Chapter 12: ONE-WAY Analysis of Variance

Analysis of variance (ANOVA)

A hypothesis-testing procedure that is used to evaluate mean differences between two or more treatments (or populations).

NULL HYPOTHESIS

• NO mean difference between the populations

  • 𝐻0: UN=UR =UU

ALTERNATIVE HYPOTHESIS

• There is at least one mean difference between the populations

• N=R =/U

• N=U R

• N=U N

• N=/R=/ U (all three means are different)

Why not conduct a bunch of t-test?

• Doing multiple t-tests creates multiple chances to make a type 1 error.

• ANOVA controls this risk by testing all groups simultaneously using variance (average squared distance from the mean).

Define Between-group Variability

Values used to measure and describe the differences between treatments (mean differences).

Between-group Variability can be due to…(2)

• Sampling error

• The Effect of the independent variable on the dependent variable

Define Within group Variability

The differences that exist inside each treatment condition.

Within group Variability is also refered to as…

ERROR TERM

Within group Variability are ….

random, individual differences

Within group Variability is used when looking…

looking for an F of 1 if there is little variability (NULL TRUE)

F is always ….

POSITIVE

The distribution of F-ratios should pile up around …

1.00

Shape of F depends on…

Degrees of freedom for between and within group variability

What is the F-ratio numerator?

Variance Between Group

What is the F-ratio denominator?

Variance Within Group

Helpful table

k symbolizes?

Total number of group you have

N symbolizes?

Total number of participants

Variance is called___________ instead of s squared

Mean Square (MS)

F-ratio test

The statistical test to use to compare variance

n2 (greek letter Eta) is the _____________________

percentage of variance explained (effect size)

Small effect, n2 (greek letter Eta)

0.10

Meduim effect, n2 (greek letter Eta)

0.25

Large effect, n2 (greek letter Eta)

0.40

What are Post hoc Tests?

Additional tests that determine which mean differences are significant and which are not.

With Tukey’s honestly significant difference (HSD) test you compute …

a single value that determines the smallest difference between the means that meets criteria that is necessary for significance.

For HSD when your df is not exactly in the table, you go with the

smaller number that is closest.

Scheffé test has the

smallest risk of Type 1 error

Anything significant with ______, is significant for ______, but ____________.

• Scheffe

• Tukey

• NOT vice versa

Assumptions

• Observations are independent

• Normal populations

• Homogeneity of variance

Both the F-ratio and the t statistic compare the ____________________________ with the _____________________

• actual differences between sample means (numerator)

• differences that would be expected if there is no treatment effect (the denominator if H₀ is true).

If the _____ is sufficiently _____ than the _______, you conclude that there is a significant difference between treatments.

• numerator

• bigger

• denominator

What is the principal reason why you should use ANOVA instead of several t tests to evaluate mean differences when an experiment consists of three or more treatment conditions?

Multiple t tests accumulate the risk of a Type I error.