A picture is worth a thousand words or in the market research world, a thousand numbers. In fact, the map below is well over a thousand numbers representing a composite view of how these grocery retailers fallout based on the strength of their customer relationships. We used a series of items, based on our own product development work, that measure the strength of a customer relationship to a corporate (or product) brand. The items cover two latent constructs or dimensions - personal and functional - critical to establishing strong customer relationships - those that produce repeat behaviors.
(By the way, this is a national data set and we've only just begun to "look under the hood" but stay tuned for more from grocery, clothing retail and cable/telecomm.)
Those retailers doing the best are in the upper right hand quadrant, their customers have a personal and functional connection to the banner level brand that delivers greater share of wallet, more trips, more store brand purchase and more recommend behavior. Conversely, those in the bottom left are falling short, relative to the competition, on both dimensions. Admittedly, much of grocery choice is driven by convenience and proximity. So while the Piggly Wiggly customers may be somewhat "trapped" by geography and limited choice there is a long-term threat of leaving completely with immediate and medium term loss from share and trips that are already going elsewhere - despite any convenience factor working in their factor. 
The numbers beside a few of the stores (e.g. Walmart with 68 and Publix at 82) are overall customer satisfaction numbers from the ACSI index. Few quick observations:
1) If these stores were ranked by satisfaction alone you'd have Publix at one end (82), Walmart at the other end as the laggard (68) and everybody else lumped right in the middle (around 75).
2) If nothing else, these relationship dimensions do a better job of discriminating among the retailers.
3) As it turns out, just creating some "spread" on the map is not their only value. The measures prove to be much more predictive of customer behavior and therefore, moving into the upper right matters for the only reason that counts, it will drive traffic, share and profit.
4) Intuitively, does anybody think Walmart is the laggard when it comes to key customer behaviors? And while they do appear to be the laggard on satisfaction, one has to ask, who cares?