Reach and Exposure-Distribution Models for Media Planning

Reproducible implementations of reach and exposure-frequency models for media planning. Provides exact reach and exposure distributions under random duplication (Sainsbury and Binomial models); the Beta-Binomial accumulation model and its multi-vehicle version of Metheringham (1964) ; the ad hoc duplication and accumulation formulas of Agostini (1961) and Hofmans (1966) ; and the multivariate exposure distribution models of Danaher (1991) (canonical expansion), Leckenby and Rice (1986) (Morgensztern sequential aggregation), Hyo Gyoo Kim (2005, doctoral dissertation, University of Texas at Austin) (canonical sequential aggregation), Heejin Kim (1994, doctoral dissertation, University of Texas at Austin) (conditional Beta distribution), and Cheong (2007, doctoral dissertation, University of Texas at Austin) and Cheong, Leckenby and Eakin (2011) (multivariate Beta-Binomial). The three dissertations have no DOI; their complete references are in the help pages. Also includes an explicitly scoped Negative-Binomial count approximation, the fitting of the Beta-Binomial model to an observed reach, and the evaluation of predicted exposure distributions against observed ones.

mediaPlanR

mediaPlanR estimates how many people a media plan reaches, and how many times, from the audience figures a planner already has.

It brings together, in one package, the classical reach and exposure-distribution models of the media-planning literature (Sainsbury, Binomial, Beta-Binomial, Metheringham, Agostini, Hofmans, CANEX, CBD, CSD, MSAD and MBD) and the tools around them: the fitting of a Beta-Binomial model to an observed reach, a Negative-Binomial approximation for count processes, and the comparison of a model with observed data.

The package is written for students and practitioners of media planning. This document is meant to be read in order: first the ideas, then a plan worked from start to finish, then how to choose a model and how to use your own data. The outputs shown below were produced by the code next to them.

Installation

# install.packages("pak")
pak::pak("majesus/mediaPlanR")

mediaPlanR requires R 4.0 or later. It contains no compiled code, but on Windows pak may stop with Could not find tools necessary to compile a package when it builds from source. You can either install Rtools (the version that matches your R) or install the package without building it:

install.packages("remotes")
remotes::install_github("majesus/mediaPlanR", build = FALSE)

This second route skips the vignette. Then load the package:

library(mediaPlanR)
#> mediaPlanR 2.0.0 | Reach and exposure-distribution models for media planning. See ?mediaPlanR

The ideas in five minutes

A media plan buys insertions (an advertisement in an issue, a spot, a post) in vehicles (a newspaper, a radio station, a website). The planner knows how many people each insertion exposes, the audience, and wants to know how many different people the plan exposes and how often. These terms are used throughout:

Take a population of 1,000,000 people and two vehicles with one insertion each, reaching 300,000 and 400,000 people:

population <- 1000000
audiences  <- c(A = 300000, B = 400000)

sum(audiences)                       # impressions
#> [1] 700000
sum(audiences) / population * 100    # GRP
#> [1] 70

The two audiences add up to 700,000 impressions, but fewer than 700,000 different people are reached, because some are in both audiences. If the two audiences overlap only by chance (random duplication), the share of the population in both is the product of the two shares, and the reach follows directly:

duplicated_people <- prod(audiences / population) * population
duplicated_people                           # people expected in both audiences
#> [1] 120000
sum(audiences) - duplicated_people          # reach, in people
#> [1] 580000

calc_sainsbury() does this calculation and adds the full exposure distribution:

calc_sainsbury(audiences, population)
#> SAINSBURY MODEL
#> ===============
#> Description: random duplication and accumulation, with heterogeneous vehicles
#>
#> HEADLINE METRICS:
#> -----------------
#> Total reach: 58.00% (580000 people)
#>
#> EXPOSURE DISTRIBUTION:
#> ----------------------
#> (Percentage of the population receiving exactly N exposures)
#> 1 exposure: 46.00% (460000 people)
#> 2 exposures: 12.00% (120000 people)
#>
#> CUMULATIVE DISTRIBUTION:
#> -------------------------
#> (Percentage of the population receiving N or more exposures)
#> >= 1 exposure: 58.00% (580000 people)
#> >= 2 exposures: 12.00% (120000 people)
#>
#> SUMMARY STATISTICS:
#> --------------------
#> Average exposures per person reached: 1.21

The result can be read by hand: 42% of the population (0.7 x 0.6) is in neither audience, 46% is in exactly one and 12% (0.3 x 0.4) is in both. Reach is therefore 58%, and the average frequency among the people reached is 700,000 / 580,000 = 1.21.

In practice duplication is observed, for example in an audience survey. If 150,000 people are in both audiences, reach is 300,000 + 400,000 - 150,000 = 550,000 people (55%), lower than under random duplication. Most models in the package exist to use such observed duplication when the plan has several vehicles and several insertions, where no simple formula is available.

A first plan, from start to finish

A plan with several insertions in each of several vehicles needs no duplication data to be evaluated under random duplication. Audience is in people per insertion and the population is in people; the functions reject inconsistent inputs (for example an audience larger than the population) instead of calculating with them. Take three vehicles with 3, 4 and 6 insertions:

audiences  <- c(TV = 250000, Radio = 150000, Digital = 100000)
insertions <- c(3, 4, 6)
population <- 1000000

sainsbury <- calc_sainsbury(audiences, population, insertions)
sainsbury
#> SAINSBURY MODEL
#> ===============
#> Description: random duplication and accumulation, with heterogeneous vehicles
#>
#> HEADLINE METRICS:
#> -----------------
#> Total reach: 88.30% (882965 people)
#>
#> EXPOSURE DISTRIBUTION:
#> ----------------------
#> (Percentage of the population receiving exactly N exposures)
#> 1 exposure: 27.77% (277671 people)
#> 2 exposures: 29.83% (298264 people)
#> 3 exposures: 19.22% (192157 people)
#> 4 exposures: 8.29% (82895 people)
#> 5 exposures: 2.53% (25305 people)
#> 6 exposures: 0.56% (5628 people)
#> 7 exposures: 0.09% (924 people)
#> 8 exposures: 0.01% (112 people)
#> 9 exposures: 0.00% (10 people)
#> 10 exposures: 0.00% (1 people)
#> 11 exposures: 0.00% (0 people)
#> 12 exposures: 0.00% (0 people)
#> 13 exposures: 0.00% (0 people)
#>
#> CUMULATIVE DISTRIBUTION:
#> -------------------------
#> (Percentage of the population receiving N or more exposures)
#> >= 1 exposure: 88.30% (882965 people)
#> >= 2 exposures: 60.53% (605295 people)
#> >= 3 exposures: 30.70% (307031 people)
#> >= 4 exposures: 11.49% (114874 people)
#> >= 5 exposures: 3.20% (31979 people)
#> >= 6 exposures: 0.67% (6675 people)
#> >= 7 exposures: 0.10% (1047 people)
#> >= 8 exposures: 0.01% (123 people)
#> >= 9 exposures: 0.00% (11 people)
#> >= 10 exposures: 0.00% (1 people)
#> >= 11 exposures: 0.00% (0 people)
#> >= 12 exposures: 0.00% (0 people)
#> >= 13 exposures: 0.00% (0 people)
#>
#> SUMMARY STATISTICS:
#> --------------------
#> Average exposures per person reached: 2.21

The result is a list. Reach is available in percent and in people, and the exposure distribution has one value for each number of exposures from one to the total number of insertions. The share of the population that is never exposed is 100 minus the reach, so the whole distribution always sums to 100:

sainsbury$reach$percent
#> [1] 88.29653
round(data.frame(contacts = seq_along(sainsbury$distribution$percent),
                 percent = sainsbury$distribution$percent)[1:5, ], 2)
#>   contacts percent
#> 1        1   27.77
#> 2        2   29.83
#> 3        3   19.22
#> 4        4    8.29
#> 5        5    2.53
100 - sainsbury$reach$percent
#> [1] 11.70347

The cumulative element gives the share exposed at least N times. The value for the effective frequency you choose is the effective reach: with an effective frequency of 2, it is the 60.5% shown for two exposures.

round(data.frame(min_contacts = 1:4,
                 percent = sainsbury$cumulative$percent[1:4],
                 people = sainsbury$cumulative$people[1:4]), 1)
#>   min_contacts percent   people
#> 1            1    88.3 882965.3
#> 2            2    60.5 605294.8
#> 3            3    30.7 307031.1
#> 4            4    11.5 114874.0

The impressions of the plan are the audience times the insertions, added over the vehicles, and the average frequency is those impressions divided by the people reached:

impressions <- sum(audiences * insertions)
impressions
#> [1] 1950000
impressions / sainsbury$reach$people
#> [1] 2.208467

Sainsbury lets each vehicle keep its own audience; Binomial gives every insertion the insertion-weighted average probability. The two models need no duplication data, and side by side they show how much the heterogeneity of the vehicles matters for this plan:

binomial <- calc_binomial(audiences, population, insertions)
c(sainsbury = sainsbury$reach$percent, binomial = binomial$reach$percent)
#> sainsbury  binomial
#>  88.29653  87.90945

Why the choice of model matters

Sainsbury and Binomial assume that every insertion is an independent opportunity, even when two insertions are in the same vehicle. For different vehicles that is the random-duplication hypothesis of the previous section. For repeated insertions in one vehicle it is a strong assumption, because the people who see the first insertion tend to be the same people who see the next one.

Consider one vehicle with a 500,000-person audience in a population of 1,000,000, and five insertions. Under random accumulation each insertion is an independent 50% chance, so almost everybody is reached:

calc_binomial(audiences = 500000, population = 1000000, insertions = 5)$reach$percent
#> [1] 96.875

Now suppose that audience research shows that the cumulative audience after two insertions is only 550,000 (a reach of 55% against 50% after one insertion): the second insertion adds just 50,000 new people. That is evidence of a loyal audience, and calc_beta_binomial() uses it. The Beta-Binomial model lets every person have a different, personal probability of exposure and estimates how those probabilities are spread from the two audience figures, written as reach proportions of the population (R1 after one insertion and R2 after two):

beta_binomial <- calc_beta_binomial(R1 = 0.50, R2 = 0.55, insertions = 5,
                                    population = 1000000)
beta_binomial$reach$percent
#> [1] 60.33654
beta_binomial$distribution$percent
#> [1]  6.009615  4.326923  4.326923  6.009615 39.663462
beta_binomial$parameters$zero_contact_probability
#> [1] 39.66346

Reach falls from 96.9% to 60.3%, and the distribution is polarized: 39.7% of the population is exposed all five times (the last element of the distribution) and 39.7% is never exposed (the zero-exposure percentage shown last). These numbers are illustrative, but the lesson is general: the same vehicles and insertions can give very different reach depending on how audiences repeat and overlap, and the model is how that behavior enters the estimate. The next section explains how to choose.

Choosing a model

The models are organized by the shape of plan they were derived for (Aldás Manzano, 1998) and by the data they need.

Some guidance for the common situations:

• You only have audience figures. Use Sainsbury or Binomial. They are exact under their assumptions, but those assumptions are strong, as the previous section showed: when audiences repeat or overlap more than chance would, they overstate reach.
• You have a vehicle's audience after one and two insertions. Use the Beta-Binomial model for that vehicle.
• You have observed duplication between vehicles. For one insertion per vehicle, the Agostini and Hofmans formulas return reach only. For several insertions, Metheringham's model is the simplest: it replaces the plan by an "average vehicle". CANEX, CBD, CSD, MSAD and MBD model each vehicle separately and combine them in different ways; they are more demanding and their differences are described in vignette("mediaPlanR-intro").
• Your exposures are unbounded counts. The Negative-Binomial functions are scoped to those processes. They are deliberately not meant for finite insertion schedules, where several insertions fall in the same vehicle.

Using your own data

Every model that needs more than audiences takes a set of arguments that mirror a small table. The quickest way to see what a function expects is its example dataset, which is already a list of arguments:

data(csd_example)     # reach and duplication figures of an illustrative scenario
str(csd_example)
#> List of 3
#>  $ vehicles_data    :'data.frame':	3 obs. of  3 variables:
#>   ..$ insertions: num [1:3] 3 2 4
#>   ..$ R1        : num [1:3] 0.35 0.18 0.1
#>   ..$ R2        : num [1:3] 0.44 0.24 0.15
#>  $ duplications     : num [1:3, 1:3] NA 0.07 0.04 0.07 NA 0.02 0.04 0.02 NA
#>  $ aggregation_order: int [1:3] 1 2 3

so that do.call() runs the model on it. Printing a result gives the full report of the model: headline metrics, parameters, the exposure distribution, the cumulative distribution and diagnostics.

csd <- do.call(calc_csd, csd_example)
csd
#> CANONICAL SEQUENTIAL AGGREGATION DISTRIBUTION (CSD)
#> ===================================================
#> Description: Beta-Binomial vehicles aggregated sequentially; the reach of every step comes from the pairwise canonical expansion
#>
#> HEADLINE METRICS:
#> -----------------
#> Total reach: 68.10%
#> Average exposures per person reached: 2.66
#>
#> MODEL PARAMETERS:
#> -----------------
#> Probability of 0 exposures (%): 31.90
#> Total insertions (N): 9
#> Aggregation order: 1 -> 2 -> 3
#> Aggregation rule: custom
#>
#> VEHICLES:
#> ---------
#>  vehicle insertions own_reach_percent aggregation_position
#>        1          3             48.83                    1
#>        2          2             24.00                    2
#>        3          4             20.67                    3
#>
#> AGGREGATION STEPS:
#> ------------------
#>  step added_vehicle target_reach_percent zero_probability
#>     1             2                60.45           0.3955
#>     2             3                68.10           0.3190
#>  random_zero_probability expansion_adjustment duplicated_reach
#>                   0.3889              0.01715           0.1239
#>                   0.3085              0.03416           0.1302
#>
#> CANONICAL CORRELATIONS BETWEEN VEHICLES:
#> ----------------------------------------
#>        V1     V2     V3
#> V1 1.0000 0.0382 0.0349
#> V2 0.0382 1.0000 0.0174
#> V3 0.0349 0.0174 1.0000
#>
#> EXPOSURE DISTRIBUTION:
#> ----------------------
#> (Percentage of the population receiving exactly N exposures)
#> 1 exposure: 16.60%
#> 2 exposures: 16.91%
#> 3 exposures: 19.35%
#> 4 exposures: 7.80%
#> 5 exposures: 4.70%
#> 6 exposures: 1.65%
#> 7 exposures: 0.80%
#> 8 exposures: 0.21%
#> 9 exposures: 0.07%
#>
#> CUMULATIVE DISTRIBUTION:
#> -------------------------
#> (Percentage of the population receiving N or more exposures)
#> >= 1 exposure: 68.10%
#> >= 2 exposures: 51.49%
#> >= 3 exposures: 34.59%
#> >= 4 exposures: 15.23%
#> >= 5 exposures: 7.43%
#> >= 6 exposures: 2.73%
#> >= 7 exposures: 1.08%
#> >= 8 exposures: 0.28%
#> >= 9 exposures: 0.07%
#>
#> DIAGNOSTICS:
#> ------------
#> Probability sum: 1.000000000000 | Smallest probability: 0.000694 | Mean error: 4.44e-16
#> Largest margin error across aggregation steps: 1.11e-16
#> Smallest eigenvalue of the correlation matrix: 0.9561

Example datasets exist for the main models (ratings_example, beta_binomial_example, hofmans_accumulation_example, duplication_example, metheringham_example, canex_example, csd_example, msad_example, mbd_example and bbd_reach_example). All of them are original illustrative scenarios: the package does not distribute the figures of published worked examples, whose redistribution basis is not established.

CSD, MSAD, CBD and MBD

These four models take the same three kinds of input. The reach and duplication figures are proportions of the population (between 0 and 1); only population is a count of people:

• vehicles_data: one row per vehicle with insertions (planned insertions, a whole number), R1 (reach after one insertion, that is, the vehicle's audience) and R2 (cumulative reach after two insertions). R2 must satisfy R1 <= R2 <= 2 * R1 - R1^2, where the upper limit is random accumulation. Use NA for vehicles with a single insertion (calc_canex() accepts the same data frame and the same NA; for it, too, R2 is needed only for vehicles with two or more insertions). Extra columns, such as a vehicle name, must be left out.
• duplications: a symmetric square matrix with the proportion of the population reached by both vehicles with one insertion each. The diagonal is ignored. Rows and columns follow the order of vehicles_data. A duplication cannot exceed the smaller of the two audiences, and cannot be smaller than the sum of the two audiences minus one.
• population: the number of people in the population, so that results are also expressed in people. The default, 1, leaves them as proportions.

vehicles <- data.frame(
  vehicle    = c("TV", "Radio", "Digital"),
  insertions = c(3, 2, 4),
  R1         = c(0.35, 0.18, 0.10),
  R2         = c(0.44, 0.24, 0.15)
)

duplications <- matrix(
  c(NA,   0.08, 0.05,
    0.08, NA,   0.03,
    0.05, 0.03, NA),
  nrow = 3, byrow = TRUE,
  dimnames = list(vehicles$vehicle, vehicles$vehicle)
)

csd <- calc_csd(
  vehicles_data = vehicles[, c("insertions", "R1", "R2")],
  duplications = duplications,
  aggregation_order = "audience_desc",
  population = 8000000
)
csd
#> CANONICAL SEQUENTIAL AGGREGATION DISTRIBUTION (CSD)
#> ===================================================
#> Description: Beta-Binomial vehicles aggregated sequentially; the reach of every step comes from the pairwise canonical expansion
#>
#> HEADLINE METRICS:
#> -----------------
#> Total reach: 65.92% (5273987 people)
#> Average exposures per person reached: 2.75
#>
#> MODEL PARAMETERS:
#> -----------------
#> Probability of 0 exposures (%): 34.08
#> Total insertions (N): 9
#> Aggregation order: 1 -> 2 -> 3
#> Aggregation rule: audience_desc
#>
#> VEHICLES:
#> ---------
#>  vehicle insertions own_reach_percent aggregation_position
#>        1          3             48.83                    1
#>        2          2             24.00                    2
#>        3          4             20.67                    3
#>
#> AGGREGATION STEPS:
#> ------------------
#>  step added_vehicle target_reach_percent zero_probability
#>     1             2                59.49           0.4051
#>     2             3                65.92           0.3408
#>  random_zero_probability expansion_adjustment duplicated_reach
#>                   0.3889              0.04164           0.1334
#>                   0.3085              0.10462           0.1424
#>
#> CANONICAL CORRELATIONS BETWEEN VEHICLES:
#> ----------------------------------------
#>        V1     V2     V3
#> V1 1.0000 0.0928 0.1048
#> V2 0.0928 1.0000 0.1041
#> V3 0.1048 0.1041 1.0000
#>
#> EXPOSURE DISTRIBUTION:
#> ----------------------
#> (Percentage of the population receiving exactly N exposures)
#> 1 exposure: 15.06% (1204889 people)
#> 2 exposures: 15.85% (1268318 people)
#> 3 exposures: 18.73% (1498537 people)
#> 4 exposures: 8.11% (649123 people)
#> 5 exposures: 5.07% (405202 people)
#> 6 exposures: 1.85% (148342 people)
#> 7 exposures: 0.91% (72930 people)
#> 8 exposures: 0.25% (20005 people)
#> 9 exposures: 0.08% (6640 people)
#>
#> CUMULATIVE DISTRIBUTION:
#> -------------------------
#> (Percentage of the population receiving N or more exposures)
#> >= 1 exposure: 65.92% (5273987 people)
#> >= 2 exposures: 50.86% (4069098 people)
#> >= 3 exposures: 35.01% (2800779 people)
#> >= 4 exposures: 16.28% (1302242 people)
#> >= 5 exposures: 8.16% (653119 people)
#> >= 6 exposures: 3.10% (247916 people)
#> >= 7 exposures: 1.24% (99575 people)
#> >= 8 exposures: 0.33% (26645 people)
#> >= 9 exposures: 0.08% (6640 people)
#>
#> DIAGNOSTICS:
#> ------------
#> Probability sum: 1.000000000000 | Smallest probability: 0.00083 | Mean error: 4.44e-16
#> Largest margin error across aggregation steps: 1.11e-16
#> Smallest eigenvalue of the correlation matrix: 0.8915

The report of these four models also lists the vehicles and the aggregation steps. Their distributions have one row per insertion of the plan, so a long one is cut after max_rows exposure levels (30 by default; use print(csd, max_rows = Inf) to list them all). For a four-line summary, use print(csd, full = FALSE).

aggregation_order can be "audience_desc" (largest audience first), "given" (the row order of vehicles_data) or a permutation such as c(2, 1, 3). The order can change the exposure distribution of CSD, MSAD and MBD, so report the one you used (CBD does not depend on it). The other three models take the same arguments; they use different formulas and therefore give different results from the same inputs (CSD and CBD obtain reach from the same canonical expansion, which is why their reach coincides here):

v <- vehicles[, c("insertions", "R1", "R2")]
models <- list(CSD = calc_csd, MSAD = calc_msad, CBD = calc_cbd, MBD = calc_mbd)
data.frame(
  model = names(models),
  reach_percent = sapply(models, function(f) f(v, duplications)$reach$percent),
  average_frequency = sapply(models, function(f) f(v, duplications)$average_frequency),
  row.names = NULL
)
#>   model reach_percent average_frequency
#> 1   CSD      65.92483          2.745551
#> 2  MSAD      64.04448          2.826161
#> 3   CBD      65.92483          2.724670
#> 4   MBD      64.69358          2.797805

The models do not accept every input. When the duplications are incompatible with the model, the function stops with a message that explains why; it does not silently alter your data. For example, calc_mbd() rejects the same plan with duplications of 0.07, 0.04 and 0.02. Each pair overlaps slightly more than chance would, but the Beta-Binomial distribution that the model imputes for three vehicles uses the average audience (0.21) and the average duplication (0.043), and that average is below the overlap expected by chance for the average audience (0.21 x 0.21 = 0.044), which that distribution cannot represent.

The other models

# Random duplication: three vehicles with 2, 1 and 3 insertions
calc_sainsbury(audiences = c(300000, 400000, 200000),
               population = 1000000, insertions = c(2, 1, 3))$reach$percent
#> [1] 84.9472

# Several vehicles, one insertion each: duplication matrix in people
duplication <- matrix(c(NA,     140000, 70000,
                        140000, NA,     90000,
                        70000,  90000,  NA), nrow = 3, byrow = TRUE)
calc_hofmans_duplication(audiences = c(300000, 400000, 200000),
                         population = 1000000,
                         duplication_matrix = duplication)$reach$percent
#> [1] 64.16971

# Several vehicles and insertions, duplication observed in people. The diagonal
# holds the duplication between two insertions of the same vehicle, so it is
# not NA.
calc_metheringham(
  audiences = c(1500000, 800000, 1200000),
  insertions = c(4, 3, 5),
  duplication_matrix = matrix(c(150000, 200000, 180000,
                                200000, 120000, 140000,
                                180000, 140000, 170000), nrow = 3),
  population = 10000000
)$reach$percent
#> [1] 74.34177

Units differ between families, and the functions check them:

Each function's help page (?calc_canex, ?calc_cbd, ...) documents its arguments, its assumptions and the published sources.

Reading the results

Results have a common core, but their shape depends on the family of models:

When you calculate with the objects yourself, two quick coherence checks are that the probabilities add up to one and that the average frequency equals the mean number of exposures per person divided by the reach.

sum(csd$distribution$probability)
#> [1] 1
csd$average_frequency
#> [1] 2.745551
sum(vehicles$insertions * vehicles$R1) / csd$reach$probability
#> [1] 2.745551

The last two numbers coincide because CSD preserves the plan's mean number of exposures per person (the sum of insertions times first-insertion reach). MSAD does too. CBD, MBD and CANEX do not always: CBD changes it through the adjustment of its zero cell, CANEX when it truncates negative probabilities, and MBD can depart from it even when no probability is negative. Each reports the difference in its diagnostics (diagnostics$mean_error, or mean_exposures_expected and mean_exposures_result for CANEX), and the average frequency of a result is always computed from the distribution that it returns.

Comparing a model with observed data

When you have a measured exposure distribution for a schedule, for example from a panel, evaluate_exposure_model() compares it with a model. The table must contain contacts and observed, include the zero-exposure cell, and declare its scale ("count", "probability" or "percent"). The counts below are hypothetical: a panel of 1,000 people classified by the number of exposures to the three-vehicle schedule of the previous sections (0 to 9 exposures).

observed <- data.frame(
  contacts = 0:9,
  observed = c(330, 150, 160, 190, 80, 50, 25, 10, 5, 0)
)

evaluation <- evaluate_exposure_model(
  observed = observed,
  predicted = csd,
  observed_scale = "count"
)
round(unlist(evaluation$summary[c("kim_aer", "kim_ape")]), 3)
#> kim_aer kim_ape
#>   0.016   0.026

kim_aer is Kim's average percentage error in reach (AER), |observed reach - predicted reach| / observed reach, averaged over schedules. kim_ape is Kim's average percentage error in the exposure distribution (APE): the sum, over the exposure levels of one or more, of the absolute differences between observed and predicted shares, divided by the observed reach (both are proposed by Kim, 2005). Both are shown as proportions. evaluation$summary also holds the total variation distance and other diagnostics. These errors are descriptive measures of fit, not statistical tests. The function requires the observed and predicted distributions to cover exactly the same exposure levels. Observed data are always supplied by the analyst; the package never infers them from model inputs.

Limits to keep in mind

• Applying a model is not validating it. The models can be applied to any mix of media, but the evidence on their accuracy comes mainly from magazine and Internet data (the sources cited in each help page). Their accuracy for a particular combination of media, such as television, radio and digital, and for a particular population has to be checked against observed data for that combination (see evaluate_exposure_model()).
• Audience is an opportunity to see, not attention. The models estimate how many people are exposed to a vehicle, not how many notice, remember or are persuaded by the message. Contacts are opportunities of exposure in the unit of the audience figures (an issue, a spot, the visits to a site in a week); they are advertising impressions in the technical sense (served, rendered or viewable) only if the figures measure that event.
• Every model is an approximation. Each one rests on stated assumptions about duplication and about how exposure probabilities vary between people. Where the assumption is wrong, so is the estimate, and the help pages state which assumption each model makes.
• Observed inputs must be coherent. The multivariate models stop when pairwise duplications are impossible given the audiences, or when some three vehicles have no compatible joint exposure probabilities. CANEX can produce small negative probabilities; it sets them to zero and reports in diagnostics how much probability was altered. MBD can do so too when it combines four or more vehicles, and warns when this is possible.
• Size is limited. The number of exposure combinations grows exponentially with vehicles and insertions. CANEX stops with an error on plans that are too large for it, and CBD and MBD are limited to 12 vehicles. CSD, MSAD, CBD and MBD also estimate the size of their tables, or their work, from the insertions before computing, and stop with an explanatory error beyond a budget that can be raised with options (section "Computational limits" of ?mediaPlanR).
• Published values are reproduced up to documented differences. Where the package differs from a published table, vignette("mediaPlanR-intro") and the file system.file("DATA-PROVENANCE.md", package = "mediaPlanR") state by how much and why.

Learn more

• vignette("mediaPlanR-intro") develops each model, its assumptions and its validation against the sources.
• Each function has a help page with its arguments, details, references and examples: start with ?mediaPlanR, ?calc_sainsbury and ?calc_csd.
• citation("mediaPlanR") gives the reference for the package. When you use a specific model, please also cite the original source of that model.
• Questions, errors and suggestions are welcome in the issue tracker.

References

• Agostini, J. M. (1961). How to estimate unduplicated audiences. Journal of Advertising Research, 1(3), 11-14. https://doi.org/10.1080/00218499.1961.12519620
• Aldás Manzano, J. (1998). Modelos de determinación de la cobertura y la distribución de contactos en la planificación de medios publicitarios impresos. Doctoral dissertation, Universidad de Valencia.
• Caffyn, J. M., & Sagovsky, M. (1963). Net audiences of British newspapers: A comparison of the Agostini and Sainsbury methods. Journal of Advertising Research, 3(1), 21-25. https://doi.org/10.1080/00218499.1963.12527134
• Chandon, J.-L. (1976). A comparative study of media exposure models. Unpublished doctoral dissertation, Northwestern University, Evanston, IL.
• Cheong, Y. (2007). Multivariate Beta Binomial Distribution Model as a Web Media Exposure Model. Doctoral dissertation, The University of Texas at Austin. Handle 2152/3215 (University of Texas at Austin repository).
• Cheong, Y., Leckenby, J. D., & Eakin, T. (2011). Evaluating the multivariate beta binomial distribution for estimating magazine and Internet exposure frequency distributions. Journal of Advertising, 40(1), 7-23. https://doi.org/10.2753/JOA0091-3367400101
• Danaher, P. J. (1991). A canonical expansion model for multivariate media exposure distributions: A generalization of the "duplication of viewing law". Journal of Marketing Research, 28(3), 361-367. https://doi.org/10.1177/002224379102800311
• Danaher, P. J. (1992). A Markov-chain model for multivariate magazine-exposure distributions. Journal of Business & Economic Statistics, 10(4), 401-407. https://doi.org/10.1080/07350015.1992.10509915
• Danaher, P. J. (2007). Modeling page views across multiple websites with an application to Internet reach and frequency prediction. Marketing Science, 26(3), 422-437. https://doi.org/10.1287/mksc.1060.0226
• Ehrenberg, A. S. C. (1959). The pattern of consumer purchases. Applied Statistics, 8(1), 26-41. https://doi.org/10.2307/2985810
• Hofmans, P. (1966). Measuring the cumulative net coverage of any combination of media. Journal of Marketing Research, 3(3), 269-278. https://doi.org/10.1177/002224376600300307
• Kim, H. (1994). A conditional beta distribution model for advertising media reach/frequency estimation. Doctoral dissertation, The University of Texas at Austin.
• Kim, H. G. (2005). A Canonical Sequential Aggregation Media Model. Doctoral dissertation, The University of Texas at Austin. Handle 2152/1590 (University of Texas at Austin repository).
• Leckenby, J. D., & Boyd, M. M. (1984). An improved beta binomial reach/frequency model for magazines. Current Issues and Research in Advertising, 7(1), 1-24.
• Leckenby, J. D., & Rice, M. D. (1986). The declining reach phenomenon in exposure distribution models. Journal of Advertising, 15(3), 13-20. https://doi.org/10.1080/00913367.1986.10673014
• Metheringham, R. A. (1964). Measuring the net cumulative coverage of a print campaign. Journal of Advertising Research, 4(4), 23-28. https://doi.org/10.1080/00218499.1964.12519751
• Morgensztern, A. (1970). Fréquentation de la presse, généralisation, effet publicitaire. Étude No. 19, Institut de Recherches et d'Études Publicitaires (IREP), Paris. Listed and attributed to the reach formula by Kim (2005); the original study was not consulted for this package.

Author and license

Manuel J. Sánchez-Franco, Universidad de Sevilla. ORCID 0000-0002-8042-3550

Released under the MIT license.