*! version 1.1.0 03Mar2013

*! version 1.0.0 31Jan2013
help rtmcii

Title

rtmcii -- Regression to the mean effects with confidence intervals

Syntax

rtmcii mean_pre sd_pre cut-off rho [, options]

mean_pre is the mean of pre-test variable, sd_pre
is the standard deviation of the pre-test variable, cut-off is the cutoff value on the pre-test variable,
and rho is the correlation between the pre- and post-test variables.

Main
periods(#) defines the number of periods used for calculating the pre-test mean and standard deviation mean_pre and sd_pre

number(#) defines the number of observations to be used in generating the artifical dataset

seed(#) sets the random-number seed to #, which will be used for both generating the artificial dataset
and for estimating bootstrapped confidence intervals

figure produces a plot of the expected pre- and post-test means and confidence intervals for values
above and below the cutoff

Bootstrapped CIs
reps(#) specifies the number of replications to be performed

size(#) draws samples of size #

level(#) sets the confidence level; default is level(95)

saving(filename, ...)
saves results to filename

Description

rtmcii calculates the regression to the mean effect for a variable that is generally measured
at two points in time (ie, "pre-test" and "post-test"), based on a defined cutoff value on the "pre-test" measure, and
estimates confidence intervals using bootstrap simulation; see bootstrap. rtmcii is an immediate form of rtmci; see immed
for more on immediate commands.

Options

Main

periods(#) defines the number of periods used for calculating
the pre-test mean_pre and sd_pre; default is per(1).

number(#) defines the number of observations to be used
in generating the artificial dataset; default is n(1000).

seed(#) sets the random-number seed to #, which will be used for both
generating the artifical dataset and for estimating bootstrapped confidence intervals. Specifying
this option is equivalent to typing . set seed # before calling rtmcii; default is seed(1234).

figure produces a plot of the expected pre- and post-test means and confidence intervals for values
above and below the cutoff. This option requires Roger Newson's xsvmat and eclplot packages to be
installed; both are downloadable from SSC

Bootstrapped CIs

reps(#) specifies the number of replications to be performed; default
is reps(1000).

size(#) sets the number of observations drawn at each replication; default is _N.

level(#) specifies the confidence level, as a percentage, for confidence
intervals. The default is level(95) or as set by set level.

saving(filename, ...)
saves the bootstrapped results of the simulations to filename. See
prefix_saving_option for details about suboptions.

Remarks

Regression to the mean (RTM), originally termed “regression toward mediocrity”, was described over a century ago by Sir Francis Galton (1886)
upon discovering that on average, tall parents had children shorter themselves and short parents had
taller children. An excellent historical review of RTM is provided by Stigler (1997).

RTM poses a major threat to the internal validity of any study in which
subjects are initially selected for their extreme values. Upon remeasurement, the sample mean
from this outlier group is likely to be much closer to the overall population mean. This
statistical phenomenon can be easily confused with a treatment effect in intervention studies.

rtmcii calculates the expected pre-test value, post-test value, and estimated RTM effects using
equations proposed by Gardner & Hardy (1973) and Davis (1976) for Normally distributed data. Importantly, rtmcii provides
confidence intervals to indicate the precision (uncertainty) for these estimates.

rtmcii uses the summary statistics provided by the user to create artificial "pre-test" and "post-test" variables using corr2data. These
artificial variables then serve as the basis for estimating the bootstrapped confidence intervals. Thus, rtmcii
can calculate all the RTM related measures and estimate confidence intervals when all that is known are summary statistics. See Linden (2013)
for the results of simulation a more comprehensive discussion.

Examples

Example 1: Taken from Yudkin and Stratton (1996) where the mean value of the single pre-test variable is 5.21, the standard deviation is 1.02,
the cutoff value is 6.50 and the correlation between the pre- and post-tests is 0.67. We set the observations for generating the artificial dataset
at 1914 to replicate their sample size. Finally, we produce a figure of the expected values.

. rtmcii 5.21 1.02 6.50 0.67, period(1) n(1914) fig

Example 2: Taken from Davis (1976) where the mean value of the single pre-test variable is 5.331, the standard deviation is 0.17,
the cutoff value is 5.580 and the correlation between the pre- and post-tests is 0.82. We set the observations for generating the artificial dataset
at 1346 to replicate the sample size in the paper. We also set the number of bootstrap repetitions to 2000 with sample draws of 50.

. rtmcii 5.331 0.17 5.580 0.82, n(1346) reps(2000) seed(4321) size(50)

Example 3: Taken from Linden (2007, data for Figure 2) where the mean value of the single pre-test variable is 53.12, the standard deviation is 8.27,
the cutoff value is 44.25 and the correlation between the pre- and post-tests is 0.742. We set the observations for generating the artificial dataset
at 118 to replicate the sample size in the paper. We follow this with estat bootstrap to display all available confidence intervals.

. rtmcii 53.12 8.27 44.25 0.742, n(118)

. estat bootstrap, all

Output tables

rtmcii produces standard bootstrap output tables. Below is a cross reference to the variables in the tables:

Variable Description

mu pre-test mean

sd pre-test sd

rho correlation between pre-test and post-test

firstval_high expected pre-test value above the cutoff

secondval_high expected post-test value above the cutoff

rtm_high RTM effect above the cutoff

pct_rtm_high percent RTM effect above the cutoff

firstval_low expected pre-test value below the cutoff

secondval_low expected post-test value below the cutoff

rtm_low RTM effect below the cutoff

pct_rtm_low percent RTM effect below the cutoff

References

Davis CE. The effect of regression to the mean in epidemiologic and clinical studies. American Journal of Epidemioliology 1976;104:493-498.

Galton F. Regression towards mediocrity in hereditary stature. Journal of the Anthropological Institute 1886;15:246-263.

Gardner MJ, Hardy JA. Some effects of within person variability in epidemiological studies. Journal of Chronic Disease 1973;26:781-795.

Linden A. Estimating the effect of regression to the mean in health management programs. Disease Management & Health Outcomes 2007;15(1):7-12.

Linden A. Assessing regression to the mean effects in health care initiatives. BMC Medical Research Methodology 2013;13(119):1-7.

Stigler SM. Regression towards the mean, historically considered. Statistical Methods in Medical Research 1997;6(2):103-14.

Yudkin PL, Stratton IM. How to deal with regression to the mean in intervention studies. Lancet 1996;347:241-243.

Citation of rtmcii

rtmcii is not an official Stata command. It is a free contribution
to the research community, like a paper. Please cite it as such:

Linden, Ariel (2013). rtmcii: Stata module for estimating regression to the mean effects with confidence intervals.
http://www.lindenconsulting.org

Author

Ariel Linden

President, Linden Consulting Group, LLC

Ann Arbor, MI, USA

alinden@lindenconsulting.org

http://www.lindenconsulting.org

Acknowledgments

I would like to thank Nicholas J. Cox for providing extremely helpful
comments. Without his help, rtmcii would provide nothing more than point estimates. I would also like to thank
Roger Newson for his guidance in creating the figure option using his xsvmat and eclplot programs.

Also see

Manual: [D] corr2data, [R] bootstrap

Online: corr2data, bootstrap, xsvmat (if installed), eclplot (if installed)