1. WELCOME TO THE WORLD OF INFERENTIAL STATISTICS
Analysis of variance – ANOVA

2. What is ANOVA? A method for comparing multiple groups.
Question: Are these groups (however many there are) significantly different from each other?
Answer: ANOVA

3. ANOVA – Null Hypothesis

4. ANOVA – Alternative Hypothesis

5. ANOVA  QUESTION????? SYSTEMATIC?

6. Basics of ANOVA Null hypothesis:
All groups are coming from the same population
Differences in means are due to normal sampling fluctuations between samples drawn from the same population
If null hypothesis is true, then certain other things must be true.
 Test if that is the case (F-test)
If yes  accept null hypothesis
If no  reject null hypothesis

7. Working with multiple samples
Under the null hypothesis, all of these are coming from the same population

8. Same idea, different picture
Population
Sample k
Sample 1
Sample 4
Sample 2
Sample 3

9. Working with multiple samples
Each of these means estimate the SAME population mean

10. Working with multiple samples: How can we estimate the population variance?
Each of these variances estimate the SAME population variance

11. Working with multiple samples: How can we estimate the population variance?
Variance estimate 1: Average the sample variances!

12. Working with multiple samples: A second estimate of the population variance
These means are the means of “all possible” samples

13. What can we compute using the means of all possible samples?
SD of “all possible” sample means is : SEM

14. Estimating the variance based on the SEM
Variance equation 2:
Variance of all possible
sample means
Variance of
the attribute of
interest in the population

15. BETWEEN GROUPS estimate of the population variance
Variance of all possible
sample means
Mean of group k
Grand mean
n subjects in each sample,
k samples
Estimate of the total variance
based on variance between
different groups.

16. G&W – For Gorkem & friends!!!!
Page 369 – comparing the logic of t-test with ANOVA
Page 372 – the components of variance
Page – a very good discussion of F statistic

17. Example
Students were grouped into low-medium and high levels of motivation.
They were asked number of hours per week they spent doing homework.
The question is, whether low-medium-high motivated students spend
time doing homework that significantly differs.

18. Example
Null and Alternative Hypotheses

19. Example
Means of each group

20. Example
Variance in means

21. Example
Between groups estimate of variance

22. WITHIN GROUPS estimate of the population variance
Variance of the attribute
in the subjects in each group
Estimate based on within group variances – average of group variances

23. Example
Within groups estimate of variance

24. F-Statistic (Fisher’s test, 1923)
Logic of the F statistic:
The two estimates would be approximately equal if samples were drawn randomly from a single population.
If the ratio of “between” samples variance to “within” samples variance is taken, it should be approximately 1.
Depending on sample size
Depending on variation of sample means among all possible samples
If the ratio of “between” samples variance is much higher than “within” samples variance, then sample means vary more than expected by chance.
This is evidence that the independent variable is associated with significant differentiation of means.

25. Formal presentation of F-statistic
dfbetween = k-1
dfwithin = k(n-1)

26. Partitioning the variance
Total variance = Between group variance + Within group variance Within group variance = Variance between subjects OR = True variance between subjects + Variance due to measurement error Between group variance = Variance due to membership of groups + True variance between subjects +
If null hyp is true

27. Understanding the F-test
Between groups
variance
Within groups
Variance due to
membership of
groups
True variance
between subjects
measurement
error + =

28. Assumptions of the F-test
Interval/ratio level measurement of the dependent variable (robust against deviations from normality)
Independence of observations (each unit of analysis is independently selected into the sample)