Overview of this module Overview of this module Course 02429 Analysis of correlated data: Mixed Linear Models The repeated measurements setup 1 Module 11: Repeated measures I, simple methods Example: Activity of rats Separate analysis for each time–point 2 Example: rats data Per Bruun Brockhoff Analysis of summary statistic 3 DTU Compute Example: rats data Building 324 - room 220 Technical University of Denmark Random effects model 4 2800 Lyngby – Denmark Example: rats data e-mail: perbb@dtu.dk Pros and cons of simple approaches 5 Per Bruun Brockhoff (perbb@dtu.dk) Mixed Linear Models, Module 11 Fall 2014 1 / 18 Per Bruun Brockhoff (perbb@dtu.dk) Mixed Linear Models, Module 11 Fall 2014 2 / 18 Aim of this module The repeated measurements setup Aim of this module The repeated measurements setup Several “individuals” Several measurements on each individual Two measurements on the same individual might be correlated Present some simple methods for dealing with repeated measurements Might even be highly correlated if “close” and less correlated if “far Easy to use and a lot better than pretending to have independent apart” observations Typical example: 30 Useful even after more advanced models are presented 20 individuals from relevant population 20 Half get drug A and half get drug B See how to specify these models in R Y Measured every week for two months 10 To pretend all observations are independent 0 can lead to wrong conclusions 1 2 3 4 5 6 7 8 Week Per Bruun Brockhoff (perbb@dtu.dk) Mixed Linear Models, Module 11 Fall 2014 3 / 18 Per Bruun Brockhoff (perbb@dtu.dk) Mixed Linear Models, Module 11 Fall 2014 5 / 18
The repeated measurements setup Example: Activity of rats Separate analysis for each time–point Example: Activity of rats Separate analysis for each time–point Summary of experiment: 3 treatments: 1, 2, 3 (concentration) Select a fixed time point 10 cages per treatment The observations at that time (one from each individual) are 10 contiguous months independent The response is activity ( log( count ) of intersections of light beam Do a separate analysis for the observations at that time during 57 hours) This is not wrong, but (possibly) a lot of information is waisted This can be done for several time–points, but 10.5 Difficult to reach a coherent conclusion 10.0 Sub–tests are not independent log(count) 9.5 Tempting to select time–points that supports out preference Mass significance: If many tests are carried out at 5% level some might 9.0 be significant by chance. (Bonferroni correction: Use significance level 8.5 0 . 05 /n instead of 0 . 05 ) 1 2 3 4 5 6 7 8 9 10 Month Per Bruun Brockhoff (perbb@dtu.dk) Mixed Linear Models, Module 11 Fall 2014 6 / 18 Per Bruun Brockhoff (perbb@dtu.dk) Mixed Linear Models, Module 11 Fall 2014 8 / 18 Separate analysis for each time–point Example: rats data Analysis of summary statistic Separate analysis of rats data Analysis of summary statistic Choose a single measure to summarize the individual curves The model at each time–step is: This again reduces the data set to independent observations ε i ∼ i.i.d. N (0 , σ 2 ) , Popular choices of summary measures: lnc i = µ + α ( treatm i ) + ε i , i = 1 . . . 30 Average over time The result of the ten tests for no treatment effect: Slope in regression with time (or higher order polynomial coefficients) Total increase (last point minus first point) Area under curve (AUC) Month 1 2 3 4 5 6 7 8 9 10 4.70 7.29 Maximum or minimum point F–value 1.22 0.27 1.02 2.30 3.87 4.10 4.09 0.88 Good method with few and easily checked assumptions Compare with F 95%;2 , 27 = 3 . 35 or F 99 . 5%;2 , 27 = 6 . 49 if Bonferroni Information may be lost correction is used Important to choose a good summary measure Per Bruun Brockhoff (perbb@dtu.dk) Mixed Linear Models, Module 11 Fall 2014 9 / 18 Per Bruun Brockhoff (perbb@dtu.dk) Mixed Linear Models, Module 11 Fall 2014 11 / 18
Analysis of summary statistic Example: rats data Random effects model Rats data analyzed via summary measure Random effects model This model uses all observations instead of reducing to one observation per individual The log of the total activity is chosen as summary measure Add “individual” as a random effect lnTot = log( Total count ) Makes measurements on same individual correlated The one way ANOVA model becomes: Unfortunately equally correlated no matter if they are “close” or “far apart” ε i ∼ i.i.d. N (0 , σ 2 ) , lnTot i = µ + α ( treatm i ) + ε i , i = 1 . . . 30 Can be considered first step in modeling the actual covariance structure The P–value for no treatment effect in this summary model is 5.23% Usually only good for short series Notice the simplicity of the model and the relative few assumptions This model is also known as the split–plot model for repeated measurements (with “individuals” as main–plots and the single measurements as sub–plots) Per Bruun Brockhoff (perbb@dtu.dk) Mixed Linear Models, Module 11 Fall 2014 12 / 18 Per Bruun Brockhoff (perbb@dtu.dk) Mixed Linear Models, Module 11 Fall 2014 14 / 18 Random effects model Example: rats data Pros and cons of simple approaches Rats data analyzed via random effects approach Pros and cons of simple approaches The model can now be enhanced to: Separate analysis for each time–point + Not wrong lnc i = µ + α ( treatm i )+ β ( month i )+ γ ( treatm i , month i )+ d ( cage i )+ ε i , – Can be confusing The covariance structure of this model is: – Difficult to reach coherent conclusion – In general not very informative 0 , if cage i 1 � = cage i 2 and i 1 � = i 2 Analysis of summary statistic σ 2 cov ( y i 1 , y i 2 ) = , if cage i 1 = cage i 2 and i 1 � = i 2 d + Good method with few and easily checked assumptions σ 2 d + σ 2 , if i 1 = i 2 – Important to choose good summary measure(s) A split-plot structure: Random effects approach treatm is whole plot factor + Good method for short series month is sub plot factor + Uses all observations The P–value for the interaction term is 0 . 0059 . Significant, but is the – Usually not good for long series model too simple? Per Bruun Brockhoff (perbb@dtu.dk) Mixed Linear Models, Module 11 Fall 2014 15 / 18 Per Bruun Brockhoff (perbb@dtu.dk) Mixed Linear Models, Module 11 Fall 2014 17 / 18
Pros and cons of simple approaches Overview of this module The repeated measurements setup 1 Example: Activity of rats Separate analysis for each time–point 2 Example: rats data Analysis of summary statistic 3 Example: rats data Random effects model 4 Example: rats data Pros and cons of simple approaches 5 Per Bruun Brockhoff (perbb@dtu.dk) Mixed Linear Models, Module 11 Fall 2014 18 / 18
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