For this week’s forum, locate a research article that used an ANOVA *and a*post-hoc analysis in their methods/results.. Please attach the article or provide a link so classmates can view it. Describe how the ANOVA and post-hoc analysis was used to answer the research (i.e. what did it compare) and where the group differences were found in the post-hoc. Please share the levels of the dependent and independent variable(s). As always, please select an article that no one else has used or discussed for this week.

SPHS500 – Statistics for

Sports and Health Sciences

Week 5

Objectives

• Analysis of Variance (ANOVA)

• One-way ANOVA

• Repeated Measures ANOVA

• Interpreting test results

• Post-hoc tests

ANOVA

• ANOVA is used when more than two groups are compared

• In order to conduct an ANOVA, several assumptions must be made:

• The population from which the samples are drawn are normally distributed

• The populations from which the samples are drawn have equal variances

• Some ask – why not multiple t-tests?

• That increases the likelihood of a Type I error

Decision Table for Inferential Statistic

Normal deviate z-test

One sample t-test

Independent t-test

Dependent t-test

One-way ANOVA and

post hoc test

Not covered

Factorial ANOVA

Not covered

Chi-square goodness of fit

Chi-square independence

Pearson r

Not covered

One Way (Between subjects) ANOVA

• Single independent variable

• IV = depression level

• 3 levels (medication, counseling, diet supplement)

ANOVA

Analysis of Variance:

• Used with a quantitative Dependent Variable

• Comparison among 3 or more groups

• A Between subjects (one-way) ANOVA is when there are 3 different groups

• Freshman, sophomore, junior, senior

• Categories of an Independent Variable (Low, Medium, High)

• A Repeated measures ANOVA is when the same group is compared at

different time points (measures are repeated)

• Baseline, 3-month, 6-month, 9-month, post

Types of variance in an ANOVA:

1. Between-groups variance

• Variance due to differences between the groups (i.e., between the

factors/levels).

2. Within-groups variance

• Variance due to differences within the groups (i.e., between the individuals).

3. Total variability

• The difference of each score from the grand mean (actually the squared

difference)

The F-Ratio

Between-subjects variability

F=

Within-subjects variability

Treatment effect + Indiv. Diff. + Exper. Error

F=

Indiv. Diff. + Exper. Error

The F-ratio

F=

Between-subjects variability

Within-subjects variability

Variance = Mean Square (MS) in ANOVA since variance is the

mean of the squared deviation scores

• For each source find:

• Sum of squares (Between-groups, Within-groups, Total)

• Degrees of freedom

• Compute Mean Sum of Squares

• F = MSB / MSW = (SSB / dfB) / (SSW / dfW)

Let’s walk through an example:

Do mini-golfers play better (i.e. lower score) using

a white, blue or red ball (3 different groups)?

Red

White

Blue

3

6

4

5

3

8

2

7

7

3

1

5

4

4

5

• What are your null and alternate hypotheses?

Our hypotheses

• Null hypothesis – there is no difference between the

means for the three different groups

• Ho: μ1 = μ2 = μ3

• example, Group 1 = Group 2 = Group 3

• Alternate hypothesis

• ANOVA is omnibus – tests for an overall difference between

means (neither one nor two-tailed)

• HA: X1 ≠ X2 ≠ … ≠ Xk

• HA: at least one of the means is different

Compute Sum of Squares

Red Ball (n1=5)

White Ball (n2=5)

Blue Ball (n3=10)

TOTAL (N=15) (k=3)

3

6

4

5

3

8

2

7

7

3

1

5

4

4

5

X =3.4

X = 4.2

X = 5.8

∑X = 17

∑X = 21

∑X = 29

∑(∑X) = 67

∑(X2) = 63

∑(X2) = 111

∑(X2) = 179

∑(∑ (X2)) = 353

Compute Sum of Squares

Red Ball (n1=5)

White Ball (n2=5)

Blue Ball (n3=5)

TOTAL (N=15) (k=3)

X = 3.4

X = 4.2

X = 5.8

∑X = 17

∑X = 21

∑X = 29

∑(∑X) = 67

∑(X2) = 63

∑(X2) = 111

∑(X2) = 179

∑(∑ (X2)) = 353

67 ) 2

((X )) 2

(2534

353

53.7 .5

SStot = (( X )) −

= 216910

−

= 2871

N

1530

sum of the scores in the column)2

((X )) 2

SSb = (

)−(

)

n of the scores in the column

N

21 2

29 2

53.7 ) 2

17 2

(766)

(852)

(916)

(2534

= (

+

+

)−(

)

10

30

510

15

5

510

= 1133.1

14.93

2

53.7 – 1133.1

14.93

38.8

SSw = SStot – SSb = 2871.5

= 1738.4

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