One way-ANOVA

Analysis of Variance

Let’s say we conduct this experiment: effects of alcohol on memory

Basic Design

• Grouping variable (IV, manipulation) with 2 or more levels

• Continuous dependent/criterion variable• Ho: 1 = 2 = ... = k

• What is Halt?

• How many levels here?

Experiment results

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

1

error

treatmenterrorF

ANOVA results

ANOVA

VAR00003

Sum of Squares df Mean Square F Sig.

Between Groups 115.600 1 115.600 14.821 .000

Within Groups 296.400 38 7.800

Total 412.000 39

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