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d_labes ★★★ Berlin, Germany, 2013-04-10 11:30 (4813 d 17:52 ago) Posting: # 10388 Views: 6,934 |
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Dear All! For some reasons I'm interested in the mse composed from the intra-subject variances of Test and Reference of the classical ANOVA used for crossover designs (same as for the 2x2x2 crossover). For the full replicate (2x2x4) design Chow and Liu mention (without an elaborate explanation) mse = 1/2*(s2wT + s2wR)For the partial replicate design (2x3x3) I have arrived with mse = 1/3*(s2wT + 2*s2wR)mostly empirical based on simulations (classical subject data sims without a subject-by-formulation interaction in the statistical model). Via these sims it could be verified that the formula for the full replicate design seems correct. My questions are: Does any body know a way to derive the formulas theoretically? Or has anybody done simulations of the 2x3x3 design to verify my finding? Or is that all bullshit since we have to assume s2wT = s2wR if we use ANOVA (homogeneous variances)? Chow, Liu "Design and Analysis of Bioavailability and Bioequivalence Studies" Third edition, Chapter 9.4 Chapman & Hall, Boca Raton 2009 — Regards, Detlew |
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ElMaestro ★★★ Denmark, 2013-04-10 11:51 (4813 d 17:31 ago) @ d_labes Posting: # 10389 Views: 5,652 |
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Hi d_labes, ❝ For the full replicate (2x2x4) design Chow and Liu mention (without an elaborate explanation) ❝ ❝ ❝ For the partial replicate design (2x3x3) I have arrived with ❝ ❝ mostly empirical based on simulations (classical subject data sims without a subject-by-formulation interaction in the statistical model). ❝ Via these sims it could be verified that the formula for the full replicate design seems correct. Did you analyse with a mixed model or with a (g)lm? And if you did a mixed model, did you simulate with balanced sequences without missing period data? ❝ My questions are: ❝ Does any body know a way to derive the formulas theoretically? In a mixed model -if that's you case cf above- you have a V where the diagonal is either a. s2wT+s2bT or s2wT+s2bT (full repl.) b. s2wT or s2wR+s2bR (partial repl.) The within's are clasically referred to as the errors. Your formula corresponds to setting the average of all those error-sigma2's equal to the mean squared error (case of no missing periods). It is more like a definition - the mean squared error is in essence the mean/average of the squared errors, right? ❝ Or has anybody done simulations of the 2x3x3 design to verify my finding? ❝ Or is that all bullshit since we have to assume s2wT = s2wR if we use ANOVA (homogeneous variances)? — Pass or fail! ElMaestro |
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d_labes ★★★ Berlin, Germany, 2013-04-10 12:28 (4813 d 16:54 ago) @ ElMaestro Posting: # 10392 Views: 5,828 |
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Dear ElMaestro, Thanks for the quick answer. I simulated with a mixed model, but without subject-by-formulation interaction. Without missings. With balanced sequence groups. I guess that you know for what end  .I analyzed with simple ANOVA all effects fixed (effects: treatment, period, subject for simplicity since the sequence doesn't change the mse) as the Mighty Oracle demands us to do so  .❝ In a mixed model -if that's you case cf above- you have a V where the diagonal is either ❝ a. s2wT+s2bT or s2wT+s2bT (full repl.) ❝ b. s2wT or s2wR+s2bR (partial repl.)  I have a V matrix (example subject in sequence TRR, i.e. 2x3x3 design) s2b+s2wT s2b s2bwhere s2b is the common between-subject variance. — Regards, Detlew |
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ElMaestro ★★★ Denmark, 2013-04-10 12:38 (4813 d 16:44 ago) @ d_labes Posting: # 10395 Views: 5,632 |
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Hi again d_labes, ❝ ❝ In a mixed model -if that's you case cf above- you have a V where the diagonal is either ❝ ❝ a. s2wT+s2bT or s2wT+s2bT (full repl.) ❝ ❝ b. s2wT or s2wR+s2bR (partial repl.) ❝ ❝ ❝ ❝ I have a V matrix (example subject in sequence TRR, i.e. 2x3x3 design) ❝ ❝ s2b s2b+s2wR s2b ❝ s2b s2b s2b+s2wR ❝ where s2b is the common between-subject variance. Sorry you're right there's something I left out. For T the error is the total var. When equal between's can be assumed for T and R you get s2b+s2wT and not s2wT. The errors are the withins on the diagonal of V; just take the average to get the mean squared error. Or did you mean something entirely different? Btw, shouldn't it be zero in cell (2,1), (3,1), (1,2) and (1,3) ? I am not sure you'd co-vary a T-observation with an R-observation. — Pass or fail! ElMaestro |
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d_labes ★★★ Berlin, Germany, 2013-04-10 13:15 (4813 d 16:07 ago) @ ElMaestro Posting: # 10398 Views: 5,652 |
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Hi again ElMaestro, ❝ The errors are the withins on the diagonal of V; just take the average to get the mean squared error. Do you have some references for that or some theoretical argument? Or is that from the category "It is obvious that ..."? All I have seen up to now assume s2wT=s2wR=s2e and argue then. ❝ Btw, shouldn't it be zero in cell (2,1), (3,1), (1,2) and (1,3) ? ❝ I am not sure you'd co-vary a T-observation with an R-observation. Frankly: no. A model without s2D is derived from the full mixed model we have just recently discussed here assuming rho=1 and s2bT=s2bR=s2b. Only then s2D = s2bT + s2bR − 2*rho*sbT*sbR is zero.You me be right (additionally removing the s2b) if we use an all effects fixed model. Then we have only the errors, distributed under T with s2wT and under R with s2wR. — Regards, Detlew |
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ElMaestro ★★★ Denmark, 2013-04-11 02:07 (4813 d 03:16 ago) @ d_labes Posting: # 10402 Views: 5,629 |
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Hello d_labes, ❝ ❝ The errors are the withins on the diagonal of V; just take the average to get the mean squared error. ❝ ❝ Do you have some references for that or some theoretical argument? ❝ Or is that from the category "It is obvious that ..."? ❝ All I have seen up to now assume s2wT=s2wR=s2e and argue then. When you do a linear model you implicitly model the error distribution via a V matrix where there is just the single sigma (intra or pseudo-intra) squared on the diagonal and zero elsewhere. It is like V=ZGZt+R where you just forget about G because there's no other random term and where Ris just sigma2*I. As I see it, this why a fixed effects model for BE with subject, sequence, treatment and period as fixed factors give the same result as a mixed model with plain subject as the error term (1|Subject). Minimisation of sums-of-squares is just a shortcut to the (restricted) maximum likelihood solution when there is just that one single error term in V. So I would say "It is obvious that..." and pretend I know, although to be frank I am just an uninformed and miserable amateur and when you scratch the paint a bit it quickly becomes ugly. — Pass or fail! ElMaestro |
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