Matlab统计工具箱1

Matlab统计工具箱

Getting Started Toolbox functions and GUIs Organizing Data Data arrays and groups Descriptive Statistics Data summaries Statistical Visualization Data patterns and trends Probability Distributions Modeling data frequency Random Number Generation Topics in random number generation Hypothesis Tests Inferences from data Analysis of Variance Modeling data variance Regression Analysis Continuous data models Multivariate Methods Visualization and reduction Cluster Analysis Identifying data categories Classification Categorical data models Markov Models Stochastic data models Design of Experiments Systematic data collection Statistical Process Control Production monitoring

方差分析 ANOVA Operations anova1 anova2 anovan aoctool dummyvar friedman kruskalwallis manova1 manovacluster multcompare 方差分析模型

方差分析是分析试验（或观测）数据的一种统计方法。在工农业生产和科学研究中，经常要分析各种因素及因素之间的交互作用对研究对象某些指标值的影 1

One-way analysis of variance Two-way analysis of variance N-way analysis of variance Interactive analysis of covariance Create dummy variables Friedman's test Kruskal-Wallis test

One-way multivariate analysis of variance

Dendrogram of group mean clusters following MANOVA Multiple comparison test

响。这时需要用到方差分析。利用方差分析，我们能推断哪些因素对所考察指标的影响是显著的，哪些是不显著的。

方差分析包括了：单因素方差分析、双因素方差分析和多因素方差分析。 单因素方差分析

单因素方差分析问题：某一因素（A）对结果（Y）的影响分析。一般情况下，假设因素A有r个不同的水平。单因素方差分析模型如下：

??yij????i??ij,1?i?r,1?j?ni ?2???ij~N?0,??,且相互独立.其中yij表示Y在第i组（水平）第j个观察值，?为总的（Y的 ）均值，?i为第

i组所的均值且满足

??ii?0，?ij表示第i组第j个样本的误差，ni为第i组的数

据个数且?ini?n。方差分析的目的是要基于数据分析确定这r组数据的均值有没有显著性差异，即假设检验问题：

H0:?1??2????r?0; v.s. H1:?1,?2,?,?r不全相等。 （1）

原假设表明因素A对Y的值没有影响。 格式

p = anova1(X)

p = anova1(X,group)

p = anova1(X,group,displayopt) [p,table] = anova1(...)

[p,table,stats] = anova1(...)

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实例 1

Create X with columns that are constants plus random normal disturbances with mean zero and standard deviation one: X = meshgrid(1:5) X =

1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

X = X + normrnd(0,1,5,5) X =

1.3550 2.0662 2.4688 5.9447 5.4897 2.0693 1.7611 1.4864 4.8826 6.3222 2.1919 0.7276 3.1905 4.8768 4.6841 2.7620 1.8179 3.9506 4.4678 4.9291 -0.3626 1.1685 3.5742 2.1945 5.9465 Perform one-way ANOVA: p = anova1(X) p =

7.9370e-006

the very small p value indicates that differences between column means are highly 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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实例 2

The following example is from a study of the strength of structural beams in Hogg. The vector strength measures deflections of beams in thousandths of an inch under 3,000 pounds of force. The vector alloy identifies each beam as steel (铁'st'), alloy 1 (合金1'al1'), or alloy 2 (合金2'al2'). (Although alloy is sorted in this example, grouping variables do not need to be sorted.) The null hypothesis is that steel beams are equal in strength to beams made of the two more expensive alloys. 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'}; p = anova1(strength,alloy) p =

1.5264e-004 双因素方差分析模型

双因素方差分析问题。一般地，不妨假设因素A有r个水平，因素B有s个水平。在双因素（多因素方差）分析中，不仅要考虑每个因素的影响，有时还要考虑因素之间的交互作用。下面是考虑有交互作用的双因素方差分析的模型如下：

??yijk????i??j??ij??ijk,1?i?r,1?j?s,1?k?s ?2???ijk~N?0,??,且相互独立.其中yijk表示A因素第i组B因素第j组第k个样本的观察值，?为总的均值，?i为A因素第i组的均值，?j为B因素第j组的均值，?ij表示A因素第i组B因素

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