Do Speciﬁc Text Features Inﬂuence Click Probabilities

in Paid Search Advertising?

Tobias Blask

Institute of Electronic Business Processes, Leuphana University of Lueneburg, Luneburg, Germany

Keywords:

Bayesian Statistics, Bayesian Analysis of Variance, Search Engine Advertising, Sponsored Search, Paid

Search Advertising, Multivariate Testing.

Abstract:

Paid Search Advertisers have only very few options to inﬂuence the user’s decision to click on one of their ads.

The textual content of the creatives seems to be one important inﬂuencing factor beneath its position on the

Search Engine Results Page (SERP) and the perceived relevance of the given ad to the present search query.

In this study we perform a non reactive multivariate test that enables us to evaluate the inﬂuence of speciﬁc

textual signals in Paid Search creatives. A Bayesian Analysis of Variance (BANOVA) is applied to evaluate

the inﬂuence of various text features on click probabilities. We conclude by ﬁnally showing that differences

in the formulation of the textual content can have inﬂuence on the click probability of Paid Search ads.

1 INTRODUCTION

Internet search engines play a key role in the modern

information society. Not only do they serve the in-

formation needs of their users but they also represent

an important source of customer acquisition for a va-

riety of companies (Jansen and Mullen, 2008; Alby

and Funk, 2011). Internet search engine companies

also generate signiﬁcant amounts of revenue through

Paid Search Advertising. While still growing rapidly,

Paid Search Advertising already dominates the online

media spending of companies that advertise on the

internet. Developed in 1998 by Overture, this form

of advertising uses text advertisements and a list of

keywords. The advertiser also usually provides at-

tributes to each of these keywords, but always indi-

cates the amount of money he is willing to pay for a

click on an ad for a speciﬁc keyword (CPC

max

)(Jansen

et al., 2009). Every time a user types a query into

a search engine, personalized result pages are gener-

ated based on the user’s location, search history and

other factors. If ads are available that could prob-

ably satisfy the need of the user, the search engine

displays these ads alongside the organic results. If

more than one advertiser is willing to pay for the dis-

play of an ad, the search engine typically auctions

the position of these ads among all interested parties

through a Generalized Second Price Auction (GSP)

(Jansen, 2011; Varian, 2009). Only the advertiser

that wins the auction by getting a click on its ad is

charged by the search engine. The effective Cost-

Per-Click (CPC

e f f

) is the maximum bid of the ad-

vertiser with the subsequent highest bid plus a small

additional fee. In practice, search engine companies

use a more robust mechanism to maximize their prof-

its by rewarding keyword/ad combinations that have

a high relevance to users (often referred to as the

quality score). Although detailed calculations are not

disclosed, the key metric is claimed to be the his-

toric Click-Through Rate (CT R) where available, oth-

erwise an expected click probability for the speciﬁc

advertiser-ad-keyword combination is used. An in-

teresting issue for advertisers is how to maximize the

probability that a given user will click on one of their

advertisements, ultimately fulﬁlling a deﬁned goal on

their website. In practice there are only limited op-

tions to do so. One is to optimize the relevance of

an advertisement by only choosing keyword / landing

page combinations that provide a suitable offer to the

respective query of a given user. Additionally, adver-

tisers can maximize click probability by inﬂuencing

the position of an ad on the SERP via the CPC

max

and by improving the ad quality. Finally, optimizing

the wording of creatives to communicate advantages

over the competition may help users with their deci-

sion on which ad to click. In the present paper we

concentrate on the impact of various signals in text ad-

vertisements. We analyze a non-reactive multivariate

test in which users are confronted with some deﬁned

variations of ads containing information on trust and

Blask T..

Do Speciﬁc Text Features Inï

Cuence Click Probabilities in Paid Search Advertising?.

DOI: 10.5220/0005048400550062

In Proceedings of the 11th International Conference on e-Business (ICE-B-2014), pages 55-62

ISBN: 978-989-758-043-7

 2014 SCITEPRESS (Science and Technology Publications, Lda.)

pricing that has been differently formulated or com-

pletely omitted. We evaluate the probability that a

user will click on a given Paid Search text ad by us-

ing a Bayesian Analysis of Variance (BANOVA). Fi-

nally, we illustrate that variations in the text of a cre-

ative have signiﬁcant inﬂuence on click probabilities

in Paid Search.

2 PAID SEARCH ADVERTISING

An interesting topic for the current research is eval-

uating the factors that inﬂuence the probability that

a given user will click on a speciﬁc advertiser’s ad.

These factors have been widely studied since Paid

Search Advertising ﬁrst began; however, due to a lack

of opportunity to observe search engine user behav-

ior, complete coverage of all factors inﬂuencing the

click probability is no easy task. Evidence suggests

that one of the most inﬂuential factors is the ad po-

sition within the Paid Search results, which depends

on the CPC

max

and quality score. The quality score

is used by search engines to determine the quality of

an advertisement and is based primarily on the histor-

ical CT R of a given keyword/ad combination. There

is a strong correlation between decreasing ad position

and a decreasing CT R (Richardson et al., 2007; Agar-

wal et al., 2011). In principle, the top positions lead

to high CT Rs. From an advertiser’s perspective, it

is appealing to be able to predict the future CT R of

a given ad or even better to ﬁnd rules for predicting

click probability in advance. As research has pro-

gressed, several models have been developed to ex-

plain the inﬂuence of the position bias on the CT R.

Crasswell, Zoeter and Taylor (Craswell et al.,

2008) present several models for predicting the CT R:

(a) baseline model, (b) mixture model, (c) examina-

tion model, and (d) cascade model . The ﬁndings were

originally based on organic search results, but they

are applicable to Paid Search results as well (Agarwal

et al., 2011). The underlying assumption of the (a)

baseline model is that a user screens every search re-

sult and then decides, which one best ﬁts the query.

As a result, the click probabilities for each individ-

ual search result are identical and independent of its

position. The (b) mixture model extends the baseline

model and divides user behavior into two groups. One

group behaves as described in the baseline model,

while the other group clicks randomly on one of the

ﬁrst search results. The (c) examination model refers

to ﬁndings from eye tracking studies, which state

that, with declining position, the probability of a click

also declines (Joachims et al., 2005; Joachims et al.,

2007). The (d) cascade model is, owing to the high

degree of explanation by click data, one of the most

applied explanation approaches. The basic assump-

tion is that the user scans each search result from top

to bottom, comparing the relevance of each ad with

the relevance of the ad before it. The user continues

scanning the results until the perceived ad relevance

reaches a certain level and the user clicks.

One challenge is to predict the CT R of keywords

or keyword combinations for potential future Paid

Search ads. A proposed solution is aggregating histor-

ical data from similar keywords (Regelson and Fain,

). Here, the CT R is represented as a function of po-

sition, independent of a bid. The resulting developed

models do not focus on a certain advertiser. The same

clustering approach can be applied in optimizing the

search engines’ proﬁt (Dave and Varma, 2010). There

are also models that take into account the quality

score (Gluhovsky, 2010). The General Click Model

model developed by Zhu et al. (Zhu et al., 2010),

focuses on the CT R prediction of long-tail queries,

based on a Bayesian network.To address the the afore-

mentioned position bias, Zhong et al. (Zhong et al.,

2010) incorporate post-click user behavior data from

the clicked ad’s landing page into the click model to

reﬁne the estimation of the perceived user relevance

after clicking on a speciﬁc ad. A similar approach

using Dynamic Bayesian networks can be found in

Chappelle and Zhang (Chapelle and Zhang, 2009).

Several models based on historical click data are lim-

ited in that they lack consideration of a possible user

learning effect. Taking Gauzente’s results as an ex-

ample, it has been shown that past user satisfaction

with Paid Search results inﬂuences current click be-

havior (Gauzente, 2009). In addition to the incorpo-

ration of position data and the perceived relevance of

presented ads, the CT R of an advertisement is also af-

fected by the relationship between organic and Paid

Search results. Listing the results in Paid and organic

search results for one company at the same time leads

to a higher CT R(Yang and Ghose, 2010; Blask et al.,

2011). What has often been overlooked is the inﬂu-

ence that speciﬁc text patterns have on click probabil-

ities.

3 CASE STUDY

It is part of a Paid Search Advertising manager’s daily

routine to test different versions of a speciﬁc ad. In

practice, at least two variations of an ad are tested

against each other in each ad group. One commonly

method is to replace the weaker performing version of

an ad with the stronger variation after enough clicks

are generated to identify which one is performing bet-

ICE-B2014-InternationalConferenceone-Business

ter. Anecdotal evidence and personal experience often

play an important role in this process, and the knowl-

edge gained from tests is often not preserved within

an organization. In this paper we present a method

for multivariate tests based on historical data that is

able to enhance the options of working with unbal-

anced study designs in A/B and multivariate tests.

This ultimately improves advertisers’ ability to rec-

ognize low performing ads sooner than with conven-

tional ANOVA methods. What makes these models

interesting is the ability to take prior knowledge into

account where only sparse data are available.

Figure 1: Variations of Paid Search ads similar the textual

ads used for this study.

For this study, selected elements have been al-

tered in a number of Paid Search text advertisements

for very similar commercial Google search queries.

These queries may lead to a business offer from the

advertiser and an online product purchase, as can be

seen in ﬁg. 1. The advertisers’ product is a major in-

vestment for the average private customer. The data

were generated directly by Google Adwords as part of

the normal conducted from 2012 to 2013. The result-

ing data set contains a large number of Paid Search

key performance indicators (KPI) for the given pe-

riod. All advertisements that had less than 100 im-

pressions in the given period were ﬁltered out. To en-

sure that only the impact of the speciﬁc text alteration

is analyzed and to exclude other factors that would

blur the results (especially the aforementioned strong

position effects), we only analyze the advertisements

that were displayed above the organic search results

and that were part of the described multivariate test.

For this study, we take almost 3 million ad impres-

sions resulting in more than 300,000 clicks and a to-

tal of 1.976 ads into account. The occurrences of the

examined text features in the ads are shown in table 1.

The aggregated CT R for each ad over the whole test

period is used for the analysis. The resulting mean

CT Rs are displayed in ﬁg 3. The nominal variables

Title, Body 1, Body 2, Display-URL of the landing

page for each ad are also included.

In this study we want to predict the metric vari-

able CT R by using the described nominal textual pre-

dictors. As such, ANOVA is a valid method of choice

for us. For the given data the model can be formulated

as written in equation (1) with the predictor variables

presence of trust seal information and presence of

pricing information denoted as⃗x

and⃗x

. β serves as

a deﬂection parameter. β

indicates the baseline value

of the prediction. For example, if x

is at the value of

, a deﬂection of β

is added to the baseline. Ulti-

mately the sum of all deﬂections of β

and β

have to

have a sum of zero for both predictors.

y = β

⃗

⃗x

⃗

⃗x

⃗

1×2

⃗x

1×2

= β

∑

j=1

∑

j=1

∑

j=1

∑

k=1

1×2, j,k

with the constraints

∑

j=1

1 j

= 0 and

∑

k=1

= 0 and

∑

j=1

1×2, j,k

= 0∀k and

∑

k=1

1×2, j,k

= 0∀k

(1)

Even a brief look at the numbers in table 1 makes

it clear that the research design is not very well bal-

anced. Only four ads contain trust seal and concrete

pricing information from the advertiser’s database,

while 345 contain no trust seal information but do

include concrete pricing information. This could

lead to serious computational difﬁculties in tradi-

tional ANOVA, which is the reason that we follow

Kruschke’s approach (Kruschke, 2010). We use a

Bayesian estimation to perform the data analysis us-

ing a hierarchical prior as illustrated in ﬁg. 2 . The

goal of this analysis is to estimate the additive and in-

teractive β values for each level of⃗x.

The following assumptions are made regarding

the hierarchical prior: The observed data y

is as-

sumed to be normally distributed around the predicted

value or central tendency µ

. As deﬁned and illus-

trated by Kruschke (Kruschke, 2010), the equation

above the observed data distribution in ﬁg 2 illustrates

that the predicted value is composed from the base-

line (β

) plus additive deﬂection (β

⃗

⃗x

⃗

⃗x

)

caused by each predictor individually, and interactive

deﬂection (

⃗

1×2

⃗x

1×2

) caused by the combination of

the given predictors. The basic assumptions about the

respective β

can be found in the distributions in the

top level of ﬁg. 2. Here, we indicate that all β

are nor-

mally distributed around zero. The variances of all β

are estimated from the given data. The hyperdistribu-

tions are applied separately to the various predictors

and interactions. This is due to the assumption that

the magnitude of the effect of the predictive variable

⃗x

is probably not informative on the magnitude of the

effect of ⃗x

(Kruschke, 2010).

As previously mentioned, we want to predict CT R

using variations of the nominal predictors ”trust seal

information” and ”pricing information”, denoted

DoSpeciﬁcTextFeaturesInﬂuenceClickProbabilitiesinPaidSearchAdvertising?

Table 1: Occurences of text features in the dataset (trust information(x1) and pricing information(x2)).

no trust info(X1A1) test winner( X 1A2 trust seal(X1A3)

no pricing( X 2B1) 789 99 330

no deposit( X 2B2) 20 35 91

save x %(X2B3) 91 84 101

real price(X2B4) 39 306 4

Figure 2: Hierarchical dependencies for two-way BANOVA (Kruschke, 2010).

as ⃗x

and ⃗x

. Both variables have several levels.

⃗x

includes ”no trust information given” (x

1A1

”unspeciﬁed test winner information given” (x

1A2

)

and ”concrete trust seal and test winner informa-

tion given”(x

1A3

). ⃗x

includes ”no pricing informa-

tion”(x

2B1

), ”no deposit”(x

2B2

), ”percentage of sav-

ings given”(x

2B3

) and ”concrete pricing informa-

tion from database given”(x

2B4

). Although that we

have an unbalanced study design and very few obser-

vations for one of the cases, we can see from ﬁg 3 that

ads with different contents gain signiﬁcantly varying

CT Rs.

The results of the Bayesian analysis concerning

the effects of the text features are shown in ﬁg 4. The

top left histogram shows that the baseline (β

) for the

given combinations of text features is at 0.114. This

is quite high in terms of average Click-Through Rates

in Paid Search Advertising in general, at least for such

a great number of queries as we observed within the

test period in the context of the given campaigns. This

could be a good hint for the fact that the campaigns

have already been very well organized and optimized

in terms of relevance to the speciﬁc queries that lead

to the display of ads of the given advertiser on Search

Engine Results Pages. Which inﬂuences selected text

features have in such an environment can be seen in

the remaining histograms in ﬁg. 4. Each histogram

illustrates deﬂections from the baseline for any given

feature combination. The third histogram in the ﬁrst

row for example indicates that 95% of the most cred-

ible values that have to be subtracted from the base-

line to describe the effect of the text feature ”unspec-

iﬁed test winner information” fall in the area between

0.0282 and 0.0164 (β1

) with the most probable value

at 0.0213. What is really interesting and makes this

kind of analysis so helpful in the case of unbalanced

study design is the additional information concerning

the 95% HDI (Highest Density Interval) for each es-

timation. This interval illustrates the area in which

95% of all credible parameter values for the respective

level of a variable are situated. This becomes espe-

cially important when it comes to levels of a variable

with only very few observed data points where the in-

group variance is estimated with the help of the prior

ICE-B2014-InternationalConferenceone-Business

0.04 0.06 0.08 0.10 0.12 0.14 0.16

TrustLevels

Click-Through Rate

A1 A2 A3

PricingLevels

Figure 3: Mean CT R values for ads with different text feature combinations.

knowledge from other levels within the same predic-

tive variable.

We estimate the effects for each of the groups

but we are also interested in the answer to the ques-

tion whether the groups are credibly different from

each other. In typical A/B test scenarios for exam-

ple this can be examined by applying an NHST t-test

or a Bayesian Parameter Estimation of t-distributions

for the comparison of two groups (Kruschke, 2012;

Blask, 2013). For the case of more than two predictive

variables this can also be done via contrast analysis in

Multifactor Analysis of Variance (MANOVA).

What we additionally want to investigate in this

study is the overall effect of having information about

a trust seal in the Paid Search Ads of the advertiser

and the effects of various levels of pricing informa-

tion. One answer to this question comes from the

analysis whether there is a credible difference in click

probabilities between ads that do not contain any spe-

ciﬁc trust information (x

1A1

) and and those that in-

clude the text feature (x

1A2

and x

1A3

). The histograms

in ﬁg 5 that the effect of having unspeciﬁed test win-

ner information in the ad does not help the advertiser

to gain a higher CTR. In fact the analysis does reveal

that having no trust information is credibly better than

the announcement of an unproven test winner state-

ment. Including a trust seal information into the ad

does not have such a negative effect. In fact about

two thirds of the credible values for the effect, includ-

ing the most probable value, indicate that this feature

may slightly help the advertiser. What is also true is

that the zero value is included in the 95% HDI what

makes it very probable that there is no credible im-

provement in the advertisers performance by includ-

ing this feature. What we can derive from the analysis

is the fact that it makes credibly more sense to include

the proven trust seal into the ad compared to unproven

test winner statement as can be seen in the right his-

togram in ﬁg 5. So, in terms of trust seal information

it becomes quite clear what the better choice might be

for the given company. It makes no mistake by taking

concrete and proven information on trust seals into

their ads. What they should not expect is a signiﬁcant

boost in terms of click probability.

What is not so clear until now is the question

which level of pricing information ﬁnally leads to the

best click probability. What we can see is that there

seem to exist two differently performing clusters. On

the one hand there are the ads that contain a concrete

pricing information from the database of the given ad-

vertiser (x

2B4

) and ads that contain a speciﬁc discount

in percent as text feature (x

2B3

) and on the other hand

there are the ads with no pricing information at all

2B1

) and those advertising that no deposit has to be

made (x

2B2

). These two groups are credibly different

from each other as can be seen in the histograms in ﬁg

DoSpeciﬁcTextFeaturesInﬂuenceClickProbabilitiesinPaidSearchAdvertising?

Baseline

0.105 0.115 0.125

mean 0.114

95% HDI

0.108 0.12

x1: A1

-0.010 0.000 0.010 0.020

mean 0.00864

95% HDI

0.000902 0.0164

x1: A2

-0.035 -0.025 -0.015 -0.005

mean -0.0213

95% HDI

-0.0282 -0.0148

x1: A3

-0.01 0.00 0.01 0.02 0.03

mean 0.0126

95% HDI

0.00291 0.0224

x2: B1

.030 -0.020 -0.010 0.000

mean -0.014

95% HDI

-0.0209 -0.00685

x2: B2

-0.04 -0.03 -0.02 -0.01 0.00

mean -0.0167

95% HDI

-0.0258 -0.00785

x2: B3

-0.010 0.000 0.010

mean 0.00177

95% HDI

-0.00612 0.00938

x2: B4

0.00 0.02 0.04 0.06

mean 0.0289

95% HDI

0.0134 0.044

x1: A1 , x2: B1

E12

1,1

-0.03 -0.02 -0.01 0.00

mean -0.0128

95% HDI

-0.0214 -0.00401

x1: A2 , x2: B1

E12

2,1

0.01 0.02 0.03 0.04

mean 0.0233

95% HDI

0.0142 0.0326

x1: A3 , x2: B1

E12

3,1

-0.03 -0.02 -0.01 0.00 0.01

mean -0.0106

95% HDI

-0.0212 0.000176

x1: A1 , x2: B2

E12

1,2

-0.01 0.01 0.03 0.05

mean 0.0289

95% HDI

0.0152 0.043

x1: A2 , x2: B2

E12

2,2

-0.06 -0.05 -0.04 -0.03 -0.02

mean -0.0403

95% HDI

-0.05 -0.0302

x1: A3 , x2: B2

E12

3,2

-0.02 0.00 0.01 0.02 0.03 0.04

mean 0.0114

95% HDI

-0.00166 0.0242

x1: A1 , x2: B3

E12

1,3

-0.04 -0.03 -0.02 -0.01 0.00

mean -0.0221

95% HDI

-0.0324 -0.0119

x1: A2 , x2: B3

E12

2,3

-0.01 0.00 0.01 0.02 0.03

mean 0.00995

95% HDI

0.000302 0.0194

x1: A3 , x2: B3

E12

3,3

-0.01 0.01 0.02 0.03 0.04

mean 0.0121

95% HDI

0.000242 0.0247

x1: A1 , x2: B4

E12

1,4

-0.02 0.00 0.02 0.04

mean 0.00589

95% HDI

-0.0118 0.0242

x1: A2 , x2: B4

E12

2,4

-0.03 -0.01 0.01 0.03

mean 0.00704

95% HDI

-0.00882 0.0224

x1: A3 , x2: B4

E12

3,4

-0.08 -0.04 0.00 0.04

mean -0.0129

95% HDI

-0.0393 0.0135

Figure 4: Beta value posterior distributions for each variation given the data (x1 = trust seal informaton (x1:A1 = no

trust info, x1:A2 = unspeciﬁed test winner information, x1:A3 = concrete trust seal and test winner information), x2 = pricing

information (x2:B1 = no pricing, x2:B2 = no deposit, x2:B3 = save x%, x2:B4 = concrete pricing information from database)).

X1 Contrast: A1vA2

1 A1 + -1 A2

0.00 0.01 0.02 0.03 0.04 0.05

mean 0.0298

0% < 0 < 100%

95% HDI

0.0194 0.0407

X1 Contrast: A1vA3

1 A1 + -1 A3

-0.04 -0.02 0.00 0.02 0.04

mean -0.00404

69.4% < 0 < 30.6%

95% HDI

-0.02 0.0127

X1 Contrast: A2vA3

1 A2 + -1 A3

-0.06 -0.04 -0.02 0.00

mean -0.0338

100% < 0 < 0%

95% HDI

-0.0488 -0.0194

Figure 5: Contrasts for various levels of trust seal information (X1).

X2 Contrast: B1vB2

1 B1 + -1 B2

-0.02 -0.01 0.00 0.01 0.02

mean 0.00282

30.7% < 0 < 69.3%

95% HDI

-0.00807 0.0139

X2 Contrast: B1vB3

1 B1 + -1 B3

-0.03 -0.02 -0.01 0.00

mean -0.0157

100% < 0 < 0%

95% HDI

-0.0243 -0.00695

X2 Contrast: B1vB4

1 B1 + -1 B4

-0.08 -0.06 -0.04 -0.02 0.00

mean -0.0427

100% < 0 < 0%

95% HDI

-0.0634 -0.0227

X2 Contrast: B2vB3

1 B2 + -1 B3

-0.04 -0.03 -0.02 -0.01 0.00 0.01

mean -0.0185

99.9% < 0 < 0.1%

95% HDI

-0.0307 -0.00668

X2 Contrast: B2vB4

1 B2 + -1 B4

-0.10 -0.08 -0.06 -0.04 -0.02 0.00

mean -0.0455

100% < 0 < 0%

95% HDI

-0.0674 -0.0238

X2 Contrast: B3vB4

1 B3 + -1 B4

-0.06 -0.04 -0.02 0.00 0.02

mean -0.027

99.4% < 0 < 0.6%

95% HDI

-0.0479 -0.00629

Figure 6: contrasts for various levels of pricing information (X2)

ICE-B2014-InternationalConferenceone-Business

6. In detail, everything, including no pricing informa-

tion at all (x

2B1

), seems to be be better than advertising

”no deposit” (x

2B2

). The next best text feature in terms

of pricing is to give an exact value for the percentage

that a user can save on the advertiser’s website (x

2B3

This feature is performing credibly better than those

mentioned above. What is the best way to communi-

cate pricing in Paid Search Ads - given the data - is

to provide exact pricing information from the adver-

tiser’s database (x

2B4

). In fact it is credibly superior to

any other feature in terms of pricing communication,

given the observed data.

4 CONCLUSIONS AND

OUTLOOK

What we applied in this paper offers a valid way to

evaluate text features and other nominal predictive

variables where tests are an essential part of the daily

business. In terms of substantive issues it is the hard

facts that the potential customers are looking for when

they research in a search engine. The more spe-

ciﬁc information on pricing is provided in an ad

- the better is the chance of winning the customers

click. Building up trust is one good feature for an ad-

vertiser to support this effect or even substitute parts

of this positive effect if they do not have competi-

tive prices or special rebates available. In this spe-

ciﬁc case this has been achieved by communicating

the existence of a credible trust seal in the ad-copy.

What we did not asses in this research but would ﬁnd

interesting for an ongoing investigation is the ques-

tion whether these ﬁndings have additional impact on

the conversion probability on the advertiser’s landing-

page as well. Applying Bayesian ANOVA to mul-

tivariate tests in Online Advertising, especially Paid

Search Advertising, has various advantages compared

to applying conventional Analysis of Variance. This

is especially true for unbalanced data like the present

one. One obvious limitation to the results is that they

should probably only be true for advertisers with com-

petitive prices. Additionally this test should be re-

peated for a number of other advertisers from various

industries to answer the question whether these obser-

vations can be generalized.

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