Repeated measures taken over time

1
Repeated measures taken over time

• Repeated measures taken over time
• We assume that we have designed an experiment
  according to one of the standard designs (CRD,
  RCB, Latin Square, Split Plot).
• For each experimental unit we take a measurement
  of some quantity of interest at time1, time2,
  etc. We want to know the effect of both
  treatments and time on the response. For
  instance, we may wish to know if there is a
  treatment by time interaction.

2
Not all experiments involving time have time as a
repeated measures factor.

• Suppose we wish to compare two types of packaging
  material that are used for hotdogs, and we wish
  to compare the packaging materials over time.
• In one experiment, each package is observed at
  one week, two weeks, and three weeks to determine
  whether the color of the hotdogs has changed.
  Time is a repeated measures factor because each
  experimental unit (the package) is measured more
  at three different times.
• In other experiment, we are interested in the
  bacteria counts of the hotdogs. Some of the
  packages are opened and tested for bacteria at
  one week. Other packages are open and tested for
  bacteria at two weeks. And other packages are
  opened and tested for bacteria at three weeks.
  This is not a repeated measures taken over time
  because each package is measured at only one time.

3
How do we make use of the information obtained
over time?

• There are many possible answers to this question
  depending on the objectives of the researcher.
• We may wish to know how big a change there has
  been from the first time the unit is measured
  until the last time. For instance, if we measure
  weights of animals at time1, time2, etc., the
  difference between the first and last time
  readings is just the weight gain.
• We may wish to know the peak response or the
  maximum reading we get over time. For instance,
  if we give a patient a medicine, we may wish to
  know what is the maximum amount of medicine that
  gets absorbed into the bloodstream.
• We may wish to know the time at which the
  response hits the maximum. For instance, we may
  measure heart rate over time as a person
  exercises, and we may wish to know how much time
  it takes for the heart rate to reach its maximum
  value.
• We may wish to compare treatments at each time.
  For instance, we may wish to compare the heights
  of corn plants at week1, week2 ,week3, etc.

4
How do we make use of the information obtained
over time?

• We may wish to know if the rate of change over
  time is the same or different for the treatments,
  in other words, is there a time by treatment
  interaction. For instance, the wear on two
  brands of automobile tires may be
• measured at 1000 miles, 2000 miles, etc., and we
  would like to know if one brand wears out faster
  than the other. (Here time is measured in miles
  rather than days or hours).

5
Analysis

• maximum and the time to reach the maximum as
  response variables. In some cases, the
  information taken over time can be reduced to a
  single meaningful response variable, and then the
  data would be analyzed as appropriate for the
  design in question.
• For instance, suppose that we assign animals
  randomly to one of two treatments, and we measure
  the weights at time1, time2, and time3. If total
  weight gain is the response of interest, then we
  would take the difference between the weights at
  time3 and time1, and use this as a response
  variable, where the design is completely random.
  In this case, we might form more that one
  response variable, for instance (time3 weight
  time1 weight) or (time3 weight time2 weight),
  or (time2 weight time1 weight).
• If the maximum response over time, or the time to
  reach the maximum response over time are of
  interest, then we could use the

6

• In some cases, we may be able to get a
  satisfactory analysis where time is regarded as a
  subplot factor.
• For instance, suppose we have two packaging
  methods and 10 randomly selected packages for
  each method, and suppose we have a color
  measurement on each package at one week, two
  weeks and three weeks. Here we could consider
  week as the subplot factor, and packaging method
  as the whole plot factor with the whole plot
  design being completely random. The sources of
  variability and degrees of freedom would be
• Source d.f.
• Method 1
• WP Error 18
• Week 2
• MethodWeek 2
• SP Error 36

7

• The split-model for measurements taken over time
  implies that observations over time within a unit
  have constant correlation. That is the
  correlation between time1 and time2 is the same
  as between time1 and time3, is the same as
  between time2 and time3. This might be o.k. when
  the subplots are randomized, but we cant
  randomly assign time so the split plot model
  may not be appropriate.
• What can we do to test for treatment effects,
  time effect, and treatment by time interaction
  when time is a repeated measures factor and split
  plot model not appropriate?
• PROC MIXED allows for several types of
  correlation structures using the REPEATED
  statement
• PROC GLM allows for a multivariate analysis of
  variance.

8

• First order autoregressive model.
• In this model, measurements that are farther
  apart in time are less correlated than
  observations close together in time.
• If the correlation for observations one unit in
  time apart is r, then for the first order
  autoregressive model the correlation two in time
  units apart is r2, and three units in time apart
  is r3, etc.
• Again suppose we have 2 packaging methods and 10
  packages per treatment, and we measure each
  package at weeks 1, 2, and 3. Here are the SAS
  statements for the AR(1) model
• PROC MIXED
• CLASS PACKAGE METHOD WEEK
• MODEL RESPONSE METHOD WEEK METHODWEEK/DDFM
  SATTERTH
• RANDOM PACKAGE(METHOD)
• REPEATED/ TYPE AR(1) SUB PACKAGE(WEEK)
• LSMEANS METHOD WEEK METHODWEEK/PDIFF

9

• If there are convergence problems, it may be that
  PACKAGE(METHOD) is negligible, and could be
  dropped from the statements.
• Unstructured covariance
• If we do not want to make any assumptions about
  the correlations we may use an unstructured
  correlation. The statements are the same as
  above except that we would use TYPE UN.
• GLM analysis.
• To do this analysis, we must have the different
  measurements on time on the same data line. For
  instance in the packaging problem above, we would
  have the week1, week2, and week3 responses on the
  same data line. Code for the above setup is as
  follows.

10

• PROC GLM
• CLASS METHOD
• MODEL WEEK1 WEEK2 WEEK3 METHOD
• REPEATED WEEK
• LSMEANS METHOD WEEK METHODWEEK/STDERR PDIFF
• Note that WEEK1, WEEK2, etc. (which are
• names given to the response variables at
  different times) are arbitrary names, and WEEK
  (which is a name given to the time variable) is
  an arbitrary name. For instance we could use W1,
  W2, etc instead of WEEK1, WEEK2, and TIME instead
  of WEEK.
• The data on the next page are pH measurements
  taken on beef carcasses at 6 different times
  under two different treatments. The measurements
  over time are repeated measures. I show both a
  GLM and a MIXED analysis using the AR(1).

11

• obs id trt t1 t2 t3
  t4 t5 t6
• 1 1 1 6.81 6.16 5.92
  5.86 5.80 5.39
• 2 2 1 6.68 6.30 6.12
  5.71 6.09 5.28
• 3 3 1 6.34 6.22 5.90
  5.38 5.20 5.46
• 4 4 1 6.68 6.24 5.83
  5.49 5.37 5.43
• 5 5 1 6.79 6.28 6.23
  5.85 5.56 5.38
• 6 6 1 6.85 6.51 5.95
  6.06 6.31 5.39
• 7 7 2 6.64 5.91 5.59
  5.41 5.24 5.23
• 8 8 2 6.57 5.89 5.32
  5.41 5.32 5.30
• 9 9 2 6.84 6.01 5.34
  5.31 5.38 5.45
• 10 10 2 6.71 5.60 5.29
  5.37 5.26 5.41
• 11 11 2 6.58 5.63 5.38
  5.44 5.17 5.62
• 12 12 2 6.68 6.04 5.62
  5.31 5.41 5.44

12

• The GLM Procedure
• Class Level
  Information
• Class
  Levels Values
• trt
  2 1 2
• Number of
  observations 12
• The GLM
  Procedure
• __________________________________________________
  ________________________________________
• Dependent Variable t1
• Sum
  of
• Source DF
  Squares Mean Square F Value Pr gt F
• Model 1
  0.00140833 0.00140833 0.06 0.8064
• Error 10
  0.22228333 0.02222833
• Corrected Total 11
  0.22369167
• R-Square Coeff Var
  Root MSE t1 Mean
• 0.006296 2.231633
  0.149092 6.680833
• Source DF Type III
  SS Mean Square F Value Pr gt F
• trt 1
  0.00140833 0.00140833 0.06 0.8064
• __________________________________________________
  _______________________________________

13

• Dependent Variable t2
• Sum
  of
• Source DF Squares Mean Square
  F Value Pr gt F
• Model 1 0.57640833 0.57640833
  23.01 0.0007
• Error 10 0.25048333 0.02504833
• Corrected Total 11 0.82689167
• R-Square Coeff Var
  Root MSE t2 Mean
• 0.697078 2.609149
  0.158267 6.065833
• Source DF Type III SS Mean
  Square F Value Pr gt F
• trt 1 0.57640833
  0.57640833 23.01 0.0007
• __________________________________________________
  __________

14

• Dependent Variable t3
• Sum of
• Source DF Squares Mean Square
  F Value Pr gt F
• Model 1 0.96900833
  0.96900833 44.37 lt.0001
• Error 10 0.21841667
  0.02184167
• Total 11 1.18742500
• R-Square Coeff Var
  Root MSE t3 Mean
• 0.816059 2.589387
  0.147789 5.707500
• Source DF Type III SS Mean Square
  F Value Pr gt F
• trt 1 0.96900833
  0.96900833 44.37 lt.0001

15

• Dependent Variable t4
• Sum
  of
• Source DF Squares Mean Square
  F Value Pr gt F
• Model 1 0.36750000
  0.36750000 10.95 0.0079
• Error 10 0.33570000
  0.03357000
• Total 11 0.70320000
• R-Square Coeff Var
  Root MSE t4 Mean
• 0.522611 3.301282
  0.183221 5.550000
• Source DF Type III SS Mean
  Square F Value Pr gt F
• trt 1 0.36750000
  0.36750000 10.95 0.0079
• __________________________________________________
  ______

16

• Dependent Variable t5
• Sum
  of
• Source DF
  Squares Mean Square F Value Pr gt F
• Model 1
  0.54187500 0.54187500 5.70 0.0381
• Error 10
  0.95081667 0.09508167
• Corrected Total 11
  1.49269167
• R-Square Coeff Var
  Root MSE t5 Mean
• 0.363019 5.597092
  0.308353 5.509167
• Source DF Type III
  SS Mean Square F Value Pr gt F
• trt 1
  0.54187500 0.54187500 5.70 0.0381
• __________________________________________________
  _____________

17

• Dependent Variable t6
• Sum of
• Source DF Squares Mean
  Square F Value Pr gt F
• Model 1 0.00120000
  0.00120000 0.11 0.7477
• Error 10 0.10976667
  0.01097667
• Total 11 0.11096667
• R-Square Coeff Var
  Root MSE t6 Mean
• 0.010814 1.940777
  0.104770 5.398333
• Source DF Type III SS
  Mean Square F Value Pr gt F
• trt 1 0.00120000
  0.00120000 0.11 0.7477
• __________________________________________________
  _____________

18

• The GLM Procedure
• Repeated Measures
  Analysis of Variance
• Repeated Measures
  Level Information
• Dependent Variable t1 t2
  t3 t4 t5 t6
• Level of time 1 2
  3 4 5 6
• __________________________________________________
  _______________
• Manova Test Criteria and Exact F
  Statistics for the Hypothesis of no time Effect
• H Type III SSCP
  Matrix for time
• E Error
  SSCP Matrix
• S1
  M1.5 N2
• Statistic Value
  F Value Num DF Den DF Pr gt F
• Wilks' Lambda
  0.00700050 170.22 5 6
  lt.0001
• Pillai's Trace
  0.99299950 170.22 5 6
  lt.0001
• Hotelling-Lawley Trace 141.84690886
  170.22 5 6 lt.0001
• Roy's Greatest Root 141.84690886
  170.22 5 6 lt.0001
• __________________________________________________
  _______________

19

• Manova Test Criteria and Exact F Statistics for
  the Hypothesis of no timetrt Effect
• H Type III SSCP
  Matrix for timetrt
• E Error
  SSCP Matrix
• S1
  M1.5 N2
• Statistic Value F
  Value Num DF Den DF Pr gt F
• Wilks' Lambda 0.18817740 5.18
  5 6 0.0348
• Pillai's Trace 0.81182260 5.18
  5 6 0.0348
• Hotelling-L Trace 4.31413429 5.18
  5 6 0.0348
• Roy's GRoot 4.31413429 5.18
  5 6 0.0348
• __________________________________________________
  ________

20

• The GLM Procedure
• Repeated Measures
  Analysis of Variance
• Tests of Hypotheses for
  Between Subjects Effects
• Source F Type III SS Mean
  Square F Value Pr gt F
• trt 1 1.59013889
  1.59013889 20.37 0.0011
• Error 10 0.78058889
  0.07805889
• The GLM
  Procedure
• Repeated Measures
  Analysis of Variance
• Univariate Tests of Hypotheses
  for Within Subject Effects
• 
  
  Adj Pr gt F
• Source DF Type III SS Mean Square F
  Value Pr gt F G - G H - F
• time 5 13.93719444 2.78743889
  106.64 lt.0001 lt.0001 lt.0001
• timetrt 5 0.86726111 0.17345222
  6.64 lt.0001 0.0020 0.0002
• Error(time) 50 1.30687778
  0.02613756
• Greenhouse-Geisser
  Epsilon 0.5538
• Huynh-Feldt Epsilon
  0.8636
• __________________________________________________
  _______________

21

• The Mixed Procedure
• Model
  Information
• Data Set
  WORK.TWO
• Dependent Variable
  ph
• Covariance Structures
  Variance Components,
• 
  Autoregressive
• Subject Effect
  id(trt)
• Estimation Method
  REML
• Residual Variance Method
  Profile
• Fixed Effects SE Method
  Model-Based
• Degrees of Freedom Method
  Satterthwaite
• Class Level
  Information
• Class Levels
  Values
• trt 2
  1 2
• time 6 1 2
  3 4 5 6
• id 12 1
  2 3 4 5 6 7 8 9 10 11 12

22

• The Mixed Procedure
• Covariance
  Parameter Estimates
• Cov Parm
  Subject Estimate
• id(trt)
  0.008373
• AR(1)
  id(trt) 0.01935
• Residual
  0.02635
• __________________________________________________
  ________
• Type 3 Tests of
  Fixed Effects
• Num
  Den
• Effect DF
  DF F Value Pr gt F
• trt 1
  9.72 20.53 0.0012
• time 5
  27.9 104.32 lt.0001
• trttime 5
  27.9 6.60 0.0004
• __________________________________________________
  ________

23

• SAS STATEMENTS FOR PREVIOUS ANALYSIS
• proc print
• proc glm
• class trt
• model t1 t2 t3 t4 t5 t6 trt
• repeated time
• These statements take the time data that are on
  the same line and put them on
• Separate lines.
• data two
• set one
• ph t1 time 1 output
• ph t2 time 2 output
• ph t3 time 3 output
• ph t4 time 4 output
• ph t5 time 5 output
• ph t6 time 6 output
• run

24

• proc mixed
• class trt time id
• model ph trt time trttime/ddfm satterth
• random id(trt)
• repeated/ type ar(1) sub id(trt)
• run