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### Re: cross-over study: 2 sample t-test vs paired t-test

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```Hi Frank,

You have two groups of subjects (maybe patients) who both receive
experimental conditions (maybe treatments) A and B. The difference between
the two groups is the order of the conditions. Group AB has condition A in
period 1 and B in period 2, while group BA receives those conditions in the
other periods.

You say you assume no carry over effects, so no effect of the order of
administering the conditions. If you are really sure about that or just do
not hypothesize them then the only efect that you probably would have
hypothesized is the condition (treatment) effect A vs. B.

Now you say you take the difference between the periods as "period 2 minus
period 1" for each of the groups and compare (test) its means. It is not
clear to me what you mean with the difference here: the individual paired
differences between period 2 and 1 and the mean of the differences or the
means of period 2 and 1 and their difference. Anyway if you really mean what
you are saying here I would interprete it as testing the order effect. You
should rather not take that difference but the difference between B and A.

Your second alternative is, as far as I understand, indeed testing the
condition (treatment) effect (negating the period difference for group BA)
and regarding it as individually paired differences that can be tested with
a one sample paired t-test.

Reasoning back to the first alternative I think that probably consists of
considering the mean scores (results) for group A on one hand and for group
B on the other hand and testing that difference with an independent two
sample t-test. I think this is not a good idea as you are not using the
repeated (paired) character of the scores, you do not use the fact that the
within variation between the periods in both order groups is (almost) nill.

So I would go for the second alternative if I have understood your first one
as you intended at all.

But, I would not exclude the order effect either. I would have hypothesized
it. And I might even have hypothesized an interaction effect between the
conditions A and B and the orders AB and BA. And I would go for analysis of
variance with main effects condition (treatment) and order, and possibly
their interaction effect.

In such a case the design to put into the AOV is:
order    A-B    B-A
condition
A               AB1    BA2
B               AB2    BA1
where order is the order of A and B, thus A-B or B-A and
AB1 is group (or rather order) AB in period 1, AB2 group AB in period 2,
BA1 is group (or rather order) BA in period 1, BA2 group BA in period 2.
So your grouping variables are A and B on one hand,
and not periods on the other hand, but orders (A-B and B-A).

And above all condition is a repeated effect in such a design, order is not.

Regards - Jim.
--
Jim Groeneveld, Netherlands
Statistician, SAS consultant
home.hccnet.nl/jim.groeneveld

On Wed, 1 Nov 2006 12:53:17 -0800, Frank <deps_bear@YAHOO.COM> wrote:

>This is a stat question/statement, simple I thought at first, but would
>like others input. If I'm doing a 2x2 randomized cross over study (so I
>have Group AB and BA for argument sake), assuming normality and no
>carry over effects, I can take the difference from period 2 - period 1
>for each group, get the means and do a 2 sample t-test to determine if
>a treatment difference between A and B exists.  True?  (I know other
>methods exist using GLM and even mixed models, but for now i just doing
>a t-test)
>
>My question now is -  what if I wanted to do a one sample paired T-test
>instead.  So for Group AB I would again just take the difference.  But
>for Group BA, I would need to 'negate' the difference, thus all
>patients would be treated as if from the same group.  With part of the
>rationale for this approach to gain a degree of freedom.  Is this
>correct?
>
>Basically, I would appreciate any affirmations to the two approaches OR
>not.  And Any references to these two approaches would be great!
>
>
>F
```
 0
Reply jim2stat (833) 11/1/2006 10:39:42 PM

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```Jim, thanks for your reply.  I would like to f/u and be more clear with
my first approach.  If I have 4 subjects enrolled in a cross-over study
with the following data:

group    pt    Baseline    Period_1    Period_2    diff    total

AB       1        8           6                4         -2       10
AB       4        4           5                9         4       14
BA       2        6           8                6        -2       14
BA       3        9           6                3        -3        9

Then to test a trt difference using a 2 sample t-test (approach 1) I
would take the mean of diff for AB (-2+4/2 = 1) and do a two sided test
of equality with mean of diff BA (-2.5) and get a test statistic of
1.15 (p=.36).  In this way, I am taking advantage of the paired nature
of the data.

Second way (paired t), I think we both agreed about negating group BA
such that all patients are treated as if they come from  group AB and
just doing a test if the mean is different from 0.  In this way, we get
the same result but gain the 1 degree of freedom.

Do these two approaches sound reasonable?

One additional question, both tests are testing for a 'dfference',
either a difference between means in (1) or a difference from 0 in (2).
How do I make a claim about the degree of difference between A and B?

Thanks,
F

Jim Groeneveld wrote:
> Hi Frank,
>
> OK, let's restate your design:
> You have two groups of subjects (maybe patients) who both receive
> experimental conditions (maybe treatments) A and B. The difference between
> the two groups is the order of the conditions. Group AB has condition A in
> period 1 and B in period 2, while group BA receives those conditions in the
> other periods.
>
> You say you assume no carry over effects, so no effect of the order of
> administering the conditions. If you are really sure about that or just do
> not hypothesize them then the only efect that you probably would have
> hypothesized is the condition (treatment) effect A vs. B.
>
> Now you say you take the difference between the periods as "period 2 minus
> period 1" for each of the groups and compare (test) its means. It is not
> clear to me what you mean with the difference here: the individual paired
> differences between period 2 and 1 and the mean of the differences or the
> means of period 2 and 1 and their difference. Anyway if you really mean what
> you are saying here I would interprete it as testing the order effect. You
> should rather not take that difference but the difference between B and A.
>
> Your second alternative is, as far as I understand, indeed testing the
> condition (treatment) effect (negating the period difference for group BA)
> and regarding it as individually paired differences that can be tested with
> a one sample paired t-test.
>
> Reasoning back to the first alternative I think that probably consists of
> considering the mean scores (results) for group A on one hand and for group
> B on the other hand and testing that difference with an independent two
> sample t-test. I think this is not a good idea as you are not using the
> repeated (paired) character of the scores, you do not use the fact that the
> within variation between the periods in both order groups is (almost) nill.
>
> So I would go for the second alternative if I have understood your first one
> as you intended at all.
>
> But, I would not exclude the order effect either. I would have hypothesized
> it. And I might even have hypothesized an interaction effect between the
> conditions A and B and the orders AB and BA. And I would go for analysis of
> variance with main effects condition (treatment) and order, and possibly
> their interaction effect.
>
> In such a case the design to put into the AOV is:
>              order    A-B    B-A
>   condition
>       A               AB1    BA2
>       B               AB2    BA1
> where order is the order of A and B, thus A-B or B-A and
> AB1 is group (or rather order) AB in period 1, AB2 group AB in period 2,
> BA1 is group (or rather order) BA in period 1, BA2 group BA in period 2.
> So your grouping variables are A and B on one hand,
> and not periods on the other hand, but orders (A-B and B-A).
>
> And above all condition is a repeated effect in such a design, order is not.
>
> Regards - Jim.
> --
> Jim Groeneveld, Netherlands
> Statistician, SAS consultant
> home.hccnet.nl/jim.groeneveld
>
>
> On Wed, 1 Nov 2006 12:53:17 -0800, Frank <deps_bear@YAHOO.COM> wrote:
>
> >This is a stat question/statement, simple I thought at first, but would
> >like others input. If I'm doing a 2x2 randomized cross over study (so I
> >have Group AB and BA for argument sake), assuming normality and no
> >carry over effects, I can take the difference from period 2 - period 1
> >for each group, get the means and do a 2 sample t-test to determine if
> >a treatment difference between A and B exists.  True?  (I know other
> >methods exist using GLM and even mixed models, but for now i just doing
> >a t-test)
> >
> >My question now is -  what if I wanted to do a one sample paired T-test
> >instead.  So for Group AB I would again just take the difference.  But
> >for Group BA, I would need to 'negate' the difference, thus all
> >patients would be treated as if from the same group.  With part of the
> >rationale for this approach to gain a degree of freedom.  Is this
> >correct?
> >
> >Basically, I would appreciate any affirmations to the two approaches OR
> >not.  And Any references to these two approaches would be great!
> >
> >
> >F

```
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Reply deps_bear (30) 11/2/2006 1:59:21 PM

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