Action Research More Crosstab Measures

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Action ResearchMore Crosstab Measures

• INFO 515
• Glenn Booker

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Nominal Crosstab Tests

• Four more measures which could apply to nominal
  data in a crosstab
• Eta
• Lambda
• Goodman and Kruskals tau
• Uncertainty coefficient

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Eta Coefficient

• Used when the dependent variable uses an interval
  or ratio scale, and the independent variable is
  nominal or ordinal
• Eta (?) squared is the proportion of the
  dependent variables variance which is explained
  by the independent variable
• Eta squared is symmetric, and ranges from 0 to 1
• This is the same eta from the end of lecture 6

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Directional vs Symmetric

• Directional measures give a different answer
  depending on whether A is dependent on B, or B is
  dependent on A
• Symmetric measures dont care which variable is
  dependent or independent
• Tests indicate whether there is a statistically
  significant relationship measures, here,
  describe the strength of association

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Directional Measures

• Directional measures help determine how much the
  dependent variable is affected by the independent
  variable
• Directional measures for nominal data
• Lambda (recommended)
• Goodman and Kruskals tau
• Uncertainty coefficient

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Directional Measures

• Directional measures generally range from 0 to 1
• A value of 0 means the independent variable
  doesnt help predict the dependent variable
• A value of 1 means the independent variable
  perfectly predicts the resulting dependent
  variable

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Directional Measures

• In this context, either variable can be
  considered dependent or independent
• Does A predict B?
• Does B predict A?
• A symmetric value is the weighted average of
  the two possible selections (A predicts B, or B
  predicts A)

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Proportional Reduction in Error

• Proportional Reduction in Error (PRE) measures
  find the fractional reduction in errors due to
  some factor (such as an independent variable)
  PRE (Error without X Error with X) /
  Error with X
• Two well look at are Lambda, and Goodman and
  Kruskals Tau

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Lambda Coefficient

• Lambda has a symmetric option for output
• Its Value is the proportion of the dependent
  variable predicted by the independent one
• The Asymptotic Std. Error allows a 95 confidence
  interval to be made
• Approx. T is the Value divided by the Std.
  Error if the parameter were zero (not the usual
  definition!)

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Goodman and Kruskals Tau

• SPSS note Goodman and Kruskals Tau is not
  directly selected it appears only when Lambda is
  checked!
• Does not have Symmetric option
• Does not approximate T
• Based on chi square
• Otherwise similar to Lambda for interpretation

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Uncertainty Coefficient

• Does have symmetric dependency option
• Does have T approximation
• Also based on chi square
• Goodman and Kruskals tau and the Uncertainty
  Coefficient may give opposite results as Lambda,
  so use them cautiously!

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Nominal Example

• Use GSS91 political.sav data set
• Use Analyze / Descriptive Statistics / Crosstabs
• Select region for Row(s), and relig for
  Column(s)
• Under Statistics select Lambda, and
  Uncertainty Coefficient

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Nominal Example
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Nominal Example - Lambda

• Focus on the Lambda (l) output first
• Lambda measures the percent of error reduction
  when using the independent variable to predict
  the dependent variable
• Calculation based on any desired outcome
  contributing to lambda
• Lambda ranges from 0 to 1

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Nominal Example

• As usual, we want Sig. lt 0.050 for the meaning of
  lambda to be statistically significant
• If Region is dependent, then we see that
  religious preference is a significant (sig.
  0.000) predictor
• relig contributes (Value) 4.8 /- (Std Error)
  1.2 of the variability of a persons region

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Lambda Example

• 95 confidence interval of that contribution is
  (not shown) 4.8 21.2 2.4 to 4.8 21.2
  7.2
• But region is not a significant predictor of
  relig (sig. 0.099)
• Ignore the value of lambda if it isnt
  significant
• The symmetric value is significant, and its
  Value is between the other two lambda values

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G and K Tau Example

• Goodman and Kruskals tau (t) is similar to
  lambda, but is based on predictions in the same
  proportion as the marginal totals (individual row
  or column subtotals)
• No symmetric value is given its only
  directional
• Same method for interpretation, but notice it
  predicts both variables can be significant as
  dependent, and relig is much stronger!

Still from slide 13
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Uncertainty Coefficient Example

• Is a measure of association that indicates the
  proportional reduction in error when values of
  one variable are used to predict values of the
  other variable
• The program calculates both symmetric and
  directional versions of it
• Here, gives results similar to G and K Tau

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Tests for 2x2 Tables

• Many special measures can be applied to a 2x2
  table, including
• Relative risk
• Odds ratio
• Look at these in the context of answering
  questions like Are people who approve of women
  working more likely to vote for a woman
  President?

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Tests for 2x2 Tables

• Use GSS91 social.sav data set
• Variables are should women work (fework) and
  vote for woman president (fepres)
• Isolate the cases using Data / Select Cases
• Use the If condition(fepres1 fepres2)
  (fework1 fework2)

means or means and
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Tests for 2x2 Tables

• Use Analyze / Descriptive Statistics / Crosstabs
• Select fework for Row(s), and fepres for
  Column(s)
• For Statistics select Risk
• For Cells select Row percentages
• This gives 947 valid cases

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Tests for 2x2 Tables
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Tests for 2x2 Tables
cohort subset
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Relative Risk

• The relative risk is a ratio of percentages
• It is very directional
• Those who (approve of voting for a woman
  president) are 1.178 times as likely to (approve
  of women working)
• Based on 93.4/79.3 1.178
• Note the 95 confidence intervals for each ratio
  are given roughly 1.09 to 1.27 for this example

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Relative Risk

• Conversely, those who do not approve of voting
  for a woman president are 0.317 times as likely
  to approve of women working (6.6/20.70.317),
  with a broader confidence interval of 0.22 to 0.47

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Odds Ratio

• The odds ratio is the ratio of (the probability
  that the event occurs) to (the probability that
  the event does not occur)
• The odds ratio that someone who (would vote for
  a woman president) also (approves of women
  working) has two terms
• One is the ratio of (those who approve of women
  working) divided by (voting for a woman
  president) (93.4/6.614.152)...

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Odds Ratio

• Divided by the ratio of (those who would NOT
  approve of women working) (voting for a woman
  president) (79.3/20.73.831)
• Hence the odds ratio is 14.152/3.831 3.694 or
  (93.420.7)/(6.679.3)
• Round off error, probably in the 6.6 value, kept
  us from getting the stated odds ratio of 3.712
  (first row of output on slide 23)

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Square Tables (RxR)

• Tables with the same number of rows as columns
  (RxR tables) also have special measures
• Cohens Kappa (k), which measures the strength of
  agreement (did two peoples measurements match
  well?)
• Applies for R values of one nominal variable

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Kappa

• Kappa is used only when the rows and columns have
  the same categories
• Set of possible diagnoses achieved by two
  different doctors
• Two sets of outcomes which are believed to be
  dependent on each other
• Kappa ranges from zero to one is one when the
  diagonal has the only non-zero values

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Kappa Example

• Example here is the educational level of ones
  parents (maeduc and paeduc as in ma and pa
  education)
• Use GSS91 social.sav data set
• Define new variables madeg and padeg, which are
  derived from maeduc and paeduc (convert years of
  education into rough levels of achievement)

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Kappa Example

• New scale for madeg and padeg is
• Education lt12 is code 1, LT High School
• Education 12-15 is code 2, High School
• Education 16 is code 3, Bachelor degree
• Education 17 is code 4, Graduate
• Use Analyze / Descriptive Statistics / Crosstabs

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Kappa Example

• Select padeg for Row(s), and madeg for
  Column(s)
• For Statistics select Kappa
• The basic crosstab just shows the data counts
  (next slide)
• Then we get the Kappa measure (slide after next)
• As usual, check to make sure the result is
  significant before going any further

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Kappa Example
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Kappa Example
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Kappa Example

• Here the significance is 0.000, very clearly
  significant (lt 0.050)
• This is confirmed by the approximate T of over 20
  - as before, this T is based on the null
  hypothesis
• The actual value of kappa and its standard error
  are 0.325 /- 0.018
• What does this mean?

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Kappa

• Kappa is judged on a fairly fixed scale
• Kappa below 0.40 indicates poor agreement beyond
  chance
• Kappa from 0.40 to 0.75 is fair to good
  agreement
• Kappa above 0.75 is strong agreement
• So in this case we are confident there is poor
  agreement between parents education

Scale from J.L. Fleiss, 1981
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Ordinal Crosstab Measures

• Several association measures can be used for a
  table with R rows and C columns which contain
  ordinal data (and presumably R ? C)
• Kendalls tau-b
• Kendalls tau-c
• (Goodman and Kruskals) Gamma (preferred)
• Somers d
• Spearmans Correlation Coefficient

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General RxC Table Measures

• Many are based on comparing adjacent pairs of
  data from the two variables
• If B increases when A increases, the pair is
  concordant
• If B decreases when A increases, the pair is
  discordant
• If A and B are equal, the pair is tied

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General RxC Table Measures

• The number of concordant pairs is P
• The number of discordant pairs is Q
• The number of ties on X are Tx
• The number of ties on Y are Ty
• The smaller of the number of rows R and columns C
  is called m m min(R,C)
• Given this vocabulary, we can define many measures

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General RxC Table Measures

• Kendalls tau-b istau-b (P-Q) / sqrt
  (PQTx)(PQTy)
• Kendalls tau-c istau-c 2m(P-Q) / N2(m-1)
• Gamma (g) isGamma (P-Q) / (PQ)
• Somers d isdy (P-Q) / (PQTy) or dx (P-Q)
  / (PQTx)

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General RxC Table Measures

• All of the RxC measures are symmetric except
  Somers d, which has both symmetric and
  directional values given
• All are evaluated by their significance, which
  also has an approximate T score
• All are expressed by a Value /- its Std Error

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RxC Measures Example

• Use GSS91 social.sav data set
• Use Analyze / Descriptive Statistics / Crosstabs
• Select paeduc for Row(s), and maeduc for
  Column(s)
• Under Statistics select Eta, Correlations,
  Gamma, Somers d, Kendalls tau-b and tau-c

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RxC Measures Example

• This compares the number of years of education of
  ones mother and father to see how strongly they
  affect one another
• The crosstab data table is very large, since it
  ranges from 0 to 20 for each category, with
  irregular gaps (were not using the simplified
  categories from the Kappa example)
• Hence were not showing it here!

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RxC Measures Example
Both measures show the mothers education is a
slightly better predictor
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RxC Measures Example

• Directional measures
• Somers d is significant
• It shows that there are about 55 /- 2 more
  concordant pairs than discordant ones, excluding
  ties on the independent variable
• The Eta measure shows that around 69 of the
  variability of one parents education is shared
  with the others

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RxC Measures Example
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RxC Measures Example

• All of the symmetric measures are statistically
  significant, with approximate t values around
  27-28
• The Kendall tau-b and tau-c measures disagree a
  little on the magnitude of the agreement
• Gamma and Spearman give fairly strong positive
  correlations

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RxC Measures Example

• Spearman, like r, ranges from -1 to 1, and
  does not require a normal distribution
• Based on ordered categories, not their values
• Even r can be calculated for this case, and it
  gives results similar to Gamma and Spearman

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Yules Q

• A special case of gamma for a 2x2 table is called
  Yules Q
• It is appropriate for ordinal data in 2x2 tables
  so values for each variable are Low/High, Yes/No,
  or similar
• Define Yules Q (ad bc) /
  (ad bc)
• See PDF page 59 of Action Research handout for
  the definition of a, b, c, and d (cell labels)

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Yules Q

• Measures the strength and direction of
  association from -1 (perfect negative
  association) to 0 (no association) to 1 (perfect
  positive association)
• Judge the results for Yules Q by the table on
  page 59 of Action Research handout and see
  pages 58-64 for other related discussion