Introduction

If you ever spent time in the field of marketing analytics, chances are that you have analyzed the existence of a causal impact from a new local TV campaign, a major PR event, or the emergence of a new local competitor. From an analytical standpoint these type of events all have one thing in common: the impact cannot be tracked at the individual customer level and hence we have to analyze the impact from a bird’s eye view using time series analysis at the market level (e.g., DMA, state, etc.). Data science may be changing at a fast pace but this is an old school use-case that is still very relevant no matter what industry you’re in.

Intervention analyses require more judgment than evaluation of randomized test/control studies. When analyzing interventions through time series analysis we typically go through two steps, each of which can involve multiple analytical decisions: 0. Identify the test market(s) where the intervention will happen. This can be based on data (optimal splitting) as well as business reasons and limitations. 1. Find matching control markets for the test market(s) where the event took place using time series matching based on historical data prior to the event (the “pre period”). If the intervention has not happened, check the accuracy of the test/control design via a fake intervention analysis.. 2. Analyze the causal impact of the event by comparing the observed data for the test and control markets following the event (the “post period”), while factoring in differences between the markets prior to the event.

The purpose of this document is to describe a robust approach to intervention analysis based on two key R packages: the CausalImpact package written by Kay Brodersen at Google and the dtw package available in CRAN. In addition, we will introduce an R package called MarketMatching which implements a simple intervention analysis workflow based on these two packages.

A Traditional Approach

For the time series matching step the most straightforward approach is to use the Euclidian distance. However, this approach implicitly over-penalizes instances where relationships between markets are temporarily shifted. Although it is preferable for test and control markets to be aligned consistently, occasional historical shifts should not eliminate viable control market candidates. Or another option is to match based on correlation, but this does not factor in size.

For the inference step, the traditional approach is a “diff in diff” analysis. This is typically a static regression model that evaluates the post-event change in the difference between the test and control markets. However, this assumes that observations are i.i.d. and that the differences between the test and control markets are constant. Both assumptions rarely hold true for time series data.

A More Flexible and Robust Approach

A better approach is to use dynamic time warping to do the time series matching (see [2]) . This technique finds the distance along the warping curve – instead of the raw data – where the warping curve represents the best alignment between two time series within some user-defined constraints. Note that the Euclidian distance is a special case of the warped distance.

For the intervention analysis the CausalImpact package provides an approach that is more flexible and robust than the “diff in diff” model (see [1]). The CausalImpact package constructs a synthetic baseline for the post-intervention period based on a Bayesian structural time series model that incorporates multiple matching control markets as predictors, as well as other features of the time series.

We can summarize this workflow as follows:

1. Pre-screening step: find the best control markets for each market in the dataset using dynamic time warping. The user can define how many matches should be retained. Note that this step merely creates a list of candidates markets; the final markets used for the post-event inference will be decided in the next step.

Note: If you don’t have a set of test markets to match, the MarketMatching can provide suggested test/control market pairs using the suggest_market_splitsoption in the best_matches() function. Also, the test_fake_lift() function provides pseudo prospective power analysis if you’re using the MarketMatching package to create your test design (i.e., not just doing the post inference).

1. Inference step: fit a Bayesian structural time series model that utilizes the control markets identified in step 1 as predictors. Based on this model, create a synthetic control series by producing a counterfactual prediction for the post period assuming that the event did not take place. We can then calculate the difference between the synthetic control and the test market for the post-intervention period – which is the estimated impact of the event – and compare to the posterior interval to gauge uncertainty.

Notes on the Workflow

As mentioned above, the purpose of the dynamic time warping step is to create a list of viable control market candidates. This is not a strictly necessary step as we can select markets directly while building the time series model during step 2. In fact, the CausalImpact package selects the most predictive markets for the structural time series model using spike-and-slab priors (for more information, see the technical details below).

However, when dealing with a large set of candidate control markets it is often prudent to trim the list of markets in advance as opposed to relying solely on the variable selection process. Creating a synthetic control based on markets that have small distances to the test market tends to boost the face-validity of the analysis as similar-sized markets are easily recognized as strong controls through simple line plots.

Ultimately, however, this is a matter of preference and the good news is that the MarketMatching package allows users to decide how many control markets should be included in the pre-screen. The user can also choose whether the pre-screening should be correlation-based or based on time-warped distances, or some mix of the two.

About MarketMatching Package

The MarketMatching package implements the workflow described above by essentially providing an easy-to-use “wrapper” for the dtw and CausalImpact. The function best_matches() finds the best control markets for each market by looping through all viable candidates in a parallel fashion and then ranking by distance and/or correlation. The resulting output object can then be passed to the inference() function which then analyzes the causal impact of an event using the pre-screened control markets.

Hence, the package does not provide any new core functionality but it simplifies the workflow of using dtw and CausalImpact together and provides charts and data that are easy to manipulate. R packages are a great way of implementing and documenting workflows.

Summary of features:

• Minimal inputs required. The only strictly necessary inputs are the name of the test market (for inference), the dates of the pre-period and post-period and, of course, the data.
• Provides a data.frame with the best matches for all markets in the input dataset. The number of matches can be defined by the user.
• Outputs all inference results as objects with intuitive names (e.g., “AbsoluteEffect” and “RelativeEffect”).
• Checks the quality of the input data and eliminates “bad” markets.
• Calculates MAPE and Durbin-Watson for the pre-period. Shows how these statistics change when you alter the prior standard error of the local level term.
• Plots and outputs the actual data for the markets selected during the initial market matching.
• Plots and outputs actual versus predicted values.
• Plots the final local level term.
• Shows the average estimated coefficients for all the markets used in the linear regression component of the structural time series model.
• Allows the user to choose how many markets are sent to the slab-and-prior model.
• All plots are done in ggplot2 and can easily be extracted and manipulated.
• Facilitates pseudo prospective power analysis for geo-based experiments (measuring causal impact of fake interventions).
• Provides suggested optimal test/control market pairs for future studies (in case the test markets have not been identified).

How to Install

## THIS PACKAGE IS IN CRAN.
## If you want to install from Github, use devtools version 1.11.1
## packageurl <- "http://cran.r-project.org/src/contrib/Archive/devtools/devtools_1.11.1.tar.gz"
## install.packages(packageurl, repos=NULL, type="source")
## library(devtools)
## install_github("klarsen1/MarketMatching", build_vignettes=TRUE)
library(MarketMatching)

Examples

The dataset supplied with the package has daily temperature readings for 20 areas (airports) for 2014. The dataset is a stacked time series (panel data) where each row represents a unique combination of date and area. It has three columns: area, date, and the average temperature reading for the day.

This is not the most appropriate dataset to demonstrate intervention inference, as humans cannot affect the weather in the short term (long term impact is a different blog post). We’ll merely use the data to demonstrate the features.

##-----------------------------------------------------------------------
## Find the best matches (default is 5) for each airport time series
##-----------------------------------------------------------------------
library(MarketMatching)
data(weather, package="MarketMatching")
mm <- best_matches(data=weather,
                   id_variable="Area",
                   date_variable="Date",
                   matching_variable="Mean_TemperatureF",
                   parallel=FALSE,
                   warping_limit=1, # warping limit=1
                   dtw_emphasis=1, # rely only on dtw for pre-screening
                   matches=5, # request 5 matches
                   start_match_period="2014-01-01",
                   end_match_period="2014-10-01")

##-----------------------------------------------------------------------
## Or just search for 5 control markets for CPH and SFO
##-----------------------------------------------------------------------
mm_only_cph <- best_matches(data=weather,
                   id_variable="Area",
                   date_variable="Date",
                   markets_to_be_matched=c"CPH", "SFO"),
                   matching_variable="Mean_TemperatureF",
                   parallel=FALSE,
                   warping_limit=1, # warping limit=1
                   dtw_emphasis=1, # rely only on dtw for pre-screening
                   matches=5, # request 5 matches
                   start_match_period="2014-01-01",
                   end_match_period="2014-10-01")


##-----------------------------------------------------------------------
## Analyze causal impact of a made-up weather intervention in Copenhagen
## Since this is weather data it is a not a very meaningful example.
## This is merely to demonstrate the functionality.
##-----------------------------------------------------------------------
results <- MarketMatching::inference(matched_markets = mm,
                                    test_market = "CPH",
                                    end_post_period = "2015-10-01")

##-----------------------------------------------------------------------
## You can also pass specific bsts model arguments (see bsts documentation)
##-----------------------------------------------------------------------
results <- MarketMatching::inference(matched_markets = mm,
                                    test_market = "CPH",
                                    analyze_betas=TRUE,
                                    bsts_modelargs = list(niter=2000, prior.level.sd=0.001),
                                    end_post_period = "2015-10-01")

A view of the best matches data.frame generated by the best_matches() function:

knitr::kable(head(mm$BestMatches))

Plot actual observations for test market (CPH) versus the expectation. It looks like CPH deviated from its expectation during the winter:

results$PlotActualVersusExpected

Plot the cumulative impact. The posterior interval includes zero as expected, which means that the cumulative deviation is likely noise:

results$PlotCumulativeEffect

Although it looks like some of the dips in the point-wise effects toward the end of the post period seem to be truly negative:

results$PlotPointEffect

Store the actual versus predicted values in a data.frame:

pred <- results$Predictions
knitr::kable(head(pred))

Plot the actual data for the test and control markets:

results$PlotActuals

Check the Durbin-Watson statistic (DW), MAPE and largest market coefficient for different values of the local level SE. It looks like it will be hard to get a DW statistic close to 2, although our model may benefit from a higher local level standard error than the default of 0.01:

results$PlotPriorLevelSdAnalysis

Store the average posterior coefficients in a data.frame. STR (Stuttgart) receives the highest weight when predicting the weather in Copenhagen:

coeff <- results$Coefficients
knitr::kable(head(coeff))

Prospective Pseudo Power Curves

In this example, the probability of a causal impact at different levels of fake interventions starting after 2014-10-01 and ending at 2015-10-01. We’re analyzing fake lifts from 0 to 5 percent in 5 steps (default is 10). This will help you evaluate if your choice of test and control markets creates a sufficient model to measure a realistic lift from a future intervention.

##-----------------------------------------------------------------------
## Find the best 5 matches for each airport time series. Matching will
## rely entirely on dynamic time warping (dtw) with a limit of 1
##-----------------------------------------------------------------------

library(MarketMatching)
data(weather, package="MarketMatching")
mm <- best_matches(data=weather,
                   id_variable="Area",
                   date_variable="Date",
                   matching_variable="Mean_TemperatureF",
                   parallel=FALSE,
                   warping_limit=1, # warping limit=1
                   dtw_emphasis=1, # rely only on dtw for pre-screening
                   matches=5, # request 5 matches
                   start_match_period="2014-01-01",
                   end_match_period="2014-10-01")
#' ##-----------------------------------------------------------------------
#' ## Power analysis for a fake intervention ending at 2015-10-01
#' ## The maximum lift analyzed is 10 percent, the minimum is 0 percent
#' ## Since this is weather data it is a not a very meaningful example. 
#' ## This is merely to demonstrate the functionality.
#' ##-----------------------------------------------------------------------
power <- MarketMatching::prospective_power(matched_markets = mm, 
                                      test_market = "CPH", 
                                      end_fake_post_period = "2015-10-01", 
                                      prior_level_sd = 0.002, 
                                      steps=5,
                                      max_fake_lift=0.05)

Inspecting the power curve:

power$ResultsGraph

Getting optimized market pairs (test/control) recommendations

This example shows how to get test/control market pair suggestions from the distances. The package stratifies the markets by size (sum of Y) and the creates pairs based on the correlation of logged values. To invoke this markets_to_matched must be NULL.

Once the optimized pairs have been generated they are passed to the pseudo power function for evaluation. The synthetic parameter in the roll_up_optimal_pairs function determines if the control markets will be aggregated (equal weights in bsts and CausalImpact) or if they’ll be left as individual markets and get separate weights (synthetic control).

##-----------------------------------------------------------------------
## Find all matches for each airport (market) time series. 
##-----------------------------------------------------------------------
library(MarketMatching)
data(weather, package="MarketMatching")
mm <- MarketMatching::best_matches(data=weather,
                   id_variable="Area",
                   date_variable="Date",
                   matching_variable="Mean_TemperatureF",
                   suggest_market_splits=TRUE,
                   parallel=FALSE,
                   warping_limit=1, # warping limit=1
                   dtw_emphasis=0, # rely only on correlation
                   start_match_period="2014-01-01",
                   end_match_period="2014-10-01")
##-----------------------------------------------------------------------
## The file that contains the suggested test/control splits
## The file is sorted from the strongest market pair to the weakest pair.
##-----------------------------------------------------------------------
head(mm$SuggestedTestControlSplits)

##-----------------------------------------------------------------------
## Pass the results to test_fake_lift to get pseudo power curves for the splits
## Not a meaningful example for this data. Just to illustrate.
## Note that the rollup() function will label the test markets "TEST"
##-----------------------------------------------------------------------
rollup <- MarketMatching::roll_up_optimal_pairs(matched_markets = mm, 
                                synthetic=FALSE)

power <- MarketMatching::test_fake_lift(matched_markets = rollup, 
                        test_market = "TEST",
                        end_fake_post_period = "2015-10-01",
                        lift_pattern_type = "constant",
                        steps=20, 
                        max_fake_lift = 0.1)

References

[1] CausalImpact version 1.0.3, Brodersen et al., Annals of Applied Statistics (2015).

[2] The vignette for the dtw package (browseVignettes(“dtw”))

[3] Predicting the Present with Bayesian Structural Time Series, Steven L. Scott and Hal Varian