This R package allows you to calculate Standardized (z-transformed) Regression Based (‘ZSRB’) change scores. Such scores permit evaluation of meaningful changes in test scores over time, while accounting for test–retest reliability, practice effects, score fluctuations due to error, and relevant clinical and demographic factors.
Calculating ZSRB scores requires a control group that serves as the baseline population for estimating ‘normal’ expected change, and one or more other groups.
There are two approaches to calculating ZSRB scores:
- Simple ZSRB (legacy option for validation of / comparison with other work)
- Complex ZSRB (preferred option that allows taking into account covariates)
More information on the differences between these methods and how the compare to other methods of change can be found in these articles:
- Duff, K. (2012). Evidence-based indicators of neuropsychological change in the individual patient: relevant concepts and methods. Archives of Clinical Neuropsychology, 27(3), 248–261. http://dx.doi.org/10.1093/arclin/acr120
- Duff, K., Atkinson, T. J., Suhrie, K. R., Dalley, B. C. A., Schaefer, S. Y., & Hammers, D. B. . (2017). Short-term practice effects in mild cognitive impairment: evaluating different methods of change. Journal of Clinical and Experimental Neuropsychology, 39(4), 396–407. http://dx.doi.org/10.1080/13803395.2016.1230596
The ‘simple’ approach to the ZSRB calculates the metric by:
- Calculating the estimated beta weight (b) by dividing the standard deviation of the reference group at Time 2 (S2) by the standard deviation of the reference group at Time 1 (S1):
b = S2/S1 - Calculating the estimated constant (c) by finding the product of b and the mean of the control group at Time 1 (M1), and subtracting the value from the mean of the control group at Time 2 (M2):
c = M2 – (bM1) - Calculating the estimated standard error of the estimate (SEE) by:
- Find the sum of the S1 squared and S2 squared:
S12+S22 - Find one minus the control group’s test-retest correlation (r12):
in 1-r12 - Multiply S12+S22 and 1-r12:
(S12+S22) * (1-r12) - Take the square root of the resulting product:
SEE = sqrt((S12+S22) * (1-r12))
- Find the sum of the S1 squared and S2 squared:
- Calculating the estimated predicted score at Time 2 (T2’):
T2’ = b * T1 + cwhere T1 is the observed score at Time 1 - Estimating the SRB z-score was by subtracting the T2’ score from the observed Time 2 score (T2), and dividing the result by the estimated SEE:
ZSRB = (T2 – T2’) / SEE
Note that this approach does not allow for adding covariates into the model. As such, the ‘complex’ zsrb function (see below) is preferred over using this option. However, this ‘simple’ approach can be used for validation of or comparison with other studies that use this approach.
zsrb_sim accepts the following input parameters:
idf: a data frame with all the variables (outcomes, predictors, covariates)t1: a string referencing the column name in `idf` of the outcome measure at baselinet2: a string referencing the column name in `idf` of the outcome measure at follow-upgroup: a string or array referencing the group variables in `idf` at baseline and follow-up. This variable is only used to select the control group that will be used to derive parameter estimates for the ZSRB calculations. If you enter a string, this single variable will be used to determine the reference group. If you enter an array with a maximum of two strings, only subjects are selected where both group variables indicate that they are control subjects. So, if a subject is a control at time point one, but have progressed to being a patient at time point two, they will not be selected into the control group. Note that group variables can be entered as either class factor or class numeric.ref(optional): a string or integer indicating with factor level (in case of a string) or numeric value (in case of an integer) represents the reference group for the ZSRB calculations. This parameter is optional. If you don’t specify this parameter, the zsrb function will automatically determine the first level of your factor variable as the reference category. If you entered your group variable(s) as class numeric and you did not specify a reference group with ‘ref’, then the minimum value is selected as the reference category.
The zsrb_sim function returns a list object with two items:
- The first element contains sample size information of the subjects used for obtaining the regression parameters and the summary statistics to calculate the ZSRB.
- The second element contains the output data frame. This data frame is a copy of input data frame with an additional column: the ZSRB score for the model. The name of this column is a concatenation of
zsrband the variable names you entered fort1andt2, separated by underscores. E.g.,zsrb_T1score_T2score.
A toy example can be found here.
- Guevara, J. E., Kurniadi, N. E., & Duff, K. (2023). Assessing longitudinal cognitive change in mild cognitive impairment using estimated standardized regression-based formulas. Journal of Alzheimer’s Disease, 95(2), 509–521. http://dx.doi.org/10.3233/jad-230160
The ‘complex’ approach to the ZSRB calculates the metric by:
- Building a linear regression model with the T2 score as output and T1 score and any specified covariates for as predictors. Only data of the reference group is used here.
- Calculate the estimated Time 2 score (T2’) for all subjects over all groups, using the intercept and beta parameters derived from the linear regression model. So, for each subject it will build this score by starting with the reference group intercept and then adding the product of the beta value for T1 score by the subject’s T1 score. If covariates were entered, this procedure is continued for all covariates, until the final T2’ score is obtained.
- The difference between the observed (T2) and estimated (T2’) is then calculated:
T2 - T2’ - This difference score is then scaled (divided by) the residual standard deviation of the linear regression model (see this documentation) to obtain the SRB z-score.
Interpretation of the ZSRB: The residual standard deviation of the regression is the average distance that the observed values fall from the regression line from the model fitted under ‘1)’. As, such dividing the difference scores by the residual standard deviation makes it such that the unit of the outcome (the ZSRB) is in standard deviations of the residual standard deviation. So, positive scores indicate observations where the subject scores x standard deviations above their expected score, while negative scores indicate observations where subject scores x standard deviations below their expected score.
zsrb_com accepts the following input parameters:
idf: a data frame with all the variables (outcomes, predictors, covariates)t1: a string referencing the column name in `idf` of the outcome measure at baselinet2: a string referencing the column name in `idf` of the outcome measure at follow-upgroup: a string or array referencing the group variables in `idf` at baseline and follow-up. This variable is only used to select the control group that will be used to derive parameter estimates for the ZSRB calculations. If you enter a string, this single variable will be used to determine the reference group. If you enter an array with a maximum of two strings, only subjects are selected where both group variables indicate that they are control subjects. So, if a subject is a control at time point one, but have progressed to being a patient at time point two, they will not be selected into the control group. Note that group variables can be entered as either class factor or class numeric.ref(optional): a string or integer indicating with factor level (in case of a string) or numeric value (in case of an integer) represents the reference group for the ZSRB calculations. This parameter is optional. If you don’t specify this parameter, the zsrb function will automatically determine the first level of your factor variable as the reference category. If you entered your group variable(s) as class numeric and you did not specify a reference group with ‘ref’, then the minimum value is selected as the reference category.covs(optional): a string or array with covariates that will be regressed out when calculating the ZSRB.
The zsrb_com function returns a list object with three items:
- The first element contains sample size information of the subjects used for obtaining the regression parameters, as well as the formula that was used to obtain the parameter estimates obtained from your control group that were used to calculate the ZSRB.
- The second element contains the summary of the regression model that was run to obtain the parameters of the control group for predicting time point 2 data. This is stored for evaluation of the estimates that were used in the ZSRB calculations.
- The third element contains the output data frame. This data frame is a copy of input data frame with an additional column: the ZSRB score for the model. The name of this column is a concatenation of
zsrband the variable names you entered fort1andt2, separated by underscores. E.g.,zsrb_T1score_T2score.
A toy example can be found here.
- Duff, K., Schoenberg, M. R., Patton, D., Mold, J., Scott, J. G., & Adams, R. L. (2004). Predicting change with the rbans in a community dwelling elderly sample. Journal of the International Neuropsychological Society, 10(6), 828–834. http://dx.doi.org/10.1017/s1355617704106048
This package does not rely on other packages.
- Make sure that devtools are installed so that you can install packages directly from github:
install.packages("devtools") - Install the zsrb package using devtools
devtools::install_github("vnckppl/zsrb")