Running Head: Quadratic Growth

1

Quadratic Growth

Supplementary Results Table for:

Statistical Power of Latent Growth Curve Models to Detect Quadratic Growth

This material is to go on an external website and not linked to the main article.

The relation between the effect size of the LCM model (R2) and the residual variances.

This section provides a formal demonstration of the relations between the effect size of the LCM model (R2 of the repeated measures) and the time-specific residual variance. This demonstration will help to make explicit how residual variances vary in relation with the R2and also why the residual variances increase over time due to increasing growth curve variance when the R2is constant over time.

For simplicity let us consider a linear LCM, , where Yt is the outcome at time t, is the intercept factor, the slope factor, the time score and the residual. Under usual assumptions, the variance of the repeated measures at each time point is defined by: , where, is the variance of the intercept growth factor, is the variance of the linear slope factor, is the covariance between the intercept and the linear slope factors.

Thus, the R2 of the repeated measure is expressed as (e.g,. Muthén & Muthén, 2002):

(1.1)

where is the residual variance for the outcome at time t and other parameters are defined as previously.

It is clear from equation 1.1 that is perfectly in line with usual effect sizes indicators. From equation 1.1, we can also obtain the relationship between the residual variance and the as:

Thus:

(1.2)

From equation 1.2 we can see that the residual variance of the repeated measures at each time point is function of two terms: a first term related to and a second term related to the variance-covariance of the LCM factors.

Hence, the residual variance at each time point can increase or decrease over time depending on the behaviour of the two terms of equation 1.2. When , and are known, the second term is a polynomial function of degree 2 in the time score . This function is a strictly increasing function as long as the time score is greater than and .

Furthermore, when is the same over time (like in this study), we can set Equation 1.2 thus becomes:

(1.3)

From Equation 1.3 we can see that when is constant over time the residual variance of the repeated measure is function of a constant (the first term:) and an increasing function due to increasing values of the time score (the second term: ). Consequently, with constant , the residual variances of the repeated measures increase over time due to increasing growth curve variance. However, when vary over time, as seen in Equation 1.2, the residual variance of the repeated measures can be monotonic or non monotonic over time. For the same reason, when the residual variances are specified as equal over time as was done in the context of previous LCM simulation studies (Fan & Fan, 2005; Zhang & Wang, 2009), then will show an increase over time.

Although we relied on a simpler linear LCM for illustrative purposes, the same logic applies to quadratic LCM and the relations can be similarly deduced from the equation 1.6 presented in the main manuscript.

Table S1. Main results regarding the effects of the design factors on the type I error rates, empirical power rates and rates of nonvergence.

Design factor / Non-Parametric Kruskal-Wallis
χ²(df) / p
Type I error rate (µS2)
Quadratic Slope Variance / χ²(2) = 0.794 / .672
Sample Size / χ²(9) = 87.167 / .000
Number of measurement points / χ²(2) = 70.914 / .000
R² / χ²(2) = 2.408 / .300
Power to detect µS2
Quadratic Slope Mean / χ²(1) = 36.324 / .000
Quadratic Slope Variance / χ²(3) = 36.735 / .000
Sample Size / χ²(9) = 199.625 / .000
Number of measurement points / χ²(2) = 123.147 / .000
R² / χ²(2) = 54.603 / .000
Rates of nonconvergence
Quadratic Slope Mean / χ²(1) = 0.007 / .933
Quadratic Slope Variance / χ²(3) = 90.164 / .000
Sample Size / χ²(9) = 260.167 / .000
Number of measurement points / χ²(2) = 249.742 / .000
R² / χ²(2) = 59.169 / .000

Note. Because of the strong ceiling effects of power data, nonparametric analyses were conducted with the Kruskal-Wallis (K-W) test. These results are in line with those obtained from parametric ANOVAS tests conducted on the same data. Using these parametric tests, we also initially probed for interaction effects (something that cannot be done with non-parametric methods) and obtained very few of them. Furthermore, these few significant effects were generally so tiny as to be meaningless and potentially due to the strong ceiling effects of power distribution and the high number of replications used here.

Table S2. Mean Power and Type I Error Rates for the Wald test and R²=.3

Sample Sizes
M(Q)/SD(Q)a / Repeated measures / 30 / 50 / 100 / 150 / 200 / 250 / 300 / 400 / 500 / 1000
0/.22 / T=4 / 0.08 / 0.06 / 0.06 / 0.06 / 0.06 / 0.05 / 0.05 / 0.06 / 0.04 / 0.05
T=6 / 0.08 / 0.05 / 0.04 / 0.05 / 0.05 / 0.05 / 0.05 / 0.05 / 0.05 / 0.04
T=10 / 0.06 / 0.04 / 0.05 / 0.04 / 0.05 / 0.05 / 0.04 / 0.05 / 0.04 / 0.05
0/.4 / T=4 / 0.08 / 0.06 / 0.05 / 0.07 / 0.06 / 0.06 / 0.06 / 0.06 / 0.04 / 0.05
T=6 / 0.07 / 0.05 / 0.04 / 0.05 / 0.04 / 0.06 / 0.05 / 0.05 / 0.05 / 0.03
T=10 / 0.06 / 0.05 / 0.05 / 0.04 / 0.04 / 0.04 / 0.04 / 0.05 / 0.04 / 0.05
0/.6 / T=4 / 0.08 / 0.06 / 0.06 / 0.07 / 0.06 / 0.06 / 0.06 / 0.06 / 0.04 / 0.04
T=6 / 0.06 / 0.06 / 0.04 / 0.05 / 0.04 / 0.06 / 0.06 / 0.04 / 0.05 / 0.04
T=10 / 0.06 / 0.05 / 0.05 / 0.05 / 0.04 / 0.04 / 0.04 / 0.04 / 0.04 / 0.05
.3/0 / T=4 / 0.22 / 0.29 / 0.5 / 0.69 / 0.79 / 0.88 / 0.92 / 0.97 / 0.99 / 1
T=6 / 0.75 / 0.92 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=10 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
.3/.22 / T=4 / 0.21 / 0.25 / 0.45 / 0.62 / 0.72 / 0.82 / 0.88 / 0.95 / 0.98 / 1
T=6 / 0.55 / 0.76 / 0.96 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=10 / 0.97 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
.3/.4 / T=4 / 0.17 / 0.2 / 0.36 / 0.51 / 0.59 / 0.72 / 0.79 / 0.88 / 0.95 / 1
T=6 / 0.38 / 0.54 / 0.82 / 0.94 / 0.98 / 1 / 1 / 1 / 1 / 1
T=10 / 0.73 / 0.91 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
.3/.6 / T=4 / 0.14 / 0.16 / 0.28 / 0.39 / 0.5 / 0.6 / 0.66 / 0.78 / 0.86 / 0.99
T=6 / 0.26 / 0.36 / 0.61 / 0.8 / 0.9 / 0.94 / 0.98 / 0.99 / 1 / 1
T=10 / 0.46 / 0.67 / 0.92 / 0.99 / 1 / 1 / 1 / 1 / 1 / 1
.5/0 / T=4 / 0.46 / 0.65 / 0.9 / 0.98 / 1 / 1 / 1 / 1 / 1 / 1
T=6 / 0.99 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=10 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
.5/.22 / T=4 / 0.41 / 0.58 / 0.84 / 0.95 / 0.99 / 1 / 1 / 1 / 1 / 1
T=6 / 0.92 / 0.99 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=10 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
.5/.4 / T=4 / 0.34 / 0.46 / 0.74 / 0.9 / 0.96 / 0.98 / 0.99 / 1 / 1 / 1
T=6 / 0.75 / 0.93 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=10 / 0.99 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
.5/.6 / T=4 / 0.27 / 0.36 / 0.61 / 0.8 / 0.88 / 0.94 / 0.97 / 0.99 / 1 / 1
T=6 / 0.53 / 0.76 / 0.98 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=10 / 0.88 / 0.98 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1

a Mean and Standard deviation of the quadratic component

Table S3. Mean Power and Type I Error Rates for the Wald test and R²=.5

Sample Sizes
M(Q)/SD(Q)a / Repeated measures / 30 / 50 / 100 / 150 / 200 / 250 / 300 / 400 / 500 / 1000
0/.22 / T=4 / 0.08 / 0.06 / 0.06 / 0.06 / 0.06 / 0.06 / 0.06 / 0.06 / 0.04 / 0.05
T=6 / 0.07 / 0.06 / 0.04 / 0.05 / 0.04 / 0.06 / 0.05 / 0.05 / 0.04 / 0.03
T=10 / 0.06 / 0.05 / 0.05 / 0.04 / 0.04 / 0.04 / 0.04 / 0.04 / 0.04 / 0.05
0/.4 / T=4 / 0.08 / 0.06 / 0.06 / 0.06 / 0.06 / 0.07 / 0.06 / 0.06 / 0.04 / 0.04
T=6 / 0.06 / 0.06 / 0.04 / 0.04 / 0.04 / 0.06 / 0.05 / 0.04 / 0.05 / 0.04
T=10 / 0.06 / 0.05 / 0.05 / 0.04 / 0.04 / 0.03 / 0.04 / 0.04 / 0.04 / 0.05
0/.6 / T=4 / 0.08 / 0.06 / 0.06 / 0.07 / 0.06 / 0.06 / 0.05 / 0.07 / 0.04 / 0.04
T=6 / 0.06 / 0.06 / 0.04 / 0.04 / 0.04 / 0.05 / 0.06 / 0.04 / 0.04 / 0.03
T=10 / 0.06 / 0.05 / 0.04 / 0.04 / 0.04 / 0.03 / 0.04 / 0.04 / 0.04 / 0.05
.3/0 / T=4 / 0.41 / 0.58 / 0.84 / 0.95 / 0.99 / 1 / 1 / 1 / 1 / 1
T=6 / 0.97 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=10 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
.3/.22 / T=4 / 0.36 / 0.49 / 0.76 / 0.91 / 0.97 / 0.99 / 1 / 1 / 1 / 1
T=6 / 0.84 / 0.97 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=10 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
.3/.4 / T=4 / 0.28 / 0.38 / 0.66 / 0.84 / 0.91 / 0.95 / 0.99 / 1 / 1 / 1
T=6 / 0.6 / 0.82 / 0.99 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=10 / 0.89 / 0.98 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
.3/.6 / T=4 / 0.22 / 0.28 / 0.51 / 0.69 / 0.79 / 0.88 / 0.92 / 0.96 / 0.99 / 1
T=6 / 0.4 / 0.6 / 0.86 / 0.97 / 1 / 1 / 1 / 1 / 1 / 1
T=10 / 0.62 / 0.83 / 0.98 / 1 / 1 / 1 / 1 / 1 / 1 / 1
.5/0 / T=4 / 0.79 / 0.95 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=6 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=10 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
.5/.22 / T=4 / 0.7 / 0.89 / 0.99 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=6 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=10 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
.5/.4 / T=4 / 0.59 / 0.8 / 0.97 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=6 / 0.95 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=10 / 0.96 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
.5/.6 / T=4 / 0.48 / 0.65 / 0.91 / 0.97 / 0.99 / 1 / 1 / 1 / 1 / 1
T=6 / 0.79 / 0.95 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=10 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1

a Mean and Standard deviation of the quadratic component

Table S4. Mean Power and Type I Error Rates for the Wald Test and R²=.75

Sample Sizes
M(Q)/SD(Q)a / Repeated measures / 30 / 50 / 100 / 150 / 200 / 250 / 300 / 400 / 500 / 1000
0/.22 / T=4 / 0.08 / 0.06 / 0.06 / 0.07 / 0.06 / 0.06 / 0.06 / 0.07 / 0.03 / 0.05
T=6 / 0.06 / 0.06 / 0.04 / 0.05 / 0.04 / 0.06 / 0.05 / 0.04 / 0.05 / 0.04
T=10 / 0.06 / 0.04 / 0.05 / 0.04 / 0.04 / 0.03 / 0.03 / 0.04 / 0.04 / 0.05
0/.4 / T=4 / 0.07 / 0.06 / 0.05 / 0.06 / 0.06 / 0.06 / 0.05 / 0.07 / 0.04 / 0.04
T=6 / 0.06 / 0.06 / 0.04 / 0.04 / 0.04 / 0.04 / 0.05 / 0.04 / 0.05 / 0.04
T=10 / 0.06 / 0.05 / 0.05 / 0.05 / 0.04 / 0.04 / 0.04 / 0.04 / 0.03 / 0.04
0/.6 / T=4 / 0.07 / 0.06 / 0.06 / 0.06 / 0.06 / 0.06 / 0.05 / 0.06 / 0.04 / 0.03
T=6 / 0.06 / 0.05 / 0.04 / 0.04 / 0.05 / 0.04 / 0.04 / 0.04 / 0.05 / 0.03
T=10 / 0.06 / 0.05 / 0.04 / 0.04 / 0.05 / 0.04 / 0.04 / 0.04 / 0.04 / 0.04
.3/0 / T=4 / 0.81 / 0.96 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=6 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=10 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
.3/.22 / T=4 / 0.7 / 0.9 / 0.99 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=6 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=10 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
.3/.4 / T=4 / 0.55 / 0.74 / 0.95 / 0.99 / 1 / 1 / 1 / 1 / 1 / 1
T=6 / 0.85 / 0.97 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=10 / 0.95 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
.3/.6 / T=4 / 0.4 / 0.57 / 0.85 / 0.96 / 0.98 / 1 / 1 / 1 / 1 / 1
T=6 / 0.6 / 0.81 / 0.98 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=10 / 0.73 / 0.89 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
.5/0 / T=4 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=6 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=10 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
.5/.22 / T=4 / 0.98 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=6 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=10 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
.5/.4 / T=4 / 0.91 / 0.99 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=6 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=10 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
.5/.6 / T=4 / 0.78 / 0.94 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=6 / 0.95 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=10 / 0.98 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1

a Mean and Standard deviation of the quadratic component

Table S5. Mean Power and Type I Error Rates for the Likelihood Ratio Test and R²=.3

Sample Sizes
M(Q)/SD(Q)a / Repeated measures / 30 / 50 / 100 / 150 / 200 / 250 / 300 / 400 / 500 / 1000
0/.22 / T=4 / 0.07 / 0.06 / 0.05 / 0.06 / 0.06 / 0.05 / 0.05 / 0.06 / 0.05 / 0.05
T=6 / 0.09 / 0.04 / 0.05 / 0.04 / 0.04 / 0.05 / 0.06 / 0.06 / 0.05 / 0.03
T=10 / 0.06 / 0.03 / 0.05 / 0.03 / 0.05 / 0.04 / 0.04 / 0.05 / 0.04 / 0.05
0/.4 / T=4 / 0.06 / 0.1 / 0.05 / 0.08 / 0.05 / 0.05 / 0.06 / 0.07 / 0.04 / 0.05
T=6 / 0.1 / 0.06 / 0.04 / 0.03 / 0.04 / 0.06 / 0.05 / 0.06 / 0.05 / 0.04
T=10 / 0.04 / 0.04 / 0.05 / 0.04 / 0.04 / 0.04 / 0.04 / 0.05 / 0.03 / 0.06
0/.6 / T=4 / 0.04 / 0.09 / 0.07 / 0.08 / 0.05 / 0.07 / 0.05 / 0.06 / 0.05 / 0.05
T=6 / 0.1 / 0.08 / 0.03 / 0.04 / 0.04 / 0.07 / 0.06 / 0.05 / 0.06 / 0.04
T=10 / 0.06 / 0.04 / 0.04 / 0.04 / 0.04 / 0.04 / 0.04 / 0.04 / 0.03 / 0.05
.3/0 / T=4 / 0.1 / 0.25 / 0.41 / 0.63 / 0.71 / 0.87 / 0.9 / 0.97 / 0.99 / 1
T=6 / 0.64 / 0.89 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=10 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
.3/.22 / T=4 / 0.11 / 0.24 / 0.39 / 0.56 / 0.65 / 0.8 / 0.85 / 0.94 / 0.97 / 1
T=6 / 0.44 / 0.69 / 0.94 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=10 / 0.94 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
.3/.4 / T=4 / 0.11 / 0.19 / 0.31 / 0.43 / 0.52 / 0.71 / 0.74 / 0.87 / 0.94 / 1
T=6 / 0.27 / 0.52 / 0.8 / 0.95 / 0.98 / 1 / 0.99 / 1 / 1 / 1
T=10 / 0.68 / 0.89 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
.3/.6 / T=4 / 0.08 / 0.15 / 0.25 / 0.34 / 0.45 / 0.57 / 0.62 / 0.75 / 0.85 / 0.99
T=6 / 0.2 / 0.36 / 0.59 / 0.8 / 0.91 / 0.93 / 0.98 / 0.99 / 1 / 1
T=10 / 0.43 / 0.61 / 0.91 / 0.99 / 1 / 1 / 1 / 1 / 1 / 1
.5/0 / T=4 / 0.41 / 0.6 / 0.84 / 0.96 / 1 / 1 / 1 / 1 / 1 / 1
T=6 / 0.99 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=10 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
.5/.22 / T=4 / 0.4 / 0.53 / 0.77 / 0.93 / 0.98 / 1 / 1 / 1 / 1 / 1
T=6 / 0.85 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=10 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
.5/.4 / T=4 / 0.31 / 0.44 / 0.69 / 0.86 / 0.92 / 0.98 / 0.99 / 1 / 1 / 1
T=6 / 0.67 / 0.89 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=10 / 0.99 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
.5/.6 / T=4 / 0.15 / 0.35 / 0.58 / 0.76 / 0.86 / 0.92 / 0.96 / 0.98 / 1 / 1
T=6 / 0.47 / 0.71 / 0.97 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=10 / 0.84 / 0.97 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1

a Mean and Standard deviation of the quadratic component

Table S6. Mean Power and Type I Error Rates for the Likelihood Ratio Test and R²=.5

Sample Sizes
M(Q)/SD(Q)a / Repeated measures / 30 / 50 / 100 / 150 / 200 / 250 / 300 / 400 / 500 / 1000
0/.22 / T=4 / 0.05 / 0.09 / 0.05 / 0.07 / 0.04 / 0.06 / 0.06 / 0.07 / 0.04 / 0.05
T=6 / 0.09 / 0.06 / 0.05 / 0.04 / 0.04 / 0.06 / 0.05 / 0.05 / 0.04 / 0.03
T=10 / 0.05 / 0.04 / 0.05 / 0.04 / 0.04 / 0.04 / 0.04 / 0.04 / 0.04 / 0.05
0/.4 / T=4 / 0.03 / 0.1 / 0.07 / 0.08 / 0.05 / 0.07 / 0.05 / 0.07 / 0.04 / 0.04
T=6 / 0.06 / 0.07 / 0.04 / 0.05 / 0.04 / 0.06 / 0.06 / 0.04 / 0.05 / 0.04
T=10 / 0.05 / 0.03 / 0.05 / 0.04 / 0.04 / 0.03 / 0.04 / 0.04 / 0.03 / 0.05
0/.6 / T=4 / 0.01 / 0.07 / 0.07 / 0.08 / 0.05 / 0.07 / 0.04 / 0.07 / 0.05 / 0.05
T=6 / 0.08 / 0.07 / 0.02 / 0.05 / 0.04 / 0.05 / 0.06 / 0.04 / 0.05 / 0.04
T=10 / 0.06 / 0.03 / 0.04 / 0.04 / 0.04 / 0.03 / 0.04 / 0.04 / 0.03 / 0.05
.3/0 / T=4 / 0.36 / 0.56 / 0.79 / 0.92 / 0.98 / 1 / 1 / 1 / 1 / 1
T=6 / 0.96 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=10 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
.3/.22 / T=4 / 0.28 / 0.46 / 0.71 / 0.88 / 0.95 / 0.98 / 1 / 1 / 1 / 1
T=6 / 0.76 / 0.96 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=10 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
.3/.4 / T=4 / 0.21 / 0.36 / 0.6 / 0.81 / 0.89 / 0.94 / 0.97 / 0.99 / 1 / 1
T=6 / 0.52 / 0.79 / 0.98 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=10 / 0.85 / 0.98 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
.3/.6 / T=4 / 0.15 / 0.25 / 0.48 / 0.65 / 0.8 / 0.87 / 0.9 / 0.95 / 0.99 / 1
T=6 / 0.34 / 0.57 / 0.83 / 0.97 / 0.99 / 1 / 1 / 1 / 1 / 1
T=10 / 0.57 / 0.8 / 0.98 / 1 / 1 / 1 / 1 / 1 / 1 / 1
.5/0 / T=4 / 0.77 / 0.96 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=6 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=10 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
.5/.22 / T=4 / 0.68 / 0.9 / 0.99 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=6 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=10 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
.5/.4 / T=4 / 0.55 / 0.77 / 0.95 / 0.99 / 1 / 1 / 1 / 1 / 1 / 1
T=6 / 0.91 / 0.99 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=10 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
.5/.6 / T=4 / 0.41 / 0.59 / 0.88 / 0.96 / 0.99 / 1 / 1 / 1 / 1 / 1
T=6 / 0.68 / 0.94 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=10 / 0.93 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1

a Mean and Standard deviation of the quadratic component

Table S7. Mean Power and Type I Error Rates for the Likelihood Ratio Test and R²=.75

Sample Sizes
M(Q)/SD(Q)a / Repeated measures / 30 / 50 / 100 / 150 / 200 / 250 / 300 / 400 / 500 / 1000
0/.22 / T=4 / 0.02 / 0.08 / 0.06 / 0.08 / 0.04 / 0.07 / 0.06 / 0.08 / 0.03 / 0.05
T=6 / 0.05 / 0.07 / 0.04 / 0.05 / 0.04 / 0.06 / 0.05 / 0.04 / 0.05 / 0.04
T=10 / 0.05 / 0.04 / 0.05 / 0.04 / 0.04 / 0.03 / 0.03 / 0.04 / 0.03 / 0.05
0/.4 / T=4 / 0.02 / 0.08 / 0.06 / 0.06 / 0.05 / 0.07 / 0.04 / 0.07 / 0.04 / 0.05
T=6 / 0.06 / 0.06 / 0.03 / 0.04 / 0.04 / 0.04 / 0.06 / 0.04 / 0.05 / 0.04
T=10 / 0.06 / 0.05 / 0.04 / 0.04 / 0.04 / 0.04 / 0.04 / 0.04 / 0.03 / 0.04
0/.6 / T=4 / 0.08 / 0.08 / 0.06 / 0.06 / 0.05 / 0.08 / 0.04 / 0.06 / 0.04 / 0.04
T=6 / 0.05 / 0.06 / 0.04 / 0.03 / 0.05 / 0.04 / 0.05 / 0.04 / 0.05 / 0.03
T=10 / 0.07 / 0.06 / 0.05 / 0.04 / 0.05 / 0.04 / 0.04 / 0.04 / 0.04 / 0.04
.3/0 / T=4 / 0.8 / 0.97 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=6 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=10 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
.3/.22 / T=4 / 0.64 / 0.9 / 0.99 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=6 / 0.99 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=10 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
.3/.4 / T=4 / 0.53 / 0.76 / 0.93 / 0.98 / 1 / 1 / 1 / 1 / 1 / 1
T=6 / 0.8 / 0.96 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=10 / 0.93 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
.3/.6 / T=4 / 0.38 / 0.58 / 0.82 / 0.93 / 0.98 / 0.99 / 1 / 1 / 1 / 1
T=6 / 0.59 / 0.77 / 0.98 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=10 / 0.7 / 0.88 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
.5/0 / T=4 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=6 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=10 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
.5/.22 / T=4 / 0.98 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=6 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=10 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
.5/.4 / T=4 / 0.93 / 0.99 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=6 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=10 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
.5/.6 / T=4 / 0.78 / 0.93 / 0.99 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=6 / 0.93 / 0.99 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1
T=10 / 0.98 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1 / 1

a Mean and Standard deviation of the quadratic component

Table S8. Mean Rates of Nonconvergence forthe condition with R²=.3

Sample Sizes
M(Q)/SD(Q)a / Repeated measures / 30 / 50 / 100 / 150 / 200 / 250 / 300 / 400 / 500 / 1000
0/.22 / T=4 / 92.5 / 88.3 / 79.8 / 76.2 / 73.5 / 73 / 71.8 / 67.8 / 66.9 / 62.7
T=6 / 85.2 / 76.6 / 65.4 / 60 / 57.3 / 52.7 / 50.9 / 48.7 / 45.1 / 37.7
T=10 / 73.4 / 60.9 / 49.1 / 41.1 / 39.6 / 33.3 / 33.5 / 27.5 / 24.3 / 13.2
0/.4 / T=4 / 92.9 / 89.4 / 79.8 / 75.4 / 73.2 / 71.7 / 70 / 65.6 / 63.1 / 59.4
T=6 / 86.6 / 75.3 / 65.9 / 58.1 / 56.6 / 50.7 / 49.3 / 48.2 / 44.3 / 36.2
T=10 / 75.9 / 63.9 / 51.8 / 44.3 / 43.4 / 36.6 / 37.2 / 34.3 / 29.8 / 18.5
0/.6 / T=4 / 93.5 / 90.2 / 81.3 / 76.9 / 74.9 / 73.1 / 70.8 / 66.6 / 63.7 / 58.4
T=6 / 87.6 / 77.9 / 67.7 / 59.9 / 59.8 / 53.6 / 52.5 / 48.8 / 46.6 / 39.3
T=10 / 78.4 / 68.1 / 56.3 / 50.5 / 47.6 / 42.5 / 43.8 / 39.5 / 34.9 / 26.9
.3/0 / T=4 / 92.7 / 88.2 / 81.3 / 77.5 / 75.6 / 76.1 / 74.3 / 71.9 / 71.3 / 67.1
T=6 / 87.9 / 81.4 / 75.1 / 71.5 / 70.8 / 67 / 66.2 / 64.5 / 63.8 / 57.8
T=10 / 81.6 / 75.7 / 70.2 / 65 / 62 / 62.5 / 61.9 / 60 / 57.7 / 56
.3/.22 / T=4 / 92.5 / 88.3 / 79.7 / 76.2 / 73.5 / 73 / 71.8 / 67.8 / 66.9 / 62.7
T=6 / 85.2 / 76.6 / 65.4 / 60 / 57.3 / 52.7 / 50.9 / 48.7 / 45.1 / 37.7
T=10 / 73.4 / 60.9 / 49.1 / 41.1 / 39.6 / 33.3 / 33.5 / 27.5 / 24.3 / 13.2
.3/.4 / T=4 / 92.9 / 89.4 / 79.8 / 75.4 / 73.2 / 71.7 / 70 / 65.6 / 63.1 / 59.4
T=6 / 86.6 / 75.3 / 65.9 / 58.1 / 56.6 / 50.7 / 49.3 / 48.2 / 44.3 / 36.2
T=10 / 75.9 / 63.9 / 51.8 / 44.3 / 43.4 / 36.6 / 37.2 / 34.3 / 29.8 / 18.5
.3/.6 / T=4 / 93.5 / 90.2 / 81.3 / 76.9 / 74.9 / 73.1 / 70.8 / 66.6 / 63.7 / 58.4
T=6 / 87.6 / 77.9 / 67.7 / 59.9 / 59.8 / 53.6 / 52.5 / 48.8 / 46.6 / 39.3
T=10 / 78.4 / 68.1 / 56.3 / 50.5 / 47.6 / 42.5 / 43.8 / 39.5 / 34.9 / 26.9
.5/0 / T=4 / 92.7 / 88.2 / 81.3 / 77.5 / 75.6 / 76.1 / 74.3 / 71.9 / 71.3 / 67.1
T=6 / 87.9 / 81.4 / 75.1 / 71.5 / 70.8 / 67 / 66.2 / 64.5 / 63.8 / 57.8
T=10 / 81.6 / 75.7 / 70.2 / 65 / 62 / 62.5 / 61.9 / 60 / 57.7 / 56
.5/.22 / T=4 / 92.5 / 88.3 / 79.7 / 76.2 / 73.5 / 73 / 71.8 / 67.8 / 66.9 / 62.7
T=6 / 85.2 / 76.6 / 65.4 / 60 / 57.3 / 52.7 / 50.9 / 48.7 / 45.1 / 37.7
T=10 / 73.4 / 60.9 / 49.1 / 41.1 / 39.6 / 33.3 / 33.5 / 27.5 / 24.3 / 13.2
.5/.4 / T=4 / 92.9 / 89.4 / 79.8 / 75.4 / 73.2 / 71.7 / 70 / 65.6 / 63.1 / 59.4
T=6 / 86.6 / 75.3 / 65.9 / 58.1 / 56.6 / 50.7 / 49.3 / 48.2 / 44.3 / 36.2
T=10 / 75.9 / 63.9 / 51.8 / 44.3 / 43.4 / 36.6 / 37.2 / 34.3 / 29.8 / 18.5
.5/.6 / T=4 / 93.5 / 90.2 / 81.3 / 76.9 / 74.9 / 73.1 / 70.8 / 66.6 / 63.7 / 58.4
T=6 / 87.6 / 77.9 / 67.7 / 59.9 / 59.8 / 53.6 / 52.5 / 48.8 / 46.6 / 39.3
T=10 / 78.4 / 68.1 / 56.3 / 50.5 / 47.6 / 42.5 / 43.8 / 39.5 / 34.9 / 26.9

a Mean and Standard deviation of the quadratic component

Table S9. Mean Rates of Nonconvergence for the condition with R²=.5

Sample Sizes
M(Q)/SD(Q)a / Repeated measures / 30 / 50 / 100 / 150 / 200 / 250 / 300 / 400 / 500 / 1000
0/.22 / T=4 / 92.2 / 87.7 / 76.5 / 73 / 70 / 68.2 / 65.7 / 60.6 / 58.3 / 54.2
T=6 / 79.1 / 68.4 / 53.6 / 45.4 / 42 / 37 / 32.7 / 31.2 / 21.1 / 16.5
T=10 / 60.4 / 46.4 / 30.1 / 19.1 / 15.6 / 12.2 / 9.9 / 7.3 / 3.7 / 0.8
0/.4 / T=4 / 91.7 / 86.6 / 76.2 / 71.7 / 68.1 / 65.7 / 63.9 / 58.7 / 56 / 47.2
T=6 / 81.3 / 69.8 / 53.7 / 45.1 / 42.3 / 36.8 / 34 / 31.7 / 26.3 / 17.4
T=10 / 66.3 / 53.7 / 35.2 / 26.2 / 21.6 / 18.8 / 16.5 / 12.6 / 8.1 / 2.3
0/.6 / T=4 / 92.4 / 88.4 / 78.4 / 74.6 / 70.2 / 67.1 / 66.7 / 61.4 / 57.4 / 46.8
T=6 / 83.4 / 74 / 59.3 / 49.7 / 47.5 / 41.9 / 40.1 / 36.8 / 30.2 / 21.6
T=10 / 70.6 / 60.1 / 42.5 / 32.8 / 29.8 / 25.3 / 23.5 / 18.9 / 15.5 / 5.6
.3/0 / T=4 / 92.9 / 88 / 79.4 / 76.5 / 73.7 / 74.2 / 70.8 / 68 / 66.2 / 63.6
T=6 / 83.3 / 77 / 67.3 / 64 / 63.3 / 61.6 / 62 / 59.8 / 58.3 / 54.8
T=10 / 76.8 / 70.9 / 64.4 / 59.2 / 59.2 / 57.7 / 56.6 / 55.5 / 54.7 / 53.3
.3/.22 / T=4 / 92.2 / 87.7 / 76.5 / 73 / 70 / 68.2 / 65.7 / 60.6 / 58.3 / 54.2
T=6 / 79.1 / 68.4 / 53.6 / 45.4 / 42 / 37 / 32.7 / 31.2 / 21.1 / 16.5
T=10 / 60.4 / 46.4 / 30.1 / 19.1 / 15.6 / 12.2 / 9.9 / 7.3 / 3.7 / 0.8
.3/.4 / T=4 / 91.7 / 86.6 / 76.2 / 71.7 / 68.1 / 65.7 / 63.9 / 58.7 / 56 / 47.2
T=6 / 81.3 / 69.8 / 53.7 / 45.1 / 42.3 / 36.8 / 34 / 31.7 / 26.3 / 17.4
T=10 / 66.3 / 53.7 / 35.2 / 26.2 / 21.6 / 18.8 / 16.5 / 12.6 / 8.1 / 2.3
.3/.6 / T=4 / 92.4 / 88.4 / 78.4 / 74.6 / 70.2 / 67.1 / 66.7 / 61.4 / 57.4 / 46.8
T=6 / 83.4 / 74 / 59.3 / 49.7 / 47.5 / 41.9 / 40.1 / 36.8 / 30.2 / 21.6
T=10 / 70.6 / 60.1 / 42.5 / 32.8 / 29.8 / 25.3 / 23.5 / 18.9 / 15.5 / 5.6
.5/0 / T=4 / 92.9 / 88 / 79.4 / 76.5 / 73.7 / 74.2 / 70.8 / 68 / 66.2 / 63.6
T=6 / 83.3 / 77 / 67.3 / 64 / 63.3 / 61.6 / 62 / 59.8 / 58.3 / 54.8
T=10 / 76.8 / 70.9 / 64.4 / 59.2 / 59.2 / 57.7 / 56.6 / 55.5 / 54.7 / 53.3
.5/.22 / T=4 / 92.2 / 87.7 / 76.5 / 73 / 70 / 68.2 / 65.7 / 60.6 / 58.3 / 54.2
T=6 / 79.1 / 68.4 / 53.6 / 45.4 / 42 / 37 / 32.7 / 31.2 / 21.1 / 16.5
T=10 / 60.4 / 46.4 / 30.1 / 19.1 / 15.6 / 12.2 / 9.9 / 7.3 / 3.7 / 0.8
.5/.4 / T=4 / 91.7 / 86.6 / 76.2 / 71.7 / 68.1 / 65.7 / 63.9 / 58.7 / 56 / 47.2
T=6 / 81.3 / 69.8 / 53.7 / 45.1 / 42.3 / 36.8 / 34 / 31.7 / 26.3 / 17.4
T=10 / 66.3 / 53.7 / 35.2 / 26.2 / 21.6 / 18.8 / 16.5 / 12.6 / 8.1 / 2.3
.5/.6 / T=4 / 92.4 / 88.4 / 78.4 / 74.6 / 70.2 / 67.1 / 66.7 / 61.4 / 57.4 / 46.8
T=6 / 83.4 / 74 / 59.3 / 49.7 / 47.5 / 41.9 / 40.1 / 36.8 / 30.2 / 21.6
T=10 / 70.6 / 60.1 / 42.5 / 32.8 / 29.8 / 25.3 / 23.5 / 18.9 / 15.5 / 5.6

a Mean and Standard deviation of the quadratic component

Table S10. Mean Rates of Nonconvergence for the condition R²=.75

Sample Sizes
M(Q)/SD(Q)a / Repeated measures / 30 / 50 / 100 / 150 / 200 / 250 / 300 / 400 / 500 / 1000
0/.22 / T=4 / 90.5 / 86.9 / 76.7 / 71 / 65.2 / 62 / 58.6 / 51.8 / 48.1 / 32.6
T=6 / 71.8 / 56.7 / 33.6 / 22.2 / 17.5 / 13.8 / 10.4 / 5.6 / 3.8 / 0.1
T=10 / 42.8 / 23 / 6.2 / 1.8 / 0.8 / 0.2 / 0 / 0.1 / 0 / 0
0/.4 / T=4 / 91.1 / 85.2 / 74.2 / 68.6 / 62.1 / 59.8 / 56 / 47 / 44.2 / 25.4
T=6 / 73.1 / 61.4 / 39.4 / 27.1 / 22.2 / 17 / 13.5 / 8.2 / 4.5 / 0.3
T=10 / 52.5 / 33.5 / 14.1 / 5.8 / 2.6 / 1 / 1 / 0.3 / 0 / 0
0/.6 / T=4 / 92.1 / 88 / 76.3 / 73.2 / 66.5 / 63.5 / 59.8 / 51.9 / 48.9 / 30.4
T=6 / 79.7 / 68.5 / 48.7 / 37 / 31.7 / 23.6 / 21.4 / 14.4 / 10.2 / 1.7
T=10 / 61.3 / 44.3 / 23.2 / 13 / 8.4 / 5.3 / 3.7 / 1.6 / 0.3 / 0
.3/0 / T=4 / 91.7 / 89.3 / 79.7 / 76.2 / 73.7 / 71.8 / 68.5 / 65.8 / 64 / 60.3
T=6 / 81.1 / 74.4 / 65 / 61.5 / 59.3 / 57.2 / 57.5 / 57 / 56.5 / 53.1
T=10 / 74.5 / 69.4 / 62.3 / 57.3 / 56.2 / 54.2 / 55 / 52.2 / 53.8 / 51.7
.3/.22 / T=4 / 90.5 / 86.9 / 76.7 / 71 / 65.2 / 62 / 58.6 / 51.8 / 48.1 / 32.6
T=6 / 71.8 / 56.7 / 33.6 / 22.2 / 17.5 / 13.8 / 10.4 / 5.6 / 3.8 / 0.1
T=10 / 42.8 / 23 / 6.2 / 1.8 / 0.8 / 0.2 / 0 / 0.1 / 0 / 0
.3/.4 / T=4 / 91.1 / 85.2 / 74.2 / 68.6 / 62.1 / 59.8 / 56 / 47 / 44.2 / 25.4
T=6 / 73.1 / 61.4 / 39.4 / 27.1 / 22.2 / 17 / 13.5 / 8.2 / 4.5 / 0.3
T=10 / 52.5 / 33.5 / 14.1 / 5.8 / 2.6 / 1 / 1 / 0.3 / 0 / 0
.3/.6 / T=4 / 92.1 / 88 / 76.3 / 73.2 / 66.5 / 63.5 / 59.8 / 51.9 / 48.9 / 30.4
T=6 / 79.7 / 68.5 / 48.7 / 37 / 31.7 / 23.6 / 21.4 / 14.4 / 10.2 / 1.7
T=10 / 61.3 / 44.3 / 23.2 / 13 / 8.4 / 5.3 / 3.7 / 1.6 / 0.3 / 0
.5/0 / T=4 / 91.7 / 89.3 / 79.7 / 76.2 / 73.7 / 71.8 / 68.5 / 65.8 / 64 / 60.3
T=6 / 81.1 / 74.4 / 65 / 61.5 / 59.3 / 57.2 / 57.5 / 57 / 56.5 / 53.1
T=10 / 74.5 / 69.4 / 62.3 / 57.3 / 56.2 / 54.2 / 55 / 52.2 / 53.8 / 51.7
.5/.22 / T=4 / 90.5 / 86.9 / 76.7 / 71 / 65.2 / 62 / 58.6 / 51.8 / 48.1 / 32.6
T=6 / 71.8 / 56.7 / 33.6 / 22.2 / 17.5 / 13.8 / 10.4 / 5.6 / 3.8 / 0.1
T=10 / 42.8 / 23 / 6.2 / 1.8 / 0.8 / 0.2 / 0 / 0.1 / 0 / 0
.5/.4 / T=4 / 91.1 / 85.2 / 74.2 / 68.6 / 62.1 / 59.8 / 56 / 47 / 44.2 / 25.4
T=6 / 73.1 / 61.4 / 39.4 / 27.1 / 22.2 / 17 / 13.5 / 8.2 / 4.5 / 0.3
T=10 / 52.5 / 33.5 / 14.1 / 5.8 / 2.6 / 1 / 1 / 0.3 / 0 / 0
.5/.6 / T=4 / 92.1 / 88 / 76.3 / 73.2 / 66.5 / 63.5 / 59.8 / 51.9 / 48.9 / 30.4
T=6 / 79.7 / 68.5 / 48.7 / 37 / 31.7 / 23.6 / 21.4 / 14.4 / 10.2 / 1.7
T=10 / 61.3 / 44.3 / 23.2 / 13 / 8.4 / 5.3 / 3.7 / 1.6 / 0.3 / 0

a Mean and Standard deviation of the quadratic component

1

Quadratic Growth

Figure S1. Empirical Power Curves for R2 of .75 and a Quadratic Mean of .3.

Figure S2. Empirical Power Curves for R2 of .75 and a Quadratic Mean of .5

1

Quadratic Growth