Estimating reach curves from one data point Georg M. Goerg Google Inc. Last update: November 21, 2014

Abstract

the reach curve rk (g): rk (0) = 0 and rk (G) = R ∈ [0, 100],

Reach curves arise in advertising and media analysis as they relate the number of content impressions to the number of people who have seen it. This is especially important for measuring the effectiveness of an ad on TV or websites (Nielsen, 2009; PricewaterhouseCoopers, 2010). For a mathematical and datadriven analysis, it would be very useful to know the entire reach curve; advertisers, however, often only know its last data point, i.e., the total number of impressions and the total reach. In this work I present a new method to estimate the entire curve using only this last data point.

where G ≥ 0 is the total GRPs and R is total reach. With this information alone one is tempted to use a (1) R linear approximation rk (g) = G g. However, reach curves are not linear and in particular, the marginal reach per GRP would equal average reach per GRP (= 1/frequency); thus (1) alone is not helpful to get a better estimate of marginal GRP (and thus cost) per reach at g = G. While the behavior of rk (g) around g = G is in general unknown, the tangent at g = 0 can be approximated quite well: starting with no exposure, adding an infinitesimally small unit of GRPs (say ) one reaches  · ι % of the population, where ι = ι(k) is the reciprocal of the expected number of impressions needed for the first person to see k impressions. One can lower bound ι by 1/k. For k = 1, the bound is tight, ι = 1; getting an exact expression of ι for k > 1 is ongoing research.1 That is, for small g the reach curve can be approximated with a line through (0, 0) with slope ι:

Furthermore, analytic derivations reveal a surprisingly simple, yet insightful relationship between marginal cost per reach, average cost per impression, and frequency. Thus, advertisers can estimate the cost of an additional reach point by just knowing their total number of impressions, reach, and cost. A comparison of the proposed one-data point method to two competing regression models on TV reach curve data, shows that the proposed methodology performs only slightly poorer than regression fits to a collection of several points along the curve.

rk (g) ≈ g · ι for small g.

1

(1)

Introduction

(2)

Thus, approximately,

∂ rk (g = G) = ι. (3) lim Let k+ reach, rk , be the percentage of the population G→0 ∂g that is exposed to a campaign at least k times. As usual, we measure impressions in gross rating points (GRPs), which is calculated as number of impressions Combining (1) with (3) allows us to estimate a divided by total (target) population multiplied by 100 two-parameter model. (measured in percent). Equipped with a functional form of the reach curve, a Section 2 reviews parametric models for reach curves. variety of quantities of interest can be computed, e.g., Section 3 derives the parameter estimates based on 1 In practice we found that ι = (k + log k)−1 gives good marginal cost per reach or maximum possible reach. 2 Advertisers, however, often only have two points of fits for several k ≥ 1.

1

2.2

Conditional Logit

2

REACH CURVE MODELS

the total GRP and reach. Simulations and compar- 2.1.1 Marginal reach isons to full least squares estimates are presented in Section 4. Finally, Section 5 summarizes the main The derivative of (7) with respect to g equals2 findings and discusses future work. Details on the  α+1 TV reach curve data and analytical derivations can ∂ α β , pk (g) = be found in the Appendix. ∂g β g+β

(8)

with

2

Reach curve models

lim

g→0

∂ ρα rk (g) = . ∂g β

(9)

Let X ≥ 0 be the number of content impressions, e.g., TV shows, websites, or commercials. For a proba- Eq. (9) has three degrees of freedom; since only two bilistic view of reach curves, it is useful to decompose data points are available, one parameters has to be fixed. Given the nested structure of the exponential k+ reach as model, it is natural to set α ≡ 1. P (X ≥ k, reachable) = (4) P (X ≥ k | reachable) · P (reachable) ⇔ rk = pk · ρ,

(5)

2.2

Conditional Logit

As an alternative we propose a logistic regression where ρ is the maximum possible reach, and pk is the probability of being reached k times, given that an inlogit(pk (g)) = β0 + β1 · log g, (10) dividual is indeed reachable. This distinction allows us to model ρ and pk with separate probabilistic mod- where logit(p) = log p , and β0 and β1 are intercept 1−p els. Since reach is usually denoted in percent, we also and slope.3 Using the logit inverse expit(x) = ex = 1+ex use percent for maximum possible reach ρ ∈ [0, 100], 1 , Eq. (10) can be rewritten as 1+e−x while we use proportions for pk ∈ [0, 1]. For further analytical derivations it is necessary to parametrize pk (g). Below we review two functional forms which are parsimonious (2 + 1 parameters), have excellent empirical fits, and lend themselves for simple analytical derivations.

2.1

pk = expit(β0 + β1 log g) = 1 1 + eβ0 · g β1 e−β0 = 1 − −β0 e + g β1 =1−

eβ0 +β1 log g 1 + eβ0 +β1 log g

(11) (12) (13)

Gamma-Mixture

which shows similarity to (7). In fact, identifying β ≡ e−β0 , both models coincide if α = 1 and β1 = 1, Jin et al. (2012) propose a Poisson distribution for the respectively. Again, this can be tested using a impressions g, with an exponential prior distribution two-sided hypothesis test for H0 : β1 = 1. with rate β on the Poisson rate λ. This yields a model of the form The Logit conditional model can also be interpreted β with α ≡ 1, . (6) as the baseline Gamma mixture model pk (g) = 1 − g+β but with transformed GRPs, g˜ = g β1 , in (7). Here β1 can be interpreted as a parameter that measures The exponential prior can be generalized to a Γ(α, β) the efficiency of GRPs: for β1 > 1 GRPs are more efficient than baseline; for β1 = 1 GRPs are spent distribution, which yields according to the baseline model; and for β1 < 1 are   α  not spent as efficiently as expected. For an empirical β rk (g) = ρ 1 − . (7) estimates see Section 4. β+g 2 See

By construction, (6) is nested in (7), which can be tested using a hypothesis test for H0 : α = 1. Google Inc.

Section B.1 for details. deliberately do not use α and β to parametrize intercept and slope, as it is prone to confusion with the (reversed) roles of α and β in (8). 3 We

2

3.1

ρb < 100 case

2.2.1

3

METHODOLOGY

Marginal reach

Solving for β and plugging in to ρ = ρ(β) gives   G·R The derivative of (11) with respect to g equals ρb = min , 100 , (20) G − R/ι ( β1 −1 ∂ g G·R/ι ρ b = G−R/ι , if ρ < 100, (14) pk (g) = eβ0 β1 2. (21) and βb = ι 100−R ∂g (eβ0 + g β1 ) if ρ = 100. G· R , Here limg→0 rk0 (g) falls into three cases:   +∞, if β1 < 1, lim rk (g) = eβρ0 , if β1 = 1, g→0   0, if β1 > 1.

R Condition ρb ≤ 100 is equivalent to G ≤ 100 ι 100−R ; thus GRPs must be less or equal to a constant times the odds ratio of reach.

(15)

Plugging them back into (16) yields expressions for reach solely as a function of R and G (details see Appendix B). According to (21) we consider the two Thus for the logit model one has to assume β1 = 1 to scenarios separately. use the linear approximation of R(g) at g = 0 for 1+ reach.4

3.1

3

Methodology

ρb < 100 case

Here

G·R·g r(g) = Equipped with the two parameter model (G − g) · R/ι + g · G   g β with derivative =ρ ∈ [0, ρ], (16) r(g; ρ, β) = ρ 1 − β+g β+g  2 G·R 1 . r0 (g) = we can use the tangent approximation in (3) and toι (G − g) · R/ι + g · G tal GRP and reach to estimate ρ and β. Note that β ≥ 0 is a saturation parameter and controls how At g = G this evaluates to efficient GRPs are: for small β reach grows quickly  2 with GRPs, for large β it grows slowly. 1 R 0 r (g = G) = . ι G Its derivative equals r0 (g; ρ, β) = ρ

β

(22)

(23)

(24)

an (17) Thus after G GRPs one additional GRP achieves  R 2 additional reach of (approximately) 1ι G . Conversely, to get one additional reach point advertisers 2 which at g = 0 evaluates to r0 (0) = βρ . need approximately ι G additional GRPs. Since R one GRP costs C/G, where C is total cost of the This gives a system of two equations (maximum GRP campaign, the marginal cost of one additional reach and reach & marginal reach at 0) with two unknowns, point is ρ ∈ [0, 100] and β > 0:  2 G C CG ρ ι × =ι . (25) = ι ⇔ ρ = β · ι, (18) R G RR β (β + g)

2,

G R(G + β) =R⇔ρ= . β+G G

(19) Both (24) and (25) give two surprisingly simple, yet insightful identities which can be computed from total GRPs, reach, and cost: first, marginal reach First note that for 1+ reach, ρ ≡ β since ι(k = 1) = 1. per GRP equals a ι times squared average frequency Moreover, ρ in (19) satisfies ρ ≥ 0 for all β, but it (= G R ); secondly, marginal cost per reach equals satisfies ρ ≤ 100 only for β ≤ G · 100−R R . c0 (r = R) = ι · cperp · frequency , (26) 4 ρ

For k > 1, the Logit model with β1 > 1 might become useful as the marginal k+ reach for the very first impression is 0. However, one then has to estimate three parameters again, which is not possible without any further assumptions or more than one data point.

where cperp is average cost per effective reach point (Rossiter and Danaher, 1998), and c0 (r) is the first derivative of cost as a function of reach, c(r). 3

4

ρb = 100 case

60%

G 100−R R

If (20) leads to ρb = 100, then βb must be set to to guarantee that r(G) = R. In this case (see Appendix B) r(g) =

1+ reach

3.2

40%

g·R G + (g − G) · R/100

● ● ● ● ● ● ● ● ●● ● ● ● ●● ●●●●●●● ● ● ● ● ●●●● ● ● ●● ●●●●●●● ●

● ●● ●

0%

● ● ●

50

100

150

GRP

(G + (g − G) · R/100)

2

(28)

80% 60%

1+ reach

G · (1 − R/100) · R

● ●● ● ● ● ● ●● ● ● ●● ●● ● ●●● ● ●●●● ●● ●●●● ●●● ● ● ●●●●● ● ● ●● ● ●●●● ● ● ● ●●● ● ● ●● ● ● ●●● ●● ● ●● ●● ●●●● ● ● ● ● ●● ●● ●

40%

(29)

20% ● ●● ● ● ● ● ● ● ● ● ● ●

0%

Again, this yields a simple identity for marginal reach as 1/frequency times the proportion of the population that has not been reached.

G C 1 1 × = cperp· . (30) R 1 − R/100 G 1 − R/100

Here, marginal cost per reach is average cost per effective reach point times a factor that is inverse proportional to the proportion of the population that has not yet been reached.

Applications

Here we compare the proposed one data point methodology to two competing regression methods using 1+ reach curves from a selection of 50 historical TV campaigns (See Appendix A for details on TV measurement and data processing). Note that for this comparison we do have several (typically hundreds of) data points along a single curve. Regression models use all the data points; the one data point methodology only uses the last data point. We evaluate the competing methods via typical model fitting metrics and ability to estimate marginal reach at g = G.

●● ●●

● ●● ● ●●●● ●● ●●

●● ● ● ●● ● ●● ●● ●

● ● ● ● ● ● ● ● ● ● ● ●

● ●●● ●● ● ● ● ●

0

100

200

300

GRP 100%

1+ reach

Thus marginal cost per reach is

4

●●●● ●● ● ● ● ● ●● ● ●●●● ●●●●

●● ● ● ●● ● ●

(27) 0

which at g = G evaluates to   R R r0 (g = G) = 1− . G 100

c0 (r) =

●●● ●● ● ● ●●

20%

with derivative r0 (g) =

APPLICATIONS

75% 50% 25% 0%

● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

0

100

200

300

400

GRP Gamma regression

Gamma tangent

Logit regression

Figure 1: Sample reach curves with different degrees of Gamma tangent fit (bottom: lowest R2 ; middle: median R2 ; top: highest R2 ). data, and colored lines are the model fits. The dashed horizontal lines represents the estimated ρb for each method. The top shows a particularly good fit of all three models, the middle has a typical (median R2 ) fit, and the bottom panel shows a reach curve where they do not coincide at all.

In fact, none of the proposed models provides a truthful representation of the GRP and reach relationship The two alternative regression methods are i) the in the bottom panel. The raw data has several inGamma mixture model, where ρ and β are estimated stances along the curve where reach flattens out alusing non-linear least squares (set α ≡ 1), and ii) the ready, but suddenly (at about 180 GRPs) it gains Logit model with logistic regression estimates for β0 momentum and reaches again more people at a faster and β1 (no restriction on β1 ). rate. One explanation could be that several creatives Figure 1 shows three reach curves with different de- within the same campaign were shown sequentially; grees of fit: black dots are historical GRP and reach or the marketing strategy might have changed as a Google Inc.

4

4

Marginal reach per GRP

cor(data, fit) 1.00

0.25%

100%

0.20% 95% 0.15%

1.0

0.05%

0.8

85%

500

^ Maximum reach ρ

40

60

80

100

−5

−4

^ Intercept β0 ^ Maximum reach ρ

40

60

80

(a) Gamma mixture model: es-(b) Logit model; red, dotted timated slope at zero ρb/βb (as-line represents β1 ≡ 1. suming α ≡ 1). Blue, dashed line represents the Tangent model assumption of slope one.

0.00%

ρ

R2

100%

0.2%

400

0.1%

300

Maximum GRPs

100.0%

200

99.5%

−6 100

Gamma

0.25

0.10%

90%

99.0%

^ Slope β1

^ ^ /β Slope at 0: ρ

1.2 0.75

0.50

APPLICATIONS

100% 80%

80%

60% 60% 40% 40%

consequence of the flattening out of the curve.

100%

80%

60%

40%

99%

100%

20% 98%

Figure 2: Parameter estimates and inference about marginal reach at 0 GRPs.

Logit regression Estimation

Regression

Tangent

The bottom panel shows that in such cases the one data point methodology will fail since the campaign in fact consists of several sub-campaigns. However, Figure 3: Comparing Gamma and Tangent estimates while they give better fit, even the regression models to – significantly better fitting – Logit model across are not really a good representation of the underly- several metrics. ing GRP to reach dependency. For such campaigns a more general model which allows for multiple sub-campaigns should be used. Figure 2 shows the estimated parameters for the Gamma mixture and the Logit model. Recall that in the Gamma Mixture model the slope at 0 equals α · βρ . Since α ≡ 1 was fixed in the estimation, the estimated slope is simply the ratio ρbb . Similarly to β Fig. 1, the slope estiamtes in Figure 2a show that the one data point assumption (slope = 1) largely overestimates reach for small GRPs.

the Logit regression estimate hits the boundary of ρb = 100% for 28% of the 50 campaigns. The upper-right panel in Figure 3 shows that the Gamma and Logit regression marginal reach estimates coincide very closely, while the Tangent model5 predicts lower marginal reach per GRP (below the 45◦ line), i.e., a flatter curve estimate. This is in agreement with the previous finding that the ι = 1 slope at g = 0 is too optimistic; since the Tangent model always goes through the point (G, R) it must compensate the slope overestimation for small g, with a flatter curve for large G. As a consequence of the minimum restriction in (20), the tangent approximation yields some of the marginal reach per GRP estimates significantly below the 45◦ line.

The Logit regression does not impose a β1 ≡ 1 constraint, but all parameters were estimated from the data. Recall that β1 can be interepreted as an efficiency parameter (see Section 2.2). According to βb1 in Fig. 2b about 80% of these campaigns do not use their GRPs as efficient as the baseline model would Apart from these deviations, the scatterplots show suggest. that the proposed tangent method provides good esFigure 3 compares the models according to several timates and useful inference. measures of fit. The Logit model stands out as a particularly good interpolation method (high R2 and cor(data, f it)). Thus we use this – presumably closest to the truth – model as the baseline (x-axis) and check how the other two fare against it. Both the 5 It is important to note that none of the reach curves have Gamma-mixture as well as the Tangent model in- ρb = 100 in (20); the Tangent model estimates are thus all based fer much lower maximum possible reach. Note that on (24) with ι = 1. Google Inc.

5

A

5

Discussion

In this work we show how to estimate the entire reach curve using only the total GRPs, reach, and cost. While a historical fit might not mimic the behavior for changes in future campaigns, it is very useful to estimate other quantities of interest from a historical campaign, such as maximum possible reach, marginal reach per GRP, or marginal cost per reach. Furthermore, we derive a simple, yet insightful equivalence between marginal cost per reach, average cost per GRP, and frequency.

A

DATA SOURCES

Data Sources

For the model fit comparison in Section 4 we use TV measurement data from the Danish TV market. The raw data is based on a panel provided by TNS Gallup Denmark (TNS Gallup Denmark, 2014).

Applications on a collection of historical TV reach A.1 Panel recruitment curves show that the proposed method has good estimation properties and performs well against regresThis panel consists of 1, 000 households in Denmark, sion methods that use several data points at a time. with approximately 2, 250 panelists. Every household in this panel has a metering box and a remote control to log in when watching TV (including possibility to Acknowledgments add guests). I would like to thank Christoph Best, Harry Case, Penny Chu, Tony Fagan, Jim Koehler, Raimundo Mirisola, Nicolas Remy, Mark Riseley, Sheethal Shobowale, and Xiaojing Wang for insightful discussions and constructive feedback. Special thanks go to Carsten Andreasen for getting the reach curve data.

Panelists have been recruited to be representative of Danish population, and weights are adjusted daily to calibrate panel for in and out-of-tab panelists. With a total population of about 5.6 million,6 one panelist represents about 2, 500 people.

References Jin, Y., Shobowale, S., Koehler, J., and Case, H. A.2 Data selection and preparation (2012). The Incremental Reach and Cost Efficiency of Online Video Ads over TV Ads. Technical reThe metering box records TV viewing among port, Google Inc. panelists, and TV-stations report airing time of Nielsen (2009). The Shifting Media Landscape - In- a campaign spots to TNS. GRPs per spot, and tegrated Measurement in a Multi-Screen World. 1+-reach (in %) amongst others is calculated by TNS. To obtain the reach curves in Section 4 we Technical report, Nielsen. compute cumulative GRPs and 1+-reach for each PricewaterhouseCoopers (2010). Measur- campaign. This data is then used to fit the presented ing the effectiveness of online advertis- models. ing. https://www.pwc.com/en_GX/gx/ entertainment-media/pdf/IAB_SRI_Online_ The data was collected on September 1, 2014 with a Advertising_Effectiveness_v3.pdf. window of ± 2 months. The 50 campaigns we use here are based on a random subsample of the top Rossiter, J. and Danaher, P. (1998). Advanced Media quartile of all campaigns in the dataset. We use the Planning. Number v. 1 in Advanced Media Plan- top quartile to get campaigns with significantly large ning. Springer US. GRPs and reach. TNS Gallup Denmark (2014). TNS TV/Radio Audience. http://tnsgallup.dk/markedsfokus/ tv-radio-audience. 6 Source:

http://denmark.dk/en/quick-facts/facts.

6

B.2

Plug-in β and ρ

B

Analytic derivations

B.1

B

B.2.2

g r(g) = 100 G·(100−R)

The derivative of (7) with respect to G equals ∂ pk (g) = α ∂g

R

=

α−1

β β · 2 g+β (g + β)  α+1 α β = β g+β α = · (1 − pK (g)). β+g

B.2

(31)

= 100 × = 100 ×

Plugging β = β(G, R, k) and ρ = ρ(G, R, k) back into (16) gives G·R g = × β+g G − R/ι

g G·R/ι G−R/ι

(34) +g

g 1 (G−g)·R/ι+g·G G−R/ι

=

G·R × G − R/ι

=

G·R·g (G − g) · R/ι + g · G

g·R , G + R/100 · (g − G)

(44)

(32) When ρ = 1 and β = G 100−R the derivative simplifies R to (33) β (45) r0 (g) = 100 × 2 (β + g)

ρ < 100% case

r(g) = ρ ×

(43) +g

with derivative

Plug-in β and ρ

B.2.1

ρ = 100% case

When ρ = 1 and β = G 100−R then R

Marginal reach



ANALYTIC DERIVATIONS

= 100 × =

G 100−R R +g G 100−R R

(46)

2

G(100 − R)R (G(100 − R) + g · R) G(100 − R)R

(47)

2

2

(G · 100 + R · (g − G)) G · (1 − R/100) · R 2

(G + (g − G) · R/100)

(48) (49)

(35) (36)

and derivative r0 (g) = ρ

2

(β + g) 

= ι  = ι

=



β

1 ι

G·R/ι G−R/ι G·R/ι G−R/ι +

β β+g

(37)

2 g

(38)

G·R/ι G−R/ι (G−g)·R/ι+g·G G−R/ι



2

2 (39)

G·R (G − g) · R/ι + g · G

2 (40)

Finally, the derivative at g = G equals



G·R (G − G) · R/ι + G · G  2 1 R = . ι G

1 r (g = G) = ι 0

2 (41) (42)

7

## Estimating reach curves from one data point - Research at Google

Nov 21, 2014 - Reach curves arise in advertising and media analysis ... curve data, shows that the proposed methodology ..... 1 in Advanced Media Plan- ning.

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1. Estimating the Size of Online Social Networks. Shaozhi Ye*. Google Inc. ... three estimators using widely available OSN functionalities/services. The first ...

Discrete Point Based Signatures and ... - Research at Google
similarity detection and image lookup from a database of indexed images. This .... 3: Signature precision/recall evaluation for the random (3a, 3b) and grid (3c,.

Feb 4, 2008 - malware distribution networks, and the different properties of these networks. ... among these is the exploitation of the web, and the services built upon it, .... Figure 1: A typical Intercation with of drive-by download victim with a

Estimating Factor Shares from Nonstationary Panel Data
The measurement of the sources of economic growth is essential for understand- ing the long-term perspective of any economy. From an empirical viewpoint, the results from any growth-accounting exercise depend both on the functional form that summariz

Functional rarefaction: estimating functional diversity from field data
With the inferential tools that we develop here, researchers will be ... conduct a functional rarefaction analysis of field data. Comparisons with ..... dimensions, it can not be easily represented on graph paper. ... Third, we used PCoA to visualize

AUTOMATIC OPTIMIZATION OF DATA ... - Research at Google
matched training speech corpus to better match target domain utterances. This paper addresses the problem of determining the distribution of perturbation levels ...

Disks for Data Centers - Research at Google
Feb 23, 2016 - 10) Optimized Queuing Management [IOPS] ... center, high availability in the presence of host failures also requires storing data on multiple ... disks to provide durability, they can at best be only part of the solution and should ...

Data-driven network connectivity - Research at Google
not made or distributed for profit or commercial advantage and that copies bear this notice and ..... A distributed routing algorithm for mobile wireless networks.

Street View Motion-from-Structure-from-Motion - Research at Google
augmented point cloud constructed with our framework and demonstrate its practical use in correcting the pose of a street-level image collection. 1. Introduction.

One Billion Word Benchmark for Measuring ... - Research at Google
amount of data involves a large amount of work, and provides a significant barrier to entry for new mod- eling techniques. By choosing one billion words as.

Data Enrichment for Incremental Reach Estimation
12.32. 0.29. Tab. 2.1: Summary statistics for 6 campaigns: web reach in SSP, TV reach in ... marketing questions of interest have to do with aggregates. Using the ...

Estimating diversification rates from phylogenetic ...
Oct 25, 2007 - Biogeography, Princeton University Press ... apply to multi-volume reference works or Elsevier Health Sciences products. For more information ...

Mathematics at - Research at Google