# Standardized and unstandardized variables yield different results for mixed regression model

I have created two mixed regression models, one with raw unstandardized variables and the same model with standardized variables. When I convert the coefficients from the standardized variables I get different coefficients, especially for the intercept.

# Raw Unstandardized Variables

The model I use for the raw unstandardized variables is:

    model <- lmer(MatchScore~ElapsedTime+UsableIrisArea+DilationChange+Sharpness+
(1|SubjectID)+(0+DilationChange|SubjectID)+(0+UsableIrisArea|SubjectID)+(0+ElapsedTime|SubjectID),
data=Data, na.action="na.fail", REML=FALSE)
summary(model)


I get the following results:

         AIC      BIC   logLik deviance df.resid
124741.4 124814.4 -62360.7 124721.4    10969

Scaled residuals:
Min      1Q  Median      3Q     Max
-8.1739 -0.5233  0.0491  0.5811  4.0564

Random effects:
Groups      Name           Variance  Std.Dev.
SubjectID   (Intercept)    16484.064 128.390
SubjectID.1 DilationChange     4.872   2.207
SubjectID.2 UsableIrisArea     2.510   1.584
SubjectID.3 ElapsedTime        6.593   2.568
Residual                    4726.140  68.747
Number of obs: 10979, groups:  SubjectID, 73

Fixed effects:
Estimate Std. Error         df t value Pr(>|t|)
(Intercept)     -162.3829    19.1380    45.9334  -8.485 5.83e-11 ***
ElapsedTime       -5.3536     0.4616    35.6739 -11.597 1.17e-13 ***
UsableIrisArea     6.4372     0.2381    46.6754  27.040  < 2e-16 ***
DilationChange    -5.3044     0.3542    42.9306 -14.974  < 2e-16 ***
Sharpness          4.8622     0.1516 10793.8131  32.063  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Correlation of Fixed Effects:
(Intr) ElpsdT UsblIA DltnCh
ElapsedTime -0.038
UsableIrsAr -0.383  0.019
DilatinChng -0.008 -0.090 -0.018
Sharpness   -0.037  0.041 -0.008  0.000


# Standardized Variables

The model I use for the standardized variables is:

    model2 <- lmer(MatchScore~ElapsedTime+UsableIrisArea+DilationChange+Sharpness+
(1|SubjectID)+(0+DilationChange|SubjectID)+(0+UsableIrisArea|SubjectID)+(0+ElapsedTime|SubjectID),
data=Data.Scaled, na.action="na.fail", REML=FALSE)

summary(model2)


I get the following results:

    AIC      BIC   logLik deviance df.resid
20216.8  20289.9 -10098.4  20196.8    10969

Scaled residuals:
Min      1Q  Median      3Q     Max
-8.1032 -0.5237  0.0489  0.5767  4.0852

Random effects:
Groups      Name           Variance Std.Dev.
SubjectID   (Intercept)    0.280539 0.52966
SubjectID.1 DilationChange 0.010866 0.10424
SubjectID.2 UsableIrisArea 0.061813 0.24862
SubjectID.3 ElapsedTime    0.008361 0.09144
Residual                   0.348526 0.59036
Number of obs: 10979, groups:  SubjectID, 73

Fixed effects:
Estimate Std. Error         df t value Pr(>|t|)
(Intercept)    -9.730e-02  6.478e-02  7.107e+01  -1.502    0.138
ElapsedTime    -1.893e-01  1.638e-02  3.960e+01 -11.562  2.9e-14 ***
UsableIrisArea  5.306e-01  3.380e-02  4.388e+01  15.698  < 2e-16 ***
DilationChange -2.558e-01  1.678e-02  4.439e+01 -15.248  < 2e-16 ***
Sharpness       3.042e-01  9.508e-03  1.079e+04  31.992  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Correlation of Fixed Effects:
(Intr) ElpsdT UsblIA DltnCh
ElapsedTime  0.083
UsableIrsAr -0.061  0.022
DilatinChng  0.023 -0.093 -0.007
Sharpness    0.017  0.040 -0.003 -0.001


Then I convert the standardized coefficients using the logic in this post https://stackoverflow.com/a/23643740/2343633

The standardized coefficients are:

       (Intercept)    ElapsedTime UsableIrisArea DilationChange      Sharpness
494.651361      -5.324217       8.430474      -3.339946       6.32240


This where it gets interesting - when converting the standardized coefficients the intercept actually makes sense, the match score should never fall below 0. Where as, the intercept from model with the unstandardized coefficients is negative - which makes absolutely no sense. Additionally, as you can see some the coefficients change for the transformed standardized coefficients, more specifically, Sharpness, UsableIrisArea, and DilationChange.

I would like to note that this data is very noisy and not all subjects have the same number of samples nor the same amount of time between samples. This is something that I can not change. Put simply, the data is what it is. My questions are:

1. Why would I get varying results with standardized and raw unstandardized variables. Is this a major issue?

2. Given the second model (the one with the standardized variables) makes more sense should this be the model I use?

3. Is there a more scientific way in determining if standardized or unstandardized variables is more appropriate for my model.

Any other suggestions, comments, or recommendations would be appreciated. If you need any additional information I more than happy to supply that.

• Guessing this happens because you're fitting random slopes as independent terms ((1|Subject) + (0+x|Subject) + (0+y|Subject)) ...thus the results are no longer invariant to linear transformations of the variables ... the tipoff that you're fitting a different model is that the log-likelihood/AIC/BIC change between models (they shouldn't if all you're doing is re-parameterizing to an equivalent model). Do you get identical answers for both data sets if your RE term is (1+x+y+z|Subject) (where x, y, z are your predictors)? ... Identical if you only scale rather than scale+center? – Ben Bolker Jun 15 '18 at 22:06
• see "Specifying uncorrelated random effects" in vignette("lmer",package="lme4") for a bit more info ... – Ben Bolker Jun 15 '18 at 22:07
• @BenBolker you are correct. I want my random effects to be uncorrelated so that means by standardizing the variables I am also changing my model. That explains the differences because they are indeed two separate models. Thank you! – thutchi Jun 15 '18 at 23:28

• I think to be precise it's not actually any linear transformation, but rather additive shifts only; that is, $x \to bx$ would be OK, but not $x \to a + bx, a \neq 0$. (You could try an experiment.) – Ben Bolker Jun 15 '18 at 23:47