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Transcript of Sociology 5811: Lecture 16: Crosstabs 2 Measures of Association Plus Differences in Proportions...
![Page 1: Sociology 5811: Lecture 16: Crosstabs 2 Measures of Association Plus Differences in Proportions Copyright © 2005 by Evan Schofer Do not copy or distribute.](https://reader035.fdocuments.us/reader035/viewer/2022062321/56649de85503460f94ae1a51/html5/thumbnails/1.jpg)
Sociology 5811:Lecture 16: Crosstabs 2Measures of Association
Plus Differences in Proportions
Copyright © 2005 by Evan Schofer
Do not copy or distribute without permission
![Page 2: Sociology 5811: Lecture 16: Crosstabs 2 Measures of Association Plus Differences in Proportions Copyright © 2005 by Evan Schofer Do not copy or distribute.](https://reader035.fdocuments.us/reader035/viewer/2022062321/56649de85503460f94ae1a51/html5/thumbnails/2.jpg)
Announcements
• Final project proposals due Nov 15• Get started now!!!
• Find a dataset
• figure out what hypotheses you might test
• Today: Wrap up Crosstabs• If time remains, we’ll discuss project ideas…
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Review: Chi-square Test
• Chi-Square test is a test of independence
• Null hypothesis: the two categorical variables are statistically independent
• There is no relationship between them
• H0: Gender and political party are independent
• Alternate hypothesis: the variables are related, not independent of each other
• H1: Gender and political party are not independent
• Test is based on comparing the observed cell values with the values you’d expect if there were no relationship between variables.
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Review: Expected Cell Values
• If two variables are independent, cell values will depend only on row & column marginals– Marginals reflect frequencies… And, if frequency is
high, all cells in that row (or column) should be high
• The formula for the expected value in a cell is:
N
fff jiij
))((ˆ
• fi and fj are the row and column marginals
• N is the total sample size
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Review: Chi-square Test
• The Chi-square formula:
R
i
C
j ij
ijij
E
OE
1 1
22 )(
• Where:
• R = total number of rows in the table
• C = total number of columns in the table
• Eij = the expected frequency in row i, column j
• Oij = the observed frequency in row i, column j
– Assumption for test: Large N (>100)– Critical value DofF: (R-1)(C-1).
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Chi-square Test of Independence
• Example: Gender and Political Views– Let’s pretend that N of 68 is sufficient
Women Men
DemocratO11: 27
E11: 23.4
O12 : 10
E12 : 13.6
RepublicanO21 : 16
E21 : 19.6
O22 : 15
E22 : 11.4
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Chi-square Test of Independence
• Compute (E – O)2 /E for each cell
Women Men
Democrat(23.4 – 27)2/23.4
= .55(13.6 – 10)2/13.6
= .95
Republican(19.6 – 16)2/19.6
= .66
(11.4 – 15)2/15
= .86
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Chi-Square Test of Independence
• Finally, sum up to compute the Chi-square
• 2 = .55 + .95 + .66 + .86 = 3.02
• What is the critical value for =.05?• Degrees of freedom: (R-1)(C-1) = (2-1)(2-1) = 1
• According to Knoke, p. 509: Critical value is 3.84
• Question: Can we reject H0?• No. 2 of 3.02 is less than the critical value
• We cannot conclude that there is a relationship between gender and political party affiliation.
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Chi-square Test of Independence
• Weaknesses of chi-square tests:
• 1. If the sample is very large, we almost always reject H0.
• Even tiny covariations are statistically significant
• But, they may not be socially meaningful differences
• 2. It doesn’t tell us how strong the relationship is• It doesn’t tell us if it is a large, meaningful difference or a
very small one
• It is only a test of “independence” vs. “dependence”
• Measures of Association address this shortcoming.
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Measures of Association
• Separate from the issue of independence, statisticians have created measures of association– They are measures that tell us how strong the
relationship is between two variables
• Weak Association Strong Association
Women Men
Dem. 51 49
Rep. 49 51
Women Men
Dem. 100 0
Rep. 0 100
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Crosstab Association:Yule’s Q
• #1: Yule’s Q– Appropriate only for 2x2 tables (2 rows, 2 columns)
• Label cell frequencies a through d: a b
c d
• Recall that extreme values along the “diagonal” (cells a & d) or the “off-diagonal” (b & c) indicate a strong relationship.
• Yule’s Q captures that in a measure
• 0 = no association. -1, +1 = strong association
adbc
adbcQ
:Formula
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Crosstab Association:Yule’s Q
• Rule of Thumb for interpreting Yule’s Q:• Bohrnstedt & Knoke, p. 150
Absolute value of Q
Strength of Association
0 to .24 “virtually no relationship”
.25 to .49 “weak relationship”
.50 to .74 “moderate relationship”
.75 to 1.0 “strong relationship”
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a b
c d
Crosstab Association:Yule’s Q• Example: Gender and Political Party Affiliation
Women Men
Dem 27 10
Rep 16 15
Calculate “bc”
bc = (10)(16) = 160
Calculate “ad”
ad = (27)(15) = 405
adbc
adbcQ
405160
405160
48.505
245
• -.48 = “weak association”, almost “moderate”
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Association: Other Measures
• Phi ()• Very similar to Yule’s Q
• Only for 2x2 tables, ranges from –1 to 1, 0 = no assoc.
• Gamma (G)• Based on a very different method of calculation
• Not limited to 2x2 tables
• Requires ordered variables
• Tau c (c) and Somer’s d (dyx)• Same basic principle as Gamma
• Several Others discussed in Knoke, Norusis.
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Crosstab Association: Gamma
• Gamma, like Q, is based on comparing “diagonal” to “off-diagonal” cases.– But, it does so differently
• Jargon:
• Concordant pairs: Pairs of cases where one case is higher on both variables than another case
• Discordant pairs: Pairs of cases for which the first case (when compared to a second) is higher on one variable but lower on another
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Crosstab Association: Gamma
• Example: Approval of candidates– Cases in “Love Trees/Love Guns” cell make
concordant pairs with cases lower on both
Hate Trees
Trees OK
Love Trees
Love Guns
1205 603 71
Guns = OK
659 1498 452
Hate Guns
431 467 1120
All 71 individuals can be a pair with everyone in the
lower cells. Just Multiply!
(71)(659+1498+ 431+467) = 216,905 conc. pairs
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Crosstab Association: Gamma
• More possible concordant pairs– The “Love Guns/Trees are OK” cell and the “Trees =
OK/Love Guns” cells also can have concordant pairs
Hate Trees
Trees = OK
Love Trees
Love Guns
1205 603 71
Guns = OK
659 1498 452
Hate Guns
431 467 1120
These 603 can pair with all those that score lower on
approval for Guns & Trees
(603)(659 + 431) = 657,270 conc. pairs
These can pair lower too!
(452)(431 + 467) = 405,896 conc. pairs
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Crosstab Association: Gamma
• Discordant pairs: Pairs where a first person ranks higher on one dimension (e.g. approval of Trees) but lower on the other (e.g., app. of Guns)
Hate Trees
Trees = OK
Love Trees
Love Guns
1205 603 71
Guns = OK
659 1498 452
Hate Guns
431 467 1120
The top-left cell is higher on Guns but lower on Trees than those in the
lower right. They make pairs:
(1205)(1498 + 452 + 467 + 1120) = 4,262,085
discordant pairs
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Crosstab Associaton: Gamma
• If all pairs are concordant or all pairs are discordant, the variables are strongly related
• If there are an equal number of discordant and concordant pairs, the variables are weakly associated.
• Formula for Gamma:ds
ds
nn
nnG
• ns = number of concordant pairs
• nd = number of discordant pairs
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Crosstab Association: Gamma
• Calculation of Gamma is typically done by computer
• Zero indicates no association
• +1 = strong positive association
• -1 = strong negative association
• It is possible to do hypothesis tests on Gamma• To determine if population gamma differs from zero
• Requirements: random sample, N > 50
• See Knoke, p. 155-6.
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Crosstab Association
• Final remarks:
• You have a variety of possible measures to assess association among variables. Which one should you use?
• Yule’s Q and Phi require a 2x2 table
• Larger ordered tables: use Gamma, Tau-c, Somer’s d
• Ideally, report more than one to show that your findings are robust.
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Odds Ratios
• Odds ratios are a powerful way of analyzing relationships in crosstabs
• Many advanced categorical data analysis techniques are based on odds ratios
• Review: What is a probability?• p(A) = # of outcomes that are “A” divided by total number
of outcomes
• To convert a frequency distribution to a probability distribution, simply divide frequency by N
• The same can be done with crosstabs: Cell frequency over N is probability.
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Odds Ratios
• If total N = 68, probability of drawing cases is:
Women Men
Dem 27 / 68 10 / 68
Rep 16 / 68 15 / 68
Women Men
Dem .397 .147
Rep .235 .220
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Odds Ratios
• Odds are similar to probability… but not quite
• Odds of A = Number of outcomes that are A, divided by number of outcomes that are not A– Note: Denominator is different that probability
• Ex: Probability of rolling 1 on a 6-sided die = 1/6
• Odds of rolling a 1 on a six-sided die = 1/5
• Odds can also be calculated from probabilities:
i
ii p
podds
1
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Odds Ratios
• Conditional odds = odds of being in one category of a variable within a specific category of another variable– Example: For women, what are the odds of being
democrat?– Instead of overall odds of being democrat, conditional
odds are about a particular subgroup in a table
Women Men
Dem 27 10
Rep 16 15
Conditional odds of being democrat are:
27 / 16 = 1.69
Note: Odds for women are different than men
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Odds Ratios
• If variables in a crosstab are independent, their conditional odds are equal
• Odds of falling into one category or another are same for all values of other variable
• If variables in a crosstab are associated, conditional odds differ
• Odds can be compared by making a ratio• Ratio is equal to 1 if odds are the same for two groups
• Ratios much greater or less than 1 indicate very different odds.
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Odds Ratios
• Formula for Odds Ratio in 2x2 table:
ad
bc
ca
dbOR XY
Women Men
Dem 27 10
Rep 16 15
• Ex: OR = (10)(16)/(27)(15) = 160 / 405 = .395
• Interpretation: men have .395 times the odds of being a democrat compared to women
• Inverted value (1/.395=2.5) indicates odds of women being democrat = 2.5 is times men’s odds
a b
c d
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Odds Ratios: Final Remarks
• 1. Cells with zeros cause problems for odds ratios
• Ratios with zero in denominator are undefined.
• Thus, you need to have full cells
• 2. Odds ratios can be used to measure assocation• Indeed, Yule’s Q is based on them
• 3. Odds ratios form the basis for most advanced categorical data analysis techniques
• For now it may be easier to use Yule’s Q, etc. But, if you need to do advanced techniques, you will use odds ratios.
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Tests for Difference in Proportions• Another approach to small (2x2) tables:
• Instead of making a crosstab, you can just think about the proportion of people in a given category
• More similar to T-test than a Chi-square test
• Ex: Do you approve of Pres. Bush? (Yes/No)
• Sample: N = 86 women, 80 men
• Proportion of women that approve: PW = .70
• Proportion of men that approve: PM = .78
• Issue: Do the populations of men/women differ?• Or are the differences just due to sampling variability
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Tests for Difference in Proportions
• Hypotheses:
• Again, the typical null hypothesis is that there are no differences between groups
• Which is equivalent to statistical independence
• H0: Proportion women = proportion men
• H1: Proportion women not = proportion men• Note: One-tailed directional hypotheses can also be used.
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Tests for Difference in Proportions
• Strategy: Figure out the sampling distribution for differences in proportions
• Statisticians have determined relevant info:
• 1. If samples are “large”, the sampling distribution of difference in proportions is normal– The Z-distribution can be used for hypothesis tests
• 2. A Z-value can be calculated using the formula:
)(
21
21σ̂
ZPP
PP
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Tests for Difference in Proportions
• Standard error can be estimated as:
21
2211
NN
PNPNPboth
21
21)( )1(σ̂
21 NN
NNPP bothbothPP
• Where:
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Difference in Proportions: Example
• Q: Do you approve of Pres. Bush? (Yes/No)
• Sample: N = 86 women, 80 men
• Women: N = 86, PW = .70
• Men: N = 80, PW = .78
• Total N is “Large”: 166 people– So, we can use a Z-test
• Use = .05, two-tailed Z = 1.96
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Difference in Proportions: Example
• Use formula to calculate Z-value
)()()(
21
212121σ̂
08.
σ̂
78.70.
σ̂Z
PPPPPP
PP
• And, estimate the Standard Error as:
21
21)( )1(σ̂
21 NN
NNPP bothbothPP
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Difference in Proportions: Example
• First: Calculate Pboth:
21
2211
NN
PNPNPboth
739.166
4.622.60
bothP
8086
)78(.80)70(.86
bothP
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Difference in Proportions: Example
• Plug in Pboth=.739:
21
21)( )739.1(739.σ̂
21 NN
NNPP
)80)(86(
8086454.σ̂ )( 21
PP
104.6880
166674.σ̂ )( 21
PP
![Page 39: Sociology 5811: Lecture 16: Crosstabs 2 Measures of Association Plus Differences in Proportions Copyright © 2005 by Evan Schofer Do not copy or distribute.](https://reader035.fdocuments.us/reader035/viewer/2022062321/56649de85503460f94ae1a51/html5/thumbnails/39.jpg)
Difference in Proportions: Example
• Finally, plug in S.E. and calculate Z:
)()()(
21
212121σ̂
08.
σ̂
78.70.
σ̂Z
PPPPPP
PP
769.104.
08.
σ̂Z
)(
21
21
PP
PP
![Page 40: Sociology 5811: Lecture 16: Crosstabs 2 Measures of Association Plus Differences in Proportions Copyright © 2005 by Evan Schofer Do not copy or distribute.](https://reader035.fdocuments.us/reader035/viewer/2022062321/56649de85503460f94ae1a51/html5/thumbnails/40.jpg)
Difference in Proportions: Example
• Results:
• Critical Z = 1.96
• Observed Z = .739
• Conclusion: We can’t reject null hypothesis– Women and Men do not clearly differ in approval of
Bush