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Concept Map For Statistics as taught in IS271 (a work in progress). Correlation: Pearson. One Predictor. Regression. Analysis of Relationships. Multiple Predictors. Multiple Regression. Interval Data. Independent Samples t-test. Independent Groups. Between Two Groups. - PowerPoint PPT Presentation
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Analysis of Differences
Between Two Groups
Between Multiple Groups
IndependentGroups
DependentGroups
IndependentGroups
DependentGroups
IndependentSamples t-test
Repeated Measures t-test
IndependentSamples ANOVA
Repeated Measures ANOVA
FrequencyCHI Square
Nominal / Ordinal
Data
Some kinds ofRegression
Correlation:Pearson
RegressionAnalysis of
Relationships MultiplePredictors
Correlation:Spearman
MultipleRegression
OnePredictor
Concept Map For Statistics as taught in IS271(a work in progress)
Rashmi Sinha
IntervalData
Type of Data
OrdinalRegression
Analysis of Variance or F testANOVA is a technique for using differences between sample means to draw inferences about the presence or absence of differences between populations means.
•The logic of ANOVA and calculation in SPSS
•Magnitude of effect: eta squared, omega squared
Note: ANOVA is equivalent to t-test in case of two group situation
Logic of Analysis of Logic of Analysis of VarianceVariance
• Null hypothesis (HNull hypothesis (Hoo): Population ): Population means from different conditions are means from different conditions are equalequal mm11 = m = m22 = m = m33 = m = m44
• Alternative hypothesis: Alternative hypothesis: HH11
Not all population means equal.Not all population means equal.
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 FThe larger the error variance, the smaller the F
LogicLogic• Create a measure of variability among group means Create a measure of variability among group means
MSMSgroupgroup
• Create a measure of variability within groups Create a measure of variability within groups MSMSerrorerror
• Form ratio of Form ratio of MSMSgroupgroup /MS /MSerrorerror Ratio approximately 1 if null trueRatio approximately 1 if null true Ratio significantly larger than 1 if null falseRatio significantly larger than 1 if null false ““approximately 1” can actually be as high as 2 or 3, but not approximately 1” can actually be as high as 2 or 3, but not
much highermuch higher Look up statistical tables to see if F ratio is significant for Look up statistical tables to see if F ratio is significant for
the specified degrees of freedomthe specified degrees of freedom
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
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 different.We want to know what groups are different.
• Such procedures are designed to control Such procedures are designed to control familywise error rate.familywise error rate. Familywise error rate definedFamilywise error rate defined
Contrast with per comparison error rateContrast with per comparison error rate
In case of multiple comparisons: In case of multiple comparisons: Bonferroni Bonferroni adjustmentadjustment
• The more tests we run the more likely we are The more tests we run the more likely we are to make Type I error.to make Type I error. Good reason to hold down number of testsGood reason to hold down number of tests
• Run Run tt tests between pairs of groups, as usual tests between pairs of groups, as usual Hold down number of Hold down number of tt tests tests Reject if Reject if tt exceeds critical value in Bonferroni table exceeds critical value in Bonferroni 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
Bonferroni Bonferroni tt--cont.--cont.• Critical value of a for each test set Critical value of a for each test set
at .05/at .05/cc, where , where cc = number of tests run = number of tests run Assuming familywise a = .05Assuming familywise a = .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
Magnitude of EffectMagnitude of Effect• Why you need to compute magnitude of Why you need to compute magnitude of
effect indiceseffect indices
• Eta squared (hEta squared (h22)) Easy to calculateEasy to calculate
Somewhat biased on the high sideSomewhat biased on the high side
Percent of variation in the data that can be Percent of variation in the data that can be attributed to treatment differencesattributed to treatment differences
Magnitude of Effect--cont.Magnitude of Effect--cont.
• Omega squared (wOmega squared (w22)) Much less biased than hMuch less biased than h22
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
hh22 and w and w22 for Foa, et al. for Foa, et al.
• hh22 = .18: 18% of variability in = .18: 18% of variability in symptoms can be accounted for by symptoms can be accounted for by treatmenttreatment
• ww22 = .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.
Factorial Analysis of Variance
• What is a factorial design?What is a factorial design?
• Main effectsMain effects
• InteractionsInteractions
• Simple effectsSimple effects
• Magnitude of effectMagnitude of effect
What is a FactorialWhat is a Factorial
• At least two independent variablesAt least two independent variables
• All combinations of each variableAll combinations of each variable
• Rows X Columns factorialRows X Columns factorial
• CellsCells2 X 2 Factorial2 X 2 Factorial
Source of Product Review
Product Type Expert Review Peer ReviewLaptopDigital Camera
There are two factors in the experiment: Source of Review and Type of Product.
•If you examine effect of Source of Review (ignoring Type of Product for the time being), you are looking at the main effect of Source of Review.
•If we look at the effect of Type of Product, ignoring Source of Review, then you are looking at the main effect of Type of Product.
Main effects
Source of Product Review
Product Type Expert Review Peer ReviewLaptopDigital Camera
If you could restrict yourself to one level of one IV for the time being, and looking at the effect of the other IV within that level.
•Effect of Source of Review at one level of Product Type (e.g. for one kind of Product), then that is a simple effect.
•Effect of Product Type at one level of Source of Review (e.g. for one kind of Source, then that is a simple effect.
Simple effects
Source of Product Review
Product Type Expert Review Peer ReviewLaptopDigital Camera
Simple of Effect of Simple of Effect of Product Type at one Product Type at one level of Source of level of Source of Review (I.e., one kind Review (I.e., one kind of Review Type, of Review Type, Expert Review)Expert Review)
Interactions (Effect of one variable on the other)
Source of Product Review
Product Type Expert Review Peer ReviewLaptop 5 12Digital Camera 10 6
Effect: Source of Review on Product Type
012
34567
89
10
Expert Review Peer Review
Sa
tis
fac
tio
n w
ith
Ad
vis
or
Laptop
Digital Camera
Source of Product Review
Product Type Expert Review Peer ReviewLaptop 8 5Digital Camera 9 6
Effect: Source of Review on Product Type
0
2
4
6
8
10
12
14
16
18
Expert Review Peer Review
Sa
tis
fac
tio
n w
ith
Ad
vis
or
Laptop
Digital Camera
0
0.5
1
1.5
2
2.5
A1 A2 A3
0
0.5
1
1.5
2
2.5
3
3.5
A1 A2 A30
0.5
1
1.5
2
2.5
3
3.5
4
4.5
A1 A2 A3
0
0 . 5
1
1. 5
2
2 . 5
3
3 . 5
4
4 . 5
A1 A2 A3
0
0 . 5
1
1. 5
2
2 . 5
3
3 . 5
4
4 . 5
A1 A2 A3
0
0 . 5
1
1. 5
2
2 . 5
3
3 . 5
4
4 . 5
A1 A2 A3
Types of Interactions
And this is when there are only two variables!And this is when there are only two variables!
F ratio is biased because it goes up with F ratio is biased because it goes up with sample size. sample size. For a true estimate for the treatment effect For a true estimate for the treatment effect size, use eta squared (the proportion of the size, use eta squared (the proportion of the treatment effect / total variance in the treatment effect / total variance in the experiment). experiment).
Eta Squared is a better estimate than F but it Eta Squared is a better estimate than F but it is still a biased estimate. A better index is is still a biased estimate. A better index is Omega Squared. Omega Squared.
Magnitude of Effect
Magnitude of EffectMagnitude of Effect
• Eta SquaredEta Squared
InterpretationInterpretation
• Omega squaredOmega squared Less biased estimateLess biased estimate
total
effect
SS
SS2
errortotal
erroreffect
MSSS
MSkSS
)1(2
k = number of levels for the effectin question
R2 is also often used. It is based on the sum of squares. For experiments use Omega Squared. For correlations use R squared.
Value of R square is greater than omega squared.
Cohen classified effects as Small Effect: .01Medium Effect: .06Large Effect: .15
The Data The Data (cell means and standard (cell means and standard
deviations)deviations) No
Instructions
Instructions
Means
Male 7.7 (4.6)
6.2 (3.5)
6.95
Female 6.5 (4.2)
5.1 (2.8)
5.80
Means 7.1 5.65 6.375
Effects to be estimatedEffects to be estimated• Differences due to instructionsDifferences due to instructions
Errors more in condition without instructionsErrors more in condition without instructions
• Differences due to genderDifferences due to gender Males appear higher than femalesMales appear higher than females
• Interaction of video and genderInteraction of video and gender What is an interaction?What is an interaction?
Do instructions effect males and females equally?Do instructions effect males and females equally?
Cont.
Estimated Effects--cont.Estimated Effects--cont.
• ErrorError average within-cell varianceaverage within-cell variance
• Sum of squares and mean squaresSum of squares and mean squares Extension of the same concepts in the Extension of the same concepts in the
one-wayone-way
CalculationsCalculations
• Total sum of squaresTotal sum of squares
• Main effect sum of squaresMain effect sum of squares 2
..XXSStotal
2..XXngSS Vvideo
2..XXnvSS Ggender
Cont.
Calculations--cont.Calculations--cont.• Interaction sum of squaresInteraction sum of squares
Calculate SSCalculate SScellscells and subtract SS and subtract SSVV and SS and SSGG
• SSSSerrorerror = SS = SStotaltotal - SS - SScellscells
or, or, MSMSerrorerror can be found as average of cell variances can be found as average of cell variances
2..)( XXnSS ijcells
Degrees of FreedomDegrees of Freedom
• dfdf for main effects = number of for main effects = number of levels - 1levels - 1
• dfdf for interaction = product of for interaction = product of dfdfmain main
effectseffects
• dfdf errorerror = = NN - - abab = = NN - # cells - # cells
• dfdftotaltotal = = NN - 1 - 1
Calculations for DataCalculations for Data
• SSSStotaltotal requires raw data. requires raw data.
It is actually = 171.50It is actually = 171.50
• SSSSvideovideo
125.105
375.665.5375.61.7250
..22
2
XXngSS Vvideo
Cont.
Calculations--cont.Calculations--cont.
• SSSSgendergender
125.66
375.680.5375.695.6)2(50
..22
2
XXnvSS Ggender
Cont.
Calculations--cont.Calculations--cont.• SSSScellscells
• SSSSVXGVXG = SS = SScellscells - SS - SSinstructioninstruction- SS- SSgendergender
== 171.375 - 105.125 - 66.125 = 0.125 171.375 - 105.125 - 66.125 = 0.125
375.171)4275.3(50
)375.61.5()375.65.6(
)375.62.6()375.67.7(50
..)(
22
22
2
XXnSS cellcells
Cont.
Calculations--cont.Calculations--cont.
• MSMSerrorerror = average of cell variances = = average of cell variances =(4.6(4.622 + 3.5 + 3.522 + 4.2 + 4.222 + 2.8 + 2.822)/4 )/4 =58.89/4 = 14.723 =58.89/4 = 14.723
• Note that this is MSNote that this is MSerrorerror and not SS and not SSerrorerror
Summary TableSummary Table
Source df SS MS F Instructions
1 105.125 105.125 7.14
Gender 1 66.125 66.125 4.49 VXG 1 0.125 0.125 .01 Error 19
6 2885.610 14.723
Total 199
3056.980
Elaborate on InteractionsElaborate on Interactions
• Diagrammed on next slide as line Diagrammed on next slide as line graphgraph
• Note parallelism of linesNote parallelism of lines Instruction differences did not depend Instruction differences did not depend
on genderon gender