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Say I sit in 2 different restaurants, Cheesecake Factory and Panda Express, that has nothing in common on their menu. I randomly order items from the menu and record how much sugar it has as well as the item is an appetizer or entree or dessert. At each restaurant, I stop ordering 30 minutes after when I order my first item.

At the end of the day I have a data set that may look like the picture below...

I am wondering how to compare the % sugar in my orders from Cheesecake Factory to that in the ones from Panda Express? I understand that the items from the two restaurants are different and may be "incomparable" in a sense, but if I wish to simply compare the average sugar content from items listed on the menu of each restaurant, what should I do?

Menu Sugar Content Compare

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    $\begingroup$ What are you trying to compare? Typical proportions of sugar in items or total amounts of sugar? Regardless of the type of comparison, because your method of selecting items from the two restaurants is not random nor even arguably representative, you cannot validly compare the menus to each other on any basis, so your options seem limited to summaries of your orders. $\endgroup$
    – whuber
    Mar 25, 2014 at 16:16
  • $\begingroup$ Hi whuber, I'm trying to compare overall percent (not amount) sugar content per item on the menu from items listed in 2 menus. The "% sugar" variable runs from 0% to 100%. I do not have any information about the amount of sugar in terms of continuous variables. Doesn't using the % sugar (instead of amount) kind of automatically "standardize" the sugar content (% sugar calculated as grams of sugar in item x divided by grams of the entire item x), so that there may be some justifiable basis of comparison between the items on the 2 menus? $\endgroup$
    – HueSX
    Mar 25, 2014 at 16:24
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    $\begingroup$ You can certainly compare the items you ordered from the menus in any way you wish (although examining only the sugar concentrations seems of rather limited utility because it does not take into account portion sizes). What is in doubt is the validity of drawing any conclusions about the menus themselves based on your data. $\endgroup$
    – whuber
    Mar 25, 2014 at 16:27
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    $\begingroup$ Hi Whuber, I agree with you that using % sugar does not take into account portion sizes, but I think perhaps it won't be too big of a problem because entree from different menus could be put on a common ground (ie I can use "one bite', for example, as the common measurement unit for % sugar in cheesecakes, salads, fried rice, noodles, etc.) What methods do you suggest could be good for doing the comparison? Thank you! $\endgroup$
    – HueSX
    Mar 25, 2014 at 18:02
  • $\begingroup$ I have nothing to suggest because I do not know the purpose of this comparison. $\endgroup$
    – whuber
    Mar 25, 2014 at 18:24

1 Answer 1

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It depends on how sugar content is distributed. If your two restaurants' items have similar distributions of sugar content, you can use a to assess whether one restaurant's items tend to contain more sugar. If the distributions differ, you may need a test, which is less exact.

You could also try on your proportion data to fit a roughly . If sugar content for items from both restaurants can be transformed to fit normal distributions, Welch's $t$-test can produce another approximate estimate of the of any .

If you're more interested in than in falsifying a of zero differences, you can calculate effect size $r$ from an exact Wilcoxon–Mann–Whitney rank sum test or using Wilcox and Tian's (2011) robust, heteroscedastic generalization of Cohen's $d$.

To answer your commented questions about the Wilcoxon–Mann–Whitney U test:

  1. The U test is , so the s don't matter as long as they're similar.
  2. It works with , but with such s, you'll have very little .
  3. Mostly close-to-zero is a distributional characteristic. Again, doesn't matter for the U test as long as it's similarly true of both restaurants. Lots of literal zeros is , which is a problem.
  4. Transformation reduces bias due to violations of a normality assumption, which only applies to a ...but your ability to avoid will be very limited with 19 observations.

Reference
- Wilcox, R. R., & Tian, T. S. (2011). Measuring effect size: A robust heteroscedastic approach for two or more groups. Journal of Applied Statistics, 38(7), 1359–1368.

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  • $\begingroup$ Hi Nick Stauner, thank you for kindly suggesting the measures and methods for comparison. I am wondering if the Mann-Whitney u test would apply if-- 1. the data on % sugar content is in proportion (not a continuous variable); and 2. I ordered 12 different menu items from Cheesecake Factory and only 5 menu items from Panda Express; and 3. % sugar content in most of the menu items is close to zero (like, 3%, 0.8%, etc.)? -- Also, pardon me for this elementary question--will the variable transformation impact the type 1 & 2 errors? $\endgroup$
    – HueSX
    Mar 25, 2014 at 18:06
  • $\begingroup$ Edited to respond. $\endgroup$ Mar 25, 2014 at 20:06
  • $\begingroup$ Hi Nick, thank you. I see the updates. I just thought of this--would a Chi-square Test of Independence work in this case? $\endgroup$
    – HueSX
    Mar 25, 2014 at 21:44
  • $\begingroup$ A $\chi^2$ test would treat sugar content as nominal data, which would waste information. I don't see anything to gain. $\endgroup$ Mar 25, 2014 at 22:39

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