Psychology 9

1
Psychology 9

• Quantitative Methods in Psychology
• Jack Wright
• Brown University
• Section 24
• (previously 26)

Note. These lecture materials are intended
solely for the private use of Brown University
students enrolled in Psychology 9, Spring
Semester, 2002-03. All other uses, including
duplication and redistribution, are unauthorized.

2
Agenda

• ANOVA
• One-way
• Two-way
• Announcements
• Chapter 11 (1-way)
• Chapter 12 (2-way)

3
Ronald Fisher (1890-1962)
4
Keys to evaluating F (review)

• If Null Hypothesis were true
• MSbetween is one estimate of the variance of the
  population
• MSwithin provides a second estimate of the
  variance of the population
• Therefore, we expect F 1
• Next
• numeric example
• working with the F distribution

5
Numeric example

• No Alc 1 drink 2 drinks
• 1 3 5
• 2 4 6
• 3 5 7
• To get MSwithin
• mean 2 4 6
• SS1 -12 0 12 2 SS2 2 SS3 2
• SSw 2 2 2 6
• DFw 2 2 2 6
• MSwithin SSw/DFw 6/6 1.0
• Write these results down

6
Numeric example MSb

• No Alc 1 drink 2 drinks
• 1 3 5
• 2 4 6
• 3 5 7
• To get MSbetween
• mean 2 4 6 grand mean 4
• SSsample means 22 02 22 8 df 3 1
  2
• s2sample means SSsample means/df 8/2 4.0
• Using LLN, get estimate of pop. variance
• MSbetween s2sample means npersample 4 3
  12.0
• (write this down too)

7
Evaluating F ratios

• We now have
• MSb 12 df 2
• MSw 1 df 6
• F MSb/MSw 12/1 12.0
• We now need to evaluate this F-ratio
• Previously, for t
• sampling distribution of t, with on one df term
• Similarly, for F
• sampling distribution for F
• with two df terms dfbetween dfwithin

8
Visualizing sampling distributions of F

• 1. Under null hypothesis, all samples from single
  population, say
• mu 4 sigma 2
• 2. Drawn k random samples, each size n
• let k 3 let n 3
• 3. Compute terms we just reviewed
• MSbetween MSwithin and F
• 4. Repeat many times record results
• 5. Get probability distribution of results
• This is the sampling distribution for F
• with DFb 2 and DFw 6

9
k3 N3 F(2,6)
MSb
MSw
F
10
k3 N10 F(2,27)
MSb
MSw
F
11
k3 N20 F(2,57)
MSb
MSw
F
12
k10 N20 F(9,190)
MSb
MSw
F
13
Our result
Our result F(2,6) 12.0
What is probability of having F(2,6) gt 12.0?
14
Using F tables

• See F table from text
• find df(2,6)
• find nearest Fcritical
• find probability beyond that Fcritical
• report
• The means were significantly different, F(2,6)
  12.0, p lt .01

15
ANOVAComputational procedure

• We could now obtain MSbetween and Mswithin using
  these familiar methods
• However, it is also useful to have computational
  approach that is
• simple as possible
• Provides internal checks
• Provides other useful information
• will generalize to more complex problems later

16
Computational procedure

• Note on text, p. 429
• The steps are useful and we will follow them
• However, computational formulas will work, but
  do not help us understand what is going on
• Our approach
• Follow the steps
• But use more conceptual methods (similar to
  those we have used already)

17
ANOVAComputational procedure

• 1. get total variation in the data, temporarily
  ignoring conditions
• total sums of squares or SStotal
• Ie, sum of squared deviations around the grand
  mean, as we have always done
• Also get total DF

18
Numeric example SStotal

• No Alc 1 drink 2 drinks
• 1 3 5
• 2 4 6
• 3 5 7
• Grand mean 4
• Dev2 -32 -22 -12 -12 0 1 1 22 32
  
• 9 4 1 1 0 1 1 4 9
• SStotal 30
• DFtotal N - 1 9 - 1 8

19
ANOVAComputational procedure

• 2. get variation in data WITHIN EACH GROUP
• within sums of squares or SSwithin
• Ie, sum of squared deviations around each
  conditions own mean, as we have already done
• Also get within DF

20
Numeric example Sswithin (repeated)

• No Alc 1 drink 2 drinks
• 1 3 5
• 2 4 6
• 3 5 7
• mean 2 4 6
• SS1 -12 0 12 2
• SS2 -12 0 12 2
• SS3 -12 0 12 2
• SSw 2 2 2 6
• DFw 2 2 2 6

21
ANOVAComputational procedure

• 3. get variation in data you would expect based
  on the differences BETWEEN groups
• Note implication
• If alternative hypothesis were true, groups
  differ
• Best estimate of true mean for each group is
  sample mean
• So, use that to predict results within each
  group
• Then get SS of these predictions
• This is SSbetween (aka SSfull model)

22
Understanding SSbetween

• No Alc 1 drink 2 drinks
• Means 2 4 6
• o p o p o p (oobs. ppredicted)
• 1 2 3 4 5 6
• 2 2 4 4 6 6
• 3 2 5 4 7 6
• Now how much do our predictions vary?
• mean prediction 4 (the grand mean)
• SSbetween -22 -22 -22 0 0 0 22 22
  22
• 24
• What is DF for our model?
• We have k 3 and only 2 ways for our predictions
  to vary, so df 3 - 1 2

23
ANOVASummary (so far)

• Source SS df-------------------------------
• Between 24 2
• Within 6 6
• Total 30 8
• ------------------------------

Dfs must Total.
SSs must Total.