Your assignment is logically flawed:

Autumn = Summer & Autumn = Winter --it follows that--> Summer = Winter,

Winter = Spring & Summer = Winter --it follows that--> Summer = Spring

but you also write that

Summer > Spring

Summer > Winter

what is logically contradicting the above conjectures. So there is something wrong.

I think the reason for the contradiction is a misconception in claiming what's similar and what not. A test (like SNK) never tells you what is different and what is similar.* You shouldn't take a low p-value as a proof of dissimilarity and you should even more not take a not-so-low p-value as a proof of similarity. That is where the logical flaw enters your system.

It would be more appropriate to make a cluster analysis and see what clusters you get and how well they separate. You might find two major clusters with Summer in one cluster and Autumn,Winter,Spring in the other, with higher values in Summer.

Ify you have the time a bit more specific than the name of the season (day of the year or maybe just the month), you may be better off fitting some (periodic) time-course model that might show a peak in Summer.

Generally, I suggest trying to find a tool/analysis that adresses your scientific question rather than applying a tool you know (like SNK) but that is not ideal for your question. Think outside of the box.

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* it is only a matter of the sample size to get an arbitrary low p-value for any comparison. The p-value thus shows you only that you have enough data to "see" a difference. The test does not help you to decide between "there is a difference" and "there is no difference". Instead, it helps you to decide between "I have enough data to see the difference" and "I don't yet have enough data to see a difference".

Hello,

Honestly, my approach would be the one described in the question and I would not see any problem that there is no group with only "c".

However, the previous answer made me question the suitability of this approach. I really hope Dr. Jochen Wilhelm will participate again in this disscussion.

If for example Autumn = 3, Summer = 4, Winter =2 and Spring =1.

Then Autumn could be statistically equal to Summer and Winter, but Summer is statistically higher than Winter (and Spring). Winter and Spring could be statistically equal. It all seems logical to me, but considering the huge difference in research experience between Dr. Jochen Wilhelm and me, I am now in doubt.

Regards,

There is nothing like "statistically equal".

I am confused. The suggestion I got from a distinguisehd professor in my field of specialty was that:

-if p0.05, then you say a=b (or a is similar to b).

The term "statistically equal" is something I used to describe values which are not mathematically equal, but which show differences that are statistically insignificant. Maybe the term does not exist, I used it only to make a point in this context.

Sorry, but that is nonsense (with my best regards to your professor). Any literature about hypothesis testing will tell you why.

I beg to differ.

if p>0.05, then H0 is accepted, and saying µa=µb (or µa is similar to µb or µa and µb do not differ) represents a simplified translation for "the differences between µa and µb are not statistically significant". So it does not mean that µa and µb are absolutely equal, but that the difference between them is not high enough to be taken into consideration and further actions/approaches could consider µa and µb to be the same.

I found that type of expression in many papers I've read.

Anyway, these are details that might be dependant on the field of research.

Now returning to the question of Barbara Loi,

The labeling she proposed seems suitable to me:

Spring "a"

Summer "b"

Autumn "bc"

Winter "ac".

That means

µSpring is significantly different from µSummer and µAutumn but not from µWInter;

µSummer is significantly different from µSpring and µWinter but not from µAutumn;

µAutumn is statistically different from µSpring but not from µSummer and µWinter;

and finally µWinter is significantly different from µSummer but not from µSpring and µWinter.

Regards

No, that's really not correct. You do not accept H0. If you can't reject it, it means that your data is not sufficient to reject it. It does not mean that H0 is plausible.

You can do an A/B test (Neyman-Pearson), specifying the hypotheses A and B in advance, and ensure a sample size that allows you to accept either of A or B with a specified confidence. This is an acceptance test.

Testing "only" H0 is not an acceptance test. The only purpose of the test is (and can only be) to see if the data is sufficient to reject H0. It should be clear if you see hat even rather large effects won't (usually) lead to a rejection of H0 when the sample size is (too) small.

If you want to be able to conclude that groups are "similar", you have to define a range of differences that would be interpreted as "similar" and you would have to set up an experiment according to A/B testing and accept one of the two alternatives (the difference is or is not within the limits being regarded as "similar") with a reasonable high confidence.

Thank you all for your answers. Now I am more confused than beforeObviously, n.

Obviously, nObviously, now I am more confused than before.

However, I feel to do a little revision of what Dr.Wilhelm r wrote in his first answer. You exactly wrote:

Your assignment is logically flawed:

Autumn = Summer & Autumn = Winter --it follows that--> Summer = Winter,

Winter = Spring & Summer = Winter --it follows that--> Summer = Spring

but you also write that

Summer > Spring

Summer > Winter

what is logically contradicting the above conjectures. So there is something wrong.

It is all correct except that Summer = Winter.

In fact, I wrote Summer > Winter.

Does it change anything in your suggestions?

That's still not consistent:

Logically, if Summer equals Winter and Winter equals Spring, then Summer must equal Spring, too. But you write that Summer > Spring.

The key reasoon why this all doesn't make sense is that you use p>0.05 as indication that two groups are equal (or similar). This conclusion does not follow from p>0.05.

Consider a witness who was aksed to say if she recoginzed a difference between the person at the crime scene and the person being presented in an identity parade. If the only thing you know is that the witness says "I am not able to make out any considerable difference", you have no reason to conclude that the person is the perpetrator. It is *relevant*, for instance, that the witness was at the right time at the crime scene and that she really saw the perpetrator in enough detail. What happens if the lawer of the suspected person asks your witness "Are you blind?" and she answers: "Yes."? -- The large p-value (p>0.05, say) is just like the sentence "I wasn't able to see a difference", but it contains no information about the reasons that lead to this result. Was there too little data? Was the variance too large? Was the model inappropriate? There could be many other reasons.

Dr. Jochen Wilhelm is technically correct and I agree that p>0.05 does not mean that H0 is accepted but rather that there is not enough information to support it. My expression in the last answer was a little unfortunate.

However, I tend to think that the interpretation of p value is highly dependent on the studied subject and on the importance of the conclusion. I agree, in the example of Dr. Wilhelm H0 cannot be accepted solely on the value of p(>0.05), because the conclusion is capital. But let's say I must fill 1000 bottles of 1 L with water. If I fill some of them with only 750 mL, then the difference would be significant (p0.05) and I'll just say all of them are filled with 1L. So in this case I'll translate p>0.05 into "no difference".

Here are a few examples of studies that indicated p>0.05 as "no difference" or at least "no significant difference". I have also encountered translations of p>0.05 as "similar" or "equal", but none of those papers came in handy at this moment.

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That being said, in my opinion the expression

Autumn = Summer & Autumn = Winter but Summer > Winter is not invalid. It just means Autumn has recorded a value somewhere between Summer and Winter and that p>0.05 between autumn and summer as well as between autumn and winter, but on the other hand p

I do know well that there are many papers published that conclude "no difference" from "p>0.05" (either explicitely or implicitely). That doesn't make it correct. Repeating other people's mistakes isn't good science.

It's ok to state that there is no statistically significant difference (or that the difference is not statistically significant). It indicates that one has not had enough data to pinpoint a difference with sufficient confidence. However, that must not be taken to mean that there is no difference, or that the difference is not relevant. There might be a huge, very relevant difference, but because of a large variance and a small sample size the test gave a large p-value. If it is important that there is only littel (irrelevant, negligible...) difference between groups, then one should use a test of equivalence rather then a test of difference (https://en.wikipedia.org/wiki/Equivalence_test).

So my advice to Barbara is to do good science and not to follow bad examples, even if such bad examples are common in her field of research.

The absence of proof is not the proof of absence.