CHAPTER OUTLINE 16.1 The F -Distribution
16.2 One-Way ANOVA: The Logic
In Chapter 10, you learned how to compare two population means, that is, the means of a single variable for two different populations. You studied various methods for making such comparisons, one being the pooledt-procedure.
Analysis of variance (ANOVA) provides methods for comparing several popu- lation means, that is, the means of a single variable for several populations. In this section and Section 16.3, we present the simplest kind of ANOVA,one-way analysis of variance.This type of ANOVA is calledone-way analysis of variance because it compares the means of a variable for populations that result from a classification by oneother variable, called thefactor.The possible values of the factor are referred to as thelevelsof the factor.
For example, suppose that you want to compare the mean energy consumption by households among the four regions of the United States. The variable under con- sideration is “energy consumption,” and there are four populations: households in the Northeast, Midwest, South, and West. The four populations result from classifying households in the United States by the factor “region,” whose levels are Northeast, Midwest, South, and West.
One-way analysis of variance is the generalization to more than two populations of the pooledt-procedure (i.e., both procedures give the same results when applied to two populations). As in the pooledt-procedure, we make the following assumptions.
KEY FACT 16.2 Assumptions (Conditions) for One-Way ANOVA
1. Simple random samples: The samples taken from the populations under consideration are simple random samples.
2. Independent samples: The samples taken from the populations under consideration are independent of one another.
3. Normal populations: For each population, the variable under consider- ation is normally distributed.
4. Equal standard deviations: The standard deviations of the variable un- der consideration are the same for all the populations.
Regarding Assumptions 1 and 2, we note that one-way ANOVA can also be used as a method for comparing several means with a designed experiment. In addition, like the pooledt-procedure, one-way ANOVA is robust to moderate violations of Assump- tion 3 (normal populations) and is also robust to moderate violations of Assumption 4 (equal standard deviations) provided the sample sizes are roughly equal.
How can the conditions of normal populations and equal standard deviations be checked? Normal probability plots of the sample data are effective in detecting gross violations of normality. Checking equal population standard deviations, however, can be difficult, especially when the sample sizes are small; as a rule of thumb, you can consider that condition met ifthe ratio of the largest to the smallest sample standard deviation is less than 2. We call that rule of thumb therule of 2.
Another way to assess the normality and equal-standard-deviations assumptions is to perform aresidual analysis.In ANOVA, the residualof an observation is the difference between the observation and the mean of the sample containing it. If the normality and equal-standard-deviations assumptions are met, a normal probability plot of (all) the residuals should be roughly linear. Moreover, a plot of the residuals against the sample means should fall roughly in a horizontal band centered and sym- metric about the horizontal axis.
The Logic Behind One-Way ANOVA
The reason for the wordvariance in analysis of variance is that the procedure for comparing the means analyzes the variation in the sample data. To examine how this
procedure works, let’s suppose that independent random samples are taken from two populations—say, Populations 1 and 2—with means μ1 andμ2. Further, let’s sup- pose that the means of the two samples arex¯1=20 andx¯2=25. Can we reasonably conclude from these statistics thatμ1 =μ2, that is, that the population means are dif- ferent? To answer this question, we must consider the variation within the samples.
Suppose, for instance, that the sample data are as displayed in Table 16.1 and depicted in Fig. 16.3.
TABLE 16.1
Sample data from Populations 1 and 2 Sample from
21 37 11 20 8 23
Population 1 Sample from
24 31 29 40 9 17
Population 2
FIGURE 16.3 Dotplots for sample data in Table 16.1
0
Sample from Population 1 (x–1= 20)
Sample from Population 2 (x–2= 25)
10 20 30 40 50
For these two samples, x¯1=20 and x¯2=25. But here we cannot infer that μ1 =μ2 because it is not clear whether the difference between the sample means is due to a difference between the population means or to the variation within the populations.
? What Does It Mean?
Intuitively speaking, because the variation between the sample means is not large relative to the variation within the samples, we cannot conclude thatμ1 =μ2.
However, suppose that the sample data are as displayed in Table 16.2 and depicted in Fig. 16.4.
TABLE 16.2
Sample data from Populations 1 and 2 Sample from
21 21 20 18 20 20
Population 1 Sample from
25 28 25 24 24 24
Population 2
FIGURE 16.4 Dotplots for sample data in Table 16.2
0
Sample from Population 1 (x–1 = 20)
Sample from Population 2 (x–2 = 25)
10 20 30 40 50
Again, for these two samples, x¯1=20 andx¯2=25. But this time, wecaninfer thatμ1 =μ2 because it seems clear that the difference between the sample means is due to a difference between the population means, not to the variation within the populations.
? What Does It Mean?
Intuitively speaking, because the variation between the sample means is large relative to the variation within the samples, we can conclude thatμ1 =μ2.
720 CHAPTER 16 Analysis of Variance (ANOVA)
The preceding two illustrations reveal the basic idea for performing a one-way analysis of variance to compare the means of several populations:
1. Take independent simple random samples from the populations.
2. Compute the sample means.
3. If the variation among the sample means is large relative to the variation within the samples, conclude that the means of the populations are not all equal.
To make this process precise, we need quantitative measures of the variation among the sample means and the variation within the samples. We also need an ob- jective method for deciding whether the variation among the sample means is large relative to the variation within the samples.
Mean Squares and F -Statistic in One-Way ANOVA
As before, when dealing with several populations, we use subscripts on parameters and statistics. Thus, for Population j, we useμj,x¯j,sj, andnjto denote the population mean, sample mean, sample standard deviation, and sample size, respectively.
We first consider the measure of variation among the sample means. In hypothe- sis tests for two population means, we measure the variation between the two sample means by calculating their difference,x¯1− ¯x2. When more than two populations are involved, we cannot measure the variation among the sample means simply by taking a difference. However, we can measure that variation by computing the standard de- viation or variance of the sample means or by computing any descriptive statistic that measures variation.
In one-way ANOVA, we measure the variation among the sample means by a weighted average of their squared deviations about the mean, x¯, of all the sample data. That measure of variation is called thetreatment mean square,MSTR,and is defined as
MSTR= SSTR k−1,
wherekdenotes the number of populations being sampled and
SSTR=n1(x¯1− ¯x)2+n2(x¯2− ¯x)2+ ã ã ã +nk(x¯k− ¯x)2. The quantitySSTRis called thetreatment sum of squares.
? What Does It Mean?
MSTRmeasures the variation among the sample means.
We note thatMSTRis similar to the sample variance of the sample means. In fact, if all the sample sizes are identical, thenMSTRequals that common sample size times the sample variance of the sample means.
? What Does It Mean?
MSEmeasures the variation within the samples.
Next we consider the measure of variation within the samples. This measure is the pooled estimate of the common population variance,σ2. It is called theerror mean square,MSE,and is defined as
MSE= SSE n−k, wherendenotes the total number of observations and
SSE=(n1−1)s12+(n2−1)s22+ ã ã ã +(nk−1)sk2. The quantitySSEis called theerror sum of squares.† ‡
? What Does It Mean?
TheF-statistic compares the variation among the sample means to the variation within the samples.
Finally, we consider how to compare the variation among the sample means,MSTR, to the variation within the samples,MSE. To do so, we use the statis- ticF=MSTR/MSE, which we refer to as theF-statistic.Large values ofFindicate
†The termstreatmentanderrorarose from the fact that many ANOVA techniques were first developed to analyze agricultural experiments. In any case, the treatments refer to the different populations, and the errors pertain to the variation within the populations.
‡For two populations (i.e.,k=2),MSEis the pooled variance,s2p, defined in Section 10.2 on page 440.
that the variation among the sample means is large relative to the variation within the samples and hence that the null hypothesis of equal population means should be rejected.