One-way Analysis of Variance

Single Independent VariableSingle Independent Variable

Between-Subjects DesignBetween-Subjects Design

  

Logic of the Analysis of Logic of the Analysis of VarianceVariance

• Null hypothesis Null hypothesis HH00 : Population : Population means equalmeans equal 11 = =

• Alternative hypothesis: Alternative hypothesis: HH11

Not all population means equal.Not all population means equal.

Cont.

Logic--cont.Logic--cont.

• Create a measure of variability Create a measure of variability among group meansamong group means MsMsgroupsgroups (accurate est. of pop. var. if null (accurate est. of pop. var. if null

true)true)

• Create a measure of variability within Create a measure of variability within groupsgroups MSMSerrorerror (accurate est. of pop. var. (accurate est. of pop. var.

regardless of regardless of whether null is true)whether null is true)

Cont.

Logic--cont.Logic--cont.

• Form ratio of MSForm ratio of MSgroupsgroups /MS /MSerrorerror

Ratio approximately 1 if null trueRatio approximately 1 if null true

Ratio significantly larger than 1 if null Ratio significantly larger than 1 if null falsefalse

““approximately 1” can actually be as approximately 1” can actually be as high as 2 or 3, but not much higherhigh as 2 or 3, but not much higher

Epinephrine and MemoryEpinephrine and Memory

• Based on Introini-Collison & McGaugh (1986)Based on Introini-Collison & McGaugh (1986) Trained mice to go Trained mice to go leftleft on on YY maze maze

Injected with 0, .1, .3, or 1.0 mg/kg epinephrineInjected with 0, .1, .3, or 1.0 mg/kg epinephrine

Next day trained to go Next day trained to go rightright in same in same YY maze maze

dep. Var. = # trials to learn reversaldep. Var. = # trials to learn reversal• More trials indicates better retention of Day 1More trials indicates better retention of Day 1

• Reflects epinephrine’s effect on memoryReflects epinephrine’s effect on memory

Grand mean = 3.78

CalculationsCalculations

• Start with Sum of Squares (SS) Start with Sum of Squares (SS) We need:We need:

• SSSStotaltotal

• SSSSgroupsgroups

• SSSSerrorerror

• Compute degrees of freedom (Compute degrees of freedom (df df ))

• Compute mean squares and Compute mean squares and FF

Cont.

Calculations--cont.Calculations--cont.

2

222

2..

222

2..

889.83

556.132444.216

556.132)364.7(18

78.389.1...78.350.478.322.318

444.216

78.31...78.33)78.31(

)(

jierror

groupstotalerror

jgroups

total

XXSS

SSSSSS

XXnSS

XXSS

Degrees of Freedom (Degrees of Freedom (df df ))• Number of “observations” free to varyNumber of “observations” free to vary

dfdftotaltotal = = NN - 1 - 1

• Variability of Variability of NN observations observations

dfdfgroupsgroups = = gg - 1 - 1

• variability of variability of gg means means

dfdferrorerror = = g g ((nn - 1) - 1)

• nn observations in each group = observations in each group = nn - 1 - 1 dfdf

• times times gg groups groups

Summary TableSummary Table

ConclusionsConclusions

• The The FF for groups is significant. for groups is significant. We would obtain an We would obtain an FF of this size, when of this size, when

HH00 true, less than one time out of 1000. true, less than one time out of 1000.

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

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

Cont.

Conclusions--cont.Conclusions--cont.

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

• High doses may have negative High doses may have negative effect.effect.

Unequal Sample SizesUnequal Sample Sizes

• With one-way, no particular problemWith one-way, no particular problem Multiply mean deviations by appropriate Multiply mean deviations by appropriate nnii

as you goas you go

The problem is more complex with more The problem is more complex with more complex designs, as shown in next chapter.complex designs, as shown in next chapter.

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

Post-Traumatic Stress Post-Traumatic Stress DisorderDisorder

• 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 Post-Traumatic Stress Disorder--cont.Disorder--cont.

Supportive counseling (SC)• Standard counseling

Waiting List Control (WL)• No treatment

SIT PE SC WL3 18 24 12

13 6 14 3013 21 21 278 34 5 20

11 26 17 179 11 17 23

12 2 23 137 5 19 28

16 5 7 1215 26 27 1318 25128

10Mean 11.071 15.400 18.091 19.500St.Dev.

3.951 11.118 7.134 7.106

SIT = Stress Inoculation Therapy

PE = Prolonged Exposure

SC = Supportive Counseling

WL = Waiting List Control

Grand mean = 15.622

Tentative ConclusionsTentative Conclusions

• Fewer symptoms with SIT and PE Fewer symptoms with SIT and PE than with other twothan with other two

• Also considerable variability within Also considerable variability within treatment groupstreatment groups

• Is variability among means just a Is variability among means just a reflection of variability of individuals?reflection of variability of individuals?

CalculationsCalculations

• Almost the same as earlierAlmost the same as earlier Note differencesNote differences

• We multiply by We multiply by nnjj as we go along. as we go along.

• MSMSerrorerror is now a is now a weighted averageweighted average. .

Cont.

Calculations--cont.Calculations--cont.

889.83

8.5076.2786

8.507

62.15500.1910)62.15091.18(1162.15118.111062.15071.1114

6.2786

62.1513...62.1513)62.153(

)(

22

22

2..

222

2..

groupstotalerror

jjgroups

total

SSSSSS

XXnSS

XXSS

Summary TableSummary Table

F.05(3,41) = 2.84

ConclusionsConclusions

• FF is significant at is significant at = .05 = .05

• The population means are not all The population means are not all equal equal

• Some therapies lead to greater Some therapies lead to greater improvement than others.improvement than others. SIT appears to be most effective.SIT appears to be most effective.

Multiple ComparisonsMultiple Comparisons

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

• Such procedures are designed to Such procedures are designed to control familywise error rate.control familywise error rate. Familywise error rate definedFamilywise error rate defined

Contrast with per comparison error rateContrast with per comparison error rate

More on Error RatesMore on Error Rates

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

• The more tests we run the more The more tests we run the more likely we are to make Type I error.likely we are to make Type I error. Good reason to hold down number of Good reason to hold down number of

teststests

Fisher’s LSD ProcedureFisher’s LSD Procedure

• Requires significant overall Requires significant overall F,F, or no tests or no tests

• Run standard Run standard tt tests between pairs of tests between pairs of groups.groups. Often we replace Often we replace s s 22

jj or pooled estimate or pooled estimate with MSwith MSerrorerror from overall analysis from overall analysis

• It is really just a pooled error term, but with It is really just a pooled error term, but with more degrees of freedom--pooled across all more degrees of freedom--pooled across all treatment groups.treatment groups.

Bonferroni Bonferroni tt Test Test

• Run Run tt tests between pairs of tests between pairs of groups, as usualgroups, as usual Hold down number of Hold down number of tt tests tests

Reject if Reject if tt exceeds critical value in exceeds critical value in Bonferroni tableBonferroni table

• Works by using a more strict value Works by using a more strict value of of for each comparison for each comparison

Cont.

Bonferroni Bonferroni tt--cont.--cont.• Critical value of Critical value of for each test set at .05/ for each test set at .05/cc, ,

where where cc = number of tests run = number of tests run Assuming familywise Assuming familywise = .05 = .05

e. g. with 3 tests, each e. g. with 3 tests, each tt must be significant must be significant at .05/3 = .0167 level.at .05/3 = .0167 level.

• With computer printout, just make sure With computer printout, just make sure calculated probability < .05/calculated probability < .05/cc

• Necessary table is in the bookNecessary table is in the book

Assumptions for Anal. of Assumptions for Anal. of Var.Var.

• Assume:Assume: Observations normally distributed Observations normally distributed

within each within each populationpopulation

Population variances are equalPopulation variances are equal• Homogeneity of variance or Homogeneity of variance or

homoscedasticityhomoscedasticity

Observations are independentObservations are independent

Cont.

Assumptions--cont.Assumptions--cont.

• Analysis of variance is generally Analysis of variance is generally robust to first tworobust to first two A robust test is one that is not greatly A robust test is one that is not greatly

affected by violations of assumptions.affected by violations of assumptions.

Magnitude of EffectMagnitude of Effect

• Eta squared (Eta squared (22)) Easy to calculateEasy to calculate

Somewhat biased on the high sideSomewhat biased on the high side

FormulaFormula• See slide #33See slide #33

Percent of variation in the data that can Percent of variation in the data that can be attributed to treatment differencesbe attributed to treatment differences

Cont.

Magnitude of Effect--cont.Magnitude of Effect--cont.

• Omega squared (Omega squared (22)) Much less biased than Much less biased than 22

Not as intuitiveNot as intuitive

We adjust both numerator and We adjust both numerator and denominator with MSdenominator with MSerrorerror

Formula on next slideFormula on next slide

12.6.556.2786)6.55(38.507)1(

18.6.2786

8.507

2

2

errortotal

errorgroups

total

groups

MSSS

MSkSS

SS

SS

22 and and 22 for Foa, et al. for Foa, et al.

• 22 = .18: 18% of variability in = .18: 18% of variability in symptoms can be accounted for by symptoms can be accounted for by treatmenttreatment

• 22 = .12: This is a less biased = .12: This is a less biased estimate, and note that it is 33% estimate, and note that it is 33% smaller.smaller.

Other Measures of Effect Other Measures of Effect SizeSize

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

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

Darley & Latene (1968)Darley & Latene (1968)

Condition:Condition:

Alone One OtherAlone One Other Four Four OthersOthers

n 13 26 13n 13 26 13

X .87 .72 .51X .87 .72 .51

Darley & Latene (1968)Darley & Latene (1968)