Linear mixed effects model standardization with Z score not giving consistent results

HI, I am running a mixed effects model with interaction terms, and when I run the model with the raw data, I essentialy get what I would expect

Linear mixed-effects model fit by ML

Model information:

Number of observations 44

Fixed effects coefficients 5

Random effects coefficients 22

Covariance parameters 2

Formula:

outcome ~ 1 + intervals + exposure*sessions + (1 | subject)

Model fit statistics:

AIC BIC LogLikelihood Deviance

264 276.49 -125 250

Fixed effects coefficients (95% CIs):

Name Estimate SE tStat DF pValue Lower Upper

'(Intercept)' 26.899 2.6919 9.9926 39 2.6128e-12 21.454 32.344

'exposure' -1.6766 1.0159 -1.6504 39 0.10689 -3.7313 0.3782

'intervals' -0.19934 0.033912 -5.8783 39 7.6441e-07 -0.26794 -0.13075

'sessions' -0.73017 0.44687 -1.634 39 0.11031 -1.6341 0.17371

'exposure:sessions' 0.4013 0.14279 2.8105 39 0.0076969 0.11249 0.69012

Random effects covariance parameters (95% CIs):

Group: subject (22 Levels)

Name1 Name2 Type Estimate Lower Upper

'(Intercept)' '(Intercept)' 'std' 3.1621 1.9751 5.0624

Group: Error

Name Estimate Lower Upper

'Res Std' 3.1433 2.3339 4.2334

But when I standardize the predictor using the matlab zscore function I get something totally different, whether I also standardize the dependent variable or not

Linear mixed-effects model fit by ML

Model information:

Number of observations 44

Fixed effects coefficients 5

Random effects coefficients 22

Covariance parameters 2

Formula:

drinking ~ 1 + intervals + exposure*sessions + (1 | subject)

Model fit statistics:

AIC BIC LogLikelihood Deviance

127.45 139.94 -56.727 113.45

Fixed effects coefficients (95% CIs):

Name Estimate SE tStat DF pValue Lower Upper

'(Intercept)' 0.0078404 0.17882 0.043845 39 0.96525 -0.35386 0.36954

'exposure' -0.020592 0.14613 -0.14091 39 0.88867 -0.31618 0.27499

'intervals' -0.1539 0.21666 -0.71034 39 0.48172 -0.59213 0.28433

'sessions' 0.3547 0.21217 1.6717 39 0.10258 -0.07446 0.78385

'exposure:sessions' -0.099309 0.24164 -0.41098 39 0.68334 -0.58807 0.38945

Random effects covariance parameters (95% CIs):

Group: subject (22 Levels)

Name1 Name2 Type Estimate Lower Upper

'(Intercept)' '(Intercept)' 'std' 0.6939 0.43148 1.1159

Group: Error

Name Estimate Lower Upper

'Res Std' 0.65418 0.4809 0.8899

When I look at plots of the individual predictors before and after z transformation, everything looks similar, just the values are (expectedly) different. I would not expect such a drastic difference in LME models. Grateful for any assistance! Thank you!

[the cyclist]

Can you upload the data and the code you used to fit the models?

[bsriv]

Hi I've attached csvs with both the unstandardized and standardized ("z") data

The code I used was simply

lme=fitlme(tbl_roc,'drinking ~ alc_cue+cbt+intervals+alc_cue*cbt+(1|subject)')

[the cyclist]

If anyone else happens to investigate, be aware that the attached files are MAT, not CSV.

Answer 1

[the cyclist]

I'm not sure why you expect the model to be unchanged (or what specific aspects of the model you expect to be unchanged).

Even with a simple linear model, it is obvious to me that the fitted parameters will be different under a zscore transform of the predictors. For example,

rng default

x1 = [-2; -1; 0; 1; 2] + 10;
x2 = zscore(x1); % new predictor is zscore of original predictor

y = 2*x1 + randn(5,1);

tbl = table(x1,x2,y);

mdl1 = fitlm(tbl,"y ~ x1") % fit original predictor

mdl1 = 
Linear regression model:
    y ~ 1 + x1

Estimated Coefficients:
                   Estimate      SE        tStat      pValue 
                   ________    _______    _______    ________

    (Intercept)     1.6682       5.552    0.30047     0.78343
    x1               1.859     0.54973     3.3817    0.043036


Number of observations: 5, Error degrees of freedom: 3
Root Mean Squared Error: 1.74
R-squared: 0.792,  Adjusted R-Squared: 0.723
F-statistic vs. constant model: 11.4, p-value = 0.043

mdl2 = fitlm(tbl,"y ~ x2") % fit new predictor

mdl2 = 
Linear regression model:
    y ~ 1 + x2

Estimated Coefficients:
                   Estimate      SE       tStat       pValue  
                   ________    _______    ______    __________

    (Intercept)     20.259     0.77744    26.058    0.00012398
    x2              2.9394      0.8692    3.3817      0.043036


Number of observations: 5, Error degrees of freedom: 3
Root Mean Squared Error: 1.74
R-squared: 0.792,  Adjusted R-Squared: 0.723
F-statistic vs. constant model: 11.4, p-value = 0.043

In this simpler case, it is pretty easy to understand why certain things (e.g. beta for the x1 vs. x2 ) is different. It may or may not be obvious why the R-squared is unchanged.

In your more complicated model, especially because of the interactions, I think it could be tricky what could be expected to remain constant across the two models.

What did you expect to remain constant after this transform? Presumably not the beta coefficients.

[bsriv]

Thanks for the response- I naturally expected the beta coefficients to change, but I did not expect the previously significant predictors (intervals, interaction term) to change

[the cyclist]

First, I have reason to believe that the zscore was not done correctly. When I zscored the ocds variable, I did not replicate the value in the other table. (I did this both with the built-in MATLAB function, and also just directly.)

When I used the correctly z-scored values on an OLS model (not LME), I get exactly what I expected (as I did in the small example I posted here). When I then used your LME model, some of the p-values were identical between the zscored and non-zscored models, but some were not. (For example, the interaction term was identical, as was the intervals term.) Note that I did zscore the response variable.

I can't say that I have figured out the entire theory behind this, but I speculate that your expectation could be wrong. In particular, I can imagine that allowing for different intercepts for each subject could result in non-significant slope betas, where they had been significant previously, given the scaling.

[the cyclist]

load("tbl_roc")
tbl_roc_z = tbl_roc;

tbl_roc_z.alc_cue   = zscore(tbl_roc.alc_cue);
tbl_roc_z.intervals = zscore(tbl_roc.intervals);
tbl_roc_z.cbt       = zscore(tbl_roc.cbt);
tbl_roc_z.drinking  = zscore(tbl_roc.drinking);

lme   = fitlme(tbl_roc,'drinking ~ alc_cue+cbt+intervals+alc_cue*cbt+(1|subject)')

lme = 
Linear mixed-effects model fit by ML

Model information:
    Number of observations              44
    Fixed effects coefficients           5
    Random effects coefficients         22
    Covariance parameters                2

Formula:
    drinking ~ 1 + intervals + alc_cue*cbt + (1 | subject)

Model fit statistics:
    AIC    BIC       LogLikelihood    Deviance
    264    276.49    -125             250     

Fixed effects coefficients (95% CIs):
    Name                   Estimate    SE          tStat      DF    pValue        Lower       Upper   
    {'(Intercept)'}          26.899      2.6919     9.9926    39    2.6128e-12      21.454      32.344
    {'alc_cue'    }         -1.6766      1.0159    -1.6504    39       0.10689     -3.7313      0.3782
    {'intervals'  }        -0.19934    0.033912    -5.8783    39    7.6441e-07    -0.26794    -0.13075
    {'cbt'        }        -0.73017     0.44687     -1.634    39       0.11031     -1.6341     0.17371
    {'alc_cue:cbt'}          0.4013     0.14279     2.8105    39     0.0076969     0.11249     0.69012

Random effects covariance parameters (95% CIs):
Group: subject (22 Levels)
    Name1                  Name2                  Type           Estimate    Lower     Upper 
    {'(Intercept)'}        {'(Intercept)'}        {'std'}        3.1621      1.9751    5.0624

Group: Error
    Name               Estimate    Lower     Upper 
    {'Res Std'}        3.1433      2.3339    4.2334

lme_z = fitlme(tbl_roc_z,'drinking ~ alc_cue+cbt+intervals+alc_cue*cbt+(1|subject)')

lme_z = 
Linear mixed-effects model fit by ML

Model information:
    Number of observations              44
    Fixed effects coefficients           5
    Random effects coefficients         22
    Covariance parameters                2

Formula:
    drinking ~ 1 + intervals + alc_cue*cbt + (1 | subject)

Model fit statistics:
    AIC       BIC      LogLikelihood    Deviance
    55.721    68.21    -20.86           41.721  

Fixed effects coefficients (95% CIs):
    Name                   Estimate    SE          tStat      DF    pValue        Lower        Upper   
    {'(Intercept)'}        0.077311    0.082032    0.94245    39       0.35176    -0.088614     0.24324
    {'alc_cue'    }        0.027453    0.074245    0.36976    39       0.71356     -0.12272     0.17763
    {'intervals'  }        -0.92989     0.15819    -5.8783    39    7.6441e-07      -1.2499    -0.60992
    {'cbt'        }        0.045009      0.1567    0.28723    39       0.77546     -0.27194     0.36196
    {'alc_cue:cbt'}         0.17342    0.061704     2.8105    39     0.0076969     0.048611     0.29823

Random effects covariance parameters (95% CIs):
Group: subject (22 Levels)
    Name1                  Name2                  Type           Estimate    Lower      Upper  
    {'(Intercept)'}        {'(Intercept)'}        {'std'}        0.29655     0.18523    0.47478

Group: Error
    Name               Estimate    Lower      Upper  
    {'Res Std'}        0.29479     0.21889    0.39703

[bsriv]

Thanks for checking! Ifixed the zscore issue and it worked.