A New View of Statistics

© 1998 Will G Hopkins

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REPEATED MEASURES WITH PROC MIXED:
INDIVIDUAL DIFFERENCES, AND COVARIATES:

OUTPUT OF THE PROGRAM

Important stats are colored, in bold, or [commented] on.

                    WITHOUT individual differences                   1
                                            21:30 Monday, June 1, 1998
 
 
                         The MIXED Procedure
 
                       Class Level Information
 
           Class     Levels  Values
 
           ATHLETE       24  1 2 3 4 5 6 7 8 9 10 11 12 13
                             14 15 16 17 18 19 20 21 22 23
                             24
           GRP            2  control exptal
           TEST           3  1 2 3
 
 
                           Parameter Search
 
    COVP1     COVP2     COVP3     COVP4     COVP5     COVP6       RLL
 
  25.0000    1.0000    0.0000   25.0000    1.0000    0.0000  -148.138
 
 
                          Parameter Search
 
                            -2RLL  Objective
 
                         296.2764   174.9765
 

                    WITHOUT individual differences                   2
                                            21:30 Monday, June 1, 1998
 
 
                  REML Estimation Iteration History
 
          Iteration  Evaluations     Objective     Criterion
 
                  1            3  173.77952015    0.00180714
                  2            1  173.60840164    0.00004607
                  3            1  173.60432225    0.00000005
                  4            1  173.60431808    0.00000000
 
                      Convergence criteria met.
 
 
                Covariance Parameter Estimates (REML)
 
  Cov Parm  Subject  Group           Estimate    Std Error       Z
 
  LIN(1)    ATHLETE  GRP control   28.37564414   12.24929742    2.32
  LIN(2)    ATHLETE  GRP control    0.93713368    0.39057869    2.40
  LIN(3)    ATHLETE  GRP control    0.32209133    0.92661491    0.35
  LIN(1)    ATHLETE  GRP exptal    29.96635541   12.89193471    2.32
  LIN(2)    ATHLETE  GRP exptal     0.71480766    0.29577343    2.42
  LIN(3)    ATHLETE  GRP exptal     0.06505026    0.64523382    0.10
 

                    WITHOUT individual differences                   3
                                            21:30 Monday, June 1, 1998
 
 
                Covariance Parameter Estimates (REML)
 
                Pr > |Z|  Alpha     Lower    Upper
 
                  0.0205   0.05    4.3675   52.3838  [between variance]
                  0.0164   0.05    0.1716    1.7027  [within variance]
                  0.7281   0.05   -1.4940    2.1382  [indiff variance]
                  0.0201   0.05    4.6986   55.2341
                  0.0157   0.05    0.1351    1.2945
                  0.9197   0.05   -1.1996    1.3297
 
[Take the square root of the above variances to get standard
deviations. If the variance is negative, make it positive, take
the square root, then make the resulting SD negative.
Note that to get the usual between-subject SD, you have to add the
between and within variances, then take the square root. You can't
combine the lower and upper limits like that, though.]
 
                Model Fitting Information for DISTANCE
 
               Description                        Value
 
               Observations                     72.0000
               Res Log Likelihood              -147.452
               Akaike's Information Criterion  -153.452
               Schwarz's Bayesian Criterion    -160.021
               -2 Res Log Likelihood           294.9042
               PARMS Model LRT Chi-Square        1.3722
               PARMS Model LRT DF                5.0000
               PARMS Model LRT P-Value           0.9273
 

                    WITHOUT individual differences                   4
                                            21:30 Monday, June 1, 1998
 
 
                       Tests of Fixed Effects
 
              Source      NDF   DDF  Type III F  Pr > F
 
              GRP           1    22      671.94  0.0001
              TEST          2    44       23.39  0.0001
              GRP*TEST     2    44       17.78  0.0001  
[P value for the test of whether the pattern of change in the mean 
distance differs between the two groups. The other two tests aren't
informative.]
 
 
                      ESTIMATE Statement Results
[These are the comparisons for the difference between specific tests
between groups.]
 
    Parameter               Estimate    Std Error    DF       t
 
    grp*test 2-1            0.33082497    0.52471283    44    0.63
    grp*test 3-2            2.73163325    0.55460378    44    4.93
 
 
                     ESTIMATE Statement Results
 
                 Pr > |t|  Alpha     Lower   Upper
 
                   0.5316   0.05   -0.7267    1.3883
                   0.0001   0.05    1.6139    3.8494

                     WITH individual differences                     5
                                            21:30 Monday, June 1, 1998
 
 
                         The MIXED Procedure
 
                       Class Level Information
 
           Class     Levels  Values
 
           ATHLETE       24  1 2 3 4 5 6 7 8 9 10 11 12 13
                             14 15 16 17 18 19 20 21 22 23
                             24
           GRP            2  control exptal
           TEST           3  1 2 3
 
 
                  REML Estimation Iteration History
 
          Iteration  Evaluations     Objective     Criterion
 
                  0            1  311.73867656
                  1            3  193.00124620    1.20211524
                  2            2  191.15202472    0.00138111
                  3            2  191.05670678    0.00014911
                  4            1  191.04158591    0.00000149
                  5            1  191.04144279    0.00000000
 
                      Convergence criteria met.
 

                     WITH individual differences                     6
                                            21:30 Monday, June 1, 1998
 
 
                Covariance Parameter Estimates (REML)
 
  Cov Parm    Subject     Estimate     Std Error       Z  Pr > |Z|
 
  INTERCEPT   ATHLETE   29.48549791    9.00663783    3.27    0.0011
  V33         ATHLETE    3.87638105    2.25949020    1.72    0.0862
  Residual               0.92523945    0.22427315    4.13    0.0001
 
 
                Covariance Parameter Estimates (REML)
 
                Alpha    Lower    Upper
 
                 0.05   17.5337   59.7021  [between variance]
                 0.05    1.5992   19.1956  [indiff variance]
                 0.05    0.6055    1.5877  [within variance]
 
 
                Model Fitting Information for DISTANCE
 
               Description                        Value
 
               Observations                     72.0000
               Res Log Likelihood              -156.171
               Akaike's Information Criterion  -159.171
               Schwarz's Bayesian Criterion    -162.455
               -2 Res Log Likelihood           312.3413
 

                     WITH individual differences                     7
                                            21:30 Monday, June 1, 1998
 
 
                       Tests of Fixed Effects
 
              Source      NDF   DDF  Type III F  Pr > F
 
              GRP           1    33      661.84  0.0001
              TEST          2    33        9.31  0.0006
              GRP*TEST      2    33        6.43  0.0044 [See above.]
 
 
                      ESTIMATE Statement Results  [See above.]
 
    Parameter                 Estimate     Std Error    DF       t
 
    grp*test 2-1            0.33082497    0.55534958    33    0.60
    grp*test 3-2            2.46693105    0.79463508    33    3.10
 
 
                     ESTIMATE Statement Results
 
                 Pr > |t|  Alpha    Lower    Upper
 
                   0.5554   0.05   -0.7990    1.4607
                   0.0039   0.05    0.8502    4.0836

                        Including a covariate                        8
                                            21:30 Monday, June 1, 1998
 
 
                         The MIXED Procedure
 
                       Class Level Information
 
           Class     Levels  Values
 
           ATHLETE       24  1 2 3 4 5 6 7 8 9 10 11 12 13
                             14 15 16 17 18 19 20 21 22 23
                             24
           GRP            2  control exptal
           TEST           3  1 2 3
 
 
                           Parameter Search
 
    COVP1     COVP2     COVP3     COVP4     COVP5     COVP6       RLL
 
  25.0000    1.0000    0.0000   25.0000    1.0000    0.0000  -140.648
 
 
                          Parameter Search
 
                            -2RLL  Objective
 
                         281.2969   171.0243
 

                        Including a covariate                        9
                                            21:30 Monday, June 1, 1998
 
 
                  REML Estimation Iteration History
 
          Iteration  Evaluations     Objective     Criterion
 
                  1            2  170.74186801    0.00516738
                  2            1  170.22319240    0.00065213
                  3            1  170.16213382    0.00001957
                  4            1  170.16043232    0.00000002
                  5            1  170.16043046    0.00000000
 
                      Convergence criteria met.
 
 
                Covariance Parameter Estimates (REML)   [See above.]
 
  Cov Parm  Subject  Group            Estimate    Std Error       Z
 
  LIN(1)    ATHLETE  GRP control   25.02509370   11.35855124    2.20
  LIN(2)    ATHLETE  GRP control    0.92168749    0.39720935    2.32
  LIN(3)    ATHLETE  GRP control    0.46476550    0.99650477    0.47
  LIN(1)    ATHLETE  GRP exptal    21.11681713    9.58293562    2.20
  LIN(2)    ATHLETE  GRP exptal     0.76907737    0.32799116    2.34
  LIN(3)    ATHLETE  GRP exptal     0.11905897    0.73138682    0.16
 

                        Including a covariate                       10
                                            21:30 Monday, June 1, 1998
 
 
                Covariance Parameter Estimates (REML)
 
                Pr > |Z|  Alpha    Lower    Upper
 
                  0.0276   0.05    2.7627   47.2874  [between]
                  0.0203   0.05    0.1432    1.7002  [within]
                  0.6409   0.05   -1.4883    2.4179  [indiff]
                  0.0276   0.05    2.3346   39.8990
                  0.0190   0.05    0.1262    1.4119
                  0.8707   0.05   -1.3144    1.5526
 
 
                Model Fitting Information for DISTANCE
 
               Description                        Value
 
               Observations                     72.0000
               Res Log Likelihood              -140.217
               Akaike's Information Criterion  -146.217
               Schwarz's Bayesian Criterion    -152.500
               -2 Res Log Likelihood           280.4331
               PARMS Model LRT Chi-Square        0.8638
               PARMS Model LRT DF                5.0000
               PARMS Model LRT P-Value           0.9728
 

                        Including a covariate                       11
                                            21:30 Monday, June 1, 1998
 
 
                       Tests of Fixed Effects
 
           Source            NDF   DDF  Type III F  Pr > F
 
           INDIFF              1    20        0.21  0.6529
           GRP                 1    20      759.27  0.0001
           INDIFF*GRP          1    20        8.60  0.0082
           TEST                2    40       21.51  0.0001
           INDIFF*TEST         2    40        7.67  0.0015
           GRP*TEST            2    40       14.13  0.0001
           INDIFF*GRP*TEST     2    40        3.89  0.0287
[Only the bottom two tests are interesting.]
 
                      ESTIMATE Statement Results
 
    Parameter              Estimate     Std Error    DF       t
 
    grp*test 2-1            0.20591257    0.54910601    40    0.37
    grp*test 3-2            2.69952322    0.59558936    40    4.53
    indiff*grp*test 2-1    -0.43357590    0.61830657    40   -0.70
    indiff*grp*test 3-2     1.83445952    0.67451441    40    2.72
[In blue are the shifts in the mean between tests, for experimental group
relative to control group.]
[Interpretation for red: subjects who differ by one standard deviation 
of the covariate differ by 1.8 units (95%CI: 0.5 to 3.2) in throwing 
distance between tests 2 and 3 (i.e. after the experimental treatment), 
relative to the control group.  Green is the difference between tests 
1 and 2, which is due to noise, rather than to a real covariance.]
 
 
                     ESTIMATE Statement Results
 
                 Pr > |t|  Alpha    Lower    Upper
 
                   0.7096   0.05   -0.9039    1.3157

                        Including a covariate                       12
                                            21:30 Monday, June 1, 1998
 
 
                     ESTIMATE Statement Results
 
                 Pr > |t|  Alpha    Lower    Upper
 
                   0.0001   0.05    1.4958    3.9033
                   0.4872   0.05   -1.6832    0.8161
                   0.0096   0.05    0.4712    3.1977


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