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Hi ElMaestro!
I think it means rank of whole X, because after: "and β is a (p × 1) vector of unknown, constant population parameters". Xi is a part of X and have same count of columns, rank Xi can be lower. Here is no contradictions I think. If look at Xiuming Zhang, 2015* p is just length of β and rank should be the same. If not, we will have rank-deficient matrix.
And if look generally - it really doesn't matter because -(N-p)*log(2π)/2 is a constant and it can be excluded from optimization. If you want find AIC ets - this will matter.
And as I know some softaware using different N for calculation (some use all observations, some only statistically independent (m)), and some include second constant part: logdet(Σim(X'X))/2
So if you want find only θ, you can minimize REML without constants at all. And add constants when you need to have real value of REML or AIC, BIC, ets.
*A Tutorial on Restricted Maximum Likelihood Estimation in Linear Regression and Linear Mixed-Effects Model
Example N=3, n=9, p=6
X:
X1
X2
X3
❝ Gurka requires me to derive Xi, the design matrix for fixed effects of subject i.
❝ They mention that "Xi is a (ni × p) known, constant design matrix for the i' th subject
❝ with rank p"
I think it means rank of whole X, because after: "and β is a (p × 1) vector of unknown, constant population parameters". Xi is a part of X and have same count of columns, rank Xi can be lower. Here is no contradictions I think. If look at Xiuming Zhang, 2015* p is just length of β and rank should be the same. If not, we will have rank-deficient matrix.
And if look generally - it really doesn't matter because -(N-p)*log(2π)/2 is a constant and it can be excluded from optimization. If you want find AIC ets - this will matter.
And as I know some softaware using different N for calculation (some use all observations, some only statistically independent (m)), and some include second constant part: logdet(Σim(X'X))/2
So if you want find only θ, you can minimize REML without constants at all. And add constants when you need to have real value of REML or AIC, BIC, ets.
*A Tutorial on Restricted Maximum Likelihood Estimation in Linear Regression and Linear Mixed-Effects Model
Example N=3, n=9, p=6
X:
β1 β2 β3 β4 β5 β6
1 1 0 1 0 0
1 1 0 1 0 0
1 1 0 1 0 0
1 0 1 0 1 0
1 0 1 0 1 0
1 0 1 0 1 0
1 1 0 0 0 1
1 1 0 0 0 1
1 1 0 0 0 1X1
β1 β2 β3 β4 β5 β6
1 1 0 1 0 0
1 1 0 1 0 0
1 1 0 1 0 0
X2
β1 β2 β3 β4 β5 β6
1 0 1 0 1 0
1 0 1 0 1 0
1 0 1 0 1 0
X3
β1 β2 β3 β4 β5 β6
1 1 0 0 0 1
1 1 0 0 0 1
1 1 0 0 0 1Complete thread:
- Semireplicated + REML in R ElMaestro 2020-07-10 13:19
![[Mix]](https://static.bebac.at/img/mix.png "open in Mix view")
- Semireplicated + REML in R Helmut 2020-07-10 19:03
- Semireplicated + REML in R ElMaestro 2020-07-10 19:24
- Avoid partial replicate designs, pleeeze! Helmut 2020-07-11 11:57
- Avoid partial replicate designs, pleeeze! ElMaestro 2020-07-11 14:43
- Braveheart! Helmut 2020-07-11 15:38
- Who can help? ElMaestro 2020-07-12 12:28
- Who can help? ElMaestro 2020-07-12 13:26
- Update II ElMaestro 2020-07-12 21:36
- Update III ElMaestro 2020-07-12 21:46
- Final update today ElMaestro 2020-07-12 22:27
- Medium rare. Helmut 2020-07-13 13:52
- took just 52 hrs to do it :-) ElMaestro 2020-07-13 14:18
- Will take much more hours still… Helmut 2020-07-13 15:34
- Negative determinant ElMaestro 2020-07-14 03:22
- Are we loosers? Helmut 2020-07-14 13:58
- "we"? Loosers? ElMaestro 2020-07-14 15:07
- Misunderstanding? Helmut 2020-07-14 15:32
- "we"? Loosers? ElMaestro 2020-07-14 15:07
- Are we loosers? Helmut 2020-07-14 13:58
- took just 52 hrs to do it :-) ElMaestro 2020-07-13 14:18
- Medium rare. Helmut 2020-07-13 13:52
- Braveheart! ElMaestro 2020-07-13 10:13
- Braveheart! Helmut 2020-07-13 14:16
- Braveheart! PharmCat 2020-07-15 14:19
- Braveheart! ElMaestro 2020-08-02 17:39
- Who can help? ElMaestro 2020-07-12 12:28
- Braveheart! Helmut 2020-07-11 15:38
- Avoid partial replicate designs, pleeeze! ElMaestro 2020-07-11 14:43
- Avoid partial replicate designs, pleeeze! Helmut 2020-07-11 11:57
- Semireplicated + REML in R ElMaestro 2020-07-10 19:24
- We were all blind (except Detlew) Helmut 2020-07-15 14:27
- It is the opposite way around for me ElMaestro 2020-07-15 16:25
- Desultory thoughts Helmut 2020-07-15 17:33
- FDA RSABE is ISC d_labes 2020-07-15 18:13
- FDA RSABE is ISC Helmut 2020-07-16 11:11
- Desultory thoughts ElMaestro 2020-07-15 23:06
- Desultory thoughts Helmut 2020-07-16 10:59
- FDA RSABE is ISC d_labes 2020-07-15 18:13
- Desultory thoughts Helmut 2020-07-15 17:33
- Phoenix - which template? mittyri 2020-07-19 00:42
- FDA RSABE Project template_ v1.4.phxproj Helmut 2020-07-19 01:45
- It is the opposite way around for me ElMaestro 2020-07-15 16:25
- "By popular demand": likelihood ElMaestro 2020-07-24 10:07
- And by the way.... ElMaestro 2020-07-24 12:52
- And by the way.... PharmCat 2020-08-03 14:24
- Not understood ElMaestro 2020-08-03 22:55
- Not understood PharmCat 2020-08-05 01:41
- Not understood ElMaestro 2020-08-05 08:13
- Not understood PharmCat 2020-08-05 16:37
- Open issues ElMaestro 2020-08-06 21:11
- Open issuesPharmCat 2020-08-07 00:02
- Open issues ElMaestro 2020-08-07 07:49
- Open issues PharmCat 2020-08-07 11:29
- Still can't make it work ElMaestro 2020-08-07 13:08
- Still can't make it work PharmCat 2020-08-07 15:42
- Still can't make it work ElMaestro 2020-08-07 16:44
- Still can't make it work PharmCat 2020-08-07 18:14
- Still can't make it work ElMaestro 2020-08-07 18:23
- And now it works ElMaestro 2020-08-07 21:31
- Still can't make it work PharmCat 2020-08-08 01:08
- Speed improvement ElMaestro 2020-08-08 12:25
- Speed improvement PharmCat 2020-08-08 17:27
- Speed improvement ElMaestro 2020-08-08 18:10
- Speed improvement PharmCat 2020-08-09 18:22
- Some tests... PharmCat 2020-08-10 11:48
- Speed improvement ElMaestro 2020-08-08 12:25
- Still can't make it work ElMaestro 2020-08-07 18:23
- Still can't make it work PharmCat 2020-08-07 18:14
- Still can't make it work ElMaestro 2020-08-07 16:44
- Still can't make it work PharmCat 2020-08-07 15:42
- Still can't make it work ElMaestro 2020-08-07 13:08
- Open issues PharmCat 2020-08-07 11:29
- Open issues ElMaestro 2020-08-07 07:49
- Open issuesPharmCat 2020-08-07 00:02
- Open issues ElMaestro 2020-08-06 21:11
- Not understood PharmCat 2020-08-05 16:37
- Not understood ElMaestro 2020-08-05 08:13
- Not understood PharmCat 2020-08-05 01:41
- Not understood ElMaestro 2020-08-03 22:55
- And by the way.... PharmCat 2020-08-03 14:24
- And by the way.... ElMaestro 2020-07-24 12:52
- Semireplicated + REML in R Helmut 2020-07-10 19:03
Ing. Helmut Schütz