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Assuming there are three methods to kill bugs, and we want to know whether their performance differs. The Tukey's HSD test gives the following grouping results:

$groups
  trt means  M
1   1  1.96  a
2   3  1.88 ab
3   2  1.68  b

If we need suggest some methods to use, can we say method 1 and 3 can both be used as they are either better than or equal to method 2 (both have the grouping letter 'a')?

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    $\begingroup$ What is the actual scientific question here? What are you going to use the treatments for? $\endgroup$ – mdewey Aug 4 '17 at 15:23
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You would not be able to say that.

Since treatment 3 shares a subscript with treatment 2, treatment 3 is not more than the expected standard error greater than treatment 2.

Basically, at an alpha of .05, we would expect a false positive 1 out of every 20 times we collect the data. Values that share a subscript mean this result would occur more than 1 out of every 20 times we collect the data.

You would be able to conclude, however, that treatment 1 is significantly greater than treatment 2.

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