求助

远行的小船儿666

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！

森之张卫东

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. 
    
39. %   Reference: Robert V. Hogg, and Johannes Ledolter, Engineering Statistics
    
40. %   Macmillan 1987 pp. 205-206.
    
41. 
    
42. %   Copyright 1993-2013 The MathWorks, Inc.
    
43. 
    
44. narginchk(1,4);
    
45. 
    
46. classical = 1;
    
47. nargs = nargin;
    
48. if (nargin>0 && strcmp(x,'kruskalwallis'))
    
49.    % Called via kruskalwallis function, adjust inputs
    
50.    classical = 0;
    
51.    if (nargin >= 2), x = group; group = []; end
    
52.    if (nargin >= 3), group = displayopt; displayopt = []; end
    
53.    if (nargin >= 4), displayopt = extra; end
    
54.    nargs = nargs-1;
    
55. end
    
56. 
    
57. if (nargs < 2), group = []; end
    
58. if (nargs < 3), displayopt = 'on'; end
    
59. % Note: for backwards compatibility, accept 'nodisplay' for 'off'
    
60. willdisplay = ~(strcmp(displayopt,'nodisplay') | strcmp(displayopt,'n') ...
    
61.                 | strcmp(displayopt,'off'));
    
62. 
    
63. % Convert group to cell array from character array, make it a column
    
64. if (ischar(group) && ~isempty(group))
    
65.     group = cellstr(group);
    
66. end
    
67. if (size(group, 1) == 1)
    
68.     group = group';
    
69. end
    
70. 
    
71. % If the input is a matrix, it may not be balanced if it contains NaNs or
    
72. % if there are repeated grouping values, so turn it into a vector.
    
73. needvector = false;
    
74. if ~isvector(x)
    
75.     if (any(isnan(x(:))))
    
76.         needvector = true;
    
77.     elseif ~isempty(group) && length(group)~=length(unique(group))
    
78.         needvector = true;
    
79.     end
    
80. end
    
81. 
    
82. % If X is a matrix with NaNs, convert to vector form.
    
83. if ~isvector(x)
    
84.    if needvector
    
85.       [n,m] = size(x);
    
86.       x = x(:);
    
87.       gi = reshape(repmat((1:m), n, 1), n*m, 1);
    
88.       if isempty(group)     % no group names
    
89.          group = gi;
    
90.       elseif (size(group,1) == m)
    
91.          group = group(gi,:);
    
92.       else
    
93.          error(message('stats:anova1:InputSizeMismatch'));
    
94.       end
    
95.    end
    
96. elseif ~isempty(group) && (size(group,1) ~= length(x))
    
97.    error(message('stats:anova1:InputSizeMismatch'));
    
98. end
    
99. 
    
100. % If X is a matrix GROUP is provided with the correct size, use GROUP
     
101. % to define groups and to label boxes
     
102. if (~isempty(group) && (length(x) < numel(x)) ...
     
103.                     && (size(x,2) == size(group,1)))
     
104.    named = 1;
     
105.    [gid,gnames] = grp2idx(group);
     
106.    gnames = gnames(gid);
     
107.    grouped = 0;
     
108. else
     
109.    named = 0;
     
110.    gnames = [];
     
111.    grouped = ~isempty(group);
     
112. end
     
113. 
     
114. if (grouped)
     
115.    % Single data vector and a separate grouping variable
     
116.    x = x(:);
     
117.    lx = length(x);
     
118.    if (lx ~= numel(x))
     
119.       error(message('stats:anova1:VectorRequired'))
     
120.    end
     
121.    nonan = ~isnan(x);
     
122.    x = x(nonan);
     
123. 
     
124.    % Convert group to indices 1,...,g and separate names
     
125.    group = group(nonan,:);
     
126.    [groupnum, gnames] = grp2idx(group);
     
127.    named = 1;
     
128. 
     
129.    % Remove NaN values
     
130.    nonan = ~isnan(groupnum);
     
131.    if (~all(nonan))
     
132.       groupnum = groupnum(nonan);
     
133.       x = x(nonan);
     
134.    end
     
135. 
     
136.    lx = length(x);
     
137.    xorig = x;                    % use uncentered version to make M
     
138.    groupnum = groupnum(:);
     
139.    maxi = size(gnames, 1);
     
140.    if isa(x,'single')
     
141.       xm = zeros(1,maxi,'single');
     
142.    else
     
143.       xm = zeros(1,maxi);
     
144.    end
     
145.    countx = xm;
     
146.    if (classical)
     
147.       mu = mean(x);
     
148.       x = x - mu;                % center to improve accuracy
     
149.       xr = x;
     
150.    else
     
151.       [xr,tieadj] = tiedrank(x);
     
152.    end
     
153.    
     
154.    for j = 1:maxi
     
155.       % Get group sizes and means
     
156.       k = find(groupnum == j);
     
157.       lk = length(k);
     
158.       countx(j) = lk;
     
159.       xm(j) = mean(xr(k));       % column means
     
160.    end
     
161. 
     
162.    gm = mean(xr);                      % grand mean
     
163.    df1 = sum(countx>0) - 1;            % Column degrees of freedom
     
164.    df2 = lx - df1 - 1;                 % Error degrees of freedom
     
165.    xc = xm - gm;                       % centered
     
166.    xc(countx==0) = 0;
     
167.    RSS = dot(countx, xc.^2);           % Regression Sum of Squares
     
168. else
     
169.    % Data in matrix form, no separate grouping variable
     
170.    [r,c] = size(x);
     
171.    lx = r * c;
     
172.    if (classical)
     
173.       xr = x;
     
174.       mu = mean(xr(:));
     
175.       xr = xr - mu;           % center to improve accuracy
     
176.    else
     
177.       [xr,tieadj] = tiedrank(x(:));
     
178.       xr = reshape(xr, size(x));
     
179.    end
     
180.    countx = repmat(r, 1, c);
     
181.    xorig = x;                 % save uncentered version for boxplot
     
182.    xm = mean(xr);             % column means
     
183.    gm = mean(xm);             % grand mean
     
184.    df1 = c-1;                 % Column degrees of freedom
     
185.    df2 = c*(r-1);             % Error degrees of freedom
     
186.    RSS = r*(xm - gm)*(xm-gm)';        % Regression Sum of Squares
     
187. end
     
188. 
     
189. TSS = (xr(:) - gm)'*(xr(:) - gm);  % Total Sum of Squares
     
190. SSE = TSS - RSS;                   % Error Sum of Squares
     
191. 
     
192. if (df2 > 0)
     
193.    mse = SSE/df2;
     
194. else
     
195.    mse = NaN;
     
196. end
     
197. 
     
198. if (classical)
     
199.    if (SSE~=0)
     
200.       F = (RSS/df1) / mse;
     
201.       p = fpval(F,df1,df2);        % Probability of F given equal means.
     
202.    elseif (RSS==0)                 % Constant Matrix case.
     
203.       F = NaN;
     
204.       p = NaN;
     
205.    else                            % Perfect fit case.
     
206.       F = Inf;
     
207.       p = 0;
     
208.    end
     
209. else
     
210.    F = (12 * RSS) / (lx * (lx+1));
     
211.    if (tieadj > 0)
     
212.       F = F / (1 - 2 * tieadj/(lx^3-lx));
     
213.    end
     
214.    p = chi2pval(F,df1);
     
215. end
     
216. 
     
217. 
     
218. Table=zeros(3,5);               %Formatting for ANOVA Table printout
     
219. Table(:,1)=[ RSS SSE TSS]';
     
220. Table(:,2)=[df1 df2 df1+df2]';
     
221. Table(:,3)=[ RSS/df1 mse Inf ]';
     
222. Table(:,4)=[ F Inf Inf ]';
     
223. Table(:,5)=[ p Inf Inf ]';
     
224. 
     
225. 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'))};
     
226. 
     
227. if (~classical)
     
228.    colheads{5} = getString(message('stats:anova1:ColHeadChisq'));
     
229.    colheads{6} = getString(message('stats:anova1:ColHeadProbGtChisq'));
     
230. end
     
231. rowheads = {getString(message('stats:anova1:RowHeadColumns')), getString(message('stats:anova1:RowHeadError')), getString(message('stats:anova1:RowHeadTotal'))};
     
232. if (grouped)
     
233.    rowheads{1} = getString(message('stats:anova1:RowHeadGroups'));
     
234. end
     
235. 
     
236. % Create cell array version of table
     
237. atab = num2cell(Table);
     
238. for i=1:size(atab,1)
     
239.    for j=1:size(atab,2)
     
240.       if (isinf(atab{i,j}))
     
241.          atab{i,j} = [];
     
242.       end
     
243.    end
     
244. end
     
245. atab = [rowheads' atab];
     
246. atab = [colheads; atab];
     
247. if (nargout > 1)
     
248.    anovatab = atab;
     
249. end
     
250. 
     
251. % Create output stats structure if requested, used by MULTCOMPARE
     
252. if (nargout > 2)
     
253.    if ~isempty(gnames)
     
254.       stats.gnames = gnames;
     
255.    else
     
256.       stats.gnames = strjust(num2str((1:length(xm))'),'left');
     
257.    end
     
258.    stats.n = countx;
     
259.    if (classical)
     
260.       stats.source = 'anova1';
     
261.       stats.means = xm + mu;
     
262.       stats.df = df2;
     
263.       stats.s = sqrt(mse);
     
264.    else
     
265.       stats.source = 'kruskalwallis';
     
266.       stats.meanranks = xm;
     
267.       stats.sumt = 2 * tieadj;
     
268.    end
     
269. end
     
270. 
     
271. if (~willdisplay), return; end
     
272. 
     
273. digits = [-1 -1 0 -1 2 4];
     
274. if (classical)
     
275.    wtitle = getString(message('stats:anova1:OnewayANOVA'));
     
276.    ttitle = getString(message('stats:anova1:ANOVATable'));
     
277. else
     
278.    wtitle = getString(message('stats:anova1:KruskalWallisOnewayANOVA'));
     
279.    ttitle = getString(message('stats:anova1:KruskalWallisANOVATable'));
     
280. end
     
281. tblfig = statdisptable(atab, wtitle, ttitle, '', digits);
     
282. set(tblfig,'tag','table');
     
283. 
     
284. f1 = figure('pos',get(gcf,'pos') + [0,-200,0,0],'tag','boxplot');
     
285. ax = axes('Parent',f1);
     
286. if (~grouped)
     
287.    boxplot(ax,xorig,'notch','on');
     
288. else
     
289.    boxplot(ax,xorig,groupnum,'notch','on');
     
290.    h = get(ax,'XLabel');
     
291.    set(h,'String',getString(message('stats:anova1:GroupNumber')));
     
292. end
     
293. 
     
294. % If there are group names, use them
     
295. if ~isempty(gnames)
     
296.    h = get(ax,'XLabel');
     
297.    if (named)
     
298.       set(h,'String','');
     
299.    end
     
300.    set(ax, 'xtick', (1:size(gnames,1)), 'xticklabel', gnames);
     
301. end
     

复制代码

森之张卫东

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。

远行的小船儿666

急需大神支招！

远行的小船儿666

有点难度

森之张卫东

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.

森之张卫东

同学，解决没，我想学习学习！

远行的小船儿666

路过的大神，如果可以给出解答，偶将不甚感激！

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