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    %% Multiple Comparisons for One-Way ANOVA
%%
% Input the sample data.

% Copyright 2015 The MathWorks, Inc.

strength = [82 86 79 83 84 85 86 87 74 82 ...
            78 75 76 77 79 79 77 78 82 79];
alloy = {'st','st','st','st','st','st','st','st',...
         'al1','al1','al1','al1','al1','al1',...
         'al2','al2','al2','al2','al2','al2'};
%%
% The data are from a study of the strength of structural beams in Hogg
% (1987). The vector strength measures deflections of beams in thousandths
% of an inch under 3000 pounds of force. The vector alloy identifies each
% beam as steel (|st|), alloy 1 (|al1|), or alloy 2 (|al2|).
% Although alloy is sorted in this example, grouping variables do not need
% to be sorted.
%%
% Perform one-way ANOVA using |anova1|. Return the
% structure |stats|, which contains the statistics <docid:stats_ug.bujc800
% multcompare> needs for performing <docid:stats_ug.bum2f3y-1 multiple
% comparisons>.
[~,~,stats] = anova1(strength,alloy);
%%
% The small _p_-value of 0.0002 suggests that the strength of the beams is
% not the same.
%%
% Perform a multiple comparison of the mean strength of the beams.
[c,~,~,gnames] = multcompare(stats);
%%
% Display the comparison results with the corresponding group names.
[gnames(c(:,1)), gnames(c(:,2)), num2cell(c(:,3:6))] 
%%
% The first two columns show the pair of groups that are compared. The
% fourth column shows the difference between the estimated group means. The third and fifth
% columns show the lower and upper limits for the 95% confidence
% intervals of the true difference of means. The sixth column shows the _p_-value for a hypothesis
% that the true difference of means for the corresponding groups is equal to zero.
%%
% The first two rows show that both comparisons involving the first group
% (steel) have confidence intervals that do not include zero. Because the
% corresponding _p_-values (1.6831e-04 and 0.0040, respectively) are
% small, those differences are significant. 
%%
% The third row shows that the differences in strength between the two
% alloys is not significant. A 95% confidence interval for the difference
% is [-5.6,1.6], so you cannot reject the hypothesis that the true
% difference is zero. The corresponding _p_-value of 0.3560 in the sixth
% column confirms this result.
%%
% In the figure, the blue bar represents the comparison interval for mean
% material strength for steel. The red bars represent the comparison
% intervals for the mean material strength for alloy 1 and alloy 2. Neither
% of the red bars overlap with the blue bar, which indicates that the mean
% material strength for steel is significantly different from that of alloy
% 1 and alloy 2. To confirm the significant difference by clicking the bars
% that represent alloy 1 and 2.