to 2 and beyond

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to ?2 and beyond

• Distribution
• Lots of tests produce statistics that are
  compared with the ?2 distribution
• Goodness of fit to compare single variables with
  a distribution
• ?2 goodness of fit
• Kolmogorov-Smirnov
• Shapiro-Wilk
• Tests of association between two variables

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Ratbert correct on 58 of the 100. Is this
different from chance?
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Pearson ?2 statistic A measure of deviation
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• Is this high enough to reject the model of no
  ESP?
• Need to know the residual degrees of freedom.
• Number of non-redundant pieces of information
  minus the number of pieces of information used in
  model.
• Two pieces of information. The model uses one (n
  used to calculate expected values), leaving one
  for the residual.
• ?2 table uses both the deviation value and the
  degrees of freedom.

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?2 distribution
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A Second Equation

• The likelihood ratio ?2 statistic
• Plugging in the numbers for this example yields
• Also non-significant
• L?2 can be partitioned like the sums of squares
  (SS) in ANOVA and regression models.

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http//glass.ed.asu.edu/stats/analysis/
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The Confidence Interval (Wald)
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• Wilson (1927) approach thought of as best.

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Men Winning More Awards

• A few good men
• Johnson, Carothers Deary (Nov, PPS) "Role of
  the X Chromosome"
• psychologicalscience.org/journals/pps/4_6_inpress
  /Johnson_final.pdf
• Wendy Northcutts (2000) approach to evolution
  Survival of the fittest only means that best
  genes remain in pool.
• Pool may still contain bad genes which just don't
  get passed onto the next gene pool (of course,
  one genetically damaged plant if propagated could
  destroy a field, or one prolific war monger).
• 131 men won Darwin Awards, but only 24 women.
• Odds is 131/24 5.46!

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Repeat Odds

• 131 men won a Darwin award
• 24 women won one
• Proportion of men winning is 131/1550.85
• Odds is 131/24 5.46.
• Odds are important for some statistics.
• In a second, as the odds ratio.
• Next term, as part of regression when the
  response variable is binary.

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Two Binary Variables Own Race Bias

• A Black confederate approached Black and White
  participants in South Africa. A few minutes
  later an RA approached the participant and asked
  them to identify the confederate from a lineup.
• 17 of the 25 Blacks (68) correct, odds 2.125
• 8 of the 25 Whites (32) correct, odds 0.471
• The odds ratio (OR) is 2.125/0.471 4.52.
• The odds ratio is a common measure of
  association, like the correlation, but for 2x2
  tables.

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• install.packages("sdtalt")
• library(sdtalt)
• sdt(17,8,8,17)

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Calculating Asymptotic CI (called the Wald CIs)

• 1. Take the ln of the observed OR. Here,
  ln(4.516)1.507.
• 2. Calculate the standard error on the log odds
  ratio
• Calculate the 95 confidence interval of ln OR
• lb ln OR - 1.96 se(ln OR) 1.507 - 1.96(0.606)
  0.319
• ub ln OR 1.96 se(ln OR) 1.507 1.96(0.606)
  2.69
• 4. Back-transform these into odds ratios
• exp(0.319) e0.319 1.376 and exp(2.69)
  e2.69 14.80

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Or write a function or find onehttps//home.comca
st.net/lthompson221/RCode.txt
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Differences due to rounding (these estimates are
more accurate)
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Calculating ?2

• E11 RT1 CT1 /n
• 12.5 (25) (25) /50
• Eij RTi CTj /n
• ln (Eij) ln(RTi) ln(CTj) - ln (n)

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?2 (1) 6.48, p .011
Pearson originally got the df wrong rc - 1
rather than (r-1)(c-1). Fisher corrected him. ?2
(1) 6.48, p .011
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Graphing Chi-Square residuals

• When the table is larger than 2x2, the Chi-Square
  value does not tell you where the effect is.
• Two approaches
• Residual statistics
• Correspondence analysis

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Two variables Multiple values(based on a UK
national survey)
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Degrees of Freedom

• 16 non-redundant pieces of information
• Equal numbers in all cells (one number)
• Accounting for column totals (3 additional)
• Accounting for column and row total (3
  additional)
• 7 df used in the model
• Leaving 9 df for the residuals. The ?2 test is
  of the residuals so use this for looking up.

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• L?2 30.56
• degrees of freedom (r-1)(c-1) 9 or calculate
  from 16-1-3-39
• Critical ?2 values for df9 are 16.92 and 21.67
  for a equal .05 and .01 respectively.
• So, an association has been detected.

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Pearson residuals (O-E)/sqrt(E)
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Association

• Find odds ratio for each 2x2 comparison.
• Useful if both variables are ordinal.

1.13
1.61
0.91
2.53
0.30
3.07
0.39
1.92
0.51
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How big is the association?

• Can look at odds ratios of each 2x2 comparison.
• Cramers V (and V2)
• SPSS gives lots of effect sizes. Cohen uses

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Where is the association? O-E Residuals
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Biggest residuals for married, but it is also the
largest row total.
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Standardized or Pearson's Residuals

• Square root of each cells contribution to the
  overall ?2

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Now widowed has the largest, but it has a small
row total.
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Correspondence Analysis(briefly and without math)

• Partitions the residual ?2 left over after the
  no association model
• If there are r rows and c columns, the program
  can use up to either (r-1) or (c-1) dimensions,
  whichever is smaller.
• Important Think about the size of association
• Analyze/Data Reduction/Correspondence Analysis in
  SPSS
• with corresp in MASS package for R (lots of
  others, too)

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?2 30 (is social class ordinal?)
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Lots of output minimum here
15 for 1st dimension, 9 2nd, and 1.5 for 3rd
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Does for class also.
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Schuman Scott (1989) Events by Age
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Three Variables

• There are techniques designed for multiple
  variables.
• Can use CA.
• Compute new variable
• Compute newvar age 6 gender
• 1 2 3 4 5 6 1
  2
• 7 8 9 10 11 12 13 14 15
  16 17 18
• CA comes to mind with newvar
• Biplot and drawing lines

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Let me repeat that

• Compute new variable
• Compute newvar age 6 gender
• 1 2 3 4 5 6 1
  2
• 7 8 9 10 11 12 13 14 15
  16 17 18

Make sure it works
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Heath et al. on social class and voting
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Summary

• Finding a significant Chi-square does not tell
  you where the effect lies.
• Graph different residuals and look at
  correspondence analysis

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Square Tables (if time allows)

• The rows and columns have the same values.
• Inter-rater reliability
• Two similar measures (two tests, two trials)
• Before-After studies
• Matched participants (eg., fathers/sons
  political party preference or social class)
• Can have more than 2 variables, but complexity
  increases.

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Special Models of Interest

• Equi-Probable (every cell equally likely)
• Independence (like what weve been doing)
• Quasi-independence
• Symmetry
• Quasi-symmetry

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Square Tables Inter-rater reliability

• Suppose a researcher was interested in the
  reliability of exam markers.
• Suppose there are first and second exam markers
  of a large number of exercises.
• Questions Are they reliable? Is one harsher
  than the other? Are non-agreements random?

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Contingency Table
Marker 1
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.067
6.25
2.40
1.00
0.40
2.50
16.67
1.00
0.12
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Equi-Probable w/ Diagonal Eij n/16
Marker 1
16 Cells. 1 used, 15 left. X2(15) 1187
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Equi-Probable w/o Diagonal Eij n/12
Marker 1
12 Cells. 1 used, 11 left. X2(11) 660
can also fit the diagonal, using those 4 df.
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Independence w/o Diagonal Eij RTi CTj /n
Marker 1
12 Cells. 7 used, 5 left. X2(5) 58, p lt .001
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Symmetry w/o Diagonal Eij Eii
Marker 1
12 Cells. 6 used, 6 left. X2(6) 149
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Quasi-Symmetry w/o Diagonal Eij Eii but
taking into account marginals
Marker 1
12 Cells. 9 used, 3 left. X2(3) 9.68 (p .02)
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Summary of Models

• Equi-probable with all data X2(15) 1187
• Equiprobably w/o diagonal X2(11) 660
• Add marginals Independence X2(5) 58
• Symmetry X2(6) 149
• Quasi-symmetry X2(3) 10
• Still significant, but small enough to accept
• Shows marginals different. Marker 1 is tougher.

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Quasi-Symmetry fits pretty wellResiduals and
standardized residuals
Marker 1
12 Cells. 9 used, 3 left. X2(3) 9.68 (p
.02) Marker 2 gives As to many that Marker 1
thinks are poor.
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Taking into account the ordinality

• Assume independence as the baseline.
• Include additional parameters to account for the
  association.
• Linear by linear
• RC model (Correspondence Analysis)

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Independence Eij RTi CTj / n
Marker 1
16 Cells. 7 used, 9 left. X2(9) 605
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Linear by Linear Term

• Eij RTi CTj / n
• ln Eij ln RTi ln CTj - ln n
• It is significant, but 9 df we have not located
  where the association is.
• If scores are interval, then including a term
  marker1marker2 tests for linear association.
• In SPSS, put interaction in as covariate and in
  model (with the two main effects).

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Linear by Linear Model
Marker 1
16 Cells. 8 used, 8 left. X2(9) 127
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Association Models

• Sometimes called uniform association model as all
  the local odds ratios are the same.
• Assume interval Row and Column values
• R models relax this for rows
• C models relax this for columns
• R C models for both
• RC(M) models and correspondence analysis

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Journal

• Be prepared to share your thoughts about your
  presentation at the next lecture.
• First Steps 10.2, 10.3