Assignment 3 slides

Single-Factor Analysis of Variance

Psychology 3800, Lab 002

In Today’s Lab…

o overview of one-way ANOVA

o example of one-way ANOVA design

o post hoc tests

o assignment #3

Single-Factor Analysis of Variance (ANOVA): Overview

(1)one independent variable (IV) or grouping variable (factor)

(2)three or more independent groups (levels)

(3)measuring all groups on one dependent variable (DV)

(4)calculating a mean score for each group

(5)testing for differences among the group means

Note: results only tell us whether there is some differencebetween the means (but not where that difference lies)

Features of the Design

H0: μ1 = μ2 = μ3 = … = μk

HA: at least two means differ significantly

Experiment: looking at level of friendliness after the consumption of different amounts of alcohol

• testing friendliness after 5 oz., 10 oz. or 15 oz. of alcohol

• 24 people randomly assigned to one of the three levels of alcohol

• each person drinks assigned amount of alcohol

• measure friendliness on a 10-point Likert scale (where 1 = not at all friendly, 10 = extremely friendly)

Example Problem

(1)one factor: amount of alcohol consumed

(2)three or more levels: 5 oz., 10 oz., 15 oz.

(3)one dependent variable: level of friendliness

… testing for differences in average friendliness across the three alcohol

consumption groups

Example Problem

Reasons why group differences may be noted:

(1)random variation / error

(2)true differences among groups due to experimenter’s

manipulation of IVANOVA tells us which one causes the observed differences

between the means.

Sources of Variance

Sources of Variance

variance within groups(error)

variance between groups(true effect)

MSbetween SSbetween

dfbetween

MSwithin SSwithin

dfwithin

F MSbetween

MSwithin

Obtained formula:

ANOVA decides if: variancebetween > variancewithin

As a general rule:

• if there is no effect then MSbetween= MSwithin and F ≈ 1

• if there is an effect, then Msbetween > Mswithin and F > 1

Interpreting the Fobtained

F MSbetween

MSwithin

(1) random sampling

(2) normality groups come from normal populations

(3) homogeneity of variance equal variances (s2) across all levels if significant: not all variances are equal just like for t-test

(4) numerator (MSbetween) and denominator (MSwithin) are independent should be true if random sampling

Assumptions

SOMETHING’S DIFFERENT…

Single-Factor Analysis of Variance (ANOVA): Example Analysis

5 oz 10 oz 15 oz

3 5 6

4 3 7

2 6 7

5 5 5

3 4 6

4 4 4

3 5 5

3 6 7

Anchors: 1 = not at all friendly; 10 = extremely friendly

24 participants(8 per condition)

The Data

5 oz.

10 oz.

15 oz.

Data Analysis in SPSS: Descriptive Statistics

Analyze Descriptive Statistics Explore…

grouping variablemeasured variable

select “Statistics”

Data Analysis in SPSS: Descriptive Statistics

For Group 1 (5 oz. of alcohol)…

o mean: M = 3.38

o standard error: SE = 0.32

o variance: s2 = 0.84

o 95% CI: 2.61 < μ1 < 4.14

Analyze General Linear Model Univariate

Data Analysis in SPSS: One-Way ANOVA

grouping variablemeasured variable

Options Menu

Data Analysis in SPSS: One-Way ANOVA

Data Analysis in SPSS: One-Way ANOVA

Data Analysis in SPSS: One-Way ANOVA

Post Hoc Menu

Output: Levene’s Test

Levene F(2, 21) = 0.24, ns

o group variances do not differ significantly

o assumption of homogeneity of variances has not be violated

Output: One-Way ANOVA

F(2, 21) = 11.84, p < .001, η2 = .53, power = .99

o significant effect observed

o at least two condition means differ significantly

o 53% of variance due to our manipulation (alcohol)

o 99% chance of finding significant main effect if experiment repeated

significant effect = at least two means differ significantly

5 oz. 10 oz. 15 oz.

3.38 4.75 5.88

• can visually see some differences between the group means

• but: ANOVA can’t reveal which means differ significantly

What Does It All Mean?

*means obtained through Explore command in SPSS

Post Hoc Tests

• performed following analysis of variance (post-hoc)

• only performed if ANOVA is significant

• tests each mean against every other mean to see which pairs differ significantly from one another

5 oz vs. 10 oz

5 oz vs. 15 oz

10 oz vs. 15 oz

Tukey’s Honestly Significant Difference (HSD)

5 oz. vs. 10 oz. 15 oz.

10 oz. vs. 5 oz. 15 oz.

15 oz. vs. 5 oz. 10 oz.

Output: Tukey’s HSD

• there is redundant information in this table (e.g., 1 vs. 2 is the same as 2 vs. 1), so read it carefully and pick out only the unique comparisons

1 vs. 2 (5 oz. vs. 10 oz) 1 vs. 3 (5 oz. vs. 15 oz.) 2 vs. 3 (10 oz. vs. 15 oz.)

Reporting Tukey HSD Results

q(df1, df2) = qobtained, significance

number of levels df value of error(“Tests of Between Subjects

Effects” table)

significance info(“Multiple Comparisons” table)

Example (5 oz. versus 15 oz.)

q(3, 21) = qobtained, p < .001

How do we find this value?

Option #1: Calculation by Hand

where…

= larger group mean

= smaller group mean

MSWITHIN = error mean squares value as provided in overall ANOVA

analysis (ANOVA output, Mean Square column, Within Groups

row)

n = number of individuals in each group (same in all groups)

q x 1 x 2MSwithin

n

x 1

x 2

(textbook: pp. 83-84)

Finding the Value of q

Option #1: Calculation by Hand

q x 1 x 2MSwithin

n

(textbook: pp. 83-84)

5.88 3.38

1.060

8

2.5

.1325

6.87

Finding the Value of q

5 oz. 10 oz. 15 oz.

3.38 4.75 5.88

Example (5 oz. versus 15 oz.)

q(3, 21) = 6.87, p < .001

Option #2: Use POSTHOC program

Accessing the program on the Social Science Network• Start My Computer L: drive Course Library Psychology• select: POSTHOC (or POSTHOC.exe)

Finding the Value of q

Option #2: Use POSTHOC program

Do you want to store your runs on disk? YType the name of the file in which to store the results. U:\name.tstWhat is the number of means to be read in? 3Mean 1? 3.38Mean 2? 4.75Mean 3? 5.88Are the sample sizes equal for each of the means? YWhat is the sample size for each of the means? 8Do you want to compute a pooled error term? NType in the mean square error. 1.060Type in the error degrees of freedom. 21Type q for q-statistic, t for t-statistic, or F for F-statistic. q

Finding the Value of q

Option #2: Use POSTHOC program

q-statistics for each pair of means

Finding the Value of q

5 oz. 10 oz. 15 oz.

3.38 4.75 5.88

15 oz. versus 10 oz. qobtained = 3.104

Assignment #3

• introductory sentence(s)• variables and coding (IV, DV, variable levels, units)• type of test being conducted• Levene’s test (hypotheses, results, conclusion)• ANOVA test (hypotheses, results, conclusion)• indication of whether any post hoc tests are necessary• all post hoc results (type of test, q-statistics)• concluding statement about the post hoc findings

What to include:

One-Way ANOVA Results Section: Overview

• discussion of confidence intervals (how they relate to the post hoc results)• additional material: figure, SPSS output, POSTHOC output (or hand calculations)

If F-test of the one-way ANOVA is significant (p < .05), reporting the results of the post hoc test is necessary.

• report all p-values, and Tukey’s q statistics for these post hoc tests (and M, SE if you have not already done so)

• as with the ANOVA tests, include a concluding statement (or statements) that involves no stats

summarize the findings and make a general conclusion

Reporting Post Hoc Results

The analyses revealed that participants in the 15 oz group (M = 5.88, SE =

0.40) were rated as being significantly more friendly than those in both the 5 oz group

(M = 3.38, SE = 0.32), q(3, 21) = 6.87, p < .001. Furthermore, participants in the 10 oz

group (M = 4.75, SE = 0.37) were rated as being significantly more friendly that those in

the 5 oz group, q(3, 21) = 3.76, p < .05. No significant differences, however, were

observed between individuals in the 10 oz group and those in the 15 oz group with

regard to average friendliness, q(3, 21) = 3.10, ns. Therefore, although greater alcohol

consumption does appear to increase perceived friendliness, noticeable differences are

not evident when comparisons are made between the two highest levels of alcohol

consumption.

Example

Reporting Post Hoc Results

The analyses revealed that participants in the 15 oz group (M = 5.88, SE =

0.40) were rated as being significantly more friendly than those in both the 5 oz group

(M = 3.38, SE = 0.32), q(3, 21) = 6.87, p < .001. Furthermore, participants in the 10 oz

group (M = 4.75, SE = 0.37) were rated as being significantly more friendly that those in

the 5 oz group, q(3, 21) = 3.76, p < .05. No significant differences, however, were

observed between individuals in the 10 oz group and those in the 15 oz group with

regard to average friendliness, q(3, 21) = 3.10, ns. Therefore, although greater alcohol

consumption does appear to increase perceived friendliness, noticeable differences are

not evident when comparisons are made between the two highest levels of alcohol

consumption.

Example

Reporting Post Hoc Results

consistent number of decimal places

The analyses revealed that participants in the 15 oz group (M = 5.88, SE =

0.40) were rated as being significantly more friendly than those in both the 5 oz group

(M = 3.38, SE = 0.32), q(3, 21) = 6.87, p < .001. Furthermore, participants in the 10 oz

group (M = 4.75, SE = 0.37) were rated as being significantly more friendly that those in

the 5 oz group, q(3, 21) = 3.76, p < .05. No significant differences, however, were

observed between individuals in the 10 oz group and those in the 15 oz group with

regard to average friendliness, q(3, 21) = 3.10, ns. Therefore, although greater alcohol

consumption does appear to increase perceived friendliness, noticeable differences are

not evident when comparisons are made between the two highest levels of alcohol

consumption.

Example

Reporting Post Hoc Results

mention descriptive statistics only once

1

2

3

The analyses revealed that participants in the 15 oz group (M = 5.88, SE =

0.40) were rated as being significantly more friendly than those in both the 5 oz group

(M = 3.38, SE = 0.32), q(3, 21) = 6.87, p < .001. Furthermore, participants in the 10 oz

group (M = 4.75, SE = 0.37) were rated as being significantly more friendly that those in

the 5 oz group, q(3, 21) = 3.76, p < .05. No significant differences, however, were

observed between individuals in the 10 oz group and those in the 15 oz group with

regard to average friendliness, q(3, 21) = 3.10, ns. Therefore, although greater alcohol

consumption does appear to increase perceived friendliness, noticeable differences are

not evident when comparisons are made between the two highest levels of alcohol

consumption.

Example

Reporting Post Hoc Results

report significant and non-significant findings

The analyses revealed that participants in the 15 oz group (M = 5.88, SE =

0.40) were rated as being significantly more friendly than those in both the 5 oz group

(M = 3.38, SE = 0.32), q(3, 21) = 6.87, p < .001. Furthermore, participants in the 10 oz

group (M = 4.75, SE = 0.37) were rated as being significantly more friendly that those in

the 5 oz group, q(3, 21) = 3.76, p < .05. No significant differences, however, were

observed between individuals in the 10 oz group and those in the 15 oz group with

regard to average friendliness, q(3, 21) = 3.10, ns. Therefore, although greater alcohol

consumption does appear to increase perceived friendliness, noticeable differences are

not evident when comparisons are made between the two highest levels of alcohol

consumption.

Example

Reporting Post Hoc Results

end off with descriptive, stats-free summary statement

• include a figure showing the group means on the DV, with confidence intervals

• make sure that figure adheres to general APA formatting even though figure is created by hand

contrary to APA requirement, please put figure and figure captions all on one page figure caption to be placed below figure

Including a Figure

• if you are not comfortable drawing this type of figure using computer software (e.g., Excel), you may draw it out by hand this week, but please adhere to APA requirements where possible

APA Figure

3 8 130

1

2

3

4

5

5 10 15

Mea

n Le

vel o

f Frie

ndlin

ess

Total Alcohol Consumed (oz.)

Figure 1. Average level of friendliness reported by participants atvarying levels of alcohol consumption.

error bars or CI indicated

no grid lines or figure frame

clear axis labels (with units if applicable)

black/white or grayscale

no title (see figure caption below)

Next Week

• unit 4: single-factor repeated measures ANOVA (essentially an extension of the repeated-measures t-test)

• readings: pp. 115-136 in textbook

• one-way ANOVA assignment due at start of lab

• will be discussing the following week (Reading Week) and the logistics around submitting assignments and receiving feedback