Module 2 - One-way ANOVA

Question 1

Comparing Two Means

Answer 1

• Review of the t-test
• Independant-samples
• Paired-samples

Question 2

Comparing Several Means

Answer 2

• Theory behind the one-way between-subjects ANOVA
• Running a one-way between-subjects ANOVA

Question 3

The t-statistic - Comparing Two Means

Answer 3

The main purpose of a t-test is to test whether two group
means are significantly (or meaningfully) different from one
another

Question 4

The t-statistic Types

Answer 4

Between-groups
- When there are two experimental conditions and different participants were assigned to each condition
- Otherwise called independent-samples, independent-measures, independent-means
Repeated-measures
- When there are two experimental conditions and the same participants took part in both conditions of the experiment
- Otherwise called paired-samples, dependent-means, matched pairs

Question 5

The t-statistic Rationale

Answer 5

Two sample means are calculated
- Under the null hypothesis we expect those means to be roughly equal
- We compare the obtained mean difference against the null hypothesis (no difference)
- We use the standard error as a gauge of the random variability expected between sample means
- If the difference between sample means is larger than expected based on the standard error then:
- There is no effect and this difference has occurred by chance
- There is an effect and the means are meaningfully different

Question 6

The t-statistic - Independent- samples t-test

Answer 6

• Level of measurement (DV interval or ratio)
• Random sampling
• Normality
• Homogeneity of variance
  (Sample Means) - (Null Hypothesis [0]) / Average Standard

Question 7

The t-statistic - Paired- samples t-test

Answer 7

• Level of measurement (DV interval or ratio)
• Random sampling
• Normality
  (Mean difference of scores) - (Null Hypothesis [0]) / Average Standard error of Differences

Question 8

One-Way ANOVA

Answer 8

Comparing several means
The main purpose of a one-way ANOVA is for situations where we want to compare more than two conditions

Question 9

Type I Error

Answer 9

False Positive

Question 10

Type II Error

Answer 10

False Negative

Question 11

Familywise error rate (FWER)

Answer 11

For a single comparison using α=.05 the probability of a type 1 error is 5%
With the addition of another comparison using α=.05

Question 12

One-way ANOVA test

Answer 12

Null hypothesis
H0: μ1 = μ2 = μ3
Alternative hypothesis
H1: At least 1 group is different from another

Question 13

One-Way ANOVA

Answer 13

The ANOVA produces an F-statistic or an F-ratio
The F-statistic represents the ratio of the model to its error
ANOVA is an omnibus test
Tests for an overall experimental effect
Significant F-statistic tells us that there is a difference somewhere between the groups but not where this difference lies

Question 14

F - test

Answer 14

F = Variability between groups / Variability within groups = (Random Error) + (Treatment Effect) / Random Error
If Null is true, treatment effect will be 0
If treatment effect increases, F value increases

Question 15

Squared Sums Total

Answer 15

All the variance in mood
𝑆𝑆𝑇 =∑ ( 𝑋 − 𝑋g𝑟𝑎𝑛𝑑 ) ²
∑ (sum of)
𝑋 (our values)
𝑋g𝑟𝑎𝑛𝑑 (Grand x-bar [mean])

Question 16

Squared Sums Between

Answer 16

Variance in mood explained by our model
𝑆𝑆𝐵 =∑ 𝑛𝑖 ( 𝑋𝑖 − 𝑋𝑔𝑟𝑎𝑛𝑑 ) ²
∑ (sum of)
𝑛𝑖 (n [the number] in I [this group])
𝑋𝑖 (mean of this group)
𝑋𝑔𝑟𝑎𝑛𝑑 (Grand mean)

Question 17

Squared Sums Within

Answer 17

Variance in mood not explained by our mode
𝑆𝑆𝑊 =∑ ( 𝑋 − 𝑋𝑔𝑟𝑜𝑢𝑝 )
∑ (sum of)
𝑋 (raw values)
𝑋𝑔𝑟𝑜𝑢𝑝 (group means)

Question 18

Degrees of freedom

Answer 18

SStotal - Entire sample -1
SSbetween - Group means -1
SSwithin - Entire sample -Number of groups (n-k)

Question 19

One-Way ANOVA Assumptions - Level of measurement

Answer 19

Dependent variable must be measured at the interval or ratio level

Question 20

One-Way ANOVA Assumptions - Random sampling

Answer 20

Scores must be obtained using a random sample from the population of interest

Question 21

One-Way ANOVA Assumptions - Independence of observations

Answer 21

The observations that make up the data must be independent of one another
Violation of this assumption is very serious as it dramatically increases the Type 1 error rate

Question 22

One-Way ANOVA Assumptions - Normal distribution

Answer 22

The populations from which the sample are taken is assumed to be normally distributed
Need to check this for each group separately in one-way ANOVA

Question 23

One-Way ANOVA Assumptions - Homogeneity of variance

Answer 23

Samples are obtained from populations of equal variances
ANOVA is fairly robust to this violation – provided the size of your groups are reasonably similar

Question 24

One-Way ANOVA Assumptions

Answer 24

Level of measurement – met (or not) by design
Random sampling – met (or not) by design
Independence of observations – met (or not) by design

Question 25

Checking assumptions in SPSS

Question 26

Reporting One-Way ANOVA - Reporting assumptions - Normality

Answer 26

The Kolmogorov-Smirnov test revealed that mood scores were not normally distributed for the low intensity exercise group, D (5) = .35, p = .046, or the high intensity exercise group, D (5) = .03, p = .026. Mood was normally distributed in the medium intensity exercise group, D (5) = .23, p = .20. This data indicates that the
assumption of normality was violated. However, ANOVA is robust to violations of normality when the sample size is large and there are roughly equal numbers of participants across experimental groups, therefore this violation was ignored (Field, 2007).

Question 27

Reporting One-Way ANOVA - Reporting assumptions - Homogeneity of Variance

Answer 27

The variances for mood across experimental groups were roughly equal as indicated by the non-significant Levene statistic, F (2, 12) = .64, p = .55. The assumption of homogeneity of variances was not violated

Question 28

Reporting ANOVA

Answer 28

A one-way ANOVA test revealed that there was a significant effect of exercise group on mood scores F (2,12) = 5.12, p = .025
Between & Within group DF
F statistic
Unless p is less than .001 in which case we report p < .001

Question 29

Module Summary

Answer 29

ANOVAs can be used to compare multiple conditions
You analyse the variance between experimental conditions and compare that to the variance within a condition (random error)
F-ratios increase in magnitude as a treatment effect increases
There are a number of assumptions that we need to check, and know how to report