Load the sample data.
The data shows the deviations from the target quality characteristic measured from the products that five operators manufacture during three shifts: morning, evening, and night. This is a randomized block design, where the operators are the blocks. The experiment is designed to study the impact of the time of shift on the performance. The performance measure is the deviation of the quality characteristics from the target value. This is simulated data.
Shift and Operator are nominal variables.
Fit a linear mixed-effects model with a random intercept grouped by operator to assess if performance significantly differs according to the time of the shift. Use the restricted maximum likelihood method and 'effects' contrasts.
'effects' contrasts indicate that the coefficients sum to 0, and fitlme creates two contrast-coded variables in the fixed-effects design matrix, $X$1 and $X$2, where
and
The model corresponds to
where ~ N(0, ) and ~ N(0, ).
lme =
Linear mixed-effects model fit by REML
Model information:
Number of observations 15
Fixed effects coefficients 3
Random effects coefficients 5
Covariance parameters 2
Formula:
QCDev ~ 1 + Shift + (1 | Operator)
Model fit statistics:
AIC BIC LogLikelihood Deviance
58.913 61.337 -24.456 48.913
Fixed effects coefficients (95% CIs):
Name Estimate SE tStat DF pValue Lower Upper
{'(Intercept)' } 3.6525 0.94109 3.8812 12 0.0021832 1.6021 5.703
{'Shift_Evening'} -0.53293 0.31206 -1.7078 12 0.11339 -1.2129 0.14699
{'Shift_Morning'} -0.91973 0.31206 -2.9473 12 0.012206 -1.5997 -0.23981
Random effects covariance parameters (95% CIs):
Group: Operator (5 Levels)
Name1 Name2 Type Estimate Lower Upper
{'(Intercept)'} {'(Intercept)'} {'std'} 2.0457 0.98207 4.2612
Group: Error
Name Estimate Lower Upper
{'Res Std'} 0.85462 0.52357 1.395
Perform an -test to determine if all fixed-effects coefficients are 0.
ans =
ANOVA MARGINAL TESTS: DFMETHOD = 'RESIDUAL'
Term FStat DF1 DF2 pValue
{'(Intercept)'} 15.063 1 12 0.0021832
{'Shift' } 11.091 2 12 0.0018721
The -value for the constant term, 0.0021832, is the same as in the coefficient table in the lme display. The -value of 0.0018721 for Shift measures the combined significance for both coefficients representing Shift.
