One-way Analysis of Variance

Single Independent Variable Between-Subjects Design

Logic of the Analysis of Variance

• Null hypothesis equal H 0 : Population means  m 1 = m 2 = m 3 = m 4 • Alternative hypothesis: H 1  Not all population means equal.

Cont.

Logic--cont.

• Create a measure of variability among group means  Ms groups (accurate est. of pop. var. if null true) • Create a measure of variability within groups  MS error (accurate est. of pop. var. regardless of whether null is true)

Cont.

Logic--cont.

• Form ratio of MS groups /MS error  Ratio approximately 1 if null true  Ratio significantly larger than 1 if null false  “approximately 1” can actually be as high as 2 or 3, but not much higher

Epinephrine and Memory

• Based on Introini-Collison & McGaugh (1986)  Trained mice to go left on Y maze  Injected with 0, .1, .3, or 1.0 mg/kg epinephrine  Next day trained to go right in same Y maze  dep. Var. = # trials to learn reversal • More trials indicates better retention of Day 1 • Reflects epinephrine’s effect on memory

Mean St. Dev.

0.0

1 3 5 3 3 5 4 5 5 2 2 3 3 3 1 3 4 3 3.22

1.26

Dosage (

m

g/kg) 0.1

0.3

5 4 2 6 5 3 4 4 3 3 4 5 6 5 5 7 4 6 4.50

1.29

6 5 5 6 7 6 6 5 5 5 5 6 6 4 6 5 5 6 5.50

0.71

1.0

2 2 2 1 1 3 5 1 3 2 3 2 1 1 1 2 1 1 1.89

1.08

Grand mean = 3.78

Calculations

• Start with Sum of Squares (SS)  We need: • SS total • SS groups • SS error • Compute degrees of freedom ( df ) • Compute mean squares and F

Cont.

Calculations--cont.

SS total

=  (

X



X

..

) 2 = ( 1  3 .

78 ) 2   3  3 .

78  2  ...

  1  3 .

78  2 =

SS

216 .

444

groups

=

n

 

X j



X

..

 2 = 18   3 .

22  3 .

78  4 .

50  3 .

78  2  ...

  1 .

89  3 .

78  2  = 18 ( 7 .

364 ) = 132 .

556

SS error

=

SS total



SS groups

= 216 .

444  132 .

556 = 83 .

889

SS error

=  

X i



X j

2 

Degrees of Freedom (

df

)

• Number of “observations” free to vary  df total = N - 1 • Variability of N observations  df groups = g - 1 • variability of g means  df error • n = g ( n - 1) observations in each group = n • times g groups - 1 df

Summary Table

Source Groups Error Total *

p

< .05

df

3 68 71

SS

132.556

83.889

MS

44.185

1.234

F 35.816*

Conclusions

• The F for groups is significant.

 We would obtain an F of this size, when true, less than one time out of 1000.

H 0  The difference in group means cannot be explained by random error.

 The number of trials to learn reversal depends on level of epinephrine.

Cont.

Conclusions--cont.

• The injection of epinephrine following learning appears to consolidate that learning.

• High doses may have negative effect.

Unequal Sample Sizes

• With one-way, no particular problem  Multiply mean deviations by appropriate you go n i as  The problem is more complex with more complex designs, as shown in next chapter.

• Example from Foa, Rothbaum, Riggs, & Murdock (1991)

Post-Traumatic Stress Disorder

• Four treatment groups given psychotherapy  Stress Inoculation Therapy (SIT) • Standard techniques for handling stress  Prolonged exposure (PE) • Reviewed the event repeatedly in their mind

Cont.

Post-Traumatic Stress Disorder -cont.

 Supportive counseling (SC) • Standard counseling  Waiting List Control (WL) • No treatment

Mean St.

Dev.

SIT 3 13 13 8 11 9 12 7 16 15 18 12 8 10 11.071

3.951

PE 18 6 21 34 26 11 2 5 5 26 11.118

24 14 21 5 17 17 23 19 7 27 25 SC 15.400 18.091

7.134

WL 12 30 27 20 17 23 13 28 12 13 19.500

7.106

SIT = Stress Inoculation Therapy PE = Prolonged Exposure SC = Supportive Counseling WL = Waiting List Control Grand mean = 15.622

Tentative Conclusions

• Fewer symptoms with SIT and PE than with other two • Also considerable variability within treatment groups • Is variability among means just a reflection of variability of individuals?

Calculations

• Almost the same as earlier  Note differences • We multiply by n j as we go along.

• MS error is now a weighted average .

Cont.

Calculations--cont.

SS total

=  (

X



X

..

) 2 = ( 3  15 .

62 ) 2   13  15 .

62  2  ...

  13  15 .

62  2 =

SS

2786 .

6

groups

= 

n j



X j



X

..

 2 =   14 11  11 .

071 ( 18 .

091   15 .

62 15 .

62 )  2 2   10 10   11 .

19 .

118 500   15 .

62 15 .

62   2 2    = 507 .

8

SS error

=

SS total



SS groups

= 2786 .

6  507 .

8 = = 83 .

889

Summary Table

Source Groups Error Total *

p

< .05

df

3 41 44

SS

507.8

2278.7

2786.6

MS

169.3

55.3

F 3.05*

F

.05

(3,41) = 2.84

Conclusions

• F is significant at a = .05

• The population means are not all equal • Some therapies lead to greater improvement than others.

 SIT appears to be most effective.

Multiple Comparisons

• Significant F only shows that not all groups are equal  We want to know what groups are different.

• Such procedures are designed to control familywise error rate.

 Familywise error rate defined  Contrast with per comparison error rate

More on Error Rates

• Most tests reduce significance level ( a ) for each t test.

• The more tests we run the more likely we are to make Type I error.

 Good reason to hold down number of tests

Fisher’s LSD Procedure

• Requires significant overall F, or no tests • Run standard groups.

t tests between pairs of  Often we replace MS error s 2 j or pooled estimate with from overall analysis • It is really just a pooled error term, but with more degrees of freedom--pooled across all treatment groups.

Bonferroni

t

Test

• Run usual t tests between pairs of groups, as  Hold down number of t tests  Reject if table t exceeds critical value in Bonferroni • Works by using a more strict value of a for each comparison

Cont.

Bonferroni

t

--cont.

• Critical value of .05/ c , where c a for each test set at = number of tests run  Assuming familywise a = .05

 e. g. with 3 tests, each at .05/3 = .0167 level.

t must be significant • With computer printout, just make sure calculated probability < .05/ c • Necessary table is in the book

Assumptions for Anal. of Var.

• Assume:  Observations normally distributed within each population  Population variances are equal • Homogeneity of variance or homoscedasticity  Observations are independent

Cont.

Assumptions--cont.

• Analysis of variance is generally robust to first two  A robust test is one that is not greatly affected by violations of assumptions.

Magnitude of Effect

• Eta squared ( h 2 )  Easy to calculate  Somewhat biased on the high side  Formula • See slide #33  Percent of variation in the data that can be attributed to treatment differences

Cont.

Magnitude of Effect--cont.

• Omega squared ( w 2 )  Much less biased than h 2  Not as intuitive  We adjust both numerator and denominator with MS error  Formula on next slide

h 2

and

w 2

for Foa, et al.

h w 2 2 = =

SS groups SS total

= 507 2786 .

.

8 6 = .

18

SS groups SS



total

(

k

 1 )

MS error



MS error

= 507 .

8  3 ( 55 .

6 ) 2786 .

6  55 .

6 = .

12 • h 2 = .18: 18% of variability in symptoms can be accounted for by treatment • w 2 = .12: This is a less biased estimate, and note that it is 33% smaller.

Other Measures of Effect Size

• We can use the same kinds of measures we talked about with t tests.

• Usually makes most sense to talk about 2 groups at a time, rather than a measure averaged over several groups.

Darley & Latene (1968)

Condition: Alone One Other Four Others n 13 26 13 X .87 .72 .51

Darley & Latene (1968)

Source Groups Error Total *

p

< .05

df

2 49 51

SS

.854 2.597 26.62

MS

F .427 8.06* .053