Mixed effect linear regression model with multiple independent observations

Hi Forum,

I am trying to implement a linear mixed effect (LME) regression model for an x-ray imaging quality metric "CNR" (contrast-to-noise ratio) for which I measured for various tube potentials (kV) and filtration materials (Filter). CNR was measured for 3 consecutive slices so I have a standard deviation of the CNR from these independent measurements as well. A representation of the data and my first attempt using fitlme is shown below. I tried looking at online resources but could not find an answer to my specific questions.

kV=[80 90 100 80 90 100 80 90 100]';
Filter={'Al','Al','Al','Cu','Cu','Cu','Ti','Ti','Ti'}';
CNR=[10 9 8 10.1 8.9 7.9 7 6 5]';
T=table(kV,Filter,CNR);

    kV     Filter    CNR
    ___    ______    ___

     80     'Al'      10
     90     'Al'       9
    100     'Al'       8
     80     'Cu'      10.1
     90     'Cu'       8.9
    100     'Cu'       7.9
     80     'Ti'       7
     90     'Ti'       6
    100     'Ti'       5

OUTPUT

Linear mixed-effects model fit by ML

Model information:
    Number of observations               9
    Fixed effects coefficients           4
    Random effects coefficients          0
    Covariance parameters                1

Formula:
    CNR ~ 1 + kV + Filter

Model fit statistics:
    AIC        BIC        LogLikelihood    Deviance
    -19.442    -18.456    14.721           -29.442 

Fixed effects coefficients (95% CIs):
    Name                 Estimate     SE           tStat       DF    pValue        Lower       Upper    
    '(Intercept)'             18.3      0.17533      104.37    5     1.5308e-09      17.849       18.751
    'kV'                  -0.10333    0.0019245     -53.694    5     4.2372e-08    -0.10828    -0.098386
    'Filter_Cu'          -0.033333      0.03849    -0.86603    5        0.42607    -0.13228     0.065608
    'Filter_Ti'                 -3      0.03849     -77.942    5     6.5868e-09     -3.0989      -2.9011

Random effects covariance parameters (95% CIs):
Group: Error
    Name             Estimate    Lower     Upper   
    'Res Std'        0.04714     0.0297    0.074821

Desired Outcome:

I want to be able to choose a reference group (e.g. 80 kV / Al filtration), and then quantify the significance of different trends relative to this (80kV/Al). I believe the reference is selected automatically in fitlme.m because I put it first in the table input. Are the following interpretations correct?

My interpretation of the output:

1. The CNR decreased with increasing kV for all filtration materials (P=4.2E-8)
2. Cu and Al achieve similar CNR (P=0.43), but Cu and Al achieved a higher CNR than Ti (P=6.6E-9)

Questions/Issues with current implementation:

1. How is the fixed effects coefficients for '(Intercept)' with P=1.53E-9 interpreted?
2. I only included fixed effects. Should the standard deviation of the ROI measurements somehow be incorporated into the random effects as well?
3. How do I incorporate the three independent measurements of CNR for three consecutive slices for a give kV/filter combination? Should I just add more rows to the table "T"? This would result in a total of 27 observations.

Thank you for your time,

AH

Answer 1

[Peng Li]

Also found this old post related to LME. Below is my opion for future readers.

My interpretation of the output:

1. The CNR decreased with increasing kV for all filtration materials (P=4.2E-8)

In the model, you don't have interaction item for kV and Filt. That means initially you don't have a hypothesis that the association between kV and CNR is modified by filteration material. If you have that hypothesis, you should include kV*Filt.

1. Cu and Al achieve similar CNR (P=0.43), but Cu and Al achieved a higher CNR than Ti (P=6.6E-9)

In your implementation, kV by default is continuous while Filt is categorical. For continuous variables, the default reference is 0. It doesn't matter too much if you don't have interaction item. But if you have that, the reference 0 may not make sense as the main effect of Filt will be the case that when kV=0. So you'd better center the kV by it's mean, or if you have a specific reference in your mind, you center your kV by it.

Questions/Issues with current implementation:

1. How is the fixed effects coefficients for '(Intercept)' with P=1.53E-9 interpreted?

Usually, it's not necessary to interpret the p value for Intercept.

1. I only included fixed effects. Should the standard deviation of the ROI measurements somehow be incorporated into the random effects as well?

You don't have repeated measurement in your current example. But if you want to have that in (next question), you should have random effect.

1. How do I incorporate the three independent measurements of CNR for three consecutive slices for a give kV/filter combination? Should I just add more rows to the table "T"? This would result in a total of 27 observations.

Using a long format, meaning that you stack the observations within a single column. But in the meantime you should have another column to indicate which three are belonging to the same image. And that variable (say image) should be used in your formula for random effect: CNR ~ kV_m*Filt + (1|image). Note in this formula I included interaction to test the hypothesis that there is interaction between kV and Filt. kV_m is a centered version of kV that you need to generate first. kV*Filt by default includes kV, Filt, kV*Filt. 1 for intercept is by default included. (1|image) indicate the random effect (random intercept for each image).

[Alfredezeo]

Thank you for your very helpful response Peng!

[Peng Li]

happy to be of help!