Tcl Library Source Code

2017-01-17  Arjen Markus <[email protected]>
	* statistics.test: Add tests for test-Tukey-range and test-Dunnett
	* statistics.tcl: Add two procedures, test-Tukey-range and test-Dunnett, and auxiliary code
	* statistics.man: Describe the two new procedures
	* pkgIndex.tcl: Bumped version of the statistics module to 1.1.0

2017-01-10  Arjen Markus <[email protected]>
	* statistics.test: Correct the tests for the test-anova-F procedure
	* statistics.tcl: Correct the namespace for the test-anova-F procedure and the comment
	* statistics.man: Correct the description of the test-anova-F procedure
	* linalg.test: Correct test solve-1.6 regarding permuted matrices
	* geometry.test: Add a simple definition of lmap for older versions of Tcl (notably 8.5)
	* interpolate.test: Correct test Interpolate-1.3 - missing keyword -result and use of existing table name

package ifneeded math::fourier           1.0.2 [list source [file join $dir fourier.tcl]]

if {![package vsatisfies [package provide Tcl] 8.3]} {return}
package ifneeded math::roman             1.0   [list source [file join $dir romannumerals.tcl]]

if {![package vsatisfies [package provide Tcl] 8.4]} {return}
# statistics depends on linearalgebra (for multi-variate linear regression).
package ifneeded math::statistics        1.1.0 [list source [file join $dir statistics.tcl]]
package ifneeded math::optimize          1.0.1 [list source [file join $dir optimize.tcl]]
package ifneeded math::calculus          0.8.1 [list source [file join $dir calculus.tcl]]
package ifneeded math::interpolate       1.1.1 [list source [file join $dir interpolate.tcl]]
package ifneeded math::linearalgebra     1.1.5 [list source [file join $dir linalg.tcl]]
package ifneeded math::bignum            3.1.1 [list source [file join $dir bignum.tcl]]
package ifneeded math::bigfloat          1.2.2 [list source [file join $dir bigfloat.tcl]]
package ifneeded math::machineparameters 0.1   [list source [file join $dir machineparameters.tcl]]

[call [cmd ::math::statistics::test-anova-F] [arg alpha] [arg args]]
Determine if two or more groups with normally distributed data have the same means.
The procedure returns 0 if the means are likely unequal, 1 if they are. This is
a one-way ANOVA test. The groups may also be stored in a nested list:

The procedure returns a list of the comparison results for each pair of groups. Each
element of this list contains: the index of the first group and that of the second group,
whether the means are likely to be different (1) or not (0) and the confidence interval
the conclusion is based on. The groups may also be stored in a nested list:

[example {
    test-anova-F 0.05 $A $B $C
    #
    # Or equivalently:
    #
    test-anova-F 0.05 [list $A $B $C]
}]
[list_begin arguments]
[arg_def float alpha] - Significance level
[arg_def list args] - Two or more groups of data to be checked
[list_end]
[para]

[call [cmd ::math::statistics::test-Tukey-range] [arg alpha] [arg args]]
Determine if two or more groups with normally distributed data have the same means,
using Tukey's range test. It is complementary to the ANOVA test.
The procedure returns a list of the comparison results for each pair of groups. Each
element of this list contains: the index of the first group and that of the second group,
whether the means are likely to be different (1) or not (0) and the confidence interval
the conclusion is based on. The groups may also be stored in a nested list, just as with
the ANOVA test.
[list_begin arguments]
[arg_def float alpha] - Significance level - either 0.05 or 0.01
[arg_def list args] - Two or more groups of data to be checked
[list_end]
[para]

[call [cmd ::math::statistics::test-Dunnett] [arg alpha] [arg control] [arg args]]
Determine if one or more groups with normally distributed data have the same means as
the group of control data, using Dunnett's test. It is complementary to the ANOVA test.
The procedure returns a list of the comparison results for each group with the control group. Each
element of this list contains: whether the means are likely to be different (1) or not (0)
and the confidence interval the conclusion is based on. The groups may also be stored in a
nested list, just as with the ANOVA test.
[nl]
Note: some care is required if there is only one group to compare the control with:
[example {
    test-Dunnett-F 0.05 $control [list $A]
}]
Otherwise the group A is split up into groups of one element - this is due to an ambiguity.

[list_begin arguments]
[arg_def float alpha] - Significance level - either 0.05 or 0.01
[arg_def list args] - One or more groups of data to be checked
[list_end]
[para]

[call [cmd ::math::statistics::quantiles] [arg data] [arg confidence]]
Return the quantiles for a given set of data
[list_begin arguments]
[para]
[arg_def list data] - List of raw data values
[para]

#                (provided by Eric Kemp-Benedict)
# version 0.7:   added Kruskal-Wallis test (by Torsten Berg)
# version 0.8:   added Wilcoxon test and Spearman rank correlation
# version 0.9:   added kernel density estimation
# version 0.9.3: added histogram-alt, corrected test-normal
# version 1.0:   added test-anova-F
# version 1.0.1: correction in pdf-lognormal and cdf-lognormal
# version 1.1:   added test-Tukey-range and test-Dunnett

package require Tcl 8.4
package provide math::statistics 1.1.0
package require math

if {![llength [info commands ::lrepeat]]} {
    # Forward portability, emulate lrepeat
    proc ::lrepeat {n args} {
	if {$n < 1} {
	    return -code error "must have a count of at least 1"

	    histogram histogram-alt interval-mean-stdev t-test-mean quantiles \
	    test-normal lillieforsFit \
	    autocorr crosscorr filter map samplescount median \
	    test-2x2 print-2x2 control-xbar test_xbar \
	    control-Rchart test-Rchart \
	    test-Kruskal-Wallis analyse-Kruskal-Wallis group-rank \
	    test-Wilcoxon spearman-rank spearman-rank-extended \
	    test-Duckworth test-anova-F test-Tukey-range test-Dunnett
    #
    # Error messages
    #
    variable NEGSTDEV   {Zero or negative standard deviation}
    variable TOOFEWDATA {Too few or invalid data}
    variable OUTOFRANGE {Argument out of range}

    set ratio [expr {$varBetween / $varWithin}]
    set nf1   [expr {[llength $args]    - 1}]
    set nf2   [expr {[llength $allData] - [llength $args]}]

    expr {[::math::statistics::cdf-F $nf1 $nf2 $ratio] <= 1.0 - $alpha}
}

# test-Tukey-range --
#     Check if two or more groups with normally distributed data have different
#     means or not, using Tukey's range test
#
# Arguments:
#     alpha            Significance level
#     args             Two or more lists containing the data for the
#                      other groups
#
# Returns:
#     For each pair of groups a list of:
#     - group indices
#     - whether the means differ (1) or not (0)
#     - limits of the confidence interval (for closer investigation)
#
# Note:
#     args may be a nested list
#
#     Implementation based on Wikipedia page on Tukey's range test
#
proc ::math::statistics::test-Tukey-range {alpha args} {
    if { [llength $args] == 1 } {
        set args [lindex $args 0]
    }

    if { [llength $args] < 2 } {
        return -code error -errorcode ARG -errorinfo "At least two groups are required" \
                                                     "At least two groups are required"
    }

    if { $alpha != 0.05 && $alpha != 0.01 } {
        return -code error -errorcode ARG -errorinfo "Alpha must 0.05 or 0.01"
    }

    #
    # Determine the mean per group and the pooled variance of the data
    #
    set meanPerGroup {}
    set allData      {}
    set sumVar       0.0
    foreach group $args {
        lappend meanPerGroup  [mean $group]
        set sumVar            [expr {$sumVar + ([llength $group]-1) * [var $group]}]
        set allData           [concat $allData $group]
    }

    set n          [llength $allData]
    set stdOverall [expr {sqrt($sumVar /($n - [llength $args]))}]

    set qcrit      [Qcrit-Tukey $alpha $n [llength $args]]

    set result {}

    for {set g 0} {$g < [llength $args]} {incr g} {
        set ggroup [lindex $args $g]
        set gmean  [mean $ggroup]
        set ng     [llength $ggroup]

        for {set h [expr {$g+1}]} {$h < [llength $args]} {incr h} {
            set hgroup    [lindex $args $h]
            set hmean     [mean $hgroup]
            set nh        [llength $hgroup]

            set halfwidth [expr {$qcrit * $stdOverall / sqrt(2.0) * sqrt( 1.0/$ng + 1.0/$nh )}]
            set lower     [expr {$hmean - $gmean - $halfwidth}]
            set upper     [expr {$hmean - $gmean + $halfwidth}]
            set unequal   1
            if { $lower < 0.0 && $upper > 0.0 } {
                set unequal 0
            }
            lappend result [list $g $h $unequal $lower $upper]
        }
    }

    return $result
}

# Qcrit-Tukey --
#     Determine the critical value for the Tukey range test
#
# Arguments:
#     alpha            Significance level
#     numberData       Total number of data
#     numberGroups     Number of groups
#
# Returns:
#     Critical value
#
# Note:
#     If there are more than 10 groups, simply use 10 groups
#
proc ::math::statistics::Qcrit-Tukey {alpha numberData numberGroups} {
    variable tukey_table_05
    variable tukey_table_01

    if { $alpha == 0.05 } {
        upvar 0 tukey_table_05 tukey_table
    } else {
        upvar 0 tukey_table_05 tukey_table
    }

    set df [expr {$numberData - $numberGroups}]

    if { $numberGroups > 10 } {
        set numberGroups 10
    }
    incr numberGroups -1 ;# Offset because of 0-based numbering

    if { $df > 120 } {
        return [lindex $tukey_table end $numberGroups]
    }

    foreach {dfe values} $tukey_table {
        if { $df <= $dfe } {
            return [lindex $values $numberGroups]
        }
    }
}

# test-Dunnett --
#     Check if one or more groups with normally distributed data have different
#     means from the control group or not, using Dunnett's test
#
# Arguments:
#     alpha            Significance level
#     control          Control group
#     args             One or more lists containing the data for the
#                      other groups
#
# Returns:
#     For each group a list of:
#     - whether the mean differs (1) from the control or not (0)
#     - the confidence interval
#
# Note:
#     args may be a nested list
#
#     Implementation based on Wikipedia page on Dunnett's test
#     The test is two-sided.
#
proc ::math::statistics::test-Dunnett {alpha control args} {
    if { [llength $args] == 1 } {
        set args [lindex $args 0]
    }

    if { [llength $args] < 1 } {
        return -code error -errorcode ARG -errorinfo "At least one additional group is required" \
                                                     "At least one additional group is required"
    }

    if { $alpha != 0.05 && $alpha != 0.01 } {
        return -code error -errorcode ARG -errorinfo "Alpha must 0.05 or 0.01"
    }

    #
    # Determine the mean per group and the pooled variance
    #
    set allData      $control
    set sumVar       [expr {([llength $control]-1)*[var $control]}]
    foreach group $args {
        set sumVar            [expr {$sumVar + ([llength $group]-1) * [var $group]}]
        set allData           [concat $allData $group]
    }

    set n          [llength $allData]
    set stdOverall [expr {sqrt($sumVar /($n - [llength $args] - 1))}]

    set tcrit [Tcrit-Dunnett $alpha $n [llength $args]]

    set result {}

    set cmean  [mean $control]
    set nc     [llength $control]

    for {set g 0} {$g < [llength $args]} {incr g} {
        set ggroup    [lindex $args $g]
        set gmean     [mean $ggroup]
        set ng        [llength $ggroup]

        set halfwidth [expr {$tcrit * $stdOverall * sqrt( 1.0/$nc + 1.0/$ng )}]
        set lower     [expr {$gmean - $cmean - $halfwidth}]
        set upper     [expr {$gmean - $cmean + $halfwidth}]
        set unequal   1
        if { $lower < 0.0 && $upper > 0.0 } {
            set unequal 0
        }
        lappend result [list $unequal $lower $upper]
    }

    return $result
}

# Tcrit-Dunnett --
#     Determine the critical value for the Dunnett test
#
# Arguments:
#     alpha            Significance level
#     numberData       Total number of data
#     numberGroups     Number of groups to compare against the control
#
# Returns:
#     Critical value
#
# Note:
#     If there are more than 10 groups, simply use 10 groups
#
proc ::math::statistics::Tcrit-Dunnett {alpha numberData numberGroups} {
    variable dunnett_table_05
    variable dunnett_table_01

    if { $alpha == 0.05 } {
        upvar 0 dunnett_table_05 dunnett_table
    } else {
        upvar 0 dunnett_table_05 dunnett_table
    }

    set df [expr {$numberData - $numberGroups - 1}]

    incr numberGroups 1 ;# Add the control group
    if { $numberGroups > 10 } {
        set numberGroups 10
    }
    incr numberGroups -2 ;# Offset because of 0-based numbering and start at 2

    if { $df > 60 } {
        return [lindex $dunnett_table end $numberGroups]
    }

    foreach {dfe values} $dunnett_table {
        if { $df <= $dfe } {
            return [lindex $values $numberGroups]
        }
    }
}

#
# Load the auxiliary scripts
#
source [file join [file dirname [info script]] pdf_stat.tcl]
source [file join [file dirname [info script]] plotstat.tcl]
source [file join [file dirname [info script]] liststat.tcl]
source [file join [file dirname [info script]] mvlinreg.tcl]
source [file join [file dirname [info script]] kruskal.tcl]
source [file join [file dirname [info script]] wilcoxon.tcl]
source [file join [file dirname [info script]] stat_kernel.tcl]

#
# Define the tables
#
namespace eval ::math::statistics {
    variable tukey_table_05
    variable tukey_table_01
    variable dunnett_table_05
    variable dunnett_table_01




    #alpha = 0.05
    #k 2 3 4 5 6 7 8 9 10
    #df
    set tukey_table_05 {
        1  {18.0 27.0 32.8 37.1 40.4 43.1 45.4 47.4 49.1}
        2  {6.08 8.33 9.80 10.88 11.73 12.43 13.03 13.54 13.99}
        3  {4.50 5.91 6.82 7.50 8.04 8.48 8.85 9.18 9.46}
        4  {3.93 5.04 5.76 6.29 6.71 7.05 7.35 7.60 7.83}
        5  {3.64 4.60 5.22 5.67 6.03 6.33 6.58 6.80 6.99}
        6  {3.46 4.34 4.90 5.30 5.63 5.90 6.12 6.32 6.49}
        7  {3.34 4.16 4.68 5.06 5.36 5.61 5.82 6.00 6.16}
        8  {3.26 4.04 4.53 4.89 5.17 5.40 5.60 5.77 5.92}
        9  {3.20 3.95 4.41 4.76 5.02 5.24 5.43 5.59 5.74}
        10 {3.15 3.88 4.33 4.65 4.91 5.12 5.30 5.46 5.60}
        11 {3.11 3.82 4.26 4.57 4.82 5.03 5.20 5.35 5.49}
        12 {3.08 3.77 4.20 4.51 4.75 4.95 5.12 5.27 5.39}
        13 {3.06 3.73 4.15 4.45 4.69 4.88 5.05 5.19 5.32}
        14 {3.03 3.70 4.11 4.41 4.64 4.83 4.99 5.13 5.25}
        15 {3.01 3.67 4.08 4.37 4.59 4.78 4.94 5.08 5.20}
        16 {3.00 3.65 4.05 4.33 4.56 4.74 4.90 5.03 5.15}
        17 {2.98 3.63 4.02 4.30 4.52 4.70 4.86 4.99 5.11}
        18 {2.97 3.61 4.00 4.28 4.49 4.67 4.82 4.96 5.07}
        19 {2.96 3.59 3.98 4.25 4.47 4.65 4.79 4.92 5.04}
        20 {2.95 3.58 3.96 4.23 4.45 4.62 4.77 4.90 5.01}
        24 {2.92 3.53 3.90 4.17 4.37 4.54 4.68 4.81 4.92}
        30 {2.89 3.49 3.85 4.10 4.30 4.46 4.60 4.72 4.82}
        40 {2.86 3.44 3.79 4.04 4.23 4.39 4.52 4.63 4.73}
        60 {2.83 3.40 3.74 3.98 4.16 4.31 4.44 4.55 4.65}
        120 {2.80 3.36 3.68 3.92 4.10 4.24 4.36 4.47 4.56}
        inf {2.77 3.31 3.63 3.86 4.03 4.17 4.29 4.39 4.47}}

    #alpha = 0.01
    #k 2 3 4 5 6 7 8 9 10
    #df
    set tukey_table_01 {
        1  {90.0 135 164 186 202 216 227 237 246}
        2  {13.90 19.02 22.56 25.37 27.76 29.86 31.73 33.41 34.93}
        3  {8.26 10.62 12.17 13.32 14.24 15.00 15.65 16.21 16.71}
        4  {6.51 8.12 9.17 9.96 10.58 11.10 11.54 11.92 12.26}
        5  {5.70 6.98 7.80 8.42 8.91 9.32 9.67 9.97 10.24}
        6  {5.24 6.33 7.03 7.56 7.97 8.32 8.61 8.87 9.10}
        7  {4.95 5.92 6.54 7.00 7.37 7.68 7.94 8.17 8.37}
        8  {4.75 5.64 6.20 6.62 6.96 7.24 7.47 7.68 7.86}
        9  {4.60 5.43 5.96 6.35 6.66 6.91 7.13 7.33 7.49}
        10 {4.48 5.27 5.77 6.14 6.43 6.67 6.87 7.05 7.21}
        11 {4.39 5.15 5.62 5.97 6.25 6.48 6.67 6.84 6.99}
        12 {4.32 5.05 5.50 5.84 6.10 6.32 6.51 6.67 6.81}
        13 {4.26 4.96 5.40 5.73 5.98 6.19 6.37 6.53 6.67}
        14 {4.21 4.89 5.32 5.63 5.88 6.08 6.26 6.41 6.54}
        15 {4.17 4.84 5.25 5.56 5.80 5.99 6.16 6.31 6.44}
        16 {4.13 4.79 5.19 5.49 5.72 5.92 6.08 6.22 6.35}
        17 {4.10 4.74 5.14 5.43 5.66 5.85 6.01 6.15 6.27}
        18 {4.07 4.70 5.09 5.38 5.60 5.79 5.94 6.08 6.20}
        19 {4.05 4.67 5.05 5.33 5.55 5.73 5.89 6.02 6.14}
        20 {4.02 4.64 5.02 5.29 5.51 5.69 5.84 5.97 6.09}
        24 {3.96 4.55 4.91 5.17 5.37 5.54 5.69 5.81 5.92}
        30 {3.89 4.45 4.80 5.05 5.24 5.40 5.54 5.65 5.76}
        40 {3.82 4.37 4.70 4.93 5.11 5.26 5.39 5.50 5.60}
        60 {3.76 4.28 4.59 4.82 4.99 5.13 5.25 5.36 5.45}
        120 {3.70 4.20 4.50 4.71 4.87 5.01 5.12 5.21 5.30}
        inf {3.64 4.12 4.40 4.60 4.76 4.88 4.99 5.08 5.16}}

    #From: http://davidmlane.com/hyperstat/table_Dunnett.html
    #Dunnett Table
    #Number of Groups Including Control Group
    #dfe     alpha = 0.05
    #         2       3       4       5       6       7       8       9       10
    set dunnett_table_05 {
    5        {2.57    3.03    3.29    3.48    3.62    3.73    3.82    3.9     3.97}
    6        {2.45    2.86    3.1     3.26    3.39    3.49    3.57    3.64    3.71}
    7        {2.36    2.75    2.97    3.12    3.24    3.33    3.41    3.47    3.53}
    8        {2.31    2.67    2.88    3.02    3.13    3.22    3.29    3.35    3.41}
    9        {2.26    2.61    2.81    2.95    3.05    3.14    3.2     3.26    3.32}
    10       {2.23    2.57    2.76    2.89    2.99    3.07    3.14    3.19    3.24}
    11       {2.2     2.53    2.72    2.84    2.94    3.02    3.08    3.14    3.19}
    12       {2.18    2.5     2.68    2.81    2.9     2.98    3.04    3.09    3.14}
    13       {2.16    2.48    2.65    2.78    2.87    2.94    3       3.06    3.1}
    14       {2.14    2.46    2.63    2.75    2.84    2.91    2.97    3.02    3.07}
    15       {2.13    2.44    2.61    2.73    2.82    2.89    2.95    3       3.04}
    16       {2.12    2.42    2.59    2.71    2.8     2.87    2.92    2.97    3.02}
    17       {2.11    2.41    2.58    2.69    2.78    2.85    2.9     2.95    3}
    18       {2.1     2.4     2.56    2.68    2.76    2.83    2.89    2.94    2.98}
    19       {2.09    2.39    2.55    2.66    2.75    2.81    2.87    2.92    2.96}
    20       {2.09    2.38    2.54    2.65    2.73    2.8     2.86    2.9     2.95}
    24       {2.06    2.35    2.51    2.61    2.7     2.76    2.81    2.86    2.9}
    30       {2.04    2.32    2.47    2.58    2.66    2.72    2.77    2.82    2.86}
    40       {2.02    2.29    2.44    2.54    2.62    2.68    2.73    2.77    2.81}
    60       {2       2.27    2.41    2.51    2.58    2.64    2.69    2.73    2.77}}

    set dunnett_table_01 {
    5        {4.03    4.63    4.98    5.22    5.41    5.56    5.69    5.8     5.89}
    6        {3.71    4.21    4.51    4.71    4.87    5       5.1     5.2     5.28}
    7        {3.5     3.95    4.21    4.39    4.53    4.64    4.74    4.82    4.89}
    8        {3.36    3.77    4       4.17    4.29    4.4     4.48    4.56    4.62}
    9        {3.25    3.63    3.85    4.01    4.12    4.22    4.3     4.37    4.43}
    10       {3.17    3.53    3.74    3.88    3.99    4.08    4.16    4.22    4.28}
    11       {3.11    3.45    3.65    3.79    3.89    3.98    4.05    4.11    4.16}
    12       {3.05    3.39    3.58    3.71    3.81    3.89    3.96    4.02    4.07}
    13       {3.01    3.33    3.52    3.65    3.74    3.82    3.89    3.94    3.99}
    14       {2.98    3.29    3.47    3.59    3.69    3.76    3.83    3.88    3.93}
    15       {2.95    3.25    3.43    3.55    3.64    3.71    3.78    3.83    3.88}
    16       {2.92    3.22    3.39    3.51    3.6     3.67    3.73    3.78    3.83}
    17       {2.9     3.19    3.36    3.47    3.56    3.63    3.69    3.74    3.79}
    18       {2.88    3.17    3.33    3.44    3.53    3.6     3.66    3.71    3.75}
    19       {2.86    3.15    3.31    3.42    3.5     3.57    3.63    3.68    3.72}
    20       {2.85    3.13    3.29    3.4     3.48    3.55    3.6     3.65    3.69}
    24       {2.8     3.07    3.22    3.32    3.4     3.47    3.52    3.57    3.61}
    30       {2.75    3.01    3.15    3.25    3.33    3.39    3.44    3.49    3.52}
    40       {2.7     2.95    3.09    3.19    3.26    3.32    3.37    3.41    3.44}
    60       {2.66    2.9     3.03    3.12    3.19    3.25    3.29    3.33    3.37}}

}

#
# Simple test code
#
if { [info exists ::argv0] && ([file tail [info script]] == [file tail $::argv0]) } {

    ::math::statistics::test-anova-F 0.05 {6 8 4 5 3 4} {8 12 9 11 6 8} {13 9 11 8 7 12}
} -result 0

test "anova-one-way-1.2" "ANOVA test from Wikipedia - using nested list" -body {
    ::math::statistics::test-anova-F 0.05 {{6 8 4 5 3 4} {8 12 9 11 6 8} {13 9 11 8 7 12}}
} -result 0

#
# Data from http://www.itl.nist.gov/div898/handbook/prc/section4/prc436.htm#example1
# See also http://www.itl.nist.gov/div898/handbook/prc/section4/prc471.htm
# Caveat: the calculation produces slightly different confidence intervals. I checked whether
# I got the calculation of the pooled variance right against the example appearing on Wikipedia
# (https://en.wikipedia.org/wiki/Pooled_variance) and that seems okay.
# No idea where the numerical difference is coming from.
#
test "Tukey-range-test-1.1" "Tukey range test" -body {
    set data {
        Group 1        {6.9     5.4     5.8     4.6     4.0}
        Group 2        {8.3     6.8     7.8     9.2     6.5}
        Group 3        {8.0     10.5    8.1     6.9     9.3}
        Group 4        {5.8     3.8     6.1     5.6     6.2}
    }

    set groupData {}
    foreach {dummy label d} $data {
        lappend groupData $d
    }

    set tukeyRange [::math::statistics::test-Tukey-range 0.05 $groupData]

    set indications {}
    foreach t $tukeyRange {
        lappend indications [lindex $t 2]
    }
    set indications
} -result {1 1 0 0 0 1}

#
# Data from https://en.wikipedia.org/wiki/Dunnett's_test
# Note that the Wikipedia uses t = 2.94, whereas it should have been 2.88
#
test "Dunnet-test-1.1" "Dunnett test" -body {
    set control {55 47 48}
    set data    {{55 64 64} {55 49 52} {50 44 41}}

    set dunnett [::math::statistics::test-Dunnett 0.05 $control $data]

    set indications {}
    foreach t $dunnett {
        lappend indications [lindex $t 0]
    }
    set indications
} -result {1 0 0}

# End of test cases
testsuiteCleanup