Analysis of variance

Analysis of variance

Analysis of variance, ANOVA, is a method of comparing means based on variations

from the mean. We begin by doing ANOVA the traditional way, but we will see that it is

a special form of the linear model discussed in the previous chapter. As such, it can be

approached in a unified way, with much of the previous work being applicable

11.1 One-way ANOVA

A one-way analysis of variance is a generalization of the t-test for two independent

samples, allowing us to compare means for several independent samples. Suppose we

have k populations of interest. From each we take a random sample. These samples are

independent if the knowledge of one sample does not effect the distribution of another.

Notationally, for the ith sample, let designate the sample values.

The one-way analysis of variance applies to normally distributed populations. Suppose

the mean of the ith population is μi and its standard deviation is σi. We use a σ if these are

all equivalent. A statistical model for the data with common standard deviation is

Yij=μi+εij,

where the error terms, εij, are independent with Normal(0, σ) distribution.

Example 11.1: Number of calories consumed by month Consider 15 subjects split

at random into three groups. Each group is assigned a month. For each group we record

the number of calories consumed on a randomly chosen day. Figure 11.1 shows the data.

We assume that the amounts consumed are normally distributed with common variance

but perhaps different means. From the figure, we see that there appears to be more

clustering around the means for each month than around the grand mean or mean for all

the data. This would indicate that the means may be different. Perhaps more calories are

consumed in the winter?

The goal of one-way analysis of variance is to decide whether the difference in the

sample means is indicative of a difference in the population means of each sample or is

attributable to sampling variation

This problem is approached as a significance test. Let the hypotheses be

H0: μ1=μ2=…=μk, HA: μi≠μj for at least one pair i and j.

A test statistic is formulated that compares the variations within a single group to

those among the groups.

Let be the grand mean, or mean of all the data, and the mean for the ith sample.

Then the total sum of squares is given by

Formula suma de cuadrados totales: sum(i)sum(j)[(yij-m(y..)}^2

This measures the amount of variation from the center of all the data.

An analysis of variance breaks this up into two sums:

: sum(i)sum(j)[(yij-m(y..)}^2=: n* sum(i)[(m(yi.-m(y..)}^2+: sum(i)sum(j)[(yij-m(yi.)}^2

Total=tratamientos+error

Ver table Anova para el calculo del estadigrafo de prueba Fc o F.observado o Fo

Example 11.2: Number of calories consumed by month, continued The one-way

test can be applied to the example on caloric intake. The two sums can be calculated

directly as follows:

> may=c(2166, 1568, 2233, 1882, 2019)

> sep=c(2279, 2075, 2131, 2009, 1793)

> dec=c(2226, 2154, 2583, 2010, 2190)

> xbar=mean(c(may,sep,dec))

> SST=5*((mean(may)-xbar)^2+(mean(sep)-xbar)^2+(mean(dec)-xbar)^2)

...