Thanks so much to all who replied to my Allstat question yesterday. It certainly provoked a good discussion.
Here's a list of some of the replies.
*************************************
My question:
Hi folks,
I have been in communication with another statistician re. the definition of the 'null hypothesis' in a one sided test. I'd appreciate your thoughts.
Say we had 2 groups of people, z1 & z2. We want to test if z1 performs better than z2, then my fellow statistician said:
Ho: z1 is equal *or worse* than z2
H1: z1 is better than z2
Shouldn't it instead be:
Ho: z1 is equal to z2
H1: z1 is better than z2
Thank you, in advance, for your opinion.
Kind Regards,
Kim
*********************************
REPLIES
You both are right. Your fellow's version is the real null of interest, but you need test only the boundary condition, which is your version. At least, that's my hypothesis...
------------------
Hi Kim
That's interesting! I thought the same as you, but I just checked the statistics reference book on my desk and to my surprise I think your colleagues are right ("All of Statistics" Wasserman 2005, p151). But I am pretty sure I've seen the form you refer to elsewhere.
----------------------
As an undergrad in australia I was taught to specify the null as your colleague does. The text we used was jerrold h zar's 'biostatistical analysis' (so I assume this habit was picked up from him). I was surprised when I arrived in the uk to see that this was not the convention, but I think it's a little pedantic
------------------------
Both are correct. Yours is more commonly used. You get the same answer from both. The most important thing is to get H1 right, which is the hypothesis you are seeking evidence from the data for. Yes, it is right from what you said.
----------------
Kim,
Both are right. Some people like to add the "or worse" part (as in the first one) just to make the total area of the two hypotheses complete. It makes no difference it the actual test.
-------------------------
Dear Kim
In statistics this step is known as "Identification of Null and Alt hypothesis". In the given example three different statements about z1 can be made. (1) z1 is better than z2 (2) z1 is same as z2 (3) z1 is worse than z2. Now in one sided test (testing 1 in this case) if one is 100% sure that z1 can never be worse than z2 (3 with zero probability) than your identification of null and Alt hypothesis is correct. But, if there is a possibility of z1 is worse than z2 then the other identification of null and Alt hypothesis is correct.
--------------------------------------
Hi,
The hypothesis you mentioned is not incorrect. You can find it in famous book "Statistical Methods" by Cochran and Snedecor.
Your hypothesis is correct only in situation when you know very well the nature of the research, so you know that mean1 could not be less than mean2. So, you should be very cautious.
---------------------------
Hi Kim,
I find it clearest when the alternative hypothesis is the complement of the null hypothesis.
Of course there are many authors that prefer your mode: null is equality while alternative is that z1 is better than z2.
However, if you model your world like this you are implicitly ruling out the possibility that z1 is worse than z2.
-----------------------------
Kim,
I think that in practical terms, in your one-tailed test if the experimental data have z1 performing worse than z2 on whatever measure is being used, there is no point doing a test anyway - you should just accept H0 and go home.
But, technically, it's my understanding that the two hypotheses should together cover all the possible outcomes of the experiment, so H0 in a one-tailed test would include the possibility that z1 is worse than z2.
As in all matters statistical, others may differ!
--------------------------------
You specify a one sided test and that is to test for a difference in one direction. The null hypothesis is what you reject in favour of an alternative. If the latter (H1) is that Z1 is better than Z2, where can 'z1 worse than z2' appear except as part of the null? In a 2 sided test the null is equality and the alternative not equal.
Some people just don't want to accept that treatments may sometimes make patients worse!
----------------------------
Hi Kim,
I think your colleague is correct in as far as my understanding is that when we write H0:mu_1=mu_2 (say) we then infer from the alternative hypothesis whether this means strict equality, or an inequality of the opposite direction to the alternative.
For example if H0:mu_1=mu_2 and H1:mu_1>mu_2 and from our observed data mu_1=0 and mu_2=100, with s=1, we would certainly fail to reject H0, but if we read it to strictly mean equality, then both the null and alternative should be rejected in my opinion.
I believe the convention of always writing mu_1=mu_2 and infering from the alternative whether this is "really" mu_1<=mu_2 comes from the fact that the test will have minimum power if one assumes mu_1=mu_2.
--------------------------------------
I totally agree with your colleague. The NH you propose does not exhaustively cover the possibilities. Anyway, if you look at what outcomes any 1-tailed test is sensitive to you will see that they lie as (s)he describes.
-------------------------------------
The Ho and H1 alternate hypotheses must include all possible conditions. If not, you need a third :)
Your formulation by implication has 3:
z1 > z2
z1 = z2
z1 < z2
Your statistician friend has combined two of them to make only 2 possible states:
z1 = or < z2
z1 > z2
The traditional two-side test is actually
z1 = z2
z1 > or < z2 (z1 more extreme/deviant than z2)
If we take the view that the H0 and H1 are written in response to someone's hoped for/hypothesized result, then a one-side test implies that we care if z1 > z2, and not if z1 = or < z2. I find that such cases are actually relatively uncommon. The user of a new paint formulation cares only if it performs better than the present paint - satisfying the one-side test criteria. The salesman of that paint, if he/she has any serious mental insight, wants to know if the new paint is better OR worse, i.e. if the new paint is different than the old paint - an interest that satisfies the two-side criteria.
-------------------------------------
The two hypotheses (of whatever kind) should strictly be mutually exclusive and exhaustive ... and thus the first set of competing hypotheses should be used. The second set does not cover the possibility space exhaustively and so can be considered incorrect (or incomplete.
--------------------------------------
We formulate
Ho:z1<=z2,
H1:z1>z2 because
we want to prove that z1 performs better than
z2 . If it were impossible/ilogical/nonsense that z1 performs
worse than z2, then your hypothesis will be the best formulation.
--------------------------------------------------
You may leave the list at any time by sending the command
SIGNOFF allstat
to [log in to unmask], leaving the subject line blank.
|
