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The Logic and the Process of
Analysis of Variance
 Chapter 12 presents the general logic and basic formulas for the hypothesis testing procedure known as analysis of variance (ANOVA) .
 The purpose of ANOVA is much the same as the t tests presented in the preceding chapters: the goal is to determine whether the mean differences that are obtained for sample data are sufficiently large to justify a conclusion that there are mean differences between the populations from which the samples were obtained.

The Logic & Process of ANOVA
 The difference between ANOVA and the t tests is that ANOVA can be used in situations where there are two or more means being compared, whereas the t tests are limited to situations where only two means are involved.
 ANOVA is necessary to protect researchers from excessive risk of a Type I error in situations where a study is comparing more than two population means.

The Logic & Process of ANOVA


These situations would require a series of several t tests to evaluate all of the mean differences.
(Remember, a t test can compare only two means at a time.)
Although each t test can be done with a specific αlevel (risk of Type I error), the α-levels accumulate over a series of tests so that the final experimentwise α-level can be quite large.
 Note: While experiment-wise Type I error does accumulate, it is not a simple additive process.

The Logic & Process of ANOVA
 ANOVA allows researcher to evaluate all of the mean differences in a single hypothesis test using a single α-level and, thereby, keeps the risk of a Type I error under control no matter how many different means are being compared.
 Although ANOVA can be used in a variety of different research situations, this chapter presents only independent-measures designs involving only one independent variable.

Typical Research Design for
Analysis of Variance

ANOVA TERMS
 In ANOVA, the variable (independent or quasiindependent) that designates the groups being compared is called a factor .
 In ANOVA, the individual groups or treatment conditions that are used to make up a factor are called levels of the factor.
 Example: A study that looks at three different telephone conditions would have three levels of the factor.

Hypotheses for ANOVA
 There are multiple means involved and so the hypotheses can read as follows:
 H
0
: µ
1
= µ
2
= µ
3
 H
1
: There is at least one mean difference among the populations - - or




H
1
: µ
1
H
1
: µ
1
H
1
: µ
1
H
1
: µ
2
≠ µ
2
= µ
3
= µ
2
= µ
3
≠ µ
3
(All three means are different) - - or
, but µ
2 is different - - or
, but µ
3 is different - - or
, but µ
1 is different .

The Logic & Process of ANOVA:
Understanding the F-Ratio
 The test statistic for ANOVA is an F -ratio , which is a ratio of two sample variances. In the context of ANOVA, the sample variances are called mean squares, or MS values .
 The top of the F -ratio, MS between
, measures the size of mean differences between samples.
The bottom of the ratio, MS within
, measures the magnitude of differences that would be expected without any treatment effects.

The Logic & Process of ANOVA:
Understanding the F-Ratio
 Thus, the F -ratio has the same basic structure as the independent-measures t statistic presented in Chapter 10.

𝑂𝑏𝑡𝑎𝑖𝑛𝑒𝑑 𝑚𝑒𝑎𝑛 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 (𝑖𝑛𝑐𝑙𝑢𝑑𝑒 𝑡𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡 𝑒𝑓𝑓𝑒𝑐𝑡𝑠)
F =
𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒𝑠 𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑏𝑦 𝑐ℎ𝑎𝑛𝑐𝑒 (𝑤𝑖𝑡ℎ𝑜𝑢𝑡 𝑡𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡 𝑒𝑓𝑓𝑒𝑐𝑡𝑠)
=
𝑀𝑆 𝑏𝑒𝑡𝑤𝑒𝑒𝑛
𝑀𝑆 𝑤𝑖𝑡ℎ𝑖𝑛
 F =
𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝐵𝑒𝑡𝑤𝑒𝑒𝑛 𝑇𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡𝑠
𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑊𝑖𝑡ℎ𝑖𝑛 𝑇𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡𝑠
=
𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒𝑠 𝐼𝑛𝑐𝑙𝑢𝑑𝑖𝑛𝑔 𝐴𝑛𝑦 𝑇𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡 𝐸𝑓𝑓𝑒𝑐𝑡𝑠
𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒𝑠 𝑤𝑖𝑡ℎ 𝑁𝑜 𝑇𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡 𝐸𝑓𝑓𝑒𝑐𝑡𝑠


F =
𝑆𝑦𝑠𝑡𝑒𝑚𝑎𝑡𝑖𝑐 𝑇𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡 𝐸𝑓𝑓𝑒𝑐𝑡𝑠+𝑅𝑎𝑛𝑑𝑜𝑚,𝑈𝑛𝑠𝑦𝑠𝑡𝑒𝑚𝑎𝑡𝑖𝑐 𝑇𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡 𝐸𝑓𝑓𝑒𝑐𝑡𝑠
𝑅𝑎𝑛𝑑𝑜𝑚,𝑈𝑛𝑠𝑦𝑠𝑡𝑒𝑚𝑎𝑡𝑖𝑐 𝑇𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡 𝐸𝑓𝑓𝑒𝑐𝑡𝑠
NOTE: The denominator
F =
0 +𝑅𝑎𝑛𝑑𝑜𝑚,𝑈𝑛𝑠𝑦𝑠𝑡𝑒𝑚𝑎𝑡𝑖𝑐 𝑇𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡 𝐸𝑓𝑓𝑒𝑐𝑡𝑠
𝑅𝑎𝑛𝑑𝑜𝑚,𝑈𝑛𝑠𝑦𝑠𝑡𝑒𝑚𝑎𝑡𝑖𝑐 𝑇𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡 𝐸𝑓𝑓𝑒𝑐𝑡𝑠
= 1.00
for an F-Ratio is called an Error Term (Variance
Caused By Random
Differences)
 NOTE: Zero Treatment Effects Gives an F-Ratio of 1.00 in This Case.

The Logic & Process of ANOVA:
Understanding Total Variability

The Logic & Process of ANOVA:
Understanding F-Ratio Values
 A large value for the F -ratio indicates that the obtained sample mean differences are greater than would be expected if the treatments had no effect.
 Each of the sample variances, MS values, in the F -ratio is computed using the basic formula for sample variance: sample variance = S
2
SS
= ── df

The Logic & Process of ANOVA:
Within & Between Treatment
Variability
 To obtain the SS and df values , you must go through an analysis that separates the total variability for the entire set of data into two basic components:
 within-treatment variability (which will be the denominator) and
 between-treatment variability (which will become the numerator of the F -ratio).

The Logic & Process of ANOVA:
Within Treatment Variability
 The two components of the F -ratio. The first component is
Within-Treatment Variability:
 Within-Treatments Variability: MS within measures the size of the differences that exist inside each of the samples.
 Because all the individuals in a sample receive exactly the same treatment, any differences (or variance) within a sample cannot be caused by different treatments.
 Thus, these differences are caused by only one source:
 Chance or Error: The unpredictable differences that exist between individual scores are not caused by any systematic factors and are simply considered to be random chance or error.

The Logic & Process of ANOVA:
Between Treatment Variability
 The two components of the F -ratio. The second component is Between-Treatment Variability:
 Between-Treatments Variability: MS between measures the size of the differences between the sample means. For example, suppose that three treatments, each with a sample of n = 5 subjects, have means of M
1
= 1, M
2
= 2, and M
3
= 3.
 Notice that the three means are different; that is, they are variable.

The Logic & Process of ANOVA
 By computing the variance for the three means we can measure the size of the differences.
 Although it is possible to compute a variance for the set of sample means, it usually is easier to use the total, T , for each sample instead of the mean, and compute variance for the set of
T values.

The Logic & Process of ANOVA:
Two Sources of Variance
 Logically, the differences (or variance) between means can be caused by two sources:
 Treatment Effects: If the treatments have different effects, this could cause the mean for one treatment to be higher
(or lower) than the mean for another treatment.
 Chance or Sampling Error: If there is no treatment effect at all, you would still expect some differences between samples. Mean differences from one sample to another are an example of random, unsystematic sampling error.

The Logic & Process of ANOVA:
Back to the F-Ratio
 Considering these sources of variability, the structure of the F -ratio becomes: treatment effect + random differences
F = ────────────────────── random differences

The Logic & Process of ANOVA:
F-Ratio Values
 When the null hypothesis is true and there are no differences between treatments, the F -ratio is balanced.
 That is, when the "treatment effect" is zero , the top and bottom of the F -ratio are measuring the same variance.
 In this case, you should expect an F -ratio near
1.00. When the sample data produce an F ratio near 1.00
, we will conclude that there is no significant treatment effect .

The Logic & Process of ANOVA:
F-Ratio Values
 On the other hand, a large treatment effect will produce a large value for the F -ratio. Thus, when the sample data produce a large F -ratio we will reject the null hypothesis and conclude that there are significant differences between treatments.
 To determine whether an F -ratio is large enough to be significant, you must select an α-level, find the df values for the numerator and denominator of the F ratio, and consult the F -distribution table to find the critical value.

The Logic & Process of ANOVA:
Structure & Sequence of ANOVA
Calculations

Analysis of Variance & Post Tests
 The null hypothesis for ANOVA states that for the general population there are no mean differences among the treatments being compared; H
0
: μ
1
= μ
2
= μ
3
= . . .
 When the null hypothesis is rejected, the conclusion is that there are significant mean differences.
 However, the ANOVA simply establishes that differences exist, it does not indicate exactly which treatments are different.

ANOVA Calculations & Notations:
Learn Your Terms!!!





 k is used to identify the number of treatment conditions n is used to identify the number of scores in each treatment condition
N is used to identify the total number scores in the entire study
 N = kn , when samples are the same size
T stands for treatment total and is calculated by ∑X , which equals the sum of the scores for each treatment condition
G stands for the sum of all scores in a study (Grand Total)

Calculate by adding up all N scores or by adding treatment total (G=∑T)
You will also need SS and M for each sample, and ∑X 2 for the entire set of all scores.

ANOVA Calculations:
 SS total
= ∑ X 2 -
𝐺
2
𝑁
 SS within treatments
= ∑SS inside each treatment
= SS
1
+SS
2
+SS
3
 SS between treatments
= SS total
SS within treatments
 ALTERNATIVE FORMULA FOR SS between
 SS between treatments
= ∑
𝑇 𝑛
2

-
𝐺
2
Use formula to check math
𝑁
Note that we use treatment totals instead of treatment means here

ANOVA Calculations:
 Calculate Total Degrees of Freedom ( df total
)
 df total
= N – 1



Calculate Within-Treatment Degrees of Freedom
(df within
)
 df within
= ∑( n 1) = ∑df in each treatment or
 df within
= N – k
Calculate Between-Treatments Degrees of Freedom
(df between
)
 df between
= k – 1
Check to see if df total
= df within
+ df between

ANOVA Calculations:
 Again, in ANOVA, the term Mean Square or MS is used in the place of the term “variance.”
 Calculate MS between Treatments
 MS between
= s 2 between
=
𝑆𝑆 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑑𝑓 𝑏𝑒𝑡𝑤𝑒𝑒𝑛
 MS within
= s 2 within
=
𝑆𝑆 𝑤𝑖𝑡ℎ𝑖𝑛 𝑑𝑓 𝑤𝑖𝑡ℎ𝑖𝑛

ANOVA Calculations:

 s 2 between
F = ------------- =
𝑀𝑆 𝑏𝑒𝑡𝑤𝑒𝑒𝑛
𝑀𝑆 𝑤𝑖𝑡ℎ𝑖𝑛 s 2 within
To determine if this is a significant F-ratio, determine if it is larger than the critical value, which can be identified in Table
B4 (F-Distribution) on p. 705 of your textbook.



Look up critical value for critical region based upon degrees of freedom in the numerator (df denominator (df within
) between
) and the degrees of freedom in the
If F ≤ F crit
, then fail to reject H
0
If F > F crit
, then reject H
0

Measuring Effect Size for an
Analysis of Variance
 As with other hypothesis tests, an ANOVA evaluates the significance of the sample mean differences; that is, are the differences bigger than would be reasonable to expect just by chance.
 With large samples, however, it is possible for relatively small mean differences to be statistically significant.
 Thus, the hypothesis test does not necessarily provide information about the actual size of the mean differences.

Measuring Effect Size for an
Analysis of Variance (cont'd.)
 To supplement the hypothesis test, it is recommended that you calculate a measure of effect size.
 For an analysis of variance the common technique for measuring effect size is to compute the percentage of variance that is accounted for by the treatment effects.

Measuring Effect Size for an
Analysis of Variance (cont'd.)
 For the t statistics, this percentage was identified as r 2 , but in the context of ANOVA the percentage is identified as η 2 (the Greek letter eta, squared) .
 The formula for computing effect size is:
SS between treatments
η 2 = ───────────
SS total

Post Hoc Tests
 With more than two treatments, this creates a problem. Specifically, you must follow the
ANOVA with additional tests, called post hoc tests , to determine exactly which treatments are different and which are not.
 The Tukey’s HSD and Scheffé test are examples of post hoc tests.
 These tests are done after an ANOVA where H
0 is rejected with more than two treatment conditions.
The tests compare the treatments, two at a time, to test the significance of the mean differences.

Post Hoc Tests

Tukey’s HSD (Honestly Significant Difference)
 Compares any two treatment conditions to determine if there is a significance difference
 Formula:
 HSD = q
𝑀𝑆 𝑤𝑖𝑡ℎ𝑖𝑛 𝑛
 For values of q, please see Table B 5 on page 708 of your textbook.
 To locate the critical q, you need to know the number of treatments in the overall experiment (k), the df for
MS within
, and the alpha level (typical to use the same as the one selected for your ANOVA)

Post Hoc Tests
 Scheffe Test
 Uses extremely cautious approach to reducing
Type I Error
 One of the safest possible post-hoc tests because it has one of the smallest risks of Type I Error
 Uses an F-Ratio for two treatments
 Numerator = MS between using two treatments
 Denominator = MS within for overall ANOVA

Post Hoc Tests
 Scheffe Test
 Formulas
 SS between treatments
= ∑
𝑇 𝑛
2
-
𝐺
2
𝑁
(compute for 2 groups)
 df between
= k - 1 (k defined by overall ANOVA study)
 MS between
=
𝑆𝑆 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑑𝑓 𝑏𝑒𝑡𝑤𝑒𝑒𝑛
 F
A versus B
=
𝑀𝑆 𝑏𝑒𝑡𝑤𝑒𝑒𝑛
𝑀𝑆 𝑤𝑖𝑡ℎ𝑖𝑛
 Note: Error term here (MS within
) is from overall ANOVA
 Note: Critical F values found in Table B4 (p.705) with df between and df within for overall ANOVA

Post Hoc Tests
 Assumptions - Independent Measures ANOVA



1. The observations within each sample must be independent (see page 254)
2. The populations from which the samples are selected must be normal
3. The populations from which the samples are selected must have equal variances (homogeneity of variances)
 Do not be too concerned about assumptions concerning normality when you study large sample, but do be concerned if there is reason to believe that the assumption has been violated.
 If you suspect that the assumption of homogeneity of variances is violated, please use Hartley’s F-max test as discussed in Ch. 10.

Post Hoc Tests
 Assumptions - Independent Measures ANOVA
 If you suspect that one or more of the 3 key assumptions for ANOVA have been violated or you have large sample variance that prevents you from finding significant results:
 Transform original scores to ranks, which leaves you with ordinal data
 Use the Kruskal-Wallis Test (see Appendix E for more details)