An Example of ANOVA using R

 In its simplest form ANOVA provides a statistical test of whether or not the means of several groups are all equal, and therefore generalizes t-test to more than two groups. The t-test tells us if the variation between two groups is "significant".

(方差分析ANOVA又称变异数分析或F检验,其目的是推断两组或多组资料的总体均数是否相同，检验两个或多个样本均数的差异是否有统计学意义)

An Example of ANOVA using R

by EV Nordheim, MK Clayton & BS Yandell, November 11, 2003

In class we handed out ”An Example of ANOVA”. Below we redo the example using R.
There are three groups with seven observations per group. We denote group i values by yi:
> y1 = c(18.2, 20.1, 17.6, 16.8, 18.8, 19.7, 19.1)
> y2 = c(17.4, 18.7, 19.1, 16.4, 15.9, 18.4, 17.7)
> y3 = c(15.2, 18.8, 17.7, 16.5, 15.9, 17.1, 16.7)

Now we combine them into one long vector, with a second vector, group, identifying group
membership:

> y = c(y1, y2, y3)
> n = rep(7, 3)
> n
[1] 7 7 7
> group = rep(1:3, n)
> group
[1] 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 3 3

Here are summaries by group and for the combined data. First we show stem-leaf diagrams.

> tmp = tapply(y, group, stem)
The decimal point is at the |
16 | 8
17 | 6
18 | 28
19 | 17
20 | 1
The decimal point is at the |
15 | 9
16 | 4
17 | 47
18 | 47
19 | 1
The decimal point is at the |
15 | 29
16 | 57
17 | 17
18 | 8
> stem(y)
The decimal point is at the |
15 | 299
16 | 4578
17 | 14677
18 | 24788
19 | 117
20 | 1

Now we show summary statistics by group and overall. We locally define a temporary
function, tmpfn, to make this easier.

> tmpfn = function(x) c(sum = sum(x), mean = mean(x), var = var(x),n = length(x))
> tapply(y, group, tmpfn)
$`1`
       sum       mean        var          n
130.300000  18.614286   1.358095   7.000000
$`2`
       sum       mean        var          n
123.600000  17.657143   1.409524   7.000000
$`3`
       sum       mean        var          n
117.900000  16.842857   1.392857   7.000000
> tmpfn(y)
       sum       mean        var          n
371.800000  17.704762   1.798476  21.000000

While we could show you how to use R to mimic the computation of SS by hand, it is
more natural to go directly to the ANOVA table. See Appendix 11 for other examples of the
use of R commands for ANOVA.

> data = data.frame(y = y, group = factor(group))
> fit = lm(y ~ group, data)
> anova(fit)
Analysis of Variance Table
Response: y
          Df Sum Sq Mean Sq F value  Pr(>F) 
group      2 11.007  5.5033  3.9683 0.03735 *
Residuals 18 24.963  1.3868                 
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

The anova(fit) object can be used for other computations on the handout and in class.
For instance, the tabled F values can be found by the following. First we extract the treatment
and error degrees of freedom. Then we use qt to get the tabled F values.

> df = anova(fit)[, "Df"]
> names(df) = c("trt", "err")
> alpha = c(0.05, 0.01)
> qf(alpha, df["trt"], df["err"], lower.tail = FALSE)
[1] 3.554557 6.012905

A confidence interval on the pooled variance can be computed as well using the anova(fit)
object. First we get the residual sum of squares, SSTrt, then we divide by the appropriate
chi-square tabled values.

> anova(fit)["Residuals", "Sum Sq"]
[1] 24.96286
> anova(fit)["Residuals", "Sum Sq"]/qchisq(c(0.025, 0.975), 18,lower.tail = FALSE)
[1] 0.7918086 3.0328790