[R] null hypothesis for two-way anova
Axel Rasche
axel_rasche at gmx.de
Sun Aug 5 13:00:53 CEST 2007
Dear R community,
Confused by some of my lab results I ask for the definition of the null
hypothesis of a two-way analysis of variance in R (anova() and aov()).
Starting with the following model
y = a_i + b_j , i in A and j in B
is the tested null hypothesis
H_0: a_i = 0 for all i in A
or
H_0: a_m = a_n for any m and n in A?
Consequently the same questions for interaction effects. Starting with
the model
y = a_i + b_j + f_ij , i in A and j in B
is the tested null hypothesis
H_0: f_ij = 0 for all i in A and j in B
or
H_0: f_ij = f_mn for any i and m in A and j and n in B?
More specific to R the questions is formulated as R code in the
appendix. Unfortunately I did not find this in the documentation.
Thanks in advance,
Axel
Appendix:
=========
> A = factor(c(rep(1, 10), rep(0, 5)))
> B = factor(c(rep(0, 5), rep(1, 10)))
> y = c(rnorm(n = 5, mean = 2, sd = 1), rnorm(n = 5, mean = 6, sd = 1),
rnorm(n = 5, mean = 4, sd = 1))
> anova(lm(formula = y ~ 1 + A + B, data = data.frame(y, A, B)))
Analysis of Variance Table
Response: y
Df Sum Sq Mean Sq F value Pr(>F)
A 1 5.041 5.041 5.5601 0.03618 *
B 1 58.643 58.643 64.6851 3.56e-06 ***
Residuals 12 10.879 0.907
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Which null hypothesis has been tested for the P-value in the fifth
column of the output ("Pr(>F)")? For example the P-value 0.03618 for
factor A.
> y = c(rnorm(n = 5, mean = 2, sd = 1), rnorm(n = 5, mean = 10, sd =
1), rnorm(n = 5, mean = 4, sd = 1))
> anova(lm(formula = y ~ 1 + A + B + A:B, data = data.frame(y, A, B)))
Analysis of Variance Table
Response: y
Df Sum Sq Mean Sq F value Pr(>F)
A 1 9.235 9.235 5.3243 0.03966 *
B 1 129.623 129.623 74.7321 1.686e-06 ***
Residuals 12 20.814 1.734
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
How do I draw a P-value for the interaction effect for the anova?
> sessionInfo()
Version 2.3.1 (2006-06-01)
i386-pc-mingw32
attached base packages:
[1] "methods" "stats" "graphics" "grDevices" "utils"
"datasets" "base"
--
*******************************************
Dipl. Math. ETH Axel Rasche
Max-Planck-Institute for Molecular Genetics
Department Lehrach (Vertebrate Genomics)
Ihnestrasse 63-73
D-14195 Berlin-Dahlem
GERMANY
Tel. ++49-30-8413-1647
Fax ++49-30-8413-1128
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