Parallel Processing Model** 
Grade 1 
87501 
0.99 

0.017 
Coefficient represents the correlation between processes and it thus not a range but a single estimate. SEM represents the standard error of the correlation coefficient. 
Parallel Processing Model 
Grade 2 
79354 
0.95 

0.088 
Parallel Processing Model 
Grade 3 
1067 
0.81 

0.189 
Parallel Processing Model** 
Grade 1 
937 
0.81 


Coefficient represents the correlation between processes and is thus not a range but a single estimate. 
Parallel Processing Model 
Grade 2 
665 
0.88 


Parallel Processing Model 
Grade 3 
122 
0.85 


**Modelbased reliability by parallelprocess structural equation growth modeling. Values represent SpearmanBrown corrected correlation coefficient between each half of the parallel process model.
See Patarapichayatham, Anderson, Irvin, Kamata, Alonzo, and Tindal (2011). easyCBM® Slope Reliability: Letter Names, Word Reading Fluency, and Passage Reading Fluency (Technical Report No. 1111). Eugene, OR: Behavioral Research and Teaching, University of Oregon.
This study aimed to estimate the reliability of the slope for three easyCBM® measures. Under a structural equation modeling (SEM) framework, a growth model with two parallel growth processes was used. Essentially, two linear growth models were simultaneously modeled. The two parallel growth processes were established by splitting the available time segments into two groups. One group of time segments was used to form one linear growth process, and another group of time segments was used to form another linear growth process. For each linear growth process, the individual slopes of growth were estimated as factor scores of the latent slope factor. Then, the correlation between individual slopes from the two parallel growth processes was computed as an estimate of the reliability of the growth slope. The SpearmanBrown formula was then used to correct the correlation coefficient.
The procedure was analogous to VanDerHeyden and Burns (2008). In order to estimate the reliability of a slope, they (1) split a series of longitudinal observations into two parallel series, (2) computed an OLS regression slope for each individual for each series, (3) computed the correlation of the individual slopes between the two parallel series, and (4) corrected the correlation by the SpearmanBrown formula. Our procedure was exactly the same as VanDerHyden and Burns’ fourstep procedure, with one exception. For step 2 VanDerHyden and Burns’s derived a direct estimate of individual slopes based only on the observed measures of each student. By contrast, our method used empirical Bayes estimates of individual slopes (e.g., Raudenbush & Bryk, 2002) that incorporated information about the estimated mean slope and the estimated variance of individual slopes from the entire sample data.
The biweekly segments were evenly split into two parallel processes in the following manner. The first biweekly segment (average of weeks 1 and 2) was labeled 1A and assigned to a group of time segments for one linear growth process (Process A). The second biweekly segment (average of weeks 3 and 4) was labeled 1B and assigned to a group of time segments for another linear growth process (Process B). Similarly, the third biweekly segment (average of weeks 5 and 6) was labeled 2A and assigned to Process A, while the fourth biweekly segment (average of weeks 7 and 8) was labeled 2B and assigned to Process B. This pattern continued for the entire available biweekly segments, totaling 20 time segments, 1A – 10B, across 38 weeks of the school year. However, in many grades there were zero or nearzero students represented in the first two time segments (1A and 1B) and the last two time segments (10A and 10B). Also, there were other time segments with very few observations for some of the data sets. As a part of data cleaning process, descriptive statistics for each time segment for each data set were examined, and time segments with zero or nearzero students represented were deleted from the data.
In each data set, the linear growth model for two parallel processes was fit. The first linear growth model (Process A) was fit with the “A” time segments (2A, 3A, 4A, 5A, 6A, 7A, 8A, and 9A), whereas the second linear growth model (Process B) was fit with the “B” time segments (2B, 3B, 4B, 5B, 6B, 7B, 8B, and 9B). For both growth processes, the time scores of the growth slope factor were fixed to 0, 1, 2, 3, 4, 5, 6, 7, and 8 to define a linear growth model with equal time intervals between time segments. The zero time score for the growth slope factor at time segment one defines the intercept, initial status factors. On the other hand, the coefficients of the growth intercept factors were fixed at one as part of the regular growth model parameterization. The residual variances of the outcome variables (observed test scores) were estimated but fixed to be the same across time segments. Also, it was assumed that the residuals were not correlated. On the other hand, the growth slope factors were assumed to be correlated. The correlation between the two growth slope factors from the two growth processes, was interpreted as the reliability of the slope of the growth. All parameters were estimated with the Mplus software, using the Maximum Likelihood estimator with robust standard error.