Analysis All Participants
Null (unconditional) model
d.Null <- lme(DV~1,random=~1|Case,data=data,
control=list(opt="optim"),na.action="na.omit")
coefficients(d.Null)
## (Intercept)
## 1 77.59026
## 2 87.60157
## 3 60.75968
## Approximate 95% confidence intervals
##
## Fixed effects:
## lower est. upper
## (Intercept) 58.50907 75.31717 92.12527
## attr(,"label")
## [1] "Fixed effects:"
##
## Random Effects:
## Level: Case
## lower est. upper
## sd((Intercept)) 5.046643 13.71001 37.24542
##
## Within-group standard error:
## lower est. upper
## 3.816190 5.290175 7.333481
## Linear mixed-effects model fit by REML
## Data: data
## AIC BIC logLik
## 140.1591 143.1463 -67.07954
##
## Random effects:
## Formula: ~1 | Case
## (Intercept) Residual
## StdDev: 13.71001 5.290175
##
## Fixed effects: DV ~ 1
## Value Std.Error DF t-value p-value
## (Intercept) 75.31717 8.000347 18 9.414238 0
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -1.68419420 -0.60105429 -0.06053865 0.36288778 2.12475331
##
## Number of Observations: 21
## Number of Groups: 3
Model with random intercept and change in level (with an autocorrelation component)
d.Level <- lme(DV~1+Phase,random=~1|Case,data=data,
correlation=corAR1(form=~1|Case),control=list(opt="optim"),na.action="na.omit")
coefficients(d.Level)
## (Intercept) Phase
## 1 75.40222 4.712859
## 2 86.22450 4.712859
## 3 60.11892 4.712859
## Approximate 95% confidence intervals
##
## Fixed effects:
## lower est. upper
## (Intercept) 56.9759896 73.915213 90.854437
## Phase -0.5146241 4.712859 9.940342
## attr(,"label")
## [1] "Fixed effects:"
##
## Random Effects:
## Level: Case
## lower est. upper
## sd((Intercept)) 4.811975 13.44052 37.54125
##
## Correlation structure:
## lower est. upper
## Phi -0.4506453 0.5158382 0.9256066
## attr(,"label")
## [1] "Correlation structure:"
##
## Within-group standard error:
## lower est. upper
## 2.396852 5.111685 10.901517
## Linear mixed-effects model fit by REML
## Data: data
## AIC BIC logLik
## 130.0597 134.7819 -60.02985
##
## Random effects:
## Formula: ~1 | Case
## (Intercept) Residual
## StdDev: 13.44052 5.111685
##
## Correlation Structure: AR(1)
## Formula: ~1 | Case
## Parameter estimate(s):
## Phi
## 0.5158382
## Fixed effects: DV ~ 1 + Phase
## Value Std.Error DF t-value p-value
## (Intercept) 73.91521 8.028770 17 9.206294 0.0000
## Phase 4.71286 2.477697 17 1.902113 0.0742
## Correlation:
## (Intr)
## Phase -0.138
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -1.6176497 -0.7255759 -0.2175843 0.3653938 1.4023217
##
## Number of Observations: 21
## Number of Groups: 3
Model with random intercept and change in slope (with an autocorrelation component)
d.Time <- lme(DV~1+Time_CTR,random=~1|Case,data=data,
correlation=corAR1(form=~1|Case),control=list(opt="optim"),na.action="na.omit")
# Rename variables and use only complete cases
coefficients(d.Time)
## (Intercept) Time_CTR
## 1 78.51794 1.318265
## 2 89.70002 1.318265
## 3 64.97939 1.318265
## Approximate 95% confidence intervals
##
## Fixed effects:
## lower est. upper
## (Intercept) 61.1423304 77.732447 94.322564
## Time_CTR -0.2819025 1.318265 2.918433
## attr(,"label")
## [1] "Fixed effects:"
##
## Random Effects:
## Level: Case
## lower est. upper
## sd((Intercept)) 4.414208 12.95666 38.0306
##
## Correlation structure:
## lower est. upper
## Phi -0.3561357 0.6619135 0.9614602
## attr(,"label")
## [1] "Correlation structure:"
##
## Within-group standard error:
## lower est. upper
## 2.229472 5.923911 15.740372
## Linear mixed-effects model fit by REML
## Data: data
## AIC BIC logLik
## 132.0507 136.7728 -61.02533
##
## Random effects:
## Formula: ~1 | Case
## (Intercept) Residual
## StdDev: 12.95666 5.923911
##
## Correlation Structure: AR(1)
## Formula: ~1 | Case
## Parameter estimate(s):
## Phi
## 0.6619135
## Fixed effects: DV ~ 1 + Time_CTR
## Value Std.Error DF t-value p-value
## (Intercept) 77.73245 7.863302 17 9.885471 0.0000
## Time_CTR 1.31827 0.758440 17 1.738128 0.1003
## Correlation:
## (Intr)
## Time_CTR 0.097
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -1.7712715 -0.5522964 -0.1364063 0.2555402 0.9086972
##
## Number of Observations: 21
## Number of Groups: 3
Model with random intercept and change in level/slope in Tx (with autocorrelation)
d.Model <- lme(DV~1+Phase+Time_PhaseCTR,random=~1|Case,data=data,
correlation=corAR1(form=~1|Case),control=list(opt="optim"),na.action="na.omit")
# Rename variables and use only complete cases
coefficients(d.Model)
## (Intercept) Phase Time_PhaseCTR
## 1 74.39644 6.007971 3.5235
## 2 85.33731 6.007971 3.5235
## 3 58.86693 6.007971 3.5235
## Approximate 95% confidence intervals
##
## Fixed effects:
## lower est. upper
## (Intercept) 55.9934443 72.866894 89.740344
## Phase 1.4569318 6.007971 10.559010
## Time_PhaseCTR 0.2448482 3.523500 6.802152
## attr(,"label")
## [1] "Fixed effects:"
##
## Random Effects:
## Level: Case
## lower est. upper
## sd((Intercept)) 4.930429 13.48576 36.88641
##
## Correlation structure:
## lower est. upper
## Phi -0.3135413 0.4175362 0.8378386
## attr(,"label")
## [1] "Correlation structure:"
##
## Within-group standard error:
## lower est. upper
## 2.437527 4.253879 7.423706
## Linear mixed-effects model fit by REML
## Data: data
## AIC BIC logLik
## 124.6657 130.008 -56.33286
##
## Random effects:
## Formula: ~1 | Case
## (Intercept) Residual
## StdDev: 13.48576 4.253879
##
## Correlation Structure: AR(1)
## Formula: ~1 | Case
## Parameter estimate(s):
## Phi
## 0.4175362
## Fixed effects: DV ~ 1 + Phase + Time_PhaseCTR
## Value Std.Error DF t-value p-value
## (Intercept) 72.86689 7.959530 16 9.154673 0.0000
## Phase 6.00797 2.146813 16 2.798554 0.0129
## Time_PhaseCTR 3.52350 1.546603 16 2.278219 0.0368
## Correlation:
## (Intr) Phase
## Phase -0.123
## Time_PhaseCTR -0.039 0.182
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -1.64953689 -0.56590973 -0.02502208 0.52730007 1.39051909
##
## Number of Observations: 21
## Number of Groups: 3
Results
| Autocorrelation |
.418 |
|
| Level |
6.01 |
.013 |
| Slope |
3.52 |
.037 |
Interpretation: The effect size isnโt particularly large, but the p-value is significant, which is helpful.The p-value for the change in slope is also significant.
SAJE
Null (unconditional) model
d.Null_saje <- lme(DV~1,random=~1|Case,data=saje,
control=list(opt="optim"),na.action="na.omit")
coefficients(d.Null_saje)
## (Intercept)
## 1 77.16667
#intervals(d.Null_saje)
summary(d.Null_saje)
## Linear mixed-effects model fit by REML
## Data: saje
## AIC BIC logLik
## 59.26448 58.09279 -26.63224
##
## Random effects:
## Formula: ~1 | Case
## (Intercept) Residual
## StdDev: 45.2989 41.60969
##
## Fixed effects: DV ~ 1
## Value Std.Error DF t-value p-value
## (Intercept) 77.16667 48.37924 5 1.595037 0.1716
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -1.01338564 -0.86918847 -0.07610406 0.89722681 1.07747327
##
## Number of Observations: 6
## Number of Groups: 1
Model with random intercept and change in level (with an autocorrelation component)
d.Level_saje <- lme(DV~1+Phase,random=~1|Case,data=saje,
correlation=corAR1(form=~1|Case),control=list(opt="optim"),na.action="na.omit")
coefficients(d.Level_saje)
## (Intercept) Phase
## 1 48.84638 64.99212
#intervals(d.Level_saje)
summary(d.Level_saje)
## Linear mixed-effects model fit by REML
## Data: saje
## AIC BIC logLik
## 38.81218 35.74365 -14.40609
##
## Random effects:
## Formula: ~1 | Case
## (Intercept) Residual
## StdDev: 20.01381 18.38386
##
## Correlation Structure: AR(1)
## Formula: ~1 | Case
## Parameter estimate(s):
## Phi
## 0.8831458
## Fixed effects: DV ~ 1 + Phase
## Value Std.Error DF t-value p-value
## (Intercept) 48.84638 26.037559 4 1.875997 0.1339
## Phase 64.99212 8.823632 4 7.365688 0.0018
## Correlation:
## (Intr)
## Phase -0.169
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -0.7531812 -0.5216787 -0.3724124 0.1038700 0.4439493
##
## Number of Observations: 6
## Number of Groups: 1
Model with random intercept and change in slope (with an autocorrelation component)
d.Time_saje <- lme(DV~1+Time_CTR,random=~1|Case,data=saje,
correlation=corAR1(form=~1|Case),control=list(opt="optim"),na.action="na.omit")
# Rename variables and use only complete cases
coefficients(d.Time_saje)
## (Intercept) Time_CTR
## 1 87.41363 17.53902
#intervals(d.Time_saje)
summary(d.Time_saje)
## Linear mixed-effects model fit by REML
## Data: saje
## AIC BIC logLik
## 49.61166 46.54313 -19.80583
##
## Random effects:
## Formula: ~1 | Case
## (Intercept) Residual
## StdDev: 28.65683 26.32298
##
## Correlation Structure: AR(1)
## Formula: ~1 | Case
## Parameter estimate(s):
## Phi
## 0.5043509
## Fixed effects: DV ~ 1 + Time_CTR
## Value Std.Error DF t-value p-value
## (Intercept) 87.41363 33.10484 4 2.640509 0.0575
## Time_CTR 17.53902 7.20741 4 2.433472 0.0717
## Correlation:
## (Intr)
## Time_CTR 0.109
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -1.1349249 -0.4985988 0.1654783 0.4591566 0.6301099
##
## Number of Observations: 6
## Number of Groups: 1
Model with random intercept and change in level/slope in Tx (with autocorrelation)
d.Model_saje <- lme(DV~1+Phase+Time_PhaseCTR,random=~1|Case,data=saje,
correlation=corAR1(form=~1|Case),control=list(opt="optim"),na.action="na.omit")
# Rename variables and use only complete cases
coefficients(d.Model_saje)
## (Intercept) Phase Time_PhaseCTR
## 1 38.54159 76.80274 9.14747
#intervals(d.Model_saje)
summary(d.Model_saje)
## Linear mixed-effects model fit by REML
## Data: saje
## AIC BIC logLik
## 31.34977 25.94145 -9.674887
##
## Random effects:
## Formula: ~1 | Case
## (Intercept) Residual
## StdDev: 4.223661 3.87968
##
## Correlation Structure: AR(1)
## Formula: ~1 | Case
## Parameter estimate(s):
## Phi
## -0.8263351
## Fixed effects: DV ~ 1 + Phase + Time_PhaseCTR
## Value Std.Error DF t-value p-value
## (Intercept) 38.54159 4.299702 3 8.96378 0.0029
## Phase 76.80274 1.181434 3 65.00810 0.0000
## Time_PhaseCTR 9.14747 1.368994 3 6.68189 0.0068
## Correlation:
## (Intr) Phase
## Phase -0.138
## Time_PhaseCTR 0.032 -0.024
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -0.91285489 -0.62326334 -0.09516797 0.60735881 1.40692384
##
## Number of Observations: 6
## Number of Groups: 1
Results
| Autocorrelation |
-0.826 |
|
| Level |
76.802 |
.000 |
| Slope |
9.147 |
.0068 |
LELI
Null (unconditional) model
d.Null_leli <- lme(DV~1,random=~1|Case,data=leli,
control=list(opt="optim"),na.action="na.omit")
coefficients(d.Null_leli)
## (Intercept)
## 1 64.375
#intervals(d.Null_leli)
summary(d.Null_leli)
## Linear mixed-effects model fit by REML
## Data: leli
## AIC BIC logLik
## 78.02054 77.85827 -36.01027
##
## Random effects:
## Formula: ~1 | Case
## (Intercept) Residual
## StdDev: 33.71566 35.76086
##
## Fixed effects: DV ~ 1
## Value Std.Error DF t-value p-value
## (Intercept) 64.375 36.00835 7 1.78778 0.117
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -1.0451370 -0.7515199 -0.3740122 0.7655017 1.6952891
##
## Number of Observations: 8
## Number of Groups: 1
Model with random intercept and change in level (with an autocorrelation component)
d.Level_leli <- lme(DV~1+Phase,random=~1|Case,data=leli,
correlation=corAR1(form=~1|Case),control=list(opt="optim"),na.action="na.omit")
coefficients(d.Level_leli)
## (Intercept) Phase
## 1 40.22831 64.41959
#intervals(d.Level_leli)
summary(d.Level_leli)
## Linear mixed-effects model fit by REML
## Data: leli
## AIC BIC logLik
## 61.31881 60.2776 -25.6594
##
## Random effects:
## Formula: ~1 | Case
## (Intercept) Residual
## StdDev: 13.08313 13.87676
##
## Correlation Structure: AR(1)
## Formula: ~1 | Case
## Parameter estimate(s):
## Phi
## -0.00723058
## Fixed effects: DV ~ 1 + Phase
## Value Std.Error DF t-value p-value
## (Intercept) 40.22831 14.46504 6 2.781071 0.0320
## Phase 64.41959 10.09062 6 6.384104 0.0007
## Correlation:
## (Intr)
## Phase -0.261
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -1.1276337 -0.6289877 -0.2117285 0.6681452 1.4666320
##
## Number of Observations: 8
## Number of Groups: 1
Model with random intercept and change in slope (with an autocorrelation component)
d.Time_leli <- lme(DV~1+Time_CTR,random=~1|Case,data=leli,
correlation=corAR1(form=~1|Case),control=list(opt="optim"),na.action="na.omit")
# Rename variables and use only complete cases
coefficients(d.Time_leli)
## (Intercept) Time_CTR
## 1 80.557 10.67289
#intervals(d.Time_leli)
summary(d.Time_leli)
## Linear mixed-effects model fit by REML
## Data: leli
## AIC BIC logLik
## 71.35846 70.31725 -30.67923
##
## Random effects:
## Formula: ~1 | Case
## (Intercept) Residual
## StdDev: 25.86805 27.43721
##
## Correlation Structure: AR(1)
## Formula: ~1 | Case
## Parameter estimate(s):
## Phi
## 0.2613086
## Fixed effects: DV ~ 1 + Time_CTR
## Value Std.Error DF t-value p-value
## (Intercept) 80.55700 29.501113 6 2.730643 0.0342
## Time_CTR 10.67289 4.878049 6 2.187943 0.0713
## Correlation:
## (Intr)
## Time_CTR 0.248
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -1.17399724 -0.46013290 -0.07531162 0.30576171 1.61980748
##
## Number of Observations: 8
## Number of Groups: 1
Model with random intercept and change in level/slope in Tx (with autocorrelation)
d.Model_leli <- lme(DV~1+Phase+Time_PhaseCTR,random=~1|Case,data=leli,
correlation=corAR1(form=~1|Case),control=list(opt="optim"),na.action="na.omit")
# Rename variables and use only complete cases
coefficients(d.Model_leli)
## (Intercept) Phase Time_PhaseCTR
## 1 40.48678 63.56558 -13.63009
#intervals(d.Model_leli)
summary(d.Model_leli)
## Linear mixed-effects model fit by REML
## Data: leli
## AIC BIC logLik
## 55.03482 52.69145 -21.51741
##
## Random effects:
## Formula: ~1 | Case
## (Intercept) Residual
## StdDev: 11.47764 12.17387
##
## Correlation Structure: AR(1)
## Formula: ~1 | Case
## Parameter estimate(s):
## Phi
## -0.2432149
## Fixed effects: DV ~ 1 + Phase + Time_PhaseCTR
## Value Std.Error DF t-value p-value
## (Intercept) 40.48678 12.298989 5 3.291878 0.0217
## Phase 63.56558 7.479762 5 8.498343 0.0004
## Time_PhaseCTR -13.63009 8.246719 5 -1.652790 0.1593
## Correlation:
## (Intr) Phase
## Phase -0.225
## Time_PhaseCTR 0.022 -0.023
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -1.2364477 -0.7382020 0.2394792 0.6595564 1.1100188
##
## Number of Observations: 8
## Number of Groups: 1
Results
| Autocorrelation |
-0.243 |
|
| Level |
63.566 |
.0004 |
| Slope |
-13.630 |
.1593 |
GIAD
Null (unconditional) model
d.Null_giad <- lme(DV~1,random=~1|Case,data=giad,
control=list(opt="optim"),na.action="na.omit")
coefficients(d.Null_giad)
## (Intercept)
## 1 163.4286
#intervals(d.Null_giad)
summary(d.Null_giad)
## Linear mixed-effects model fit by REML
## Data: giad
## AIC BIC logLik
## 76.32274 75.69802 -35.16137
##
## Random effects:
## Formula: ~1 | Case
## (Intercept) Residual
## StdDev: 72.74827 72.17769
##
## Fixed effects: DV ~ 1
## Value Std.Error DF t-value p-value
## (Intercept) 163.4286 77.69519 6 2.103458 0.0801
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -1.28056981 -0.76101869 -0.08906591 0.74221588 1.40724132
##
## Number of Observations: 7
## Number of Groups: 1
Model with random intercept and change in level (with an autocorrelation component)
d.Level_giad <- lme(DV~1+Phase,random=~1|Case,data=giad,
correlation=corAR1(form=~1|Case),control=list(opt="optim"),na.action="na.omit")
coefficients(d.Level_giad)
## (Intercept) Phase
## 1 136.5865 114.3772
#intervals(d.Level_giad)
summary(d.Level_giad)
## Linear mixed-effects model fit by REML
## Data: giad
## AIC BIC logLik
## 59.62525 57.67244 -24.81262
##
## Random effects:
## Formula: ~1 | Case
## (Intercept) Residual
## StdDev: 58.18389 57.72754
##
## Correlation Structure: AR(1)
## Formula: ~1 | Case
## Parameter estimate(s):
## Phi
## 0.8183866
## Fixed effects: DV ~ 1 + Phase
## Value Std.Error DF t-value p-value
## (Intercept) 136.5865 75.65526 5 1.805381 0.1308
## Phase 114.3772 34.03763 5 3.360316 0.0201
## Correlation:
## (Intr)
## Phase -0.211
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -1.13613970 -0.74098335 -0.43244078 0.00389515 0.35361730
##
## Number of Observations: 7
## Number of Groups: 1
Model with random intercept and change in slope (with an autocorrelation component)
d.Time_giad <- lme(DV~1+Time_CTR,random=~1|Case,data=giad,
correlation=corAR1(form=~1|Case),control=list(opt="optim"),na.action="na.omit")
# Rename variables and use only complete cases
coefficients(d.Time_giad)
## (Intercept) Time_CTR
## 1 226.3155 18.00746
#intervals(d.Time_giad)
summary(d.Time_giad)
## Linear mixed-effects model fit by REML
## Data: giad
## AIC BIC logLik
## 66.72318 64.77037 -28.36159
##
## Random effects:
## Formula: ~1 | Case
## (Intercept) Residual
## StdDev: 321.5464 319.0244
##
## Correlation Structure: AR(1)
## Formula: ~1 | Case
## Parameter estimate(s):
## Phi
## 0.9831512
## Fixed effects: DV ~ 1 + Time_CTR
## Value Std.Error DF t-value p-value
## (Intercept) 226.31553 448.0850 5 0.5050727 0.6350
## Time_CTR 18.00746 23.4107 5 0.7691983 0.4765
## Correlation:
## (Intr)
## Time_CTR 0.052
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -0.373954519 -0.256408649 -0.057434433 -0.024521589 0.008508131
##
## Number of Observations: 7
## Number of Groups: 1
Model with random intercept and change in level/slope in Tx (with autocorrelation)
d.Model_giad <- lme(DV~1+Phase+Time_PhaseCTR,random=~1|Case,data=giad,
correlation=corAR1(form=~1|Case),control=list(opt="optim"),na.action="na.omit")
# Rename variables and use only complete cases
coefficients(d.Model_giad)
## (Intercept) Phase Time_PhaseCTR
## 1 125.2049 141.9503 28.2272
#intervals(d.Model_giad)
summary(d.Model_giad)
## Linear mixed-effects model fit by REML
## Data: giad
## AIC BIC logLik
## 52.24512 48.56288 -20.12256
##
## Random effects:
## Formula: ~1 | Case
## (Intercept) Residual
## StdDev: 257.158 255.1411
##
## Correlation Structure: AR(1)
## Formula: ~1 | Case
## Parameter estimate(s):
## Phi
## 0.9911537
## Fixed effects: DV ~ 1 + Phase + Time_PhaseCTR
## Value Std.Error DF t-value p-value
## (Intercept) 125.2049 361.0692 4 0.346761 0.7462
## Phase 141.9503 41.4083 4 3.428063 0.0266
## Time_PhaseCTR 28.2272 23.9428 4 1.178945 0.3038
## Correlation:
## (Intr) Phase
## Phase -0.076
## Time_PhaseCTR -0.065 0.573
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -0.21245055 -0.14180403 -0.12121920 -0.06386127 0.12461789
##
## Number of Observations: 7
## Number of Groups: 1
Results
| Autocorrelation |
0.991 |
|
| Level |
141.950 |
.0266 |
| Slope |
28.2272 |
.3038 |
Analysis Digit Span
Null (unconditional) model
d.Null_ds <- lme(DV~1,random=~1|Case,data=data_ds,
control=list(opt="optim"),na.action="na.omit")
coefficients(d.Null_ds)
## (Intercept)
## 1 4.276191
## 2 4.241883
## 3 3.331209
## Approximate 95% confidence intervals
##
## Fixed effects:
## lower est. upper
## (Intercept) 3.200901 3.949761 4.698622
## attr(,"label")
## [1] "Fixed effects:"
##
## Random Effects:
## Level: Case
## lower est. upper
## sd((Intercept)) 0.1876599 0.5752461 1.763339
##
## Within-group standard error:
## lower est. upper
## 0.4253815 0.5895381 0.8170434
## Linear mixed-effects model fit by REML
## Data: data_ds
## AIC BIC logLik
## 48.72156 51.70876 -21.36078
##
## Random effects:
## Formula: ~1 | Case
## (Intercept) Residual
## StdDev: 0.5752461 0.5895381
##
## Fixed effects: DV ~ 1
## Value Std.Error DF t-value p-value
## (Intercept) 3.949761 0.3564437 18 11.08102 0
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -2.1065359 -0.5618106 -0.4684877 1.1344326 1.2859505
##
## Number of Observations: 21
## Number of Groups: 3
Model with random intercept and change in level (with an autocorrelation component)
d.Level_ds <- lme(DV~1+Phase,random=~1|Case,data=data_ds,
correlation=corAR1(form=~1|Case),control=list(opt="optim"),na.action="na.omit")
coefficients(d.Level_ds)
## (Intercept) Phase
## 1 3.880750 0.8579849
## 2 3.896431 0.8579849
## 3 2.969476 0.8579849
## Approximate 95% confidence intervals
##
## Fixed effects:
## lower est. upper
## (Intercept) 2.869102 3.5822188 4.295336
## Phase 0.491807 0.8579849 1.224163
## attr(,"label")
## [1] "Fixed effects:"
##
## Random Effects:
## Level: Case
## lower est. upper
## sd((Intercept)) 0.1927352 0.5503671 1.571607
##
## Correlation structure:
## lower est. upper
## Phi -0.4697066 0.03833955 0.5273091
## attr(,"label")
## [1] "Correlation structure:"
##
## Within-group standard error:
## lower est. upper
## 0.2688373 0.3837592 0.5478075
## Linear mixed-effects model fit by REML
## Data: data_ds
## AIC BIC logLik
## 37.39326 42.11545 -13.69663
##
## Random effects:
## Formula: ~1 | Case
## (Intercept) Residual
## StdDev: 0.5503671 0.3837592
##
## Correlation Structure: AR(1)
## Formula: ~1 | Case
## Parameter estimate(s):
## Phi
## 0.03833955
## Fixed effects: DV ~ 1 + Phase
## Value Std.Error DF t-value p-value
## (Intercept) 3.582219 0.3379996 17 10.598294 0e+00
## Phase 0.857985 0.1735592 17 4.943471 1e-04
## Correlation:
## (Intr)
## Phase -0.223
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -2.33592002 0.07953901 0.26988088 0.44960207 0.68080589
##
## Number of Observations: 21
## Number of Groups: 3
Model with random intercept and change in slope (with an autocorrelation component)
d.Time_ds <- lme(DV~1+Time_CTR,random=~1|Case,data=data_ds,
correlation=corAR1(form=~1|Case),control=list(opt="optim"),na.action="na.omit")
# Rename variables and use only complete cases
coefficients(d.Time_ds)
## (Intercept) Time_CTR
## 1 4.390155 0.1869227
## 2 4.435741 0.1869227
## 3 3.695434 0.1869227
## Approximate 95% confidence intervals
##
## Fixed effects:
## lower est. upper
## (Intercept) 3.50718262 4.1737767 4.8403709
## Time_CTR 0.06128407 0.1869227 0.3125614
## attr(,"label")
## [1] "Fixed effects:"
##
## Random Effects:
## Level: Case
## lower est. upper
## sd((Intercept)) 0.1312738 0.4726437 1.701727
##
## Correlation structure:
## lower est. upper
## Phi -0.2222999 0.4384268 0.8231912
## attr(,"label")
## [1] "Correlation structure:"
##
## Within-group standard error:
## lower est. upper
## 0.2731961 0.4653590 0.7926872
## Linear mixed-effects model fit by REML
## Data: data_ds
## AIC BIC logLik
## 40.40173 45.12393 -15.20087
##
## Random effects:
## Formula: ~1 | Case
## (Intercept) Residual
## StdDev: 0.4726437 0.465359
##
## Correlation Structure: AR(1)
## Formula: ~1 | Case
## Parameter estimate(s):
## Phi
## 0.4384268
## Fixed effects: DV ~ 1 + Time_CTR
## Value Std.Error DF t-value p-value
## (Intercept) 4.173777 0.31594900 17 13.210286 0.000
## Time_CTR 0.186923 0.05954959 17 3.138942 0.006
## Correlation:
## (Intr)
## Time_CTR 0.19
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -2.2818838 -0.5346795 0.1122929 0.5071339 1.2125248
##
## Number of Observations: 21
## Number of Groups: 3
Model with random intercept and change in level/slope in Tx (with autocorrelation)
d.Model_ds <- lme(DV~1+Phase+Time_PhaseCTR,random=~1|Case,data=data_ds,
correlation=corAR1(form=~1|Case),control=list(opt="optim"),na.action="na.omit")
# Rename variables and use only complete cases
coefficients(d.Model_ds)
## (Intercept) Phase Time_PhaseCTR
## 1 3.883105 0.8602356 0.3351175
## 2 3.898722 0.8602356 0.3351175
## 3 2.958187 0.8602356 0.3351175
## Approximate 95% confidence intervals
##
## Fixed effects:
## lower est. upper
## (Intercept) 2.86859841 3.5800044 4.2914105
## Phase 0.53576638 0.8602356 1.1847048
## Time_PhaseCTR 0.04271431 0.3351175 0.6275207
## attr(,"label")
## [1] "Fixed effects:"
##
## Random Effects:
## Level: Case
## lower est. upper
## sd((Intercept)) 0.1971253 0.5539434 1.556641
##
## Correlation structure:
## lower est. upper
## Phi -0.4516962 0.03930146 0.5120266
## attr(,"label")
## [1] "Correlation structure:"
##
## Within-group standard error:
## lower est. upper
## 0.2344441 0.3382198 0.4879314
## Linear mixed-effects model fit by REML
## Data: data_ds
## AIC BIC logLik
## 36.20836 41.55059 -12.10418
##
## Random effects:
## Formula: ~1 | Case
## (Intercept) Residual
## StdDev: 0.5539434 0.3382198
##
## Correlation Structure: AR(1)
## Formula: ~1 | Case
## Parameter estimate(s):
## Phi
## 0.03930146
## Fixed effects: DV ~ 1 + Phase + Time_PhaseCTR
## Value Std.Error DF t-value p-value
## (Intercept) 3.580004 0.3355839 16 10.667987 0.0000
## Phase 0.860236 0.1530584 16 5.620311 0.0000
## Time_PhaseCTR 0.335117 0.1379322 16 2.429581 0.0273
## Correlation:
## (Intr) Phase
## Phase -0.198
## Time_PhaseCTR -0.004 0.010
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -2.6572115 -0.2319723 0.1236261 0.3456198 1.7035084
##
## Number of Observations: 21
## Number of Groups: 3
Results
| Autocorrelation |
.039 |
|
| Level |
0.860 |
.000 |
| Slope |
0.335 |
.0273 |
Interpretation: Neither the change in level or slope have large values, but they are both significant.
Analysis Sentences
Null (unconditional) model
d.Null_sen <- lme(DV~1,random=~1|Case,data=data_sen,
control=list(opt="optim"),na.action="na.omit")
coefficients(d.Null_sen)
## (Intercept)
## 1 5.317938
## 2 5.661048
## 3 4.933130
## Approximate 95% confidence intervals
##
## Fixed effects:
## lower est. upper
## (Intercept) 4.439292 5.304038 6.168785
## attr(,"label")
## [1] "Fixed effects:"
##
## Random Effects:
## Level: Case
## lower est. upper
## sd((Intercept)) 0.07850796 0.5097337 3.309581
##
## Within-group standard error:
## lower est. upper
## 0.9491056 1.3140811 1.8194068
## Linear mixed-effects model fit by REML
## Data: data_sen
## AIC BIC logLik
## 78.15763 81.14483 -36.07882
##
## Random effects:
## Formula: ~1 | Case
## (Intercept) Residual
## StdDev: 0.5097337 1.314081
##
## Fixed effects: DV ~ 1
## Value Std.Error DF t-value p-value
## (Intercept) 5.304038 0.4116035 18 12.88628 0
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -1.0029348 -0.7101009 -0.5030493 0.5190413 1.7799149
##
## Number of Observations: 21
## Number of Groups: 3
Model with random intercept and change in level (with an autocorrelation component)
d.Level_sen <- lme(DV~1+Phase,random=~1|Case,data=data_sen,
correlation=corAR1(form=~1|Case),control=list(opt="optim"),na.action="na.omit")
coefficients(d.Level_sen)
## (Intercept) Phase
## 1 4.403356 2.013627
## 2 4.960095 2.013627
## 3 4.114123 2.013627
## Approximate 95% confidence intervals
##
## Fixed effects:
## lower est. upper
## (Intercept) 3.644317 4.492525 5.340732
## Phase 1.267507 2.013627 2.759747
## attr(,"label")
## [1] "Fixed effects:"
##
## Random Effects:
## Level: Case
## lower est. upper
## sd((Intercept)) 0.1039863 0.5254244 2.654877
##
## Correlation structure:
## lower est. upper
## Phi -0.6479984 0.369898 0.9135294
## attr(,"label")
## [1] "Correlation structure:"
##
## Within-group standard error:
## lower est. upper
## 0.3561243 0.7115617 1.4217510
## Linear mixed-effects model fit by REML
## Data: data_sen
## AIC BIC logLik
## 54.38267 59.10487 -22.19134
##
## Random effects:
## Formula: ~1 | Case
## (Intercept) Residual
## StdDev: 0.5254244 0.7115617
##
## Correlation Structure: AR(1)
## Formula: ~1 | Case
## Parameter estimate(s):
## Phi
## 0.369898
## Fixed effects: DV ~ 1 + Phase
## Value Std.Error DF t-value p-value
## (Intercept) 4.492525 0.4020290 17 11.174628 0
## Phase 2.013627 0.3536422 17 5.693965 0
## Correlation:
## (Intr)
## Phase -0.389
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -1.58489381 -0.56685989 -0.16038408 0.05608141 1.44229062
##
## Number of Observations: 21
## Number of Groups: 3
Model with random intercept and change in slope (with an autocorrelation component)
d.Time_sen <- lme(DV~1+Time_CTR,random=~1|Case,data=data_sen,
correlation=corAR1(form=~1|Case),control=list(opt="optim"),na.action="na.omit")
# Rename variables and use only complete cases
coefficients(d.Time_sen)
## (Intercept) Time_CTR
## 1 5.997233 0.4967177
## 2 6.002565 0.4967177
## 3 5.996993 0.4967177
## Approximate 95% confidence intervals
##
## Fixed effects:
## lower est. upper
## (Intercept) 5.1780939 5.9989302 6.8197664
## Time_CTR 0.2450553 0.4967177 0.7483802
## attr(,"label")
## [1] "Fixed effects:"
##
## Random Effects:
## Level: Case
## lower est. upper
## sd((Intercept)) 9.994779e-18 0.046682 2.180348e+14
##
## Correlation structure:
## lower est. upper
## Phi 0.1822145 0.6708665 0.8937705
## attr(,"label")
## [1] "Correlation structure:"
##
## Within-group standard error:
## lower est. upper
## 0.5594165 0.9419209 1.5859650
## Linear mixed-effects model fit by REML
## Data: data_sen
## AIC BIC logLik
## 56.90522 61.62741 -23.45261
##
## Random effects:
## Formula: ~1 | Case
## (Intercept) Residual
## StdDev: 0.046682 0.9419209
##
## Correlation Structure: AR(1)
## Formula: ~1 | Case
## Parameter estimate(s):
## Phi
## 0.6708665
## Fixed effects: DV ~ 1 + Time_CTR
## Value Std.Error DF t-value p-value
## (Intercept) 5.998930 0.3890559 17 15.419197 0e+00
## Time_CTR 0.496718 0.1192817 17 4.164239 6e-04
## Correlation:
## (Intr)
## Time_CTR 0.316
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -1.593037447 -1.058467764 -0.009692116 0.516599430 1.593252269
##
## Number of Observations: 21
## Number of Groups: 3
Model with random intercept and change in level/slope in Tx (with autocorrelation)
d.Model_sen <- lme(DV~1+Phase+Time_PhaseCTR,random=~1|Case,data=data_sen,
correlation=corAR1(form=~1|Case),control=list(opt="optim"),na.action="na.omit")
# Rename variables and use only complete cases
coefficients(d.Model_sen)
## (Intercept) Phase Time_PhaseCTR
## 1 4.238451 2.185726 0.8314312
## 2 5.000405 2.185726 0.8314312
## 3 3.855836 2.185726 0.8314312
## Approximate 95% confidence intervals
##
## Fixed effects:
## lower est. upper
## (Intercept) 3.561847 4.3648972 5.167947
## Phase 1.752451 2.1857262 2.619001
## Time_PhaseCTR 0.468935 0.8314312 1.193927
## attr(,"label")
## [1] "Fixed effects:"
##
## Random Effects:
## Level: Case
## lower est. upper
## sd((Intercept)) 0.2089304 0.6094887 1.777992
##
## Correlation structure:
## lower est. upper
## Phi -0.3663893 0.1668545 0.6175932
## attr(,"label")
## [1] "Correlation structure:"
##
## Within-group standard error:
## lower est. upper
## 0.2869010 0.4267684 0.6348226
## Linear mixed-effects model fit by REML
## Data: data_sen
## AIC BIC logLik
## 42.90278 48.24501 -15.45139
##
## Random effects:
## Formula: ~1 | Case
## (Intercept) Residual
## StdDev: 0.6094887 0.4267684
##
## Correlation Structure: AR(1)
## Formula: ~1 | Case
## Parameter estimate(s):
## Phi
## 0.1668545
## Fixed effects: DV ~ 1 + Phase + Time_PhaseCTR
## Value Std.Error DF t-value p-value
## (Intercept) 4.364897 0.3788141 16 11.522530 0e+00
## Phase 2.185726 0.2043841 16 10.694210 0e+00
## Time_PhaseCTR 0.831431 0.1709964 16 4.862273 2e-04
## Correlation:
## (Intr) Phase
## Phase -0.237
## Time_PhaseCTR -0.025 0.052
##
## Standardized Within-Group Residuals:
## Min Q1 Med Q3 Max
## -2.440579422 -0.558735282 -0.000949844 0.337804235 1.907049516
##
## Number of Observations: 21
## Number of Groups: 3
Results
| Autocorrelation |
.167 |
|
| Level |
2.186 |
< .001 |
| Slope |
0.831 |
< .001 |
Interpretation: Change in level is stronger for sentences than for digit span. The change in slope is still relatively low. Both values are significiant. Overall, the effect for change in sentences appears to be greater than for change in digit span.
