. However, their exact values are sometimes far from the ac- tual ones and depend noticeably on the difference observed at the X-day for the actual metric. This finding correlates with the fact that the information from the last days of the observation period has a large impact on the prediction quality (see Sec. 5). Overall detection of the treatment effect. Next, we discuss the results of application of our approach to our six user engagement metrics M in order to improve sensitivity of 32 studied A/B experiments with 14-day duration. Ta- ble 4 summarizes the number of A/B experiments, whose treatment effect is detected (i.e., pval � 0:05) by each of the actual key metrics and each of its FUBPA modifications19. The best result in each column is underlined. Additionally, the number of A/B experiments, whose treatment effect is detected by a considered modification and is not detected by the corresponding baseline metric, is indicated in brack- ets. First, we see that the metrics based on the number of sessions measure S (i.e., the actual per-user metric and its FUBPA modifications) detect the treatment effects in a fewer number of A/B tests than the ones based on the ab- sence time measure ATpS, which is assumed to be a novel alternative to S [9]. Thus, the absence time appears to be more sensitive than the state-of-the-art number of sessions per user. Second, the click-based metrics (i.e., the number of clicks per user C and the number of clicks per query per user CpQ) are more sensitive than the others. This obser- vation is expected and exactly correlates with the \rule of thumb"#5 in [16], which states that clicks are easy to shift, while the number of sessions is hard to change. Third, one could note that, for instance, the actual met- ric C detects the treatment effects in 8 A/B tests, while its modification FUBPApost 14!21 detects the ones in 8 A/B tests also, but one of them is new (i.e., it is not detected by C). It means that a FUBPA modification does not always de- tect the treatment effect in a test, where the baseline metric 18The absence time should decrease for an improvement [9]. 19We present several representative forecast period lengths Y , because the results for all the other are similar to them. Table 4: The number of A/B tests whose treatment effect is detected by each UE metric and its FUBPA modifications. Metric S Q C PT CpQ ATpS Actual@14 day 2 1 8 4 15 4 FUBPA14!15 3 (+1) 1 9 (+1) 4 14 4 FUBPA14!18 3 (+1) 0 9 (+1) 4 14 4 FUBPA14!21 3 (+1) 0 9 (+1) 4 15 (+1) 4 FUBPA14!25 3 (+1) 0 9 (+1) 4 15 4 FUBPA14!28 3 (+1) 0 9 (+1) 3 14 4 FUBPApost 14!21 2 0 8 (+1) 2 8 1 FUBPApost 14!28 2 0 6 3 8 3 detects it. On the one hand, if the change of the service is already detected in such experiments, then we do not need to utilize sensitivity improvement approaches. On the other hand, one could use the FUBPA technique to ensure that the treatment effect will not disappear in the future (see further in this section). Nonetheless, in all those treatment effects that were detected both by a baseline metric and its FUBPA modifications, the sign and the magnitude of the relative difference value Di� are the same. This means that our modifications at least do not harm the decision taken af- ter a successful experiment: to accept the evaluated change of the web service, or to reject it. In total, the baseline met- rics detected the treatment effects in 17 A/B experiments (i.e., pval � 0:05 for at least one of the metrics), while, after applying the FUBPA, we additionally detect the treatment effect in 3 tests. Thereby, the FUBPA metrics increase this number from 53:125% to 62:5%, w.r.t. the number of all A/B tests. Thus, we conclude that our FUBPA technique improves the sensitivity of the studied user engagement met- rics and helps us to detect the treatment effect in more online controlled experiments. Discover of future metric sensitivity. Let us see how the FUBPA detects the actual future sensitivity of a base- line metric. We consider the metric CpQ, which detected the largest number of treatment effects. For these purposes, we consider our 32 A/B experiments as if they ran only for the �rst 7 days of their duration, and, based on these obser- vations, we apply the FUBPA to our baseline metric. Fur- ther, we consider only those A/B tests that have no detected treatment effect w.r.t. the baseline metric over 7 days. For each FUBPA modification and for each A/B test, we look at the FUBPA modification and check if it expects appear- ing of the treatment effect in the future measurement of the baseline metric (i.e., in the future 7 days, 14 days, etc., de- pending on the type of the FUBPA). After that, we check it against the actual effect in the baseline metric measured for the considered A/B test over the 14 days of its duration. Fi- nally, Table 5 summarizes this information over considered A/B tests and the baseline metric CpQ obtaining the statis- tics for each type of the FUBPA modification. Thus, we are interested in how good our approach is at prediction of the detection of the treatment effect in the future (i.e., at the 14-th day of an A/B test) in the case of the non-significant difference observed at the current moment (i.e., at the 7-th day of an A/B test). From Table 5, one sees that almost all FUBPA modifications have very good positive and nega- tive predictive values (i.e., Col. 1 and Col. 4). So, one could conclude that our FUBPA technique can be used in prac- tice as an additional ag to make the decision (e.g., at a half of time period of an A/B test which had not detected a 264 Table 5: The prediction of appearing of the treat- ment effect of 17 A/B tests w.r.t. CpQ. Actual 14-day effect: appears (1) none (16) FUBPA expects: appears none appears none FUBPA7!8 0 (0%) 1 1 15 (94%) FUBPA7!11 0 (0%) 1 1 15 (94%) FUBPA7!14 0 1 0 16 (94%) FUBPA7!21 0 1 0 16 (94%) FUBPA7!28 0 (0%) 1 1 15 (94%) FUBPApost 7!14 0 (0%) 1 1 15 (94%) FUBPApost 7!21 1 (50%) 0 1 15 (100%) FUBPApost 7!28 1 (50%) 0 1 15 (100%) treatment effect at this time): to stop the experiment (and, hence, save some experimentation platform's resources for other A/B tests, e.g., a fraction of an experimentation traf- �c) or to continue it. Control of the current effect persistence. On the contrary, we can consider the opposite situation. Suppose that a key metric detects the treatment effect at the 7-th day of an A/B test, then we are interested to know if this effect remains at the 14-th day or disappears. This case is very important in practice, because the statistically significance difference in the first days of an A/B experiment may be caused by the Primacy and Novelty effects [18, 14, 16], and, as a result, the true treatment effect might be delayed or be absent at all. In order to check the persistence of the treat- ment effect observed at the 7-th day to the 14-th day, we apply our FUBPA technique in the same way as in the pre- vious paragraph, but we consider only those A/B tests that have detected treatment effect w.r.t. to the baseline metric over 7 days. We present these results for predicting the cur- rent treatment effect persistence in Table 6 for the metric CpQ. First, one sees that all types of the FUBPA have a very good precision (i.e., � 93%, see Col. 1), however the negative predictive values (i.e., Col. 4) are very low, i.e. the biggest one is 17%. The recall values (Col. 1 divided by the sum of Col. 1 and Col. 2) for all types of the FUBPA are higher than 57%, and the best recall value 93% is observed for FUBPA7!Y , Y = 8; 11; and 14. So, one could conclude that our FUBPA technique can be used in practice as an additional ag to make the decision after obtaining a treat- ment effect earlier than the predefined experiment duration: to continue the experiment (e.g., to ensure the result against the Primacy and Novelty effects), or to stop the experiment (and, hence, save some experimentation platform's resources for other A/B tests, e.g., a fraction of an experimentation tra�c). Possible extensions and combinations with other techniques. Our future user behavior prediction approach could be applied in many cases and could be combined with other sensitivity improvement techniques. For example, we could combine the FUBPA with the stratification technique, which is proposed in [7]. On the one hand, we could directly apply stratification of users (e.g., w.r.t. the used browser) to the FUBPA modification of the key metric. On the other hand, we could stratify users in the training set of the user behavior predictor and obtain an individual predictor for each stratum (e.g., own predictor for each user preference to a particular browser). After that, during an A/B test, we would apply the proper predictor for each stratum. Besides, a FUBPA modification of a metric could be re- garded as a self-su�cient evaluation metric, because it is Table 6: The persistence of the treatment effect of 15 A/B tests w.r.t. CpQ. Actual 14-day effect: remains (14) disappears (1) FUBPA expects: remains disappears remains disappears FUBPA7!8 13 (93%) 1 1 0 (0%) FUBPA7!11 13 (93%) 1 1 0 (0%) FUBPA7!14 13 (93%) 1 1 0 (0%) FUBPA7!21 14 (93%) 0 1 0 FUBPA7!28 14 (93%) 0 1 0 FUBPApost 7!14 8 (100%) 6 0 1 (14%) FUBPApost 7!21 9 (100%) 5 0 1 (17%) FUBPApost 7!28 9 (100%) 5 0 1 (17%) the average value of a user's feature, which is calculated based on the data observed in the experiment period solely (as it is shown in Sec. 3). The only difference between this feature and a simple measure, like the number of sessions, consists in that the first one is calculated in a sophisticated way by means of the predictor. Therefore, we can apply any previously proposed approaches of sensitivity improve- ment, treating a FUBPA modification as a key metric. For instance, one can (a) remove those users from the treatment group, who were not a�ected by the evaluated change [18, 5]; (b) transform or cap this metric [16]; and (c) consider this metric in two-stage controlled experiments, applying proper variation reduction techniques, that are proposed in [6]. Fi- nally, note, that some techniques (e.g., the point (b)) could be applied during the prediction training stage (i.e., train a future predictor of an improved metric).

7. CONCLUSIONS AND FUTURE WORK

In our work, we consider the problem of prediction of user engagement in terms of exact values of several state-of-the- art UE metrics. To the best of our knowledge, this problem is assumed to be novel and never investigated in existing studies. We performed a deep study of the in uence on the

prediction quality of utilized engagement measures and their transformations. Then, we applied the obtained predictor for the purpose of improving the sensitivity of online con- trolled experiments. We found that this approach increases the number of online controlled experiments with detected a treatment effect. The evaluations on real online experi- ments run at Yandex show that the approach can be used to detect the treatment effect of an A/B test faster (up to 50% of saved time) with the same level of statistical signif- icance. Our technique can also be used in practice as an additional

ag to make the decision during an A/B test to continue or to stop it based on the probabilities of obtain- ing the significant treatment effect. Hence, the results of our study coincide with the emerging needs of modern web companies to run more and fast controlled experiments on a limited number of their users.

Future work. We believe that our sensitivity improve- ment approach will be of interest to researchers and prac- titioner in online controlled experiments. As future work we can, first, extend the set of user engagement measures and other user behavior data for further improvement of the user engagement prediction quality and, hence, the con- trolled experiment sensitivity. Second, we can study more sophisticated prediction models to be applied in our sensi- tivity improvement approach. Finally, it would be interest- ing to combine different sensitivity improvement techniques with our approach.

References

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 AuthorvolumeDate ValuetitletypejournaltitleUrldoinoteyear
2015 FutureUserEngagementPredictionaAlexey Drutsa
Gleb Gusev
Pavel Serdyukov
Future User Engagement Prediction and Its Application to Improve the Sensitivity of Online Experiments10.1145/2736277.27411162015
AuthorAlexey Drutsa +, Gleb Gusev + and Pavel Serdyukov +
doi10.1145/2736277.2741116 +
titleFuture User Engagement Prediction and Its Application to Improve the Sensitivity of Online Experiments +
year2015 +