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    1#
    发表于 2015-10-29 20:51 |只看该作者 |倒序浏览
    |招呼Ta 关注Ta |邮箱已经成功绑定
    这张图,是我在MATLAB中调试这个程序得到的
    x=[2.48 2.90 2.5 2.5 2.3    2.39 3.30 2.7 2.8 2.5
        2.62 2.90 2.6 2.9 2.4
        2.51 2.95 2.6 2.2 2.4]
    p=anova1(x)
    但是,不知道怎样解读,麻烦各位大神,指点迷津!


    捕获.PNG (5.32 KB, 下载次数: 266)

    捕获.PNG

    zan
    转播转播0 分享淘帖0 分享分享0 收藏收藏0 支持支持0 反对反对0 微信微信

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    1. function [p,anovatab,stats] = anova1(x,group,displayopt,extra)
    2. %ANOVA1 One-way analysis of variance (ANOVA).
    3. %   ANOVA1 performs a one-way ANOVA for comparing the means of two or more
    4. %   groups of data. It returns the p-value for the null hypothesis that the
    5. %   means of the groups are equal.
    6. %
    7. %   P = ANOVA1(M) for a matrix M treats each column as a separate group,
    8. %   and determines whether the population means of the columns are equal.
    9. %   This form of ANOVA1 is appropriate when each group has the same number
    10. %   of elements (balanced ANOVA).
    11. %
    12. %   P = ANOVA1(V,GROUP) groups elements in the vector V according to values
    13. %   in the grouping variable GROUP. GROUP must be a categorical variable,
    14. %   numeric vector, logical vector, string array, or cell array of strings
    15. %   with one group name for each element of X.  X values corresponding to
    16. %   the same value of GROUP are placed in the same group.
    17. %
    18. %   P = ANOVA1(M,GROUP) accepts a character array or cell array of strings,
    19. %   with one group name for each column of M. Columns with the same group
    20. %   name are treated as part of the same group.
    21. %
    22. %   P = ANOVA1(X,GROUP,DISPLAYOPT) controls the display. DISPLAYOPT can be
    23. %   'on' (the default) to display figures containing a standard one-way
    24. %   anova table and a boxplot, or 'off' to omit these displays.  Note that
    25. %   the notches in the boxplot provide a test of group medians (see HELP
    26. %   BOXPLOT), and this is not the same as the F test for different means in
    27. %   the anova table. X can be either a vector or matrix. If X is a matrix
    28. %   and there are no group names, specify GROUP as [].
    29. %
    30. %   [P,ANOVATAB] = ANOVA1(...) returns the ANOVA table values as the
    31. %   cell array ANOVATAB.
    32. %
    33. %   [P,ANOVATAB,STATS] = ANOVA1(...) returns an additional structure
    34. %   of statistics useful for performing a multiple comparison of means
    35. %   with the MULTCOMPARE function.
    36. %
    37. %   See also ANOVA2, ANOVAN, BOXPLOT, MANOVA1, MULTCOMPARE.

    38. %   Reference: Robert V. Hogg, and Johannes Ledolter, Engineering Statistics
    39. %   Macmillan 1987 pp. 205-206.

    40. %   Copyright 1993-2013 The MathWorks, Inc.

    41. narginchk(1,4);

    42. classical = 1;
    43. nargs = nargin;
    44. if (nargin>0 && strcmp(x,'kruskalwallis'))
    45.    % Called via kruskalwallis function, adjust inputs
    46.    classical = 0;
    47.    if (nargin >= 2), x = group; group = []; end
    48.    if (nargin >= 3), group = displayopt; displayopt = []; end
    49.    if (nargin >= 4), displayopt = extra; end
    50.    nargs = nargs-1;
    51. end

    52. if (nargs < 2), group = []; end
    53. if (nargs < 3), displayopt = 'on'; end
    54. % Note: for backwards compatibility, accept 'nodisplay' for 'off'
    55. willdisplay = ~(strcmp(displayopt,'nodisplay') | strcmp(displayopt,'n') ...
    56.                 | strcmp(displayopt,'off'));

    57. % Convert group to cell array from character array, make it a column
    58. if (ischar(group) && ~isempty(group))
    59.     group = cellstr(group);
    60. end
    61. if (size(group, 1) == 1)
    62.     group = group';
    63. end

    64. % If the input is a matrix, it may not be balanced if it contains NaNs or
    65. % if there are repeated grouping values, so turn it into a vector.
    66. needvector = false;
    67. if ~isvector(x)
    68.     if (any(isnan(x(:))))
    69.         needvector = true;
    70.     elseif ~isempty(group) && length(group)~=length(unique(group))
    71.         needvector = true;
    72.     end
    73. end

    74. % If X is a matrix with NaNs, convert to vector form.
    75. if ~isvector(x)
    76.    if needvector
    77.       [n,m] = size(x);
    78.       x = x(:);
    79.       gi = reshape(repmat((1:m), n, 1), n*m, 1);
    80.       if isempty(group)     % no group names
    81.          group = gi;
    82.       elseif (size(group,1) == m)
    83.          group = group(gi,:);
    84.       else
    85.          error(message('stats:anova1:InputSizeMismatch'));
    86.       end
    87.    end
    88. elseif ~isempty(group) && (size(group,1) ~= length(x))
    89.    error(message('stats:anova1:InputSizeMismatch'));
    90. end

    91. % If X is a matrix GROUP is provided with the correct size, use GROUP
    92. % to define groups and to label boxes
    93. if (~isempty(group) && (length(x) < numel(x)) ...
    94.                     && (size(x,2) == size(group,1)))
    95.    named = 1;
    96.    [gid,gnames] = grp2idx(group);
    97.    gnames = gnames(gid);
    98.    grouped = 0;
    99. else
    100.    named = 0;
    101.    gnames = [];
    102.    grouped = ~isempty(group);
    103. end

    104. if (grouped)
    105.    % Single data vector and a separate grouping variable
    106.    x = x(:);
    107.    lx = length(x);
    108.    if (lx ~= numel(x))
    109.       error(message('stats:anova1:VectorRequired'))
    110.    end
    111.    nonan = ~isnan(x);
    112.    x = x(nonan);

    113.    % Convert group to indices 1,...,g and separate names
    114.    group = group(nonan,:);
    115.    [groupnum, gnames] = grp2idx(group);
    116.    named = 1;

    117.    % Remove NaN values
    118.    nonan = ~isnan(groupnum);
    119.    if (~all(nonan))
    120.       groupnum = groupnum(nonan);
    121.       x = x(nonan);
    122.    end

    123.    lx = length(x);
    124.    xorig = x;                    % use uncentered version to make M
    125.    groupnum = groupnum(:);
    126.    maxi = size(gnames, 1);
    127.    if isa(x,'single')
    128.       xm = zeros(1,maxi,'single');
    129.    else
    130.       xm = zeros(1,maxi);
    131.    end
    132.    countx = xm;
    133.    if (classical)
    134.       mu = mean(x);
    135.       x = x - mu;                % center to improve accuracy
    136.       xr = x;
    137.    else
    138.       [xr,tieadj] = tiedrank(x);
    139.    end
    140.    
    141.    for j = 1:maxi
    142.       % Get group sizes and means
    143.       k = find(groupnum == j);
    144.       lk = length(k);
    145.       countx(j) = lk;
    146.       xm(j) = mean(xr(k));       % column means
    147.    end

    148.    gm = mean(xr);                      % grand mean
    149.    df1 = sum(countx>0) - 1;            % Column degrees of freedom
    150.    df2 = lx - df1 - 1;                 % Error degrees of freedom
    151.    xc = xm - gm;                       % centered
    152.    xc(countx==0) = 0;
    153.    RSS = dot(countx, xc.^2);           % Regression Sum of Squares
    154. else
    155.    % Data in matrix form, no separate grouping variable
    156.    [r,c] = size(x);
    157.    lx = r * c;
    158.    if (classical)
    159.       xr = x;
    160.       mu = mean(xr(:));
    161.       xr = xr - mu;           % center to improve accuracy
    162.    else
    163.       [xr,tieadj] = tiedrank(x(:));
    164.       xr = reshape(xr, size(x));
    165.    end
    166.    countx = repmat(r, 1, c);
    167.    xorig = x;                 % save uncentered version for boxplot
    168.    xm = mean(xr);             % column means
    169.    gm = mean(xm);             % grand mean
    170.    df1 = c-1;                 % Column degrees of freedom
    171.    df2 = c*(r-1);             % Error degrees of freedom
    172.    RSS = r*(xm - gm)*(xm-gm)';        % Regression Sum of Squares
    173. end

    174. TSS = (xr(:) - gm)'*(xr(:) - gm);  % Total Sum of Squares
    175. SSE = TSS - RSS;                   % Error Sum of Squares

    176. if (df2 > 0)
    177.    mse = SSE/df2;
    178. else
    179.    mse = NaN;
    180. end

    181. if (classical)
    182.    if (SSE~=0)
    183.       F = (RSS/df1) / mse;
    184.       p = fpval(F,df1,df2);        % Probability of F given equal means.
    185.    elseif (RSS==0)                 % Constant Matrix case.
    186.       F = NaN;
    187.       p = NaN;
    188.    else                            % Perfect fit case.
    189.       F = Inf;
    190.       p = 0;
    191.    end
    192. else
    193.    F = (12 * RSS) / (lx * (lx+1));
    194.    if (tieadj > 0)
    195.       F = F / (1 - 2 * tieadj/(lx^3-lx));
    196.    end
    197.    p = chi2pval(F,df1);
    198. end


    199. Table=zeros(3,5);               %Formatting for ANOVA Table printout
    200. Table(:,1)=[ RSS SSE TSS]';
    201. Table(:,2)=[df1 df2 df1+df2]';
    202. Table(:,3)=[ RSS/df1 mse Inf ]';
    203. Table(:,4)=[ F Inf Inf ]';
    204. Table(:,5)=[ p Inf Inf ]';

    205. colheads = {getString(message('stats:anova1:ColHeadSource')), getString(message('stats:anova1:ColHeadSS')), getString(message('stats:anova1:ColHeadDf')), getString(message('stats:anova1:ColHeadMS')), getString(message('stats:anova1:ColHeadF')), getString(message('stats:anova1:ColHeadProbGtF'))};

    206. if (~classical)
    207.    colheads{5} = getString(message('stats:anova1:ColHeadChisq'));
    208.    colheads{6} = getString(message('stats:anova1:ColHeadProbGtChisq'));
    209. end
    210. rowheads = {getString(message('stats:anova1:RowHeadColumns')), getString(message('stats:anova1:RowHeadError')), getString(message('stats:anova1:RowHeadTotal'))};
    211. if (grouped)
    212.    rowheads{1} = getString(message('stats:anova1:RowHeadGroups'));
    213. end

    214. % Create cell array version of table
    215. atab = num2cell(Table);
    216. for i=1:size(atab,1)
    217.    for j=1:size(atab,2)
    218.       if (isinf(atab{i,j}))
    219.          atab{i,j} = [];
    220.       end
    221.    end
    222. end
    223. atab = [rowheads' atab];
    224. atab = [colheads; atab];
    225. if (nargout > 1)
    226.    anovatab = atab;
    227. end

    228. % Create output stats structure if requested, used by MULTCOMPARE
    229. if (nargout > 2)
    230.    if ~isempty(gnames)
    231.       stats.gnames = gnames;
    232.    else
    233.       stats.gnames = strjust(num2str((1:length(xm))'),'left');
    234.    end
    235.    stats.n = countx;
    236.    if (classical)
    237.       stats.source = 'anova1';
    238.       stats.means = xm + mu;
    239.       stats.df = df2;
    240.       stats.s = sqrt(mse);
    241.    else
    242.       stats.source = 'kruskalwallis';
    243.       stats.meanranks = xm;
    244.       stats.sumt = 2 * tieadj;
    245.    end
    246. end

    247. if (~willdisplay), return; end

    248. digits = [-1 -1 0 -1 2 4];
    249. if (classical)
    250.    wtitle = getString(message('stats:anova1:OnewayANOVA'));
    251.    ttitle = getString(message('stats:anova1:ANOVATable'));
    252. else
    253.    wtitle = getString(message('stats:anova1:KruskalWallisOnewayANOVA'));
    254.    ttitle = getString(message('stats:anova1:KruskalWallisANOVATable'));
    255. end
    256. tblfig = statdisptable(atab, wtitle, ttitle, '', digits);
    257. set(tblfig,'tag','table');

    258. f1 = figure('pos',get(gcf,'pos') + [0,-200,0,0],'tag','boxplot');
    259. ax = axes('Parent',f1);
    260. if (~grouped)
    261.    boxplot(ax,xorig,'notch','on');
    262. else
    263.    boxplot(ax,xorig,groupnum,'notch','on');
    264.    h = get(ax,'XLabel');
    265.    set(h,'String',getString(message('stats:anova1:GroupNumber')));
    266. end

    267. % If there are group names, use them
    268. if ~isempty(gnames)
    269.    h = get(ax,'XLabel');
    270.    if (named)
    271.       set(h,'String','');
    272.    end
    273.    set(ax, 'xtick', (1:size(gnames,1)), 'xticklabel', gnames);
    274. end
    复制代码

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    在matlab中的ANOVA分为anova1,anova2,anovan共三个函数。
    详细说明见官网解释:http://www.mathworks.cn/help/toolbox/stats/anova1.html
    anova1: one-way analysis of variance 单因素方差分析
    anova2: two-way analysis of variance 两因素方差分析
    anovan: n-way analysis of variance 多因素方差分析
    一般使用anova1足矣。
    特别注意,给出的矩阵X要每个列含有相同个数的元素,做anova时结果会给出p-value,p-value越小,说明各组样品的均值中不全相同,至少有一组的均值有显著性差异。一般的显著性等级(Common significance level)为0.05或者0.01。
    若p-value大于Common significance level,则说明各组均值之间没有统计学差异。

    点评

    森之张卫东  The standard ANOVA table has six columns: 1. The source of the variability. 2. The sum of squares (SS) due to each source. 3. The degrees of freedom (df) associated with each source. 4. The mean s  详情 回复 发表于 2015-11-9 23:02
      收起(4)
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    森之张卫东 发表于 2015-10-29 22:45
    在matlab中的ANOVA分为anova1,anova2,anovan共三个函数。
    详细说明见官网解释:http://www.mathworks.cn ...

    The standard ANOVA table has six columns:
    1.        The source of the variability.
    2.        The sum of squares (SS) due to each source.
    3.        The degrees of freedom (df) associated with each source.
    4.        The mean squares (MS) for each source, which is the ratio SS/df.
    5.        The F-statistic, which is the ratio of the mean squares.
    6.        The p-value, which is derived from the cdf of F.
    The box plot of the columns of X suggests the size of the F-statistic and the p-value. Large differences in the center lines of the boxes correspond to large values of F and correspondingly small values of p.
    标准方差分析表有六列:
    1。变化的来源。
    2。平方和(SS)由于每个源。
    3所示。自由度(df)与每个源有关。
    4所示。平均为每个源广场(MS),SS / df的比率。
    5。的f统计量的比例意味着广场。
    6。假定值,这是来自F的运作。
    X的列的箱线图表明f统计量和假定值的大小。大箱子对应的中心线的差异大的F值和相应的小p值。

    The anova1 function displays two figures, the standard ANOVA table and a box plot of the columns of X.
    The standard ANOVA table divides the variability of the data into two parts:
    •        Variability due to the differences among the column means (variability between groups)
    •        Variability due to the differences between the data in each column and the column mean (variability within groups)
    anova1函数显示两个数据,标准方差分析表和列的箱线图的X。
    标准方差分析表中数据的变化分为两个部分:
    •可变性由于差异列意味着(组)之间的差异
    •可变性将在每一列数据之间的差异和列意味着(组内差异

    The small p -value indicates that differences between column means are significant. The probability of this outcome under the null hypothesis (that samples drawn from the same population would have means differing by the amounts seen in X ) is equal to the p -value.

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