Using Linear Mixed Models with two fixed factors and a random factor

I am trying to make a Linear Mixed Model to see if there is a statistical significance with two fixed factors ('ANTS and LABEL') with the ('STATE') as a random factor. Can anyone suggest to me how to proceed with such a model in Matlab?

Data file is attached alongside a sample code. Is the code right?

m = fitlme(data, 'RATE ~ (ANT*LABEL) + (1 | STATE)');

Fixed effects coefficients (95% CIs):
    Name                            Estimate       SE            tStat      DF     pValue        Lower         Upper     
    {'(Intercept)'         }           0.033027     0.0056879     5.8065    294    1.6533e-08      0.021832      0.044221
    {'ANTS'                }        -0.00012262    0.00065256    -0.1879    294       0.85108    -0.0014069     0.0011617
    {'LABEL_Poor nest'     }          -0.015508     0.0080189     -1.934    294      0.054076      -0.03129    0.00027342
    {'ANTS:LABEL_Poor nest'}         0.00036122     0.0009203     0.3925    294       0.69497      -0.00145     0.0021724

Answer 1

[the cyclist]

You can fit such a model using the fitlme function from the Statistics and Machine Learning Toolbox.

I think the following code fits the model you mentioned:

data = readtable('https://www.mathworks.com/matlabcentral/answers/uploaded_files/764231/Entry_rates.xlsx'); % You can just read in your local file

mdl = fitlme(data,'RATE ~ ANTS + LABEL + (1|STATE)')

mdl = 
Linear mixed-effects model fit by ML

Model information:
    Number of observations             298
    Fixed effects coefficients           3
    Random effects coefficients         11
    Covariance parameters                2

Formula:
    RATE ~ 1 + ANTS + LABEL + (1 | STATE)

Model fit statistics:
    AIC        BIC        LogLikelihood    Deviance
    -1158.9    -1140.4    584.45           -1168.9 

Fixed effects coefficients (95% CIs):
    Name                       Estimate      SE            tStat      DF     pValue        Lower          Upper     
    {'(Intercept)'    }          0.031647     0.0044726     7.0757    295    1.0864e-11       0.022844      0.040449
    {'ANTS'           }        5.8999e-05    0.00046026    0.12819    295       0.89809    -0.00084681    0.00096481
    {'LABEL_Poor nest'}         -0.012768     0.0039439    -3.2373    295     0.0013442      -0.020529    -0.0050059

Random effects covariance parameters (95% CIs):
Group: STATE (11 Levels)
    Name1                  Name2                  Type           Estimate      Lower    Upper
    {'(Intercept)'}        {'(Intercept)'}        {'std'}        7.5587e-18    NaN      NaN  

Group: Error
    Name               Estimate    Lower       Upper   
    {'Res Std'}        0.034041    0.031415    0.036887

But I strongly recommend you read the documentation carefully, to understand how to specify the model you want to fit.

[Hari krishnan]

In the model you have suggested, does the intercept means that their is a significant difference between the 'LABEL', while no significant difference between the 'ANTS' in each label?

I was wondering tha the output contains two P values for multiple labels..

[the cyclist]

You really should make sure you understand Wilkinson notiation, and how to specify the exact model you want.

The difference between your formula and mine is that your formula will include the interaction term ANTS*LABEL.

[Hari krishnan]

Thank you for your comment.

There is an interaction between the predictor variable (ANTS) and Category (LABEL) as they are related. But I should also get a P value for each 'LABEL' category as there are multiple groups in the each 'LABEL', right?

[the cyclist]

I have two questions, trying to clarify:

1. Is ANTS a continuous variable, or categorical?
2. Are you trying to predict RATE from ANTS and LABEL (with STATE as random)? Or are you trying to predict LABEL from ANTS (with STATE as random, and ignoring RATE)? Or something else?

[Hari krishnan]

1. ANTS is a categorical variable
2. I am trying to predict RATE from ANTS and LABEL (with STATE as random)

[the cyclist]

The first model I suggested treats ANTS as continuous numeric by default, so we need to convert it to categorical before putting it in the model.

This model has the interaction term as well:

data = readtable('https://www.mathworks.com/matlabcentral/answers/uploaded_files/764231/Entry_rates.xlsx'); % You can just read in your local file
data.ANTS = categorical(data.ANTS);
m = fitlme(data, 'RATE ~ (ANTS*LABEL) + (1 | STATE)')

m = 
Linear mixed-effects model fit by ML

Model information:
    Number of observations             298
    Fixed effects coefficients          32
    Random effects coefficients         11
    Covariance parameters                2

Formula:
    RATE ~ 1 + ANTS*LABEL + (1 | STATE)

Model fit statistics:
    AIC        BIC        LogLikelihood    Deviance
    -1102.9    -977.18    585.44           -1170.9 

Fixed effects coefficients (95% CIs):
    Name                               Estimate       SE          tStat        DF     pValue       Lower        Upper   
    {'(Intercept)'            }            0.03472    0.010729       3.2361    266    0.0013653     0.013595    0.055845
    {'ANTS_2'                 }        -0.00018056    0.014824     -0.01218    266      0.99029    -0.029369    0.029007
    {'ANTS_3'                 }         -0.0037793    0.014824     -0.25494    266      0.79897    -0.032967    0.025409
    {'ANTS_4'                 }          -0.005943    0.014824     -0.40089    266      0.68882    -0.035131    0.023245
    {'ANTS_5'                 }          -0.003487    0.014824     -0.23522    266      0.81422    -0.032675    0.025701
    {'ANTS_6'                 }          0.0032408    0.014824      0.21861    266      0.82712    -0.025947    0.032429
    {'ANTS_7'                 }         -0.0024946    0.014824     -0.16828    266      0.86649    -0.031683    0.026693
    {'ANTS_8'                 }         -0.0020313    0.014824     -0.13703    266      0.89111    -0.031219    0.027157
    {'ANTS_9'                 }         -0.0075855    0.014824      -0.5117    266      0.60929    -0.036773    0.021602
    {'ANTS_10'                }         -0.0038417    0.014824     -0.25915    266      0.79572     -0.03303    0.025346
    {'ANTS_11'                }         -0.0023158    0.015589     -0.14855    266      0.88202    -0.033009    0.028378
    {'ANTS_12'                }         -0.0037099     0.01672     -0.22188    266      0.82458     -0.03663    0.029211
    {'ANTS_13'                }         -0.0022144     0.01672     -0.13244    266      0.89474    -0.035135    0.030706
    {'ANTS_14'                }         0.00054688     0.01752     0.031214    266      0.97512     -0.03395    0.035043
    {'ANTS_15'                }         -0.0039004     0.01752     -0.22262    266        0.824    -0.038397    0.030596
    {'ANTS_16'                }         -0.0043122    0.018583     -0.23205    266      0.81668    -0.040901    0.032277
    {'LABEL_Poor nest'        }          -0.011158    0.014824     -0.75269    266       0.4523    -0.040346     0.01803
    {'ANTS_2:LABEL_Poor nest' }         -0.0077307    0.020714     -0.37322    266      0.70928    -0.048514    0.033053
    {'ANTS_3:LABEL_Poor nest' }         9.4789e-05    0.020714    0.0045762    266      0.99635    -0.040689    0.040878
    {'ANTS_4:LABEL_Poor nest' }         -0.0033669    0.020965      -0.1606    266      0.87253    -0.044645    0.037911
    {'ANTS_5:LABEL_Poor nest' }         -0.0050268    0.020714     -0.24268    266      0.80844     -0.04581    0.035757
    {'ANTS_6:LABEL_Poor nest' }         -0.0076487    0.020714     -0.36926    266      0.71223    -0.048432    0.033135
    {'ANTS_7:LABEL_Poor nest' }         -0.0012082    0.020714     -0.05833    266      0.95353    -0.041992    0.039575
    {'ANTS_8:LABEL_Poor nest' }         0.00011595    0.020714    0.0055978    266      0.99554    -0.040668      0.0409
    {'ANTS_9:LABEL_Poor nest' }          0.0022824    0.020714      0.11019    266      0.91234    -0.038501    0.043066
    {'ANTS_10:LABEL_Poor nest'}         0.00078623    0.020714     0.037957    266      0.96975    -0.039997     0.04157
    {'ANTS_11:LABEL_Poor nest'}         -0.0023263    0.021807     -0.10667    266      0.91513    -0.045264    0.040611
    {'ANTS_12:LABEL_Poor nest'}        -0.00027695    0.023423    -0.011824    266      0.99058    -0.046396    0.045842
    {'ANTS_13:LABEL_Poor nest'}         -0.0039373    0.023423     -0.16809    266      0.86664    -0.050056    0.042182
    {'ANTS_14:LABEL_Poor nest'}         0.00067151    0.024566     0.027335    266      0.97821    -0.047696    0.049039
    {'ANTS_15:LABEL_Poor nest'}          0.0014948    0.024566      0.06085    266      0.95152    -0.046873    0.049863
    {'ANTS_16:LABEL_Poor nest'}          0.0053685    0.026081      0.20584    266      0.83707    -0.045983     0.05672

Random effects covariance parameters (95% CIs):
Group: STATE (11 Levels)
    Name1                  Name2                  Type           Estimate    Lower    Upper
    {'(Intercept)'}        {'(Intercept)'}        {'std'}        0           NaN      NaN  

Group: Error
    Name               Estimate    Lower       Upper   
    {'Res Std'}        0.033928    0.031311    0.036764

Does that align better with the output you would expect, and can interpret?