{smcl}
{hline}
help for {cmd:multproc} and {cmd:smileplot7} {right:(SJ3-2: st0035)}
{hline}


{title:Multiple test procedures and Stata 7 smile plots}

{p 8 15 2} {cmd:multproc} [{cmd:if} {it:exp}] [{cmd:in} {it:range}],
 [ {cmdab:pu:ncor}{cmd:(}{it:#} | {it:scalarname} | {it:varname}{cmd:)}
 {cmdab:pc:or}{cmd:(}{it:#} | {it:scalarname} | {it:varname}{cmd:)}
 {cmdab:me:thod}{cmd:(}{it:method_name}{cmd:)} {cmdab:pv:alue}{cmd:(}{it:varname}{cmd:)}
 {cmdab:ra:nk}{cmd:(}{it:newvarname}{cmd:)} {cmdab:gpu:ncor}{cmd:(}{it:newvarname}{cmd:)}
 {cmdab:cr:itical}{cmd:(}{it:newvarname}{cmd:)} {cmdab:gpc:or}{cmd:(}{it:newvarname}{cmd:)}
 {cmdab:nh:cred}{cmd:(}{it:newvarname}{cmd:)} {cmdab:rej:ect}{cmd:(}{it:newvarname}{cmd:)}
 {cmd:float} {cmd:fast}

{p 8 15 2}{cmd:smileplot7} [{cmd:if} {it:exp}] [{cmd:in} {it:range}],
 {cmdab:es:timate}{cmd:(}{it:varname}{cmd:)}
 {cmdab:logb:ase}{cmd:(}{it:#}{cmd:)}
 {cmdab:nl:ine}{cmd:(}{it:#}{cmd:)}
 {cmdab:pts:ymbol}{cmd:(}{it:symbol}{cmd:)} {cmdab:ptl:abel}{cmd:(}{it:varname}{cmd:)}
 {cmd:by(}{it:varname}{cmd:)}
 {it:multproc_options} {it:graph_options} ]

{p 4 4 2}
{cmd:by} {it:varlist}{cmd::} can be used with {cmd:multproc} and
{cmd:smileplot7}.  (See help for {help by}.) If {cmd:by} {it:varlist}{cmd::}
is used, the log output and all generated variables are calculated
using the specified multiple-test procedure within each by-group defined by
the variables in the {it:varlist}.


{title:Description}

{p 4 4 2}
{cmd:smileplot7} provides access to the Stata 7 version of {help smileplot}
and is provided for users of Stata 8 or above who want to produce
smile plots using the "quick and dirty" {help graph7:Stata 7 graphics}.  (Note
that Stata 7 users can still 
{net "describe http://www.kcl-phs.org.uk/rogernewson/stata7/smileplot":click here} to
download the Stata 7 version of {help smileplot} under its old name from 
{net "from http://www.kcl-phs.org.uk/rogernewson/":Roger Newson's web site at http://www.kcl-phs.org.uk/rogernewson/}.  This may save them from having to
modify their Stata programs if and when they upgrade to a higher version of
Stata.) {cmd:multproc} takes, as input, a dataset with one value for each of
a set of multiple statistical tests of multiple null hypotheses, including a
variable containing {it:p}-values for these tests and an uncorrected overall
critical {it:p}-value specified by the user, and carries out a multiple-test
procedure. A multiple-test procedure calculates a corrected overall critical
{it:p}-value, which has the feature that an individual null hypothesis is
considered to be acceptable if and only if its corresponding {it:p}-value is
greater than the corrected overall critical {it:p}-value.  {cmd:smileplot7}
takes, as input, a dataset with one observation for each of a set of
estimated parameters and data on their estimates and {it:p}-values.
{cmd:smileplot7} calls {cmd:multproc} to carry out a multiple-test procedure
and creates a smile plot, with data points corresponding to estimated
parameters, the corresponding {it:p}-values (on a reverse log scale) on the
{it:y}-axis, and another variable (usually the corresponding parameter
estimates) on the {it:x}-axis. There are {it:y}-axis reference lines
corresponding to the uncorrected and corrected overall critical {it:p}-values.
The {it:y}-axis reference line corresponding to the corrected overall critical
{it:p}-value is known as the parapet line. Data points on or above the parapet
line correspond to rejected null hypotheses.  There may be a reference line on
the {it:x}-axis corresponding to the value of the parameter under a null
hypothesis (defaulting to 1 if the {it:x}-axis is logged, 0 otherwise). The
user can therefore see, at a glance, both the statistical significance and the
practical significance of each parameter estimate and can also see the
parapet line as an "upper confidence bound" on the {it:y}-axis for how many of
the corresponding null hypotheses are true. {cmd:multproc} and
{cmd:smileplot7} are usually used on datasets with one observation per
parameter estimate and data on estimates and their {it:p}-values.  Such
datasets may be created (directly or indirectly) by {help postfile}, 
{help statsby}, {help parmby}, or {help parmest}.


{title:Options for {cmd:multproc} and {cmd:smileplot7}}

{p 4 8 2}{cmd:puncor(}{it:#} | {it:scalarname} | {it:varname}{cmd:)} specifies the uncorrected overall
critical {it:p}-value for statistical significance.  This option may be
specified as a number, as a scalar, or as a variable (in which case
the variable is expected to contain only one nonmissing value in the sample or
by-group).  If absent, this option is set to 1-{hi:$S_level}/100, where
{hi:$S_level} is the value of the currently set default confidence level.

{p 4 8 2}{cmd:pcor(}{it:#} | {it:scalarname} | {it:varname}{cmd:)} specifies the corrected overall
critical {it:p}-value for statistical significance.  This option may be
specified as a number, as a scalar, or as a variable (in which case
the variable is expected to contain only one nonmissing value in the sample or
by-group).  If absent, this option is set by the method specified in the
{cmd:method()} option (see below).

{p 4 8 2}{cmd:method(}{it:method_name}{cmd:)} specifies the multiple-test
procedure method to be used for deriving the corrected {it:p}-value threshold
from the uncorrected {it:p}-value threshold. This option is ignored and set
to {cmd:userspecified} if the {cmd:pcor()} option is specified and is in the range
0 <= {cmd:pcor()} <= 1.  Otherwise, if {cmd:method()} is absent, it is set to
{cmd:bonferroni}.

{p 4 8 2}{cmd:pvalue(}{it:varname}{cmd:)} is the name of the variable
containing the {it:p}-values.  If this option is absent, {cmd:multproc}
looks for a variable named {hi:p} (as created by {help parmby} or 
{help parmest}). {cmd:multproc} carries out a multiple-test procedure on all
observations selected by the {help if} and {help in} qualifiers which also
have nonmissing values for the variable containing the {it:p}-values.

{p 4 8 2}{cmd:rank(}{it:newvarname}{cmd:)} is the name of a new variable to be
generated, containing, in each observation, the rank of the corresponding
{it:p}-value, from the lowest to the highest. Tied {it:p}-values are ranked
according to their position in the input dataset. If {cmd:by}
{it:varlist}{cmd::} is specified with {cmd:multproc}, the ranks are
defined within the by-group.

{p 4 8 2}{cmd:gpuncor(}{it:newvarname}{cmd:)} is the name of a new variable to
be generated that contains, in each observation, the uncorrected overall
critical {it:p}-value, as specified by the {cmd:puncor()} option, or by the
standard default if the {cmd:puncor()} option is not specified. This new
variable will have the same value for all observations in the sample of
observations used by {cmd:multproc} or {cmd:smileplot7}.

{p 4 8 2}{cmd:critical(}{it:newvarname}{cmd:)} is the name of a new variable
to be generated that contains, in each observation, an individual critical
{it:p}-value corresponding to the original {it:p}-value in the variable
specified by {cmd:pvalue()}.  The values of the individual critical
{it:p}-values are defined by a nondecreasing function (specified by the
{cmd:method()} option) of the ranks of the corresponding original {it:p}-values
(generated by the {cmd:rank()} option). The corrected overall critical
{it:p}-value is selected from the individual critical {it:p}-values in a way
specified by the {cmd:method()} option, depending on whether the method
specified is a one-step method, a step-down method, or a step-up method.

{p 4 8 2}{cmd:gpcor(}{it:newvarname}{cmd:)} is the name of a new variable to
be generated that contains, in each observation, the corrected overall critical
{it:p}-value, as specified by the {cmd:pcor()} option, or by the {cmd:method()}
option if the {cmd:pcor()} option is not specified. If {cmd:by}
{it:varlist}{cmd::} is specified with {cmd:multproc}, the value of this
new variable will be the same in all observations within each by-group, but
may be different for observations in different by-groups, if a step-up or
step-down procedure is specified by the {cmd:method()} option.

{p 4 8 2}{cmd:nhcred(}{it:newvarname}{cmd:)} is the name of a new variable to
be generated that contains, for each observation, an indicator of the
credibility of the corresponding null hypothesis under the method specified by
the {cmd:method()} option.  This indicator is 1 if the null hypothesis is
credible, and 0 otherwise. A null hypothesis is said to be credible if its
{it:p}-value is greater than the corrected overall critical {it:p}-value. The
set of observations with a value of 1 corresponds to a set of credible null
hypotheses. The exact interpretation of the set of credible null hypotheses
depends on whether the method specified controls the family-wise error rate
(FWER) or the false discovery rate (FDR).

{p 4 8 2}{cmd:reject(}{it:newvarname}{cmd:)} is the name of a new variable to
be generated that contains, for each observation, an indicator of the rejection
of the corresponding null hypothesis under the method specified by the
{cmd:method()} option.  This indicator is 1 if the null hypothesis is rejected,
and 0 otherwise. The new variable generated by the {cmd:reject()} option is
therefore the negation of the new variable generated by the {cmd:nhcred()}
option.

{p 4 8 2}{cmd:float} specifies that the individual critical {it:p}-value
variable specified by {cmd:critical()} (if requested) will be created as a
{cmd:float} variable. If {cmd:float} is absent, the {cmd:critical()}
variable is created as a {cmd:double} variable. Whether or not {cmd:float} is
specified, all generated variables are stored to the lowest precision possible
without loss of information.

{p 4 8 2}{cmd:fast} is an option for programmers. It specifies that
{cmd:multproc} and {cmd:smileplot7} will not take any action so that it can
restore the original data if the user presses {cmd:Break}.


{title:Options available for {cmd:smileplot7} only}

{p 4 8 2}{cmd:estimate(}{it:varname}{cmd:)} is the name of the variable to be
plotted on the {it:x}-axis, which usually contains the parameter estimates
corresponding to the {it:p}-values specified by the {cmd:pvalue()} option.  If
this option is absent, {cmd:smileplot7} looks for a variable named
{hi:estimate} (as created by {help parmby} or {help parmest}).
{cmd:smileplot7} carries out a multiple-test procedure by calling
{cmd:multproc} for observations with nonmissing values for the variables
specified by the {cmd:estimate()} and {cmd:pvalue()} options, using the {help if}
or {help in} qualifiers if these are supplied by the user.  Note that the
variable specified by {cmd:estimate()} may contain values that are not parameter
estimates.  For instance, the observations may correspond to genes in a genome
scan, the {it:p}-values may be derived from tests for associations of those
genes with a disease, and the {it:x}-axis variable specified by the
{cmd:estimate()} option may contain the positions of those genes on a chromosome
map.

{p 4 8 2}{cmd:logbase(}{it:#}{cmd:)} specifies a log base used to define the
{it:y}-axis labels.  This log base is a factor by which each {it:y}-axis label
is divided to arrive at the next {it:y}-axis label, where the {it:y}-axis
labels are ordered from the highest {it:p}-value to the lowest {it:p}-value.
If absent, this option is set to 10, so the y-labels are set to nonpositive
powers of 10.  If this rule defines too many {it:y}-axis labels, the
{it:y}-axis labels are set to be every {it:k}th member of the logarithmic
series, where {it:k} is the minimum positive integer such that the number of
{it:y}-axis labels defined in this way is not too large.

{p 4 8 2}{cmd:nline(}{it:#}{cmd:)} specifies the position, on the {it:x}-axis,
of the reference line indicating the value of the estimated parameters under
the null hypothesis. If {cmd:nline()} is unspecified, it is set to 1 if
{cmd:xlog} is specified and to 0 otherwise. This option allows the user to
plot odds ratios and geometric mean ratios on a linear scale, instead of on
the more usual log scale.  If {cmd:nline()} is set to a missing value by
specifying {cmd:nline(.)}, the null reference line is suppressed.  This
is useful for creating "smile plots" for which the {it:x}-axis variable
specified by the {cmd:estimate()} option contains values other than parameter
estimates, such as positions of genes on a chromosome map.

{p 4 8 2}{cmd:ptsymbol(}{it:symbol}{cmd:)} specifies a graph symbol for the
data points of the smile plot. If absent, it is set to {hi:T} (triangles).

{p 4 8 2}{cmd:ptlabel(}{it:varname}{cmd:)} specifies a variable to be used to
label the data points.  If absent, there are no data point labels, only
unlabeled data points.

{p 4 8 2}{cmd:by(}{it:varname}{cmd:)} is a {help graph7:graph} option, and
works as for {help graph7:graph}, creating one plot for each by-group,
arranged in a square array. The corrected overall critical {it:p}-value,
indicated by the parapet line, is calculated for all the {it:p}-values from
all the by-groups pooled together, not for the subset of {it:p}-values in each
by-group individually.  (This is in contrast to the use of {cmd:by}
{it:varlist}{cmd::}, which causes corrected individual and overall critical
{it:p}-values to be calculated only from the subset of {it:p}-values in each
by-group.)


{title:Remarks}

{p 4 4 2}
The smile plot is so named because, if the standard errors of the parameters
are similar, the data points fall along a curve shaped like a smile.  It
summarizes a set of multiple parameter estimates graphically, in the way that
a Cochrane forest plot summarizes a meta-analysis. The {it:y}-axis reference
line corresponding to the corrected overall critical {it:p}-value is known as
the parapet line. Data points on or above the parapet line correspond to
parameters for which we can reject the null hypotheses under the specified
multiple-test procedure.  Data points below the parapet line correspond to
parameters for which the null hypotheses are credible (acceptable).

{p 4 4 2}
The methods specified by the {cmd:method()} option are multiple-test procedures
for defining an upper confidence bound for the set of null hypotheses that are
true, given multiple parameter estimates with multiple {it:p}-values. More
formally, each method defines a set of credible (or acceptable) null
hypotheses and a set of incredible (rejected) null hypotheses, whose exact
interpretation depends on the method.  The uncorrected overall {it:p}-value
may either be treated as an upper bound for the family-wise error rate (FWER),
or be treated as an upper bound for the false discovery rate (FDR).

{p 4 4 2}
The FWER is the probability that at least one true null hypothesis is
rejected.  If a method controls the FWER, the power set of the set of
credible null hypotheses is a power-set-valued confidence region for a
set-valued parameter, namely the set of null hypotheses which are true. We can
therefore say, with a confidence level of 100*(1-{cmd:puncor}) percent, that
the set of null hypotheses that are true is some subset (possibly empty) of
the set of credible null hypotheses. In other words, we are
100*(1-{cmd:puncor}) percent confident that all the rejected null hypotheses
are false.

{p 4 4 2}
The FDR is defined as follows. Let {it:V} denote the number of true null
hypotheses rejected, and let {it:R} denote the total number of null hypotheses
rejected. Then the FDR is equal to the expectation of {it:Q}, where {it:Q} is
defined to be equal to {it:V/R} if {it:R>0}, and equal to zero if {it:R=0}.
The probability that {it:Q=1} can be no more than the FDR. Therefore, if the
method controls the FDR, we can say, with 100*(1-{cmd:puncor}) percent
confidence, that the set of null hypotheses that are true is a subset of null
hypotheses (possibly empty) which does not contain the rejected set as a
non-empty subset.  In other words, we are 100*(1-{cmd:puncor}) percent
confident that at least some of the rejected null hypotheses are false. If the
number of null hypotheses tested is very large indeed, arguably, we may
be 100 percent confident that 100*(1-{cmd:puncor}) percent of the rejected
null hypotheses are false.

{p 4 4 2}
The methods may also be classified into one-step, step-down and step-up
procedures.  All three classes of methods work by defining a list of {it:m}
individual critical {it:p}-values {it:C_1,...,C_m}, one for each of the {it:m}
individual input {it:p}-values {it:P_1,...,P_m}, ranked from the lowest to the
highest. These individual critical {it:p}-values can be saved as output using
the {cmd:critical()} option, and are defined as a nondecreasing function of the
ranks of the original {it:p}-values, which can be saved as output using the
{cmd:rank()} option. An overall corrected critical {it:p}-value {cmd:pcor()} is
selected from the individual critical {it:p}-values. A null hypothesis is
acceptable if and only if its {it:p}-value is greater than the overall
corrected critical {it:p}-value.  For a one-step procedure, the {it:C_i} are
all equal to the overall corrected critical {it:p}-value {cmd:pcor()}, which is
defined as a function of the uncorrected critical {it:p}-value {cmd:puncor()}.
For a step-down procedure, {cmd:pcor()} is equal to the lowest {it:C_i} such
that {it:P_i > Ci}, if such a {it:C_i} exists, and equal to {it:C_m}
otherwise. For a step-up procedure, {cmd:pcor()} is equal to the highest
{it:C_i} such that {it:P_i <= C_i}, if such a {it:C_i} exists, and equal to
{it:C_1} otherwise.

{p 4 4 2}
The different methods use different assumptions. Some assume that the
different {it:p}-values are statistically independent, others allow the
different {it:p}-values to be nonnegatively correlated, and others allow the
different {it:p}-values to be arbitrarily correlated. The more recently
developed methods are documented in their original source papers. The
available methods are as follows:


{col 12}{cmd:method()}{col 30}{ul:Step type}{col 43}{ul:FWER/FDR}{col 55}{ul:Definition or source}
{col 12}{cmd:userspecified}{col 30}One-step{col 43}Either{col 55}{cmd:pcor()} option
{col 12}{cmd:bonferroni}{col 30}One-step{col 43}FWER{col 55}{cmd:pcor=puncor/m}
{col 12}{cmd:sidak}{col 30}One-step{col 43}FWER{col 55}{cmd:pcor=1-(1-puncor)^(1/m)}
                                        {col 55}(or Sidak, 1967)
{col 12}{cmd:holm}{col 30}Step-down{col 43}FWER{col 55}Holm, 1979
{col 12}{cmd:holland}{col 30}Step-down{col 43}FWER{col 55}Holland and Copenhaver, 1987
{col 12}{cmd:liu1}{col 30}Step-down{col 43}FDR{col 55}Benjamini and Liu, 1999a
{col 12}{cmd:liu2}{col 30}Step-down{col 43}FDR{col 55}Benjamini and Liu, 1999b
{col 12}{cmd:hochberg}{col 30}Step-up{col 43}FWER{col 55}Hochberg, 1988
{col 12}{cmd:rom}{col 30}Step-up{col 43}FWER{col 55}Rom, 1990
{col 12}{cmd:simes}{col 30}Step-up{col 43}FDR{col 55}Benjamini and Hochberg, 1995 (or
                                        {col 55}Benjamini and Yekutieli, 2001
                                        {col 55}(first method))
{col 12}{cmd:yekutieli}{col 30}Step-up{col 43}FDR{col 55}Benjamini and Yekutieli, 2001
                                        {col 55}(second method)
{col 12}{cmd:krieger}{col 30}Step-up{col 43}FDR{col 55}Benjamini, Krieger and Yekutieli, 2001

{p 4 4 2}
Note that, in the case of the {cmd:holland} method, the procedure used is the
simplified (and less powerful) version of the procedure of Holland and
Copenhaver (1987), which takes no account of logical dependencies between the
null hypotheses, although it takes advantage of nonnegative dependencies
between the {it:p}-values. The {cmd:simes} method is so named because it was
proposed in Simes (1986), although its justification in terms of the FDR was
presented in the references indicated above.


{title:Examples}

{p 4 4 2}
If we type the following example in the auto data, a smile plot will be
produced with 1 observation per parameter of the fitted model. The corrected
{it:p}-value defines an upper confidence bound for how many of these
parameters are 0 in the population from which these cars were sampled.

{p 16 20 2}{cmd:. parmby "xi:regress mpg i.rep78 i.foreign",label norestore}{p_end}
{p 16 20 2}{cmd:. smileplot7 if parm!="_cons",me(holm) ptl(label)}{p_end}

{p 4 4 2}
If we type the following example in the auto data, a pair of smile plots
will be created, one for US-made cars and one for nonUS cars, with one data
point for each parameter of the model (other than the intercept). The
corrected {it:p}-value is corrected for the total number of parameters for
both car types (US and nonUS).

{p 16 20 2}{cmd:. parmby "xi:regress mpg weight i.rep78",label norestore by(foreign)}{p_end}
{p 16 20 2}{cmd:. smileplot7 if parm!="_cons",ptl(parm) by(foreign)}{p_end}

{p 4 4 2}
The following advanced example demonstrates the use of {cmd:by}
{it:varlist}{cmd::} together with the {cmd:by} option of {cmd:smileplot7}. The
example assumes that there is a dataset in memory, with 1 observation per
parameter estimate. The dataset contains variables {hi:or} and {hi:siglev},
containing estimated odds ratios and {it:p}-values respectively, and also
identifier variables {hi:outcome}, {hi:exposure}, {hi:subset} and
{hi:adjusted}. The program {cmd:multproc} is used to carry out the Simes
method on each subset defined by the variable {hi:adjusted}, storing the
uncorrected and corrected overall critical {it:p}-values in new variables
{hi:uncp} and {hi:corp}, and a hypothesis rejection indicator in a new
variable {hi:signif}.  We then use {cmd:smileplot7} to create, for each
combination of values of {hi:adjusted} and {hi:outcome}, an array of smile
plots for each value of {hi:subset}, with data points labeled by the value of
{hi:exposure}.  Finally, the rejected null hypotheses are listed.

{p 16 20 2}{cmd:. sort adjusted outcome subset exposure}{p_end}
{p 16 20 2}{cmd:. by adjusted:multproc,pval(siglev) meth(simes) gpunc(uncp) gpcor(corp) rej(signif)}{p_end}
{p 16 20 2}{cmd:. by adjusted outcome:smileplot7,est(or) pval(siglev) punc(uncp) pcor(corp) by(subset) ptl(exposure) xlog t1(" ")}{p_end}
{p 16 20 2}{cmd:. by adjusted outcome:list if signif,nodisp}{p_end}


{title:Author}

{p 4 4 2}
Roger Newson, King's College, London, UK{break}
Email: {browse "mailto:roger.newson@kcl.ac.uk":roger.newson@kcl.ac.uk}


{title:References} 

{p 4 8 2}Benjamini, Y. and Y. Hochberg. 1995. Controlling the false discovery rate: a practical and powerful
approach to multiple testing. {it:Journal of the Royal Statistical Society B} 57: 289-300.

{p 4 8 2}Benjamini Y., A. Krieger, and D. Yekutieli. 2001. Two staged linear step-up FDR controlling procedure.
Downloadable from
{browse "http://www.math.tau.ac.il/~ybenja/":Yoav Benjamini's web site at http://www.math.tau.ac.il/~ybenja/}.

{p 4 8 2}Benjamini, Y. and W. Liu. 1999a. A step-down multiple hypotheses testing procedure that controls
the false discovery rate under independence. {it:Journal of Statistical Planning and Inference}
82: 163-170. Also downloadable from
{browse "http://www.math.tau.ac.il/~ybenja/":Yoav Benjamini's web site at http://www.math.tau.ac.il/~ybenja/}.

{p 4 8 2}Benjamini, Y. and W. Liu. 1999b. A distribution-free multiple-test procedure that controls
the false discovery rate. Report, Dept. of Statistics and OR, Tel Aviv University, RP-SOR-99-3.
Downloadable from
{browse "http://www.math.tau.ac.il/~ybenja/":Yoav Benjamini's web site at http://www.math.tau.ac.il/~ybenja/}.

{p 4 8 2}Benjamini, Y. and D. Yekutieli. 2001. The control of the false discovery rate in multiple testing
under dependency. {it:Annals of Statistics} 29: 1165-1188. Also downloadable from
{browse "http://www.math.tau.ac.il/~ybenja/":Yoav Benjamini's web site at http://www.math.tau.ac.il/~ybenja/}.

{p 4 8 2}Hochberg, Y. 1988. A sharper Bonferroni procedure for multiple tests of significance.
{it:Biometrika} 75: 800-802.

{p 4 8 2}Holland, B. S. and Copenhaver, M. D. 1987. An improved sequentially rejective
Bonferroni test procedure. {it:Biometrics} 43: 417-423.

{p 4 8 2}Holm, S. 1979. A simple sequentially rejective multiple test procedure.
{it:Scandinavian Journal of Statistics} 6: 65-70.

{p 4 8 2}Rom, D. M. 1990. A sequentially rejective test procedure based on a modified Bonferroni
inequality. {it:Biometrika} 77: 663-665.

{p 4 8 2}Sidak, Z. 1967. Rectangular confidence regions for the means of multivariate normal
distributions. {it:Journal of the American Statistical Association} 62: 626-633.

{p 4 8 2}Simes, R. J. 1986. An improved Bonferroni procedure for multiple tests of significance.
{it:Biometrika} 73: 751-754.

{p 4 8 2}Wright, S. P. 1992. Adjusted {it:p}-values for simultaneous inference.
{it:Biometrics} 48: 1005-1013.


{title:Also see}

{p 4 10 2}
{bind: }Manual:   {hi:[R] by}, {hi:[R] statsby}, {hi:[G] graph7}
{p_end}
{p 4 10 2}
Online:  help for {help by}, {help statsby}, {help postfile}, {help graph7}
 {break} help for {help smileplot}, {help parmby} and {help parmest} if installed