Analysis of

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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
Repeated Measures t-test
Dependent Groups
Analysis of Differences
Independent Samples ANOVA
Independent Groups
Type of Data
Between Multiple Groups
Repeated Measures ANOVA
Dependent Groups
Correlation Spearman
Nominal / Ordinal Data
Ordinal
Regression
CHI Square
Frequency
Some kinds of Regression
Rashmi Sinha
2
Analysis of Variance or F test
ANOVA 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

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Logic of Analysis of Variance

• Null hypothesis (Ho) Population means from
  different conditions are equal
• m1 m2 m3 m4
• Alternative hypothesis H1
• Not all population means equal.

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Lets visualize total amount of variance in an
experiment
Total Variance Mean Square Total
Between Group Differences (Mean Square Group)
Error Variance (Individual Differences Random
Variance) Mean Square Error
F ratio is a proportion of the MS group/MS
Error. The larger the group differences, the
bigger the F The larger the error variance, the
smaller the F
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Logic

• Create a measure of variability among group means
  MSgroup
• Create a measure of variability within groups
  MSerror
• Form ratio of MSgroup /MSerror
• 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
• Look up statistical tables to see if F ratio is
  significant for the specified degrees of freedom

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Grand mean 3.78
Hypothetical Data
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Calculations

• Start with Sum of Squares (SS)
• We need
• SStotal
• SSgroups
• SSerror
• Compute degrees of freedom (df )
• Compute mean squares and F

Cont.
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Calculations--cont.
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Degrees of Freedom (df )

• Number of observations free to vary
• dftotal N - 1
• N observations
• dfgroups g - 1
• g means
• dferror g (n - 1)
• n observations in each group n - 1 df
• times g groups

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Summary Table
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When there are more than two groups

• 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

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In case of multiple comparisons Bonferroni
adjustment

• The more tests we run the more likely we are to
  make Type I error.
• Good reason to hold down number of tests
• Run t tests between pairs of groups, as usual
• Hold down number of t tests
• Reject if t exceeds critical value in Bonferroni
  table
• Works by using a more strict level of
  significance for each comparison

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Bonferroni t--cont.

• Critical value of a for each test set at .05/c,
  where c number of tests run
• Assuming familywise a .05
• e. g. with 3 tests, each t must be significant at
  .05/3 .0167 level.
• With computer printout, just make sure calculated
  probability lt .05/c
• Necessary table is in the book

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Magnitude of Effect

• Why you need to compute magnitude of effect
  indices
• Eta squared (h2)
• Easy to calculate
• Somewhat biased on the high side
• Percent of variation in the data that can be
  attributed to treatment differences

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Magnitude of Effect--cont.

• Omega squared (w2)
• Much less biased than h2
• Not as intuitive
• We adjust both numerator and denominator with
  MSerror
• Formula on next slide

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h2 and w2 for Foa, et al.

• h2 .18 18 of variability in symptoms can be
  accounted for by treatment
• w2 .12 This is a less biased estimate, and
  note that it is 33 smaller.

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Factorial Analysis of Variance

• What is a factorial design?
• Main effects
• Interactions
• Simple effects
• Magnitude of effect

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What is a Factorial

• At least two independent variables
• All combinations of each variable
• Rows X Columns factorial
• Cells

2 X 2 Factorial
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Main effects

• 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.

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Simple effects

• 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 of Effect of Product Type at one level of
Source of Review (I.e., one kind of Review Type,
Expert Review)
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Interactions (Effect of one variable on the other)
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Types of Interactions
And this is when there are only two variables!
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Magnitude of Effect
F ratio is biased because it goes up with sample
size. For a true estimate for the treatment
effect size, use eta squared (the proportion of
the treatment effect / total variance in the
experiment). Eta Squared is a better estimate
than F but it is still a biased estimate. A
better index is Omega Squared.
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Magnitude of Effect

• Eta Squared
• Interpretation
• Omega squared
• Less biased estimate

k number of levels for the effect in question
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Omega Squared

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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 .01 Medium
Effect .06 Large Effect .15
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The Data (cell means and standard deviations)
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Plotting Results
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Effects to be estimated

• Differences due to instructions
• Errors more in condition without instructions
• Differences due to gender
• Males appear higher than females
• Interaction of video and gender
• What is an interaction?
• Do instructions effect males and females equally?

Cont.
30
Estimated Effects--cont.

• Error
• average within-cell variance
• Sum of squares and mean squares
• Extension of the same concepts in the one-way

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Calculations

• Total sum of squares
• Main effect sum of squares

Cont.
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Calculations--cont.

• Interaction sum of squares
• Calculate SScells and subtract SSV and SSG
• SSerror SStotal - SScells
• or, MSerror can be found as average of cell
  variances

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Degrees of Freedom

• df for main effects number of levels - 1
• df for interaction product of dfmain effects
• df error N - ab N - cells
• dftotal N - 1

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Calculations for Data

• SStotal requires raw data.
• It is actually 171.50
• SSvideo

Cont.
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Calculations--cont.

• SSgender

Cont.
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Calculations--cont.

• SScells
• SSVXG SScells - SSinstruction- SSgender
  171.375 - 105.125 - 66.125 0.125

Cont.
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Calculations--cont.

• MSerror average of cell variances (4.62
  3.52 4.22 2.82)/4 58.89/4 14.723
• Note that this is MSerror and not SSerror

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Summary Table
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Elaborate on Interactions

• Diagrammed on next slide as line graph
• Note parallelism of lines
• Instruction differences did not depend on gender

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Line Graph of Interaction