t =t =t =t =

df = df =

==--

(( ++11 11

))

nnMS

XXt

jierror

ji

observed

11

-

df: dfMS error

pv =pv =

Compute

Clear Me

Post Hoc Analysis:Post Hoc Analysis:Which groups differ?Which groups differ?

If there are If there are kk groups, groups,

how many pairs (possible t how many pairs (possible t

tests) are there?tests) are there?

Risk of at least one Risk of at least one

Type I error ( ‘family-wise Type I error ( ‘family-wise

error rate’ ): error rate’ ):

kk = =

mm αα

mm

reset

Post Hoc Analysis:Post Hoc Analysis:Which groups differ?Which groups differ?

If there are If there are kk

groups, how many pairs are groups, how many pairs are

there?there?

The Bonferroni The Bonferroni

Procedure Procedure If you want the chance of 1 or

more Type I errors to be less than 0.05, use

0.05/m for each post hoc comparison, where m is

the number of comparisons to be made.

kk = =

mm

If you perform m tests using α, your family-wise risk is (approximately) m*α,

αFW = m*α

So, if you perform m tests, use

for each test.

Example: I performed 25 tests using α = 0.01 for each. My family-wise risk is (approximately) 25*0.01 = 0.25.

mFW

BonferroniBonferroniForwards and BackwardsForwards and Backwards

AssumptionsAssumptionsof ANOVA (Dream Land)of ANOVA (Dream Land)

NormalityNormalityThe scores in eThe scores in each population (each level

of the IV) have a normal distribution.

HHomogeneity of omogeneity of

VarianceVariance The scores in each

population have the same standard deviation.

Violations of Violations of AssumptionsAssumptions

If enough If enough

observations are taken, observations are taken,

the assumptions are the assumptions are

safely ignored…unless: safely ignored…unless:

Really bad:Really bad: Very Very unequal unequal population variances (4 to 1 ratios

or more) AND unequal sample sizes.

Types of ANOVA Designs

One-way ANOVA Completely Randomized Design

Goal: Compare three different fertilizers to see whether there is any difference in their effectiveness.

Approach: Divide growing regions into 9 fields, randomly assign a fertilizer to each field:

Fertilizer 2 Fertilizer 2

Fertilizer 2

Fertilizer 1

Fertilizer 1

Fertilizer 1

Fertilizer 3

Fertilizer 3 Fertilizer 3

Mojave Desert

San Luis Obispo

MontrealCanada

Goal: Compare three different fertilizers to see whether there is any difference in their effectiveness.

Approach: Divide farm into 3 fields that vary according to some measurable quantity (growing region). Subdivide these 3 fields into 3 parts, randomly assign a fertilizer to each part of the field.

Growing Region is the "blocking variable".

Two-Way ANOVAThe Randomized Block Design (RBD)

Purpose: Increase your ‘power’ to reject the null.

Mojave Desert

San Luis Obispo

MontrealCanada

Fertilizer 2 Fertilizer 1

Fertilizer 2

Fertilizer 1

Fertilizer 1

Fertilizer 2

Fertilizer 3

Fertilizer 3 Fertilizer 3

Example ProblemA researcher wishes to determine whether the stress levels of students depends on their housing condition. It is known from previous research that the stress level of students depends strongly on their academic major. The researcher therefore decides to use the academic major as a blocking variable.

Analyze the data below and determine whether the idea to block on major was successful (increased power).

Housing Major Stress

OCP Sci 1.3OCP Hum 13.5

OWNA Sci 1.7OWNA Hum 11.9OWNR Sci 2.2OWNR Hum 12.2DORM Sci 9.2DORM Hum 19.2

One-way ANOVA: Stress versus Housing Source DF SS MS F PHousing 3 75.3 25.1 0.44 0.735Error 4 226.4 56.6Total 7 301.7

Two-way ANOVA: Stress versus Housing, Major Source DF SS MS F PHousing 3 75.28 25.093 43.77 0.006Major 1 224.72 224.720 391.95 0.000Error 3 1.72 0.573Total 7 301.72

It Worked!