Growth Curve Analysis, Part 3
Dan Mirman
22 May 2020
Random effects
- Keep it maximal: A full or maximal random effect structure is when all of the factors that could hypothetically vary across individual observational units are allowed to do so.
- Incomplete random effects can inflate false alarms
- Full random effects can produce convergence problems (we’ll come back to this issue in a little while)
Within-subject effects
Example: Target fixation in spoken word-to-picure matching (VWP)
ggplot(TargetFix, aes(Time, meanFix, color=Condition, fill=Condition)) +
stat_summary(fun=mean, geom="line") +
stat_summary(fun.data=mean_se, geom="ribbon", color=NA, alpha=0.3) +
theme_bw() + expand_limits(y=c(0,1)) + legend_positioning(c(1,1)) +
labs(y="Fixation Proportion", x="Time since word onset (ms)")
Within-subject effects: prep for GCA
## 'data.frame': 300 obs. of 11 variables:
## $ Subject : Factor w/ 10 levels "708","712","715",..: 1 1 1 1 1 1 1 1 1 1 ...
## $ Time : num 300 300 350 350 400 400 450 450 500 500 ...
## $ timeBin : num 1 1 2 2 3 3 4 4 5 5 ...
## $ Condition : Factor w/ 2 levels "High","Low": 1 2 1 2 1 2 1 2 1 2 ...
## $ meanFix : num 0.1944 0.0286 0.25 0.1143 0.2778 ...
## $ sumFix : num 7 1 9 4 10 5 13 5 14 6 ...
## $ N : int 36 35 36 35 36 35 36 35 36 35 ...
## $ Time.Index: num 1 1 2 2 3 3 4 4 5 5 ...
## $ poly1 : num -0.418 -0.418 -0.359 -0.359 -0.299 ...
## $ poly2 : num 0.4723 0.4723 0.2699 0.2699 0.0986 ...
## $ poly3 : num -0.4563 -0.4563 -0.0652 -0.0652 0.1755 ...
Within-subject effects: fit full GCA model
## boundary (singular) fit: see ?isSingular
## Estimate Std..Error df t.value Pr...t.. p.normal
## (Intercept) 0.4773228 0.01385 19.87 34.457856 3.383e-19 0.000e+00
## poly1 0.6385604 0.05994 17.27 10.654065 5.152e-09 0.000e+00
## poly2 -0.1095979 0.03849 19.99 -2.847535 9.954e-03 4.406e-03
## poly3 -0.0932612 0.02330 18.07 -4.001954 8.305e-04 6.282e-05
## ConditionLow -0.0581122 0.01879 16.22 -3.093146 6.891e-03 1.980e-03
## poly1:ConditionLow 0.0003188 0.06579 15.70 0.004846 9.962e-01 9.961e-01
## poly2:ConditionLow 0.1635455 0.05393 19.08 3.032574 6.826e-03 2.425e-03
## poly3:ConditionLow -0.0020869 0.02705 16.67 -0.077163 9.394e-01 9.385e-01
## p.normal.star
## (Intercept) ***
## poly1 ***
## poly2 **
## poly3 ***
## ConditionLow **
## poly1:ConditionLow
## poly2:ConditionLow **
## poly3:ConditionLow
Within-subject effects: random effects
## (Intercept) poly1 poly2 poly3
## 708 -0.0001522 0.01085 -0.0035635 -0.004929
## 712 0.0110222 0.10693 -0.0093253 -0.036517
## 715 0.0113775 0.11482 -0.0109566 -0.039652
## 720 -0.0020717 -0.01300 -0.0003714 0.003738
## 722 0.0138362 0.16223 -0.0200919 -0.058180
## 725 -0.0178395 -0.20733 0.0253549 0.074204
## (Intercept) poly1 poly2 poly3
## 708:High 0.012273 -0.13120 -0.12985 0.01515
## 708:Low -0.061222 0.17039 0.06228 0.01232
## 712:High 0.021229 0.08290 0.02803 0.02002
## 712:Low -0.014453 0.04483 0.03255 -0.01730
## 715:High 0.012361 0.05971 0.05517 -0.02500
## 715:Low -0.008687 0.10599 0.13031 -0.03948
Within-subject effects: random effects
## Groups Name Std.Dev. Corr
## Subject:Condition (Intercept) 0.0405
## poly1 0.1404 -0.43
## poly2 0.1124 -0.33 0.72
## poly3 0.0417 0.13 -0.49 -0.43
## Subject (Intercept) 0.0124
## poly1 0.1195 0.91
## poly2 0.0165 -0.42 -0.76
## poly3 0.0421 -0.85 -0.99 0.83
## Residual 0.0438
What is being estimated?
- Random variance and covariance
- Unit-level random effects (but constrained to have mean = 0)
This is why df for parameter estimates are poorly defined in MLR
Alternative random effects structure
## boundary (singular) fit: see ?isSingular
## Estimate Std..Error df t.value Pr...t.. p.normal
## (Intercept) 0.4773228 0.01679 10.003 28.436524 6.691e-11 0.000e+00
## poly1 0.6385604 0.05146 10.000 12.407713 2.133e-07 0.000e+00
## poly2 -0.1095979 0.03717 10.004 -2.948277 1.457e-02 3.196e-03
## poly3 -0.0932612 0.02063 10.039 -4.521487 1.095e-03 6.141e-06
## ConditionLow -0.0581122 0.02110 10.005 -2.754110 2.033e-02 5.885e-03
## poly1:ConditionLow 0.0003188 0.07479 9.994 0.004263 9.967e-01 9.966e-01
## poly2:ConditionLow 0.1635455 0.06274 10.000 2.606797 2.618e-02 9.139e-03
## poly3:ConditionLow -0.0020869 0.03334 10.676 -0.062602 9.512e-01 9.501e-01
## p.normal.star
## (Intercept) ***
## poly1 ***
## poly2 **
## poly3 ***
## ConditionLow **
## poly1:ConditionLow
## poly2:ConditionLow **
## poly3:ConditionLow
Alternative random effects structure
## Groups Name Std.Dev. Corr
## Subject (Intercept) 0.0519
## poly1 0.1570 0.18
## poly2 0.1094 -0.29 0.03
## poly3 0.0491 -0.28 -0.37 0.63
## ConditionLow 0.0649 -0.89 -0.13 0.49 0.34
## poly1:ConditionLow 0.2286 0.38 -0.46 -0.62 0.10 -0.56
## poly2:ConditionLow 0.1889 0.20 0.08 -0.81 -0.22 -0.43 0.74
## poly3:ConditionLow 0.0862 -0.08 -0.40 -0.06 -0.47 0.06 -0.27 -0.43
## Residual 0.0430
This random effect structure makes fewer assumptions:
- Allows unequal variances across conditions
- Allows more flexible covariance structure between random effect terms
Alternative random effects structure requires more parameters
Convergence problems
Warning message:
In (function (fn, par, lower = rep.int(-Inf, n), upper = rep.int(Inf, :
failure to converge in 10000 evaluations
Need to simplify random effects
- Remove random effects of higher-order terms
Outcome ~ (poly1+poly2+poly3)*Condition + (poly1+poly2+poly3 | Subject)
Outcome ~ (poly1+poly2+poly3)*Condition + (poly1+poly2 | Subject)
- Remove correlation between random effects
Outcome ~ (poly1+poly2+poly3)*Condition + (1 | Subject) +
(0+poly1 | Subject) + (0+poly2 | Subject) + (0+poly3 | Subject)
Participants as fixed vs. random effects
In general, participants should be treated as random effects.
This captures the typical assumption of random sampling from some population to which we wish to generalize.
Pooling
Participants as fixed vs. random effects
In general, participants should be treated as random effects.
This captures the typical assumption of random sampling from some population to which we wish to generalize.
Shrinkage
Participants as fixed vs. random effects
In general, participants should be treated as random effects.
This captures the typical assumption of random sampling from some population to which we wish to generalize.
Exceptions are possible: e.g., neurological/neuropsychological case studies where the goal is to characterize the pattern of performance for each participant, not generalize to a population.
Lunch break; Exercise 3
Az: Use natural (not orthogonal) polynomials to analyze decline in performance of 30 individuals with probable Alzheimer’s disease on three different kinds of tasks - Memory, complex ADL, and simple ADL.
- Plot the observed data
- (Why are natural polynomials more useful for these data?)
- Prep the data for GCA
- Fit the GCA model(s)
- Interpret the results:
- Which terms show significant effects of experimental factors?
- To what extent do model comparisons and the parameter-specific p-values yield the same results?
- Plot observed and model fit data
