MGMT 8525M Week 5 Sampling Analysis Example

Members Rarely Buy It, but Who Else Does? A Sampling Analysis of a Grocery Chain's Loyalty-Card File, Estimating the Shoppers Outside the Frame Before Cutting a Product Line on Members' Behavior Alone

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Walden University

MGMT 8525M: Big Data Decision Making and Management

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Month Day, Year

Members Rarely Buy It, but Who Else Does? A Sampling Analysis of a Grocery Chain's Loyalty-Card File, Estimating the Shoppers Outside the Frame Before Cutting a Product Line on Members' Behavior Alone

The Decision's Population

Fairhaven Markets, a composite chain of 38 grocery stores in the Ohio River valley, plans to drop its line of prepaid phone cards and money orders from the customer service counter, a line that occupies staff time and counter space. The case for cutting it rests on the loyalty-card file, which covers about 11 million transactions a year and shows that fewer than 1 in 1,000 member baskets include the product. The decision affects everyone who shops at Fairhaven, not only members. The question is whether the members' file can stand for them.

How Records Enter the File

A transaction enters the loyalty file only when the shopper scans a card or enters a phone number. The frame therefore excludes shoppers who pay cash and decline to enroll, occasional visitors, shoppers who forget or refuse to give a number and households that share one card. Fairhaven's point-of-sale data, which records all transactions without identifying shoppers, shows that about 71 percent of transactions carry a loyalty number. The other 29 percent are invisible to the file.

Coverage Against the Trade Area

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Member figures come from enrollment forms and payment records; trade-area figures come from the market study Fairhaven commissioned in 2024, which draws on federal survey estimates for its three counties.

Table 1

Loyalty Members Compared With the Chain's Trade-Area Profile

When a Coverage Gap Becomes Bias

A gap in coverage does not by itself bias an estimate. It does so when whether someone is in the file is related to the quantity being measured. Meng (2018) showed formally that in that situation the size of a dataset does not protect it: the error from such selection does not shrink as the dataset grows, so a very large biased file can produce a precise and confident wrong answer. He called this the big data paradox. The 1936 Literary Digest poll is the classic illustration. The magazine polled millions of people and predicted that Alfred Landon would defeat Franklin Roosevelt; Squire (1988), using a 1937 survey that asked people whether they had taken part in the poll, concluded that both the mailing list and the pattern of returns were biased and together produced the error, and that had everyone polled replied, the magazine would at least have picked the right winner.

For prepaid phone cards and money orders, the link between coverage and purchase is direct. These products are used disproportionately by households without bank accounts or with limited credit, who are more likely to pay cash and less likely to enroll in a loyalty program. The members' purchase rate is therefore likely to understate the overall rate, perhaps substantially. Groves (2006) showed that nonresponse bias depends not on the response rate itself but on whether the propensity to respond is correlated with the variable of interest. The same logic applies to enrollment.

A Rough Estimate of the Missing Shoppers

Fairhaven's point-of-sale data does record the product line's total sales, without shopper identity. Those sales are about 2,900 transactions a month. The loyalty file attributes only about 900 of them to members. The remaining 2,000, about 69 percent of the line's transactions, come from shoppers outside the frame, even though non-members make up only 29 percent of all transactions. Non-member shoppers are thus roughly five times as likely as members to buy the product.

Bounds, Not a Verdict

The finding is a bound on the decision rather than a verdict. The members' file can tell Fairhaven that members rarely buy the product. It cannot tell Fairhaven how many shoppers depend on the counter service, and the point-of-sale totals suggest the number is larger than the file implies, concentrated among households with the fewest alternatives. Before cutting the line, Fairhaven should count the product's total customers, ask counter staff how many are regular, and consider whether the loss of those shoppers' other purchases, which the file also cannot see, outweighs the counter space saved. A simple first step would be a two-week tally at the counter of every prepaid card and money order sold, with a note of whether the buyer also bought groceries. That count, unlike the loyalty file, would include everyone, and it would tell Fairhaven how many shopping trips the counter service brings into its stores.

References

Groves, R. M. (2006). Nonresponse rates and nonresponse bias in household surveys. Public Opinion Quarterly, 70(5), 646-675. https://doi.org/10.1093/poq/nfl033

Meng, X.-L. (2018). Statistical paradises and paradoxes in big data (I): Law of large populations, big data paradox, and the 2016 US presidential election. Annals of Applied Statistics, 12(2), 685-726. https://doi.org/10.1214/18-AOAS1161SF

Squire, P. (1988). Why the 1936 Literary Digest poll failed. Public Opinion Quarterly, 52(1), 125-133. https://doi.org/10.1086/269085