Starting out in Statistics by de Winter Patricia Cahusac Peter M. B

Author:de Winter, Patricia, Cahusac, Peter M. B. [de Winter, Patricia, Cahusac, Peter M. B.]
Language: eng
Format: epub
Published: 0101-01-01T00:00:00+00:00

ina

25

4.24

3.38

2.99

2.76

2.60

m

26

4.22

3.37

2.98

2.74

2.59

eno

27

4.21

3.35

2.96

2.73

2.57

D

28

4.20

3.34

2.95

2.71

2.56

29

4.18

3.33

2.93

2.70

2.54

The shaded value is the critical value of F for the one-way ANOVA performed on data for the effect of placebo pills on blood pressure.

software) and the magnitude of the critical value of F for at 𝛼 = .05 varies

with the degrees of freedom, both those for the treatments and error. As we

anticipate that F will increase if the null hypothesis is false, the numerator is

always the treatment MS and the denominator the error MS. Unlike the t-test

we do not have a choice of one-tailed or two-tailed tests depending on the null

hypothesis being tested, the null hypothesis is already determined for us.

The pertinent section of an F table for 𝛼 = .05 below shows that with our

degrees of freedom (3 for the numerator or treatment MS, and 28 for the

denominator or error MS) the critical value of F is 2.95 (Table 6.5). So in this

case to reject the null hypothesis and conclude that the treatments had an

effect we require approximately three times the variance associated with the

treatments (fixed factor) compared with the error. Our calculated F ratio is

much bigger than this at 24.64 so p < .05 and we would reject the null hypothe-

sis that every sample is drawn from a population(s) with the same population

mean. Note that this does not tells us whether the samples are drawn from two

or more populations with the same population mean or whether they are all

samples from the same population with a single population mean. Note also

that the smaller the sample size, and therefore error degrees of freedom, the

larger the critical value of F. For smaller samples the effect of the treatments

on the variance has to be much larger to attain statistical significance. We

could report this F ratio as F 3,28 = 24.64, where by convention the subscripted numbers are the treatment and error degrees of freedom respectively.

The results of our ANOVA tell us that there is an effect of treatment, at

least one of the placebos alters blood pressure, but it does not tell us which

particular treatments are different from each other. To find this out we need

to compare the groups using a comparison test. There are several tests that

can be used in conjunction with an ANOVA and which we use depends

partly on the comparisons we wish to make. We will briefly describe how to

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CH 6

COMPARING GROUPS USING t-TESTS AND ANOVA

interpret two commonly used tests, Tukey’s and Dunnett’s. Both are based on

the t- test, but unlike Bonferroni’s correction, these tests correct for multiple

comparisons by adjusting 𝛼 in a less conservative way.

6.11.1 Tukey’s honest significant difference test

Tukey’s post hoc test can compare all possible pairs of means, where the over-

all (family) value for 𝛼 is .05 and the value of 𝛼 for each comparison is determined by the number of treatments, k, the error degrees of freedom, v, and a

test statistic called q. As long as the test assumptions are met (homogeneity

of variances, independence and normality) it is a robust test that maintains 𝛼

at intended values.

The output from Minitab for a one-way ANOVA is reported in Box 6.

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