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I have a very large data set with repeated measurements of same blood value (co) (1 to 7 measurements per patient). Each measurement is coupled with time which is the time interval between surgical operation and blood level measurement.

My aim is to show that this blood value correlates positively with time.

Blood level measurements are highly skewed to right and hence I am using a log-transformation and linear mixed effect regression model (lmer in lme4 package).

I have constructed a null model:

fit1<-(lmer(lgco~(1|id),data=ASR))

Model 2 includes time as independent variable:

fit2<-(lmer(lgco~time+(1|id),data=ASR))

Id is the patient number in th dataset.

By using the anova() function I see that fit2 is significantly better than fit1:

> anova(fit1,fit2)
refitting model(s) with ML (instead of REML)

Data: ASR
Models:
fit1: lgco ~ (1 | id)
fit2: lgco ~ time + (1 | id)
     Df    AIC    BIC  logLik deviance  Chisq Chi Df Pr(>Chisq)    
fit1  3 342.77 357.50 -168.39   336.77                             
fit2  4 320.64 340.27 -156.32   312.64 24.135      1  8.983e-07 ***

However I have other data which suggests that the correlation between time and blood value might even more profound, for example quadratic. This would be Model 3.

I tried the following: first I took the square root of the blood value and after that I made the transformation using log.

fit3<-(lmer(lgsqrtco~time+(1|id),data=ASR))

My question is that can I compare models 2 and 3 in anyway now after the dependent variable has two different transformations in these models. In fit1 and fit2 the transformation is identical, only the independent is added. I assume that with different dependent variable transformation the use of anova() is not allowed:

anova(fit2,fit3)
refitting model(s) with ML (instead of REML)
Data: ASR
Models:
fit2: lgco ~ time + (1 | id)
fit3: lgsqrtco ~ time + (1 | id)
     Df      AIC      BIC  logLik deviance  Chisq Chi Df Pr(>Chisq)    
fit2  4   320.64   340.27 -156.32   312.64                             
fit3  4 -1065.66 -1046.03  536.83 -1073.66 1386.3      0  < 2.2e-16 ***
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    $\begingroup$ Why are you using a log-transformation for left-skewed data? Wouldn't you take power transformations? $\endgroup$ – missingdataguy May 22 '15 at 13:48
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    $\begingroup$ @Missing is correct. There are many related issues. First, the distribution of responses is irrelevant: what matters is the distribution of residuals. Second, if the residuals indeed are "left skewed," then taking logs will only make them worse, not better. Instead, you would need to use a power (greater than $1$). Third, perhaps your understanding of "left skewed" is the opposite of the conventional meaning? Fourth, taking the log of a square root has exactly the same effect on distributional shape as taking the log. These all suggest studying transformations further before applying them. $\endgroup$ – whuber May 22 '15 at 14:17
  • $\begingroup$ @whuber The distribution is in fact skewed to RIGHT not to left. Sorry about that. I took resid(fit2) and those are distributed normally. $\endgroup$ – arkiaamu May 23 '15 at 9:34
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    $\begingroup$ Before you move to the third model, you should really consider the possibility that the slope for time is not constant across individuals. That calls for a model with random slopes. In particular: fit2 <- lmer(lgco~time+(time|id),data=ASR). $\endgroup$ – Wolfgang May 23 '15 at 9:57
  • $\begingroup$ @Wolfgang You are absolutelly correct. That was originally my idea but i lost to this model comparison problem. Random slopes are very reasonable for my data and I intend to use them in my models. $\endgroup$ – arkiaamu May 23 '15 at 15:53
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While you can compare model 1 and model 2, and choose among them by ordinary likelihood ratio tests or F tests (e.g. anova in R), you cannot compare model 1 with 3 or model 2 with 3 by likelihood ratio tests or F tests. Nor you can compare 1 vs 3 and 2 vs 3 by information criteria, as the response variables are on different scales.

Hence, pvalues form anova(fit2,fit3) and anova(fit1,fit3) are misleading. The reason of this is that the model for $\log y$ and the model for, say, $\log (y)^3$ are not nested and the likelihood ratio and F tests do not have any more the usual asymptotic distributions. There are some special tests for such difficult cases (see MacKinnon 1983, Model Specification Tests Against Non-Nested Alternatives, Econometric Reviews 2, 5-110 link). Hope this helps.

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  • $\begingroup$ Thank you very much. I was quite sure I was missing a point there and that was the fact that models are not nested. $\endgroup$ – arkiaamu May 23 '15 at 9:36
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    $\begingroup$ Actually, information criteria cannot be used either. The DV must be the same for meaningful comparisons based on ICs. $\endgroup$ – Wolfgang May 23 '15 at 9:55
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    $\begingroup$ yes @Wolfgang you are right, also ICs are cannot be used as the models are on different scales. I'll update my answer. $\endgroup$ – utobi May 23 '15 at 10:52

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