## Slightly off topic, but related :-) [Design Issues]

Hi ElMaestro!

» This. I think, is around the "effective zero" for fits in R at default settings on 64- and 32-bit systems.

Anyway residual "can't" be zero.

» Is this a fact? How do we actually know this? Do you have a reference I coud learn from (not Pinheiro and Bates, I don't understand a word of it).

For ML: Yes, it's a fact. ML biased always "by definition". proof

For REML: this is a more difficult question. We can see following:

* link 1

* link 2

* link 3

* link 4

* link 5

* link 6

Some links behind the paywall, but this problem can be solved with sci-hub.

What can we say: "REML does effectively reduce bias.", "The REML estimates are typically less biased than the ML methods."

Hm... I understand that REML not always unbiased. May be I'm wrong, but all above make me think this.

» Does "less biased" apply to both the fixed effects and to the variance components?

Mmm. If we call model coefficients as fixed effects - they are unbiased, no problems with them. Variance component in LM if say strictly is a model error. LM have analytical solution and ubiased estimator of variance. But model should be fitted correctly: each level of any factor should have at least 2 observations.

» This. I think, is around the "effective zero" for fits in R at default settings on 64- and 32-bit systems.

Anyway residual "can't" be zero.

» Is this a fact? How do we actually know this? Do you have a reference I coud learn from (not Pinheiro and Bates, I don't understand a word of it).

For ML: Yes, it's a fact. ML biased always "by definition". proof

For REML: this is a more difficult question. We can see following:

* link 1

* link 2

* link 3

* link 4

* link 5

* link 6

Some links behind the paywall, but this problem can be solved with sci-hub.

What can we say: "REML does effectively reduce bias.", "The REML estimates are typically less biased than the ML methods."

*REML does not always eliminate all of the bias in parameter estimation, since many methods for obtaining REML estimates cannot return negative estimates of a variance component. However, this source of bias also exists with ML, so REML is clearly the preferred method for analyzing large data sets with complex structure.*Hm... I understand that REML not always unbiased. May be I'm wrong, but all above make me think this.

» Does "less biased" apply to both the fixed effects and to the variance components?

Mmm. If we call model coefficients as fixed effects - they are unbiased, no problems with them. Variance component in LM if say strictly is a model error. LM have analytical solution and ubiased estimator of variance. But model should be fitted correctly: each level of any factor should have at least 2 observations.

### Complete thread:

- Should those subjects have only one period data be included in BE analysis? ssussu 2019-12-06 10:58 [Design Issues]
- No way! But... Beholder 2019-12-06 12:36
- No way! But... ElMaestro 2019-12-06 13:14
- No way! But... Helmut 2019-12-06 14:47
- No way! But... PharmCat 2019-12-06 23:46
- No way! But... wienui 2019-12-07 04:01
- EMA guideline: no way...: mittyri 2019-12-07 20:58
- EMA guideline: no way...: wienui 2019-12-09 05:25

- EMA guideline: no way...: mittyri 2019-12-07 20:58
- Slightly off topic, but related :-) ElMaestro 2019-12-08 01:31
- Slightly off topic, but related :-) Shuanghe 2019-12-09 12:00
- Slightly off topic, but related :-) PharmCat 2019-12-09 15:13
- Slightly off topic, but related :-) ElMaestro 2019-12-21 15:02
- Slightly off topic, but related :-)PharmCat 2019-12-22 01:12
- 2.220446e-16 ≈ 0 Helmut 2019-12-22 10:37
- The optional tolerance argument ElMaestro 2019-12-23 14:37
- 2.220446e-16 ≈ 0 PharmCat 2019-12-24 14:18
- Sum of residuals ~ ε Helmut 2019-12-24 14:54
- Sum of residuals ~ ε ElMaestro 2019-12-24 15:10
- Wrong terminology Helmut 2019-12-28 13:55

- Sum of residuals ~ ε PharmCat 2019-12-24 18:40

- Sum of residuals ~ ε ElMaestro 2019-12-24 15:10

- Sum of residuals ~ ε Helmut 2019-12-24 14:54

- Slightly off topic, but related :-) ElMaestro 2019-12-21 15:02

- No way! But... wienui 2019-12-07 04:01

- No way! But... PharmCat 2019-12-06 23:46

- No way! But... Helmut 2019-12-06 14:47

- No way! But... ElMaestro 2019-12-06 13:14

- No way! But... Beholder 2019-12-06 12:36