A few spline models (also known as piecewise models). As in previous posts, ‘affect’ is the name given to values of \(y\) throughout.
1) Growth and Even More Growth
A model that captures a process that increases initially and then increases at an even greater rate once it reaches time point 5. The data generating process:
\[\begin{equation} y_{it} = \begin{cases} 4 + 0.3t + error_{t}, & \text{if time < 5}\\ 8 + 0.9t + error_{t}, & \text{otherwise} \end{cases} \end{equation}\]
The data generating code and plot
library(tidyverse)
library(lavaan)
library(ggplot2)
library(MASS)
N <- 400
time <- 10
intercept_1 <- 4
intercept_2 <- 8
growth1 <- 0.3
growth2 <- 0.9
df_matrix <- matrix(, ncol = 3, nrow = N*time)
count <- 0
for(i in 1:N){
unob_het_y <- rnorm(1,0,1)
for(j in 1:time){
count <- count + 1
if(j < 5){
df_matrix[count, 1] <- i
df_matrix[count, 2] <- j
df_matrix[count, 3] <- intercept_1 + growth1*j + unob_het_y + rnorm(1,0,1)
}else{
df_matrix[count, 1] <- i
df_matrix[count, 2] <- j
df_matrix[count, 3] <- intercept_2 + growth2*j + unob_het_y + rnorm(1,0,1)
}
}
}
df <- data.frame(df_matrix)
names(df) <- c('id', 'time', 'affect')
df1 <- df %>%
filter(time < 5)
df2 <- df %>%
filter(time >= 5)
df_sum1 <- df1 %>%
group_by(time) %>%
summarise(
affect = mean(affect)
)
df_sum2 <- df2 %>%
group_by(time) %>%
summarise(
affect = mean(affect)
)
ggplot() +
geom_point(data = df1, aes(x = time, y = affect, group = id), color = 'gray85') +
geom_line(data = df1, aes(x = time, y = affect, group = id), color = 'gray85') +
geom_point(data = df2, aes(x = time, y = affect, group = id), color = 'gray85') +
geom_line(data = df2, aes(x = time, y = affect, group = id), color = 'gray85') +
geom_line(data = df_sum1, aes(x = time, y = affect)) +
geom_line(data = df_sum2, aes(x = time, y = affect))
Estimating the parameters using SEM:
library(lavaan)
df_wide <- reshape(df, idvar = 'id', timevar = 'time', direction = 'wide')
spline_string <- [1687 chars quoted with ''']
spline_model <- growth(spline_string, data = df_wide)
summary(spline_model, fit.measures = T)
lavaan 0.6-9 ended normally after 83 iterations
Estimator ML
Optimization method NLMINB
Number of model parameters 24
Number of equality constraints 9
Number of observations 400
Model Test User Model:
Test statistic 40.920
Degrees of freedom 50
P-value (Chi-square) 0.816
Model Test Baseline Model:
Test statistic 1854.037
Degrees of freedom 45
P-value 0.000
User Model versus Baseline Model:
Comparative Fit Index (CFI) 1.000
Tucker-Lewis Index (TLI) 1.005
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -6174.770
Loglikelihood unrestricted model (H1) -6154.310
Akaike (AIC) 12379.539
Bayesian (BIC) 12439.411
Sample-size adjusted Bayesian (BIC) 12391.815
Root Mean Square Error of Approximation:
RMSEA 0.000
90 Percent confidence interval - lower 0.000
90 Percent confidence interval - upper 0.021
P-value RMSEA <= 0.05 1.000
Standardized Root Mean Square Residual:
SRMR 0.032
Parameter Estimates:
Standard errors Standard
Information Expected
Information saturated (h1) model Structured
Latent Variables:
Estimate Std.Err z-value P(>|z|)
level1_affect =~
affect.1 1.000
affect.2 1.000
affect.3 1.000
affect.4 1.000
affect.5 0.000
affect.6 0.000
affect.7 0.000
affect.8 0.000
affect.9 0.000
affect.10 0.000
level2_affect =~
affect.1 0.000
affect.2 0.000
affect.3 0.000
affect.4 0.000
affect.5 1.000
affect.6 1.000
affect.7 1.000
affect.8 1.000
affect.9 1.000
affect.10 1.000
slope1_affect =~
affect.1 1.000
affect.2 2.000
affect.3 3.000
affect.4 4.000
affect.5 0.000
affect.6 0.000
affect.7 0.000
affect.8 0.000
affect.9 0.000
affect.10 0.000
slope2_affect =~
affect.1 0.000
affect.2 0.000
affect.3 0.000
affect.4 0.000
affect.5 5.000
affect.6 6.000
affect.7 7.000
affect.8 8.000
affect.9 9.000
affect.10 10.000
Covariances:
Estimate Std.Err z-value P(>|z|)
level1_affect ~~
level2_affect 0.941 0.183 5.147 0.000
slope1_affect -0.093 0.052 -1.783 0.075
slope2_affect 0.006 0.021 0.292 0.770
level2_affect ~~
slope1_affect 0.041 0.052 0.802 0.423
slope2_affect -0.028 0.037 -0.748 0.454
slope1_affect ~~
slope2_affect -0.004 0.006 -0.581 0.561
Intercepts:
Estimate Std.Err z-value P(>|z|)
.affect.1 0.000
.affect.2 0.000
.affect.3 0.000
.affect.4 0.000
.affect.5 0.000
.affect.6 0.000
.affect.7 0.000
.affect.8 0.000
.affect.9 0.000
.affect.10 0.000
level1_affect 3.929 0.083 47.407 0.000
level2_affect 7.849 0.107 73.630 0.000
slope1_affect 0.294 0.024 12.166 0.000
slope2_affect 0.909 0.013 72.612 0.000
Variances:
Estimate Std.Err z-value P(>|z|)
lvl1_ff 1.234 0.199 6.198 0.000
lvl2_ff 1.135 0.336 3.375 0.001
slp1_ff 0.032 0.018 1.839 0.066
slp2_ff 0.005 0.005 1.068 0.285
.affct.1 (rs_v) 1.009 0.029 34.641 0.000
.affct.2 (rs_v) 1.009 0.029 34.641 0.000
.affct.3 (rs_v) 1.009 0.029 34.641 0.000
.affct.4 (rs_v) 1.009 0.029 34.641 0.000
.affct.5 (rs_v) 1.009 0.029 34.641 0.000
.affct.6 (rs_v) 1.009 0.029 34.641 0.000
.affct.7 (rs_v) 1.009 0.029 34.641 0.000
.affct.8 (rs_v) 1.009 0.029 34.641 0.000
.affct.9 (rs_v) 1.009 0.029 34.641 0.000
.affc.10 (rs_v) 1.009 0.029 34.641 0.000The structure of the basis coefficients is the important piece that allows us to capture the change in slope:
'
# latent slope for first half basis coefficients
slope1_affect =~ 1*affect.1 + 2*affect.2 + 3*affect.3 + 4*affect.4 + 0*affect.5 + 0*affect.6 + 0*affect.7 + 0*affect.8 + 0*affect.9 + 0*affect.10
# latent slope for second half basis coefficients
slope2_affect =~ 0*affect.1 + 0*affect.2 + 0*affect.3 + 0*affect.4 + 5*affect.5 + 6*affect.6 + 7*affect.7 + 8*affect.8 + 9*affect.9 + 10*affect.10
'
2) Growth and Negative Growth
A model that captures a process that goes up and then goes down. The data generating process:
\[\begin{equation} y_{it} = \begin{cases} 4 + 0.5t + error_{t}, & \text{if time < 5}\\ 4 - 0.5t + error_{t}, & \text{otherwise} \end{cases} \end{equation}\]
The data generating code and plot
library(tidyverse)
library(lavaan)
library(ggplot2)
library(MASS)
N <- 400
time <- 10
intercept_1 <- 4
intercept_2 <- 4
growth1 <- 0.8
growth2 <- -0.8
df_matrix_b <- matrix(, ncol = 3, nrow = N*time)
count <- 0
for(i in 1:N){
unob_het_y <- rnorm(1,0,1)
for(j in 1:time){
count <- count + 1
if(j < 5){
df_matrix_b[count, 1] <- i
df_matrix_b[count, 2] <- j
df_matrix_b[count, 3] <- intercept_1 + growth1*j + unob_het_y + rnorm(1,0,1)
}else{
df_matrix_b[count, 1] <- i
df_matrix_b[count, 2] <- j
df_matrix_b[count, 3] <- intercept_2 + growth2*j + unob_het_y + rnorm(1,0,1)
}
}
}
df_b <- data.frame(df_matrix_b)
names(df_b) <- c('id', 'time', 'affect')
df1_b <- df_b %>%
filter(time < 5)
df2_b <- df_b %>%
filter(time >= 5)
df_sum1_b <- df1_b %>%
group_by(time) %>%
summarise(
affect = mean(affect)
)
df_sum2_b <- df2_b %>%
group_by(time) %>%
summarise(
affect = mean(affect)
)
ggplot() +
geom_point(data = df1_b, aes(x = time, y = affect, group = id), color = 'gray85') +
geom_line(data = df1_b, aes(x = time, y = affect, group = id), color = 'gray85') +
geom_point(data = df2_b, aes(x = time, y = affect, group = id), color = 'gray85') +
geom_line(data = df2_b, aes(x = time, y = affect, group = id), color = 'gray85') +
geom_line(data = df_sum1_b, aes(x = time, y = affect)) +
geom_line(data = df_sum2_b, aes(x = time, y = affect))
Estimating the parameters using SEM:
library(lavaan)
df_wide_b <- reshape(df_b, idvar = 'id', timevar = 'time', direction = 'wide')
spline_string_b <- [1687 chars quoted with ''']
spline_model_b <- growth(spline_string_b, data = df_wide_b)
summary(spline_model_b, fit.measures = T)
lavaan 0.6-9 ended normally after 79 iterations
Estimator ML
Optimization method NLMINB
Number of model parameters 24
Number of equality constraints 9
Number of observations 400
Model Test User Model:
Test statistic 60.348
Degrees of freedom 50
P-value (Chi-square) 0.150
Model Test Baseline Model:
Test statistic 1931.249
Degrees of freedom 45
P-value 0.000
User Model versus Baseline Model:
Comparative Fit Index (CFI) 0.995
Tucker-Lewis Index (TLI) 0.995
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -6219.675
Loglikelihood unrestricted model (H1) -6189.501
Akaike (AIC) 12469.349
Bayesian (BIC) 12529.221
Sample-size adjusted Bayesian (BIC) 12481.625
Root Mean Square Error of Approximation:
RMSEA 0.023
90 Percent confidence interval - lower 0.000
90 Percent confidence interval - upper 0.041
P-value RMSEA <= 0.05 0.995
Standardized Root Mean Square Residual:
SRMR 0.059
Parameter Estimates:
Standard errors Standard
Information Expected
Information saturated (h1) model Structured
Latent Variables:
Estimate Std.Err z-value P(>|z|)
level1_affect =~
affect.1 1.000
affect.2 1.000
affect.3 1.000
affect.4 1.000
affect.5 0.000
affect.6 0.000
affect.7 0.000
affect.8 0.000
affect.9 0.000
affect.10 0.000
level2_affect =~
affect.1 0.000
affect.2 0.000
affect.3 0.000
affect.4 0.000
affect.5 1.000
affect.6 1.000
affect.7 1.000
affect.8 1.000
affect.9 1.000
affect.10 1.000
slope1_affect =~
affect.1 1.000
affect.2 2.000
affect.3 3.000
affect.4 4.000
affect.5 0.000
affect.6 0.000
affect.7 0.000
affect.8 0.000
affect.9 0.000
affect.10 0.000
slope2_affect =~
affect.1 0.000
affect.2 0.000
affect.3 0.000
affect.4 0.000
affect.5 5.000
affect.6 6.000
affect.7 7.000
affect.8 8.000
affect.9 9.000
affect.10 10.000
Covariances:
Estimate Std.Err z-value P(>|z|)
level1_affect ~~
level2_affect 1.110 0.180 6.151 0.000
slope1_affect -0.029 0.049 -0.594 0.553
slope2_affect 0.016 0.020 0.788 0.431
level2_affect ~~
slope1_affect -0.041 0.046 -0.893 0.372
slope2_affect 0.036 0.034 1.037 0.300
slope1_affect ~~
slope2_affect -0.003 0.005 -0.576 0.565
Intercepts:
Estimate Std.Err z-value P(>|z|)
.affect.1 0.000
.affect.2 0.000
.affect.3 0.000
.affect.4 0.000
.affect.5 0.000
.affect.6 0.000
.affect.7 0.000
.affect.8 0.000
.affect.9 0.000
.affect.10 0.000
level1_affect 3.886 0.084 46.497 0.000
level2_affect 3.740 0.103 36.398 0.000
slope1_affect 0.813 0.023 36.105 0.000
slope2_affect -0.778 0.012 -65.684 0.000
Variances:
Estimate Std.Err z-value P(>|z|)
lvl1_ff 1.241 0.203 6.124 0.000
lvl2_ff 0.721 0.315 2.289 0.022
slp1_ff -0.004 0.016 -0.290 0.772
slp2_ff -0.003 0.004 -0.709 0.478
.affct.1 (rs_v) 1.035 0.030 34.641 0.000
.affct.2 (rs_v) 1.035 0.030 34.641 0.000
.affct.3 (rs_v) 1.035 0.030 34.641 0.000
.affct.4 (rs_v) 1.035 0.030 34.641 0.000
.affct.5 (rs_v) 1.035 0.030 34.641 0.000
.affct.6 (rs_v) 1.035 0.030 34.641 0.000
.affct.7 (rs_v) 1.035 0.030 34.641 0.000
.affct.8 (rs_v) 1.035 0.030 34.641 0.000
.affct.9 (rs_v) 1.035 0.030 34.641 0.000
.affc.10 (rs_v) 1.035 0.030 34.641 0.000Notice that the string syntax is the exact same because the process changes at the same point in time, it does not matter if the process changes to ‘more positive’ or ‘more negative.’
3) Negative Growth, Growth, and Negative Growth
Now a process that goes down, goes up, and then goes back down. The data generating process:
\[\begin{equation} y_{it} = \begin{cases} 4 - 0.5t + error_{t}, & \text{if time < 5}\\ 4 + 0.5t + error_{t}, & \text{if 5 < time < 10}\\ 4 - 0.5t + error_{t}, & \text{otherwise} \end{cases} \end{equation}\]
The data generating code and plot
library(tidyverse)
library(lavaan)
library(ggplot2)
library(MASS)
N <- 400
time <- 15
intercept_1 <- 4
intercept_2 <- 4
intercept_3 <- 4
growth1 <- -0.5
growth2 <- 0.5
growth3 <- -0.5
df_matrix_c <- matrix(, ncol = 3, nrow = N*time)
count <- 0
for(i in 1:N){
unob_het_y <- rnorm(1,0,1)
for(j in 1:time){
count <- count + 1
if(j < 5){
df_matrix_c[count, 1] <- i
df_matrix_c[count, 2] <- j
df_matrix_c[count, 3] <- intercept_1 + growth1*j + unob_het_y + rnorm(1,0,1)
}else if(j >= 5 && j < 10){
df_matrix_c[count, 1] <- i
df_matrix_c[count, 2] <- j
df_matrix_c[count, 3] <- intercept_2 + growth2*j + unob_het_y + rnorm(1,0,1)
}else{
df_matrix_c[count, 1] <- i
df_matrix_c[count, 2] <- j
df_matrix_c[count, 3] <- intercept_3 + growth3*j + unob_het_y + rnorm(1,0,1)
}
}
}
df_c <- data.frame(df_matrix_c)
names(df_c) <- c('id', 'time', 'affect')
df1_c <- df_c %>%
filter(time < 5)
df2_c <- df_c %>%
filter(time >= 5 & time < 10)
df3_c <- df_c %>%
filter(time >= 10)
df_sum1_c <- df1_c %>%
group_by(time) %>%
summarise(
affect = mean(affect)
)
df_sum2_c <- df2_c %>%
group_by(time) %>%
summarise(
affect = mean(affect)
)
df_sum3_c <- df3_c %>%
group_by(time) %>%
summarise(
affect = mean(affect)
)
ggplot() +
geom_point(data = df1_c, aes(x = time, y = affect, group = id), color = 'gray85') +
geom_line(data = df1_c, aes(x = time, y = affect, group = id), color = 'gray85') +
geom_point(data = df2_c, aes(x = time, y = affect, group = id), color = 'gray85') +
geom_line(data = df2_c, aes(x = time, y = affect, group = id), color = 'gray85') +
geom_line(data = df_sum1_c, aes(x = time, y = affect)) +
geom_line(data = df_sum2_c, aes(x = time, y = affect)) +
geom_point(data = df3_c, aes(x = time, y = affect, group = id), color = 'gray85') +
geom_line(data = df3_c, aes(x = time, y = affect, group = id), color = 'gray85') +
geom_line(data = df_sum3_c, aes(x = time, y = affect))
Now estimate the parameters using SEM:
library(lavaan)
df_wide_c <- reshape(df_c, idvar = 'id', timevar = 'time', direction = 'wide')
spline_string_c <- [2843 chars quoted with ''']
spline_model_c <- growth(spline_string_c, data = df_wide_c)
summary(spline_model_c, fit.measures = T)
lavaan 0.6-9 ended normally after 167 iterations
Estimator ML
Optimization method NLMINB
Number of model parameters 42
Number of equality constraints 9
Number of observations 400
Model Test User Model:
Test statistic 120.754
Degrees of freedom 102
P-value (Chi-square) 0.099
Model Test Baseline Model:
Test statistic 3055.629
Degrees of freedom 105
P-value 0.000
User Model versus Baseline Model:
Comparative Fit Index (CFI) 0.994
Tucker-Lewis Index (TLI) 0.993
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -9115.240
Loglikelihood unrestricted model (H1) -9054.863
Akaike (AIC) 18296.479
Bayesian (BIC) 18428.198
Sample-size adjusted Bayesian (BIC) 18323.486
Root Mean Square Error of Approximation:
RMSEA 0.021
90 Percent confidence interval - lower 0.000
90 Percent confidence interval - upper 0.035
P-value RMSEA <= 0.05 1.000
Standardized Root Mean Square Residual:
SRMR 0.040
Parameter Estimates:
Standard errors Standard
Information Expected
Information saturated (h1) model Structured
Latent Variables:
Estimate Std.Err z-value P(>|z|)
level1_affect =~
affect.1 1.000
affect.2 1.000
affect.3 1.000
affect.4 1.000
affect.5 0.000
affect.6 0.000
affect.7 0.000
affect.8 0.000
affect.9 0.000
affect.10 0.000
affect.11 0.000
affect.12 0.000
affect.13 0.000
affect.14 0.000
affect.15 0.000
level2_affect =~
affect.1 0.000
affect.2 0.000
affect.3 0.000
affect.4 0.000
affect.5 1.000
affect.6 1.000
affect.7 1.000
affect.8 1.000
affect.9 1.000
affect.10 0.000
affect.11 0.000
affect.12 0.000
affect.13 0.000
affect.14 0.000
affect.15 0.000
level3_affect =~
affect.1 0.000
affect.2 0.000
affect.3 0.000
affect.4 0.000
affect.5 0.000
affect.6 0.000
affect.7 0.000
affect.8 0.000
affect.9 0.000
affect.10 1.000
affect.11 1.000
affect.12 1.000
affect.13 1.000
affect.14 1.000
affect.15 1.000
slope1_affect =~
affect.1 1.000
affect.2 2.000
affect.3 3.000
affect.4 4.000
affect.5 0.000
affect.6 0.000
affect.7 0.000
affect.8 0.000
affect.9 0.000
affect.10 0.000
affect.11 0.000
affect.12 0.000
affect.13 0.000
affect.14 0.000
affect.15 0.000
slope2_affect =~
affect.1 0.000
affect.2 0.000
affect.3 0.000
affect.4 0.000
affect.5 5.000
affect.6 6.000
affect.7 7.000
affect.8 8.000
affect.9 9.000
affect.10 0.000
affect.11 0.000
affect.12 0.000
affect.13 0.000
affect.14 0.000
affect.15 0.000
slope3_affect =~
affect.1 0.000
affect.2 0.000
affect.3 0.000
affect.4 0.000
affect.5 0.000
affect.6 0.000
affect.7 0.000
affect.8 0.000
affect.9 0.000
affect.10 10.000
affect.11 11.000
affect.12 12.000
affect.13 13.000
affect.14 14.000
affect.15 15.000
Covariances:
Estimate Std.Err z-value P(>|z|)
level1_affect ~~
level2_affect 1.113 0.209 5.311 0.000
level3_affect 1.354 0.280 4.844 0.000
slope1_affect -0.046 0.051 -0.902 0.367
slope2_affect -0.004 0.025 -0.169 0.866
slope3_affect -0.023 0.020 -1.142 0.254
level2_affect ~~
level3_affect 0.729 0.399 1.828 0.067
slope1_affect -0.062 0.058 -1.074 0.283
slope2_affect 0.076 0.055 1.391 0.164
slope3_affect 0.021 0.030 0.725 0.468
level3_affect ~~
slope1_affect -0.074 0.077 -0.963 0.336
slope2_affect 0.048 0.050 0.960 0.337
slope3_affect -0.028 0.060 -0.462 0.644
slope1_affect ~~
slope2_affect 0.005 0.007 0.626 0.531
slope3_affect 0.003 0.006 0.511 0.609
slope2_affect ~~
slope3_affect -0.004 0.004 -1.151 0.250
Intercepts:
Estimate Std.Err z-value P(>|z|)
.affect.1 0.000
.affect.2 0.000
.affect.3 0.000
.affect.4 0.000
.affect.5 0.000
.affect.6 0.000
.affect.7 0.000
.affect.8 0.000
.affect.9 0.000
.affect.10 0.000
.affect.11 0.000
.affect.12 0.000
.affect.13 0.000
.affect.14 0.000
.affect.15 0.000
level1_affect 3.876 0.083 46.673 0.000
level2_affect 3.993 0.122 32.835 0.000
level3_affect 3.964 0.163 24.272 0.000
slope1_affect -0.491 0.024 -20.791 0.000
slope2_affect 0.496 0.015 32.376 0.000
slope3_affect -0.501 0.012 -41.367 0.000
Variances:
Estimate Std.Err z-value P(>|z|)
lvl1_ff 1.167 0.201 5.816 0.000
lvl2_ff 0.505 0.448 1.127 0.260
lvl3_ff 1.470 0.815 1.804 0.071
slp1_ff 0.011 0.017 0.666 0.505
slp2_ff -0.012 0.007 -1.658 0.097
slp3_ff 0.001 0.005 0.253 0.801
.affct.1 (rs_v) 1.061 0.032 33.639 0.000
.affct.2 (rs_v) 1.061 0.032 33.639 0.000
.affct.3 (rs_v) 1.061 0.032 33.639 0.000
.affct.4 (rs_v) 1.061 0.032 33.639 0.000
.affct.5 (rs_v) 1.061 0.032 33.639 0.000
.affct.6 (rs_v) 1.061 0.032 33.639 0.000
.affct.7 (rs_v) 1.061 0.032 33.639 0.000
.affct.8 (rs_v) 1.061 0.032 33.639 0.000
.affct.9 (rs_v) 1.061 0.032 33.639 0.000
.affc.10 (rs_v) 1.061 0.032 33.639 0.000
.affc.11 0.958 0.076 12.593 0.000
.affc.12 0.939 0.072 12.961 0.000
.affc.13 0.896 0.069 12.940 0.000
.affc.14 0.964 0.076 12.657 0.000
.affc.15 1.001 0.087 11.487 0.000Again, the basis coefficients are the important piece here:
'
# latent slope for first third basis coefficients
slope1_affect =~ 1*affect.1 + 2*affect.2 + 3*affect.3 + 4*affect.4 +
0*affect.5 + 0*affect.6 + 0*affect.7 + 0*affect.8 +
0*affect.9 + 0*affect.10 + 0*affect.11 + 0*affect.12 +
0*affect.13 + 0*affect.14 + 0*affect.15
# latent slope for second third basis coefficients
slope2_affect =~ 0*affect.1 + 0*affect.2 + 0*affect.3 + 0*affect.4 +
5*affect.5 + 6*affect.6 + 7*affect.7 + 8*affect.8 +
9*affect.9 + 0*affect.10 + 0*affect.11 + 0*affect.12 +
0*affect.13 + 0*affect.14 + 0*affect.15
# latent slope for final third basis coefficients
slope3_affect =~ 0*affect.1 + 0*affect.2 + 0*affect.3 + 0*affect.4 +
0*affect.5 + 0*affect.6 + 0*affect.7 + 0*affect.8 +
0*affect.9 + 10*affect.10 + 11*affect.11 + 12*affect.12 +
13*affect.13 + 14*affect.14 + 15*affect.15
'
[1] "\n\n\n# latent slope for first third basis coefficients\n\nslope1_affect =~ 1*affect.1 + 2*affect.2 + 3*affect.3 + 4*affect.4 + \n 0*affect.5 + 0*affect.6 + 0*affect.7 + 0*affect.8 + \n 0*affect.9 + 0*affect.10 + 0*affect.11 + 0*affect.12 + \n 0*affect.13 + 0*affect.14 + 0*affect.15\n\n# latent slope for second third basis coefficients\n\nslope2_affect =~ 0*affect.1 + 0*affect.2 + 0*affect.3 + 0*affect.4 + \n 5*affect.5 + 6*affect.6 + 7*affect.7 + 8*affect.8 + \n 9*affect.9 + 0*affect.10 + 0*affect.11 + 0*affect.12 + \n 0*affect.13 + 0*affect.14 + 0*affect.15\n\n# latent slope for final third basis coefficients\n\nslope3_affect =~ 0*affect.1 + 0*affect.2 + 0*affect.3 + 0*affect.4 +\n 0*affect.5 + 0*affect.6 + 0*affect.7 + 0*affect.8 + \n 0*affect.9 + 10*affect.10 + 11*affect.11 + 12*affect.12 + \n 13*affect.13 + 14*affect.14 + 15*affect.15\n\n\n\n"Bo\(^2\)m =)