One-way Analysis of Variance Single Independent Variable Between-Subjects Design

Logic of the Analysis of Variance Null hypothesis H 0 : Population means equalNull hypothesis H 0 : Population means equal   1 =       Alternative hypothesis: H 1Alternative hypothesis: H 1 XNot all population means equal. Cont.

Logic--cont. Create a measure of variability among group meansCreate a measure of variability among group means XMs groups (accurate est. of pop. var. if null true) Create a measure of variability within groupsCreate a measure of variability within groups XMS error (accurate est. of pop. var. regardless of whether null is true) Cont.

Logic--cont. Form ratio of MS groups /MS errorForm ratio of MS groups /MS error XRatio approximately 1 if null true XRatio significantly larger than 1 if null false X“approximately 1” can actually be as high as 2 or 3, but not much higher

Epinephrine and Memory Based on Introini-Collison & McGaugh (1986)Based on Introini-Collison & McGaugh (1986) XTrained mice to go left on Y maze XInjected with 0,.1,.3, or 1.0 mg/kg epinephrine XNext day trained to go right in same Y maze Xdep. 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

Calculations Start with Sum of Squares (SS)Start with Sum of Squares (SS) XWe need: SS totalSS total SS groupsSS groups SS errorSS error Compute degrees of freedom (df )Compute degrees of freedom (df ) Compute mean squares and FCompute mean squares and F Cont.

Calculations--cont.

Degrees of Freedom (df ) Number of “observations” free to varyNumber of “observations” free to vary Xdf total = N - 1 Variability of N observationsVariability of N observations Xdf groups = g - 1 variability of g meansvariability of g means Xdf error = g (n - 1) n observations in each group = n - 1 dfn observations in each group = n - 1 df times g groupstimes g groups

Summary Table

Conclusions The F for groups is significant.The F for groups is significant. XWe would obtain an F of this size, when H 0 true, less than one time out of XThe difference in group means cannot be explained by random error. XThe 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.The injection of epinephrine following learning appears to consolidate that learning. High doses may have negative effect.High doses may have negative effect.

Unequal Sample Sizes With one-way, no particular problemWith one-way, no particular problem XMultiply mean deviations by appropriate n i as you go XThe problem is more complex with more complex designs, as shown in next chapter. Example from Foa, Rothbaum, Riggs, & Murdock (1991)Example from Foa, Rothbaum, Riggs, & Murdock (1991)

Post-Traumatic Stress Disorder Four treatment groups given psychotherapy X XStress Inoculation Therapy (SIT) Standard techniques for handling stress X XProlonged exposure (PE) Reviewed the event repeatedly in their mind Cont.

Post-Traumatic Stress Disorder- -cont. XSupportive counseling (SC) Standard counseling XWaiting List Control (WL) No treatment

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

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

Calculations Almost the same as earlierAlmost the same as earlier XNote differences We multiply by n j as we go along.We multiply by n j as we go along. MS error is now a weighted average.MS error is now a weighted average. Cont.

Calculations--cont.

Summary Table F.05 (3,41) = 2.84

Conclusions F is significant at  =.05F is significant at  =.05 The population means are not all equalThe population means are not all equal Some therapies lead to greater improvement than others.Some therapies lead to greater improvement than others. XSIT appears to be most effective.

Multiple Comparisons Significant F only shows that not all groups are equalSignificant F only shows that not all groups are equal XWe want to know what groups are different. Such procedures are designed to control familywise error rate.Such procedures are designed to control familywise error rate. XFamilywise error rate defined XContrast with per comparison error rate

More on Error Rates Most tests reduce significance level (  ) for each t test.Most tests reduce significance level (  ) for each t test. The more tests we run the more likely we are to make Type I error.The more tests we run the more likely we are to make Type I error. XGood reason to hold down number of tests

Fisher’s LSD Procedure Requires significant overall F, or no testsRequires significant overall F, or no tests Run standard t tests between pairs of groups.Run standard t tests between pairs of groups. XOften we replace s 2 j or pooled estimate with MS error from overall analysis It is really just a pooled error term, but with more degrees of freedom--pooled across all treatment groups.It is really just a pooled error term, but with more degrees of freedom--pooled across all treatment groups.

Bonferroni t Test Run t tests between pairs of groups, as usualRun t tests between pairs of groups, as usual XHold down number of t tests XReject if t exceeds critical value in Bonferroni table Works by using a more strict value of  for each comparisonWorks by using a more strict value of  for each comparison Cont.

Bonferroni t--cont. Critical value of  for each test set at.05/c, where c = number of tests runCritical value of  for each test set at.05/c, where c = number of tests run  Assuming familywise  =.05 Xe. g. with 3 tests, each t must be significant at.05/3 =.0167 level. With computer printout, just make sure calculated probability <.05/cWith computer printout, just make sure calculated probability <.05/c Necessary table is in the bookNecessary table is in the book

Assumptions for Anal. of Var. Assume:Assume: XObservations normally distributed within each population XPopulation variances are equal Homogeneity of variance or homoscedasticityHomogeneity of variance or homoscedasticity XObservations are independent Cont.

Assumptions--cont. Analysis of variance is generally robust to first twoAnalysis of variance is generally robust to first two XA robust test is one that is not greatly affected by violations of assumptions.

Magnitude of Effect Eta squared (  2 )Eta squared (  2 ) XEasy to calculate XSomewhat biased on the high side XFormula See slide #33See slide #33 XPercent of variation in the data that can be attributed to treatment differences Cont.

Magnitude of Effect--cont. Omega squared (  2 )Omega squared (  2 )  Much less biased than  2 XNot as intuitive XWe adjust both numerator and denominator with MS error XFormula on next slide

 2 and  2 for Foa, et al.  2 =.18: 18% of variability in symptoms can be accounted for by treatment  2 =.18: 18% of variability in symptoms can be accounted for by treatment  2 =.12: This is a less biased estimate, and note that it is 33% smaller.  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.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.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 X

Darley & Latene (1968)