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### One way-ANOVA

Analysis of Variance

Basic Design

- Grouping variable (IV, manipulation) with 2 or more levels
- Continuous dependent/criterion variable
- Ho: 1 = 2 = ... = k
- WhatisHalt?
- How many levels here?

Analysis

- Q: How do you know the effect was caused by the manipulation (vodka) rather than chance factors (e.g. brainier people happened to be in group B)?
- Or: Do these two samples differ enough from each other to reject the null hypothesis that alcohol has no effect on mean memory?
- A: A statistical test (such as ANOVA or a t-test) is usually applied to decide this.

What does ANOVA do?

- ANOVA assesses the extent to which the distributions of two or more variables overlap
- The more the distributions overlap the less likely it is that they are different
- What is 2.6? 3.2? What should it be in our case?

F-ratio

- ANOVA involves calculating a statistic called the “F ratio”
- (the between groups variance=MSb/ the within groups variance=MSw)
- The F ratio gets larger as the distribution overlap gets smaller (i.e. a larger F indicates a difference in the group means )

F

- F = MSb/ MSw
- If H is true, expect F = error/error = 1.
- If H is false, expect

What does ANOVA do?

- You have calculated F - what next?
- Someone somewhere ran numerous ANOVAs on random data and worked out what values of F occur by chance alone
- We check our calculated F ratio statistic against this chance value; if it is greater than the tabulated value we reject chance and argue that the manipulation is the most likely explanation for the data

The p-value is the probability of obtaining an F value as extreme or even more extreme than the one actually observed. So, p-value = P(F > Fobs).

Writing up ANOVA results

- A one-way ANOVA was calculated on participants\' memory rating. The analysis was significant or n.s?, F( , ) = , p = .xxx .

ANOVA doesn’t always give a true result

- ANOVA can only be applied under certain conditions, i.e….
- Certain assumptions must be met:
- Homogeneity of variance of the measured variable (e.g. memory score)
- Normal distribution of the measured variable

Assumption of homogeneity of variance

- The dependent variable scores show the same degree of variability across the treatments, i.e.
- The treatment variances are of similar magnitude
- This diagram represents data from two treatments that meet the assumption of homogeneity of variance

- The spread of data within each treatment is similar hence the variances of the treatments are similar also

Assumption of normality

- The normal distribution..

- Symmetrical about its mean therefore the mean is a good estimate of central tendency
- There are fixed percentages of scores falling between points that can be defined using the SD (e.g. 68.26% of scores fall within 1 SD of the mean) therefore the SD and/or the variance are good estimates of spread around the mean
- Sensible to employ ANOVA, i.e. to analyse for differences in treatment means using estimates of variance

Consequences of violating assumption of normality

- A common violation of the normal distribution is skew
- Here is a figure showing a positively skewed distribution

- Not symmetrical about its mean therefore the mean is NOT a good estimate of central tendency
- The relationship between the percentages of scores falling between SD points is NOT FIXED therefore the SD/ variance is NOT a good estimate of spread around the mean
- NO LONGER sensible to employ ANOVA, i.e. to analyse for differences in treatment means using estimates of variance

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