3Single-Factor ANOVASingle-factor ANOVA focuses on a comparison of more than two population or treatment means. Letl = the number of populations or treatments being compared1 = the mean of population 1 or the true average response when treatment 1 is applied.I = the mean of population I or the true average response when treatment I is applied
4Single-Factor ANOVA The relevant hypotheses are H0: 1 = 2 = ··· = I versusHa: at least two the of the i’s are differentIf I = 4, H0 is true only if all four i’s are identical. Ha would be true, for example, if 1 = 2 3 = 4, if 1 = 3 = 4 2, or if all four i’s differ from one another.
5The Idea of ANOVAThe sample means for the three samples are the same for each set.The variation among sample means for (a) is identical to (b).The variation among the individuals within the three samples is much less for (b).CONCLUSION: the samples in (b) contain a larger amount of variation among the sample means relative to the amount of variation within the samples, so ANOVA will find more significant differences among the means in (b)assuming equal sample sizes here for (a) and (b).Note: larger samples will find more significant differences.
6Comparing Several Means Do SUVs, trucks and midsize cars have same gas mileage?Response variable: gas mileage (mpg)Groups: vehicle classification31 midsize cars31 SUVs14 standard-size pickup trucksData from the Environmental Protection Agency’s Model Year 2003 Fuel Economy Guide,
7Comparing Several Means Midsize:SUV:Pickup:Mean gas mileage for SUVs and pickups appears less than for midsize cars.Are these differences statistically significant?
8Comparing Several Means Midsize:SUV:Pickup:Null hypothesis: The true means (for gas mileage) are the same for all groups (the three vehicle classifications).We could look at separate t tests to compare each pair of means to see if they are different:vs , vs , & vsH0: μ1 = μ H0: μ1 = μ H0: μ2 = μ3However, this gives rise to the problem of multiple comparisons!
9The One-Way ANOVA Model Random sampling always produces chance variations. Any “factor effect” would thus show up in our data as the factor-driven differences plus chance variations (“error”):Data = fit + residualThe one-way ANOVA model analyzes situations where chance variations are normally distributed N(0,σ) such that:
10The ANOVA F TestTo determine statistical significance, we need a test statistic that we can calculate:The ANOVA F StatisticThe analysis of variance F statistic for testing the equality of several means has this form:Difference in means small relative to overall variabilityDifference in means large relative to overall variability F tends to be small F tends to be largeLarger F-values typically yield more significant results. How large depends on the degrees of freedom (I− 1 and N− I).
11The ANOVA F TestThe measures of variation in the numerator and denominator are mean squares:Numerator: Mean Square for Treatments (MSTr)Denominator: Mean Square for Error (MSE)
12NotationThe individual sample means will be denoted by X1, X2, . . ., XI.That is,for i=1,…,ISimilarly, the average of all N observations, called the grand mean, is
13NotationAdditionally, let , denote the sample variances:for i=1,…,I
14The ANOVA TableThe computations are often summarized in a tabular format, called an ANOVA table in below Table.Tables produced by statistical software customarily include a P-value column to the right of f.Source of variationSum of squaresDfMean squareFP valueF critTreatmentsSSTrI -1SSTr/(I -1)MSTr/MSETail area above FValue of F for aErrorSSEN – ISSE/(N – I)TotalSST=SSTr+SSEN – 1An ANOVA Table
15F Distributions and the F Test Both v1 and v2 are positive integers. Figure 10.3 pictures an F density curve and the corresponding upper-tail critical value Appendix Table A.9 gives these critical values for = .10, .05, .01, and .001.Values of v1 are identified with different columns of the table, and the rows are labeled with various values of v2.An F density curve and critical valueFigure 10.3
16Nematodes and plant growth Do nematodes affect plant growth? A botanist prepares 16 identical planting pots and adds different numbers of nematodes into the pots. Seedling growth (in mm) is recorded two weeks later.Hypotheses: All mi are the same (H0)versus not All mi are the same (Ha)Any attempt to modify the 3.2, the third seedling for 10,000 nematodes whitens the background.
17Output for the one-way ANOVA numeratordenominatorHere, the calculated F-value (12.08) is larger than Fcritical (3.49) for a=0.05.Thus, the test is significant at a 5% Not all mean seedling lengths are the same; the number of nematodes is an influential factor.
18Using F-tableThe F distribution is asymmetrical and has two distinct degrees of freedom. This was discovered by Fisher, hence the label “F.”Once again, what we do is calculate the value of F for our sample data and then look up the corresponding area under the curve in F-Table.
19Fcritical for a 5% is 3.49F = > 10.80Thus p< 0.001