Growth Curve Analysis, Part 2
Dan Mirman
22 May 2020
Modeling non-linear change over time
- Choosing a functional form
- Function must be adequate to data
- Dynamic consistency
- Prediction
- Higher-order polynomials
- Orthogonal polynomials
Function must be adequate to data
- Easy plotting residuals vs. fitted:
ggplot(fortify(m, dat), aes(.fitted, .resid)) + geom_point()
Two kinds of non-linearity
- Non-linear in variables (Time): \(Y_{ij} = \beta_{0i} + \beta_{1i} \cdot Time_{j} + \beta_{2i} \cdot Time^2_{j} + \epsilon_{ij}\)
- Non-linear in parameters (b): \(Y = \frac{p-b}{1+exp(4 \cdot \frac{s}{p-b} \cdot (c-t))} + b\)
- Dynamic Consistency: The model of the average is equal to the average of the individual models
- Recall: random effects must have a mean of 0
- Average of individual deviations is 0
- So, the average of the individual models will have 0 deviation from the model of the average
- Not true of some tempting functional forms
Example of lack of dynamic consistency
Logistic power peak function (Scheepers, Keller, & Lapata, 2008) fit to semantic competition data (Mirman & Magnuson, 2009).
- Standard statistical inference assumes that the group average represents central tendency of individuals
- For dynamically inconsistent models this is not true
- Can’t make inferences about central tendencies without dynamic consistency
Prediction: Two kinds
Fits: Statistical models
- Example: Mean and SD
- Describe the observed data
- No new predictions
- Not falsifiable
Forecasts: Theoretical models
- Example: Interactive Activation
- Match the observed data
- Makes new predictions
- Falsifiable
Using higher-order polynomials
- Can model any curve shape
- Dynamically consistent
- Bad at capturing asymptotic behavior
- Don’t extrapolate
- Try to avoid long flat sections
- How to choose polynomial order?
- Curve shape
- Statistical: include only terms that statistically improve model fit
- Theoretical: include only terms that are predicted to matter
Natural vs. Orthogonal polynomials
- Natural Polynomials: Correlated time terms
- Orthogonal Polynomials: Uncorrelated time terms
- Need to specify range and order
Interpreting orthogonal polynomial terms
Intercept (\(\beta_0\)): Overall average
Interpreting orthogonal polynomial terms
Intercept (\(\beta_0\)): Overall average
Linear (\(\beta_1\)): Overall slope
Quadratic (\(\beta_2\)): Centered rise and fall rate
Cubic, Quartic, … (\(\beta_3, \beta_4, ...\)): Inflection steepness
Example
Effect of transitional probability on word learning
## Subject TP Block Accuracy
## 244 : 10 Low :280 Min. : 1.0 Min. :0.000
## 253 : 10 High:280 1st Qu.: 3.0 1st Qu.:0.667
## 302 : 10 Median : 5.5 Median :0.833
## 303 : 10 Mean : 5.5 Mean :0.805
## 305 : 10 3rd Qu.: 8.0 3rd Qu.:1.000
## 306 : 10 Max. :10.0 Max. :1.000
## (Other):500
Example: Prepare for GCA
Create 2nd-order orthogonal polynomial
## Subject TP Block Accuracy Block.Index
## 244 : 10 Low :280 Min. : 1.0 Min. :0.000 Min. : 1.0
## 253 : 10 High:280 1st Qu.: 3.0 1st Qu.:0.667 1st Qu.: 3.0
## 302 : 10 Median : 5.5 Median :0.833 Median : 5.5
## 303 : 10 Mean : 5.5 Mean :0.805 Mean : 5.5
## 305 : 10 3rd Qu.: 8.0 3rd Qu.:1.000 3rd Qu.: 8.0
## 306 : 10 Max. :10.0 Max. :1.000 Max. :10.0
## (Other):500
## poly1 poly2
## Min. :-0.495 Min. :-0.348
## 1st Qu.:-0.275 1st Qu.:-0.261
## Median : 0.000 Median :-0.087
## Mean : 0.000 Mean : 0.000
## 3rd Qu.: 0.275 3rd Qu.: 0.174
## Max. : 0.495 Max. : 0.522
##
Example: Compare the model fits
## Data: WordLearn.gca
## Models:
## m.base: Accuracy ~ (poly1 + poly2) + (poly1 + poly2 | Subject)
## m.0: Accuracy ~ (poly1 + poly2) + TP + (poly1 + poly2 | Subject)
## m.1: Accuracy ~ (poly1 + poly2) + TP + TP:poly1 + (poly1 + poly2 |
## m.1: Subject)
## m.2: Accuracy ~ (poly1 + poly2) * TP + (poly1 + poly2 | Subject)
## npar AIC BIC logLik deviance Chisq Df Pr(>Chisq)
## m.base 10 -331 -288 175 -351
## m.0 11 -330 -283 176 -352 1.55 1 0.213
## m.1 12 -329 -277 176 -353 0.36 1 0.550
## m.2 13 -333 -276 179 -359 5.95 1 0.015 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Example: Inspect model
## Linear mixed model fit by maximum likelihood . t-tests use Satterthwaite's
## method [lmerModLmerTest]
## Formula: Accuracy ~ (poly1 + poly2) * TP + (poly1 + poly2 | Subject)
## Data: WordLearn.gca
##
## AIC BIC logLik deviance df.resid
## -332.6 -276.4 179.3 -358.6 547
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -3.618 -0.536 0.126 0.567 2.616
##
## Random effects:
## Groups Name Variance Std.Dev. Corr
## Subject (Intercept) 0.01076 0.1037
## poly1 0.01542 0.1242 -0.33
## poly2 0.00628 0.0792 -0.28 -0.82
## Residual 0.02456 0.1567
## Number of obs: 560, groups: Subject, 56
##
## Fixed effects:
## Estimate Std. Error df t value Pr(>|t|)
## (Intercept) 0.77853 0.02173 56.02039 35.83 < 2e-16 ***
## poly1 0.28632 0.03779 62.51319 7.58 2.1e-10 ***
## poly2 -0.05085 0.03319 93.21826 -1.53 0.129
## TPHigh 0.05296 0.03073 56.02039 1.72 0.090 .
## poly1:TPHigh 0.00108 0.05344 62.51319 0.02 0.984
## poly2:TPHigh -0.11645 0.04693 93.21826 -2.48 0.015 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Correlation of Fixed Effects:
## (Intr) poly1 poly2 TPHigh p1:TPH
## poly1 -0.183
## poly2 -0.114 -0.229
## TPHigh -0.707 0.129 0.081
## poly1:TPHgh 0.129 -0.707 0.162 -0.183
## poly2:TPHgh 0.081 0.162 -0.707 -0.114 -0.229
## convergence code: 0
## boundary (singular) fit: see ?isSingular
Example: Plot the model fit
Exercise 2
CP (d’ peak at category boundary): Compare categorical perception along spectral vs. temporal dimensions using second-order orthogonal polynomials.
- Plot the observed data
- (Why are orthogonal 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