Confidence Limits on Confidence Limits on MeanMean

• Sample mean is a point estimateSample mean is a point estimate

• We want interval estimateWe want interval estimate Probability that interval computed this Probability that interval computed this

way includes way includes = 0.95 = 0.95

XstXCI 025.95.

For Our DataFor Our Data

97.723.687.01.7

44.098.11.7025.95.

XstXCI

Confidence IntervalConfidence Interval

• The interval does not include 5.65--The interval does not include 5.65--the population mean without a the population mean without a violent videoviolent video

• Consistent with result of Consistent with result of tt test test

• What can we conclude from What can we conclude from confidence interval?confidence interval?

Analysis of Variance

ANOVA is a technique for using differences between sample means to draw inferences about the presence or absence of differences between populations means.

Major PointsMajor Points

• The logicThe logic

• Calculations in SPSSCalculations in SPSS

• Magnitude of effectMagnitude of effect eta squaredeta squared

omega squaredomega squared

Cont.

Major Points--Major Points--AssumeptionsAssumeptions

• 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

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.

Logic of the Analysis of Logic of the Analysis of VarianceVariance

• Null hypothesis: Population means Null hypothesis: Population means from different conditions are equalfrom different conditions are equal 11 = =

• Alternative hypothesis: Alternative hypothesis: HH11

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

Cont.

Lets visualize total amount of Lets visualize total amount of variance in an experimentvariance in an experiment

Between Group Differences(Mean Square Group)

Error Variance (Individual Differences + Random Variance) Mean Square Error

Total Variance = Mean Square Total

F ratio is a proportion of the MS group/MS Error.The larger the group differences, the bigger the F

Logic--cont.Logic--cont.

• Create a measure of variability Create a measure of variability among group meansamong group means MSMSgroupgroup

• Create a measure of variability Create a measure of variability within groupswithin groups MSMSerrorerror

Cont.

Logic--cont.Logic--cont.

• Form ratio of MSForm ratio of MSgroupgroup /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

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.

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(

)(

222

2..

222

2..

groupstotalerror

jgroups

total

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

• NN observations observations

dfdfgroupsgroups = = gg - 1 - 1

• 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

When there are more than When there are more than two groupstwo groups

• 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 We want to know what groups are

different.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

Multiple ComparisonsMultiple Comparisons

• 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 or no teststests

• Run standard Run standard tt tests between pairs tests between pairs of groups.of 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 level of Works by using a more strict level of significance for each comparison significance 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

AssumptionsAssumptions

• 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.

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.