Type: Package
Title: Regression Toward the Mean
Version: 1.2
License: MIT + file LICENSE
Encoding: UTF-8
LazyData: true
RoxygenNote: 7.1.1
Depends: R (>= 3.4.0)
NeedsCompilation: no
Repository: CRAN
Date/Publication: 2022-10-26 13:52:37 UTC
In repeated measures studies with extreme large or small values, it is common for the subjects’ measurements on average to be closer to the mean of the basic population. Interpreting possible changes in the mean in such situations can lead to biased results since the values were not randomly selected, they come from truncated sampling. This method allows estimating the range of means where treatment effects are likely to occur when regression toward the mean is present.
Ostermann, T., Willich, Stefan N. & Luedtke, Rainer. (2008). Regression toward the mean - a detection method for unknown population mean based on Mee and Chua’s algorithm. BMC Medical Research Methodology.
Daniela Recchia daniela.rodriguesrecchia@uni-wh.de
We would like to acknowledge Lena Roth and Nico Steckhan for the package’s initial updates (Q3 2024) and continued supervision and guidance. Both have contributed to discussing and integrating these methods into the package, ensuring they are up-to-date and contextually relevant.
To install this package, use:
install.packages("regtomean")A dataset with scores from 8 students who failed a high school test and could not get their diploma. They repeated the exam and got new scores.
data("language_test")A data frame with 8 observations on the following 9 variables:
Student: a numeric vectorBefore: a numeric vectorAfter: a numeric vectorTotal N: a numeric vectorCross: a numeric vectorPre-treatment Mean: a numeric vectorPre-treatment Std: a numeric vectorPost-treatment Mean: a numeric vectorPost-treatment Std: a numeric vectorMcClave, J.T; Dietrich, F.H.: “Statistics”; New York, Dellen Publishing; 1988.
This function calculates the correlation for the data and Cohen’s d effect sizes, both based on pooled and on treatment standard deviations.
cordata(Before, After, data)Before: a numeric vector giving the data values for the
first (before) measure.After: a numeric vector giving the data values for the
second (after) measure.data: an optional data frame containing the variables
in the formula. By default, the variables are taken from the environment
(formula).This function computes the correlation between both measures as also both effect sizes based on Cohen’s d statistic.
The inputs must be numeric.
Returns a table containing the correlation, effect size pooled, and effect size based on treatment.
Daniela Recchia, Thomas Ostermann.
Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). New York: Academic Press.
cohen.d, cor
cordata("Before", "After", data=language_test)This function replicates 100 times the before and after values giving a start and end reference.
replicate_data(start, end, Before, After, data)start: a start value for µ.end: an end value for µ.Before: a numeric vector giving the data values for the
first (before) measure.After: a numeric vector giving the data values for the
second (after) measure.data: an optional data frame containing the before and
after variables in the formula. By default, the variables are taken from
the environment (formula).In order to overcome the limitation of Mee and Chua’s test regarding the population mean µ, a replication of the data is performed.
After replicating the data, the unknown population mean µ is systematically estimated over a range of values. Further estimations will be based on this new dataset.
Returns a data frame we could call mee_chua containing
the values for µ, before, and after.
Daniela Recchia, Thomas Ostermann.
Ostermann, T., Willich, Stefan N. & Luedtke, Rainer. (2008). Regression toward the mean - a detection method for unknown population mean based on Mee and Chua’s algorithm. BMC Medical Research Methodology.
Galton, F. (1886). Regression towards mediocrity in hereditary stature. Journal of the Anthropological Institute (15: 246-263).
rep
replicate_data(0, 100, "Before", "After", data=language_test)This function fits linear models for a subset of data frames.
meechua_reg(x)x: Data to be used in the regression.The data used for the regression must be sorted by mu.
A set of linear models will be estimated and model coefficients are
saved and stored in mod_coef.
The estimated standard error for the after measure is also stored in
se_after to be used further in other functions.
A table containing the estimations for each mu. Global variables
models, mod_coef, se_after are
stored for further analysis. The models are saved in an object called
mee_chua, which is not automatically printed but is saved
in the environment.
Daniela Recchia, Thomas Ostermann.
Ostermann, T., Willich, Stefan N. & Luedtke, Rainer. (2008). Regression toward the mean - a detection method for unknown population mean based on Mee and Chua’s algorithm. BMC Medical Research Methodology.
lm, dlply
## get the values ##
mee_chua <- replicate_data(0, 100, "Before", "After", data=language_test)
meechua_reg(mee_chua)This function calculates and plots treatment and regression effects of both before and after measures as also its p-values.
meechua_eff.CI(x, n, se_after)x: a data frame containing the results from
meechua_reg. It is stored as mod_coef.n: the original sample size (number of observations)
from data.se_after: the estimated standard error from
meechua_reg. It is stored as se_after.After performing the meechua_reg, the model coefficients
mod_coef and the global variable se_after are
used as input in this function to estimate treatment and regression
effects.
Two plots are performed: the first “Treatment Effect and p-value” and the second “Confidence Intervals” for µ.
Daniela Recchia, Thomas Ostermann
Ostermann, T., Willich, Stefan N. & Luedtke, Rainer. (2008). Regression toward the mean - a detection method for unknown population mean based on Mee and Chua’s algorithm. BMC Medical Research Methodology.
# First perform replicate_data and meechua_reg
replicate_data(0, 100, "Before", "After", data=language_test)
meechua_reg(mee_chua)
# Model coefficients (mod_coef) and se_after are stored in the environment
# as a result from the function meechua_reg
meechua_eff.CI(mod_coef, 8, se_after)Based on the data before and after the intervention and the regression models of the function meechua_reg, this function plots for a given range of µ the t-statistics and p-values of one sided tests, wether the intervention is having an significant impact on the measurements accounting for regression to the mean.
For each µ the t-statistic and p-value correspond to the one sided
test, if the intercept of the regression model from
meechua_reg is significantly different from µ in the
specified direction. Respecting the assumptions of the method, this is
equivalent to the intervention having an significant impact accounting
for regression to the mean. If for a concrete µ the p-value is below the
specified threshold -visible as a blue dashed line- the impact of the
intervention is significant under the assumption that µ is the real
population mean.
plot_mu(x, n, se_after, lower = F, alpha = 0.05)| Argument | Description |
|---|---|
x |
A data frame containing the results from meechua_reg.
It is stored as mod_coef. |
n |
The original sample size (number of observations) of the data. |
se_after |
The estimated standard error from meechua_reg. It is
stored as se_after. |
lower |
Boolean value specifying the direction of the one sided tests. For
lower = F (the default) it is testing, wether the
intervention is increasing the measurements, for lower = T,
wether the second measurements are lower than expected. |
alpha |
Specifies the significance threshold for the p-values of
corresponding one sided tests. The default is
alpha = 0.05. |
Plot for a range of µ the p-values and t-values of the corresponding tests against µ and prints some relevant values:
The value of µ, for which the treatment effect is the most statistically significant, and the corresponding t-statistic and p-value. The highest and lowest µ, for which the treatment impact is significant.
Those variables will be returned as a list as well.
Julian Stein
Ostermann, T., Willich, Stefan N. & Luedtke, Rainer. (2008). Regression toward the mean - a detection method for unknown population mean based on Mee and Chua’s algorithm. BMC Medical Research Methodology.
# First perform replicate_data and meechua_reg
replicate_data(0, 100, "Before", "After", data=language_test)
meechua_reg(mee_chua)
#mod_coef and se_after are stored in the environment. The parameters lower = F and alpha = 0.05 can be omitted
plot_mu(mod_coef, 8, se_after)
#Alternative usage: Testing for decreased values due to the intervention with significance threshold alpha = 0.1
plot_mu(mod_coef, 8, se_after, lower=T, alpha = 0.1)Similar to plot_mu, this function plots for a given
range of µ the t-statistics and p-values of one sided tests, wether the
intervention is having an significant impact on the measurements
accounting for regression to the mean. The difference is, that this
function is only based on some statistics of the samples before and
after the treatment, like the mean, standard deviation and
covariance/correlation.
For each µ the t-statistic and p-value correspond to the one sided test, if the intervention has an significant impact on the second measurements accounting for regression to the mean. If for a concrete µ the p-value is below the specified threshold -visible as a blue dashed line- the impact of the intervention is significant under the assumption that µ is the real population mean.
plot_t(mu_start, mu_end, n, y1_mean, y2_mean, y1_std, y2_std, cov, lower = F, alpha = 0.05, r_insteadof_cov = F)| Argument | Description |
|---|---|
mu_start |
Lower end for the range of µ to be considered. |
mu_end |
Upper end for the range of µ to be considered. |
n |
The number of observations. |
y1_mean |
Mean of the first measurement. |
y2_mean |
Mean of the second measurement. |
y1_std |
Standard deviation of the first measurement. |
y2_std |
Standard deviation of the second measurement. |
cov |
Covariance between the first and second measurements. If
r_insteadof_cov = T this argument represents the
correlation instead. |
lower |
Boolean value specifying the direction of the one sided tests. For
lower = F (the default) it is testing, wether the
intervention is increasing the measurements, for lower = T,
wether the second measurements are lower than expected. |
alpha |
Specifies the significance threshold for the p-values of
corresponding one sided tests. The default is
alpha = 0.05. |
r_insteadof_cov |
Boolean value for the alternative usage of correlation instead of
covariance. If r_insteadof_cov = T, the input
cov is interpreted as the correlation. |
Plot for a range of µ the p-values and t-values of the corresponding tests against µ and prints some relevant values:
The value of µ, for which the treatment effect is the most statistically significant, and the corresponding t-statistic and p-value. The highest and lowest µ, for which the treatment impact is significant.
Those variables will be returned as a list as well.
Julian Stein
Ostermann, T., Willich, Stefan N. & Luedtke, Rainer. (2008). Regression toward the mean - a detection method for unknown population mean based on Mee and Chua’s algorithm. BMC Medical Research Methodology.
#Using the parameters corresponding to the example of the function plot_mu
plot_t(mu_start = 0, mu_end = 100, n = 8 , y1_mean = 57.375, y2_mean = 60.375, y1_std = 7.0, y2_std = 8.8, cov = 54.268)This function plots all 4 diagnostics plots for each linear regression model: “Residuals vs Fitted”, “Normal Q-Q”, “Scale-Location” and “Residuals vs Leverage”.
meechua_plot(x)x: List containing the estimated linear models from
meechua_reg. It is stored as models.For each model from models, 4 diagnostic plots are
performed. For the first model, the numbers 1 to 4 should be given, for
the second model numbers from 5 to 8, and so on.
Diagnostics plots for the set of models from
meechua_reg.
Daniela Recchia, Thomas Ostermann.
Ostermann, T., Willich, Stefan N. & Luedtke, Rainer. (2008). Regression toward the mean - a detection method for unknown population mean based on Mee and Chua’s algorithm. BMC Medical Research Methodology.
plot.lm, meechua_reg
# models are an output from meechua_reg
replicate_data(0, 100, "Before", "After", data=language_test)
meechua_reg(mee_chua)
# models are the output from meechua_reg saved in the environment after running the function
meechua_plot(models)