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In , the two-way analysis of variance (ANOVA) is an extension of the
that examines the influence of two different
. The two-way ANOVA not only aims at assessing the
of each independent variable but also if there is any
between them.
mentions the two-way ANOVA in his celebrated book from 1925,
(chapters 7 and 8). In 1934,
published procedures for the unbalanced case. Since then, an extensive literature has been produced, reviewed in 1993 by Fujikoshi. In 2005,
proposed a different approach of ANOVA, viewed as a .
Let us imagine a
for which a dependent variable may be influenced by two factors which are potential sources of variation. The first factor has
levels () and the second has
levels (). Each combination
defines a treatment, for a total of
treatments. We represent the number of replicates for treatment
by , and let
be the index of the replicate in this treatment ().
From these data, we can build a , where
and , and the total number of replicates is equal to .
is balanced if each treatment has the same number of replicates, . In such a case, the design is also said to be orthogonal, allowing to fully distinguish the effects of both factors. We hence can write , and .
Upon observing variation among all
data points, for instance via a , " may be used to describe such variation". Let us hence denote by
which observed value
is the -th measure for treatment . The two-way ANOVA models all these variables as varying
around a mean, , with a constant variance,
Specifically, the mean of the response variable is modeled as a
of the explanatory variables:
is the grand mean,
is the additive main effect of level
from the first factor (i-th row in the contigency table),
is the additive main effect of level
from the second factor (j-th column in the contigency table) and
is the non-additive interaction effect of treatment
from both factors (cell at row i and column j in the contigency table).
An other, equivalent way of describing the two-way ANOVA is by mentioning that, besides the variation explained by the factors, there remains some . This amount of unexplained variation is handled via the introduction of one random variable per data point, , called . These
random variables are seen as deviations from the means, and are assumed to be independent and normally distributed:
Following Gelman and Hill, the assumptions of the ANOVA, and more generally the , are, in decreasing order of importance:
the data points are relevant with respect to the scientific question
the mean of the response variable is influenced additively (if not interaction term) and lin
the errors ha
the errors are normally distributed.
of parameters, we can add the following "sum-to-zero" constraints:
In the classical approach,
(that the factors have no effect) is achieved via their
which requires calculating .
Testing if the interaction term is significant can be difficult because of the potentially-large number of .
(Includes a one-way ANOVA example)
(18 April 2008). . .  .
Yates, Frank (March 1934). . Journal of the American Statistical Association (American Statistical Association) 29 (185): 51–66 2014.
Fujikoshi, Yasunori (1993). "Two-way ANOVA models with unbalanced data". Discrete Mathematics (Elsevier) 116 (1): 315–334. :.
Gelman, Andrew (February 2005). "Analysis of variance? why it is more important than ever". The Annals of Statistics 33 (1): 1–53. :.
Kass, Robert E (1 February 2011). .
() 26 (1): 1–9. :.
Gelman, A Hill, Jennifer (18 December 2006). . . pp. 45–46.  .
Yi-An Ko et al. (September 2013). "Novel Likelihood Ratio Tests for Screening Gene-Gene and Gene-Environment Interactions with Unbalanced Repeated-Measures Data". Genetic epidemiology 37 (6): 581–591. :.
: Hidden categories:求助SPSS如何使用two way ANOVA想要分析药物A和药物B对小鼠的作用,分四组,给生理盐水,给药物A,给药物B,给药物A+B,求助如何用SPSS分析two way ANOVA,_百度作业帮
求助SPSS如何使用two way ANOVA想要分析药物A和药物B对小鼠的作用,分四组,给生理盐水,给药物A,给药物B,给药物A+B,求助如何用SPSS分析two way ANOVA,
如果要做two way anova,那你这个研究设计就有问题了估计你数据还没收集好,实验还没开始做,你现在还有时间去重新设计的我替别人做这类的数据分析蛮多的
方差分析就行啊我还是很擅长spss的
能不能详细点，不是太明白方差分析与&R&软件
以下内容译自网络，参考了《The R
Book》。仅用于学习、交流。
models 方差分析模型
R 提供了模型句法用于处理各种情况——多个因子的情况，因子之间有交互作用的情况。
当进行 anova 时，我们有一个因（应）变量（dependent
variable）和多个解释变量（explanatory factors）。
可以建立一个通式：dependent ~ explanatory1...
explanatory2...
模型有多种形式：
y is explained by x1 only, a one-way anova
y ~ x1 + x2
y is explained by x1 and x2, a two-way anova
y ~ x1 + x2 + x3
y is explained by x1, x2 and x3, a 3-way anova
y ~ x1 * x2
y is explained by x1, x2 and also by the interaction between them
2 ANOVA Step
by Step 方差分析的步骤
首先创建你的数据文件。在工作表中每列代表一个变量，每行代表一个重复。第一行包含变量名。保存为.CSV格式文件。
把文件读入到 R 中，赋值给一个变量，这样便于选择、调用你的文件。
read.csv(file.choose())
允许 R 读数据文件中的某个或某几个变量
attach(your.data)
确定模型，运行anova分析
= aov(dependent ~
explanatory)
summary(your.aov)
若 anova 结果差异显著，就进行后续的两两（成对）检验——所谓的multiple
comparisons
TukeyHSD(your.aov)
函数 pairwise.t.test(
)也可用于成对检验，具体用法见该函数的技术文档，不再累述。
另一个有用的 R PACKAGE：multcomp，里面的
comand：glht，有空看看吧，管用。
Simultaneous inference is a common problem in many areas of
application. If
multiple null hypotheses are tested simultaneously, the probability
of rejecting erroneously
at least one of them increases beyond the pre-specified
significance level.
Simultaneous inference procedures have to be used which adjust for
multiplicity and
thus control the overall type I error rate. In this paper we
describe simultaneous inference
procedures in general parametric models, where the experimental
questions are
specified through a linear combination of elemental model
parameters. The framework
described here is quite general and extends the canonical theory of
comparison procedures in ANOVA models to linear regression
problems, generalized
linear models, linear mixed effects models, the Cox model, robust
linear models, etc.
Several examples using a variety of different statistical models
illustrate the breadth
of the results. For the analyses we use the R add-on package
multcomp, which
a convenient interface to the general approach adopted here.
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——Torsten Hothorn, Frank Bretz & Peter
Westfall,2008
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