MEASURES OF ASSOCIATION CHI SQUARE, CORRELATION, REGRESSION Research Methods.

MEASURES OF ASSOCIATIONCHI SQUARE, CORRELATION, REGRESSION

Research Methods

Relationships BetweenTwo (or More) Variables I

Association:The relationship to which two variables

covaryDirection: Positive, Negative, or No

relationshipMagnitude: Coefficient of Association: (0,1)

or (-1,1)Significance (Statistical Significance)

Relationships BetweenTwo (or More) Variables II

Significance (Statistical Significance):The likelihood that the association is due to

chance and is not true of the populationOr (inversely) level of confidence in the

resultsWe don’t expect to find the exact value of a

parameter, but something close (better as the random sample is larger)

Measures of Association(very abbreviated)

Nominal data: Lambda(Statistical significance: Chi-square)(Reduction in error: Tau-b)

Ordinal data: Gamma(Statistical significance: Chi-square or Gamma)(Reduction in error: Tau)

Interval or Ratio data: Pearson Correlation Coefficient(Statistical significance: t or Z)(Explained variance or reduction in error: R2)

Multivariate Analysis:Three or More Variables

Control for the effects of other factors:Value of DV is probably not influenced solely by the IV of interest. What else influences DV?

Alternative rival hypotheses:To persuade your audience, show that your theory and evidence is better than, or adds to, other possible explanations

Causality (earlier in semester)

Regression Example:Size of Legislatures

How large should a legislature be? “…no political problem is less susceptible of

a precise solution than that which relates to the number most convenient for a representative legislature” The Federalist, No. 55

Theory suggests legislature size should be a function of population

Stigler (1972): logged populationTaagepera (1972), Taagepera and Shughart

(1989): cube root of population

Population and Size of National Assemblies

0

100

200

300

400

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700

0 200 400 600 800 1000 1200

Cube Root of Population

Siz

e o

f L

egis

latu

re

Simple Regression

^L The SAS System 5 19:20 Tuesday, November 28, 2006 The REG Procedure Model: MODEL1 Dependent Variable: seats Number of Observations Read 140 Number of Observations Used 139 Number of Observations with Missing Values 1 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 1 2491897 2491897 315.10 <.0001 Error 137 1083429 7908.24296 Corrected Total 138 3575326 Root MSE 88.92830 R-Square 0.6970 Dependent Mean 172.03653 Adj R-Sq 0.6948 Coeff Var 51.69152 Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > |t| Intercept 1 -15.25735 12.96996 -1.18 0.2415 cubrtpop 1 0.91992 0.05182 17.75 <.0001

How Large Should aState Legislature Be?

H0: Si = ‾Ŝ Representative bodies fulfill functions and, for a certain level or type of government, all legislative chambers should be of the same size (S). Regardless of the polity (i) represented, an optimal size for a legislature exists for given functions.

H1: Si = f( Pi ) Chamber size is positively correlated with population (P) of the polity (through some transformation).

H2: Ci = f( Xi ) Chamber size varies with regard to characteristics of the polity represented. These characteristics include population, political culture, geography, ethnic or other cleavages, social mobilization, the economy, and interest diversity.

Size of Legislatures

The range over assembly size is rather small. National

Assembly Lower

House (US) Upper

House (US) Smallest 12 40 20 Largest 659 400 (203) 67 Mean 171 110 39 Median 109 100 38

Population and Size of Lower House - U.S. States

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0 50 100 150 200 250 300 350

Cube Root of Population

Sea

ts in

Ch

amb

er

Population and Size of Lower House - U.S. States (Excluding NH)

0

50

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0 50 100 150 200 250 300 350

Cube Root of Population

Sea

ts in

Ch

amb

er

Simple Regression

The SAS System 1 00:19 Wednesday, November 29, 2006 The REG Procedure Model: MODEL1 Dependent Variable: lower03 Number of Observations Read 49 Number of Observations Used 49 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 1 12938 12938 10.88 0.0019 Error 47 55882 1188.97866 Corrected Total 48 68820 Root MSE 34.48157 R-Square 0.1880 Dependent Mean 103.26531 Adj R-Sq 0.1707 Coeff Var 33.39124 Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > |t| Intercept 1 54.73439 15.51501 3.53 0.0009 cubrtpop 1 0.30105 0.09126 3.30 0.0019

Goodness of Fit between Population and Chamber Size – U.S. States

Transformation Lower Upper Combined

None .10 * .08 .11 * Cube Root .19 ** .15** .21 *** Log10 .21 *** .18** .23 *** Cube Root of .21 *** .18** .24 *** Log 10 Log/ Log (seats & .24 *** .20** .24 *** population) Values are r2 for one-tailed test (positive relationship), n=49. * Significant at .025 level. ** Significant at .005 level. *** Significant at .0005 level. All tests exclude New Hampshire.

Factors that May Contributeto Legislature Size

1. Population “…however small the republic may be, the representatives must be raised to

a certain number, in order to guard against the cabals of a few; and that, however large it may be, they must be limited to a certain number, in order to guard against the confusion of a multitude.”

2. Population Change

Responsiveness Lag, Status Quo Stability

3. Wealth “But the most common and durable source of factions has been the various

and unequal distribution of property.”

4. Geography

“The smaller the society, the fewer probably will be the distinct parties and interests composing it…. Extend the sphere, and you take in a greater variety of parties and interests; you make it less probable that a majority of the whole will have a common motive to invade the rights of other citizens….”

5. Interest Diversity

“A landed interest, a manufacturing interest, a mercantile interest, a moneyed interest, with many lesser interests, grow up of necessity in civilized nations, and divide them into different classes, actuated by different sentiments and views. The regulation of these various and interfering interests forms the principal task of modern legislation….”

Multiple Regression

Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 5 21290 4257.96811 4.07 0.0048 Error 37 38748 1047.25258 Corrected Total 42 60038 Root MSE 32.36128 R-Square 0.3546 Dependent Mean 107.25581 Adj R-Sq 0.2674 Coeff Var 30.17205 Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > |t| Intercept 1 8.42148 68.23537 0.12 0.9024 cubrtpop 1 0.26674 0.09810 2.72 0.0099 popchg90 1 -1.28636 0.63504 -2.03 0.0501 pci00 1 -0.38773 1.24462 -0.31 0.7572 area 1 -0.10661 0.05748 -1.85 0.0716 igconc 1 7.10605 5.04406 1.41 0.1672

Chamber Size Predicted by Constituency Characteristics – U. S. State Legislatures

Lower House Upper House Both Chambers Cube Root Log / Log Cube Root Log / Log Cube Root Log / Log

Population (trans-formed)

.267*** (.098)

.181*** (.051)

.076** (.028)

.123*** (.036)

.398*** (.117)

.168*** (.048)

Pop. Change from 1990 (%)

-1.296** (.635)

-.00668*** (.00265)

-.244 (.189)

-.00311 (.00195)

-1.458 (.754)

-.00531 (.00251)

Per Capita Income ($000s)

-.388 (1.24)

.00459 (.00513)

-.347 (.362)

-.00537 (.00369)

-.668 (1.479)

-.00444 (.00487)

Area (000s sq. mi.)

0.107* (.057)

.00491* (.00241)

-.0236 (.0170)

-.288 (.175)

-.134 (.068)

.445 (.229)

Interest Group Concen-tration

7.106 (5.044)

1.286 (2.086)

-1.328 (1.485)

-.017 (.152)

6.387 (5.992)

.0081 (.0198)

Intercept 8.421 (68.235)

.897** (.390)

59.579 (20.022)

1.216*** (.290)

56.297 (81.061)

1.161*** (.370)

R2 (Adjusted)

.355 (.267)

.431 (.354)

.249 (.148)

.356 (.269)

.360 (.273)

.398 (.317)

F 4.07*** 5.59*** 2.45** 4.09*** 4.16*** 4.89*** Values are parameter estimates, standard errors in parentheses. Significance for two-tailed tests: * .10 level; ** .05 level; *** .01 level.

Crosstabulation Example:Gender and Party

Is there a “Gender Gap” between two parties?

Which is the dependent variable and which is the independent variable?

Compute percentages across the dependent variable, that is, sum to 100% for each independent variable.

Partisanship by Gender

1996 American National Election Study 1 PS 3000 Contingency Table Analysis 16:44 Tuesday, November 28, 2006 The FREQ Procedure Table of gender by party gender party Frequency| Percent | Row Pct | Col Pct |Demo |Ind |Repub | Total ---------+--------+--------+--------+ Female | 409 | 228 | 230 | 867 | 25.89 | 14.43 | 14.56 | 54.87 | 47.17 | 26.30 | 26.53 | | 61.69 | 51.12 | 48.83 | ---------+--------+--------+--------+ Male | 254 | 218 | 241 | 713 | 16.08 | 13.80 | 15.25 | 45.13 | 35.62 | 30.58 | 33.80 | | 38.31 | 48.88 | 51.17 | ---------+--------+--------+--------+ Total 663 446 471 1580 41.96 28.23 29.81 100.00 Frequency Missing = 134

Partisanship by Gender:Significance and Association

Statistics for Table of gender by party Statistic DF Value Prob ------------------------------------------------------ Chi-Square 2 21.9160 <.0001 Likelihood Ratio Chi-Square 2 22.0215 <.0001 Mantel-Haenszel Chi-Square 1 19.7065 <.0001 Phi Coefficient 0.1178 Contingency Coefficient 0.1170 Cramer's V 0.1178 Statistics for Table of gender by party Statistic Value ASE ------------------------------------------------------ Gamma 0.1863 0.0403 Kendall's Tau-b 0.1072 0.0236 Stuart's Tau-c 0.1221 0.0269 Somers' D C|R 0.1233 0.0271 Somers' D R|C 0.0932 0.0205 Pearson Correlation 0.1117 0.0250 Spearman Correlation 0.1135 0.0249 Lambda Asymmetric C|R 0.0000 0.0000 Lambda Asymmetric R|C 0.0154 0.0302 Lambda Symmetric 0.0067 0.0133 Uncertainty Coefficient C|R 0.0064 0.0027 Uncertainty Coefficient R|C 0.0101 0.0043 Uncertainty Coefficient Symmetric 0.0079 0.0033

Crosstabulation Example:Ideology and Party

Are the two parties ideological distinct?Which is the dependent variable and which is

the independent variable?[Here, party=f(ideology), not all would agree!]

Again, compute percentages across the dependent variable, that is, sum to 100% for each independent variable.

Partisanship by Ideology

1996 American National Election Study 3 PS 3000 Contingency Table Analysis 16:44 Tuesday, November 28, 2006 The FREQ Procedure Table of party by ideology party ideology Frequency| Percent | Row Pct | Col Pct |1Lib |2Mod |3Con | Total ---------+--------+--------+--------+ Demo | 226 | 154 | 104 | 484 | 18.12 | 12.35 | 8.34 | 38.81 | 46.69 | 31.82 | 21.49 | | 70.19 | 41.96 | 18.64 | ---------+--------+--------+--------+ Ind | 75 | 137 | 133 | 345 | 6.01 | 10.99 | 10.67 | 27.67 | 21.74 | 39.71 | 38.55 | | 23.29 | 37.33 | 23.84 | ---------+--------+--------+--------+ Repub | 21 | 76 | 321 | 418 | 1.68 | 6.09 | 25.74 | 33.52 | 5.02 | 18.18 | 76.79 | | 6.52 | 20.71 | 57.53 | ---------+--------+--------+--------+ Total 322 367 558 1247 25.82 29.43 44.75 100.00 Frequency Missing = 467

Statistics for Table of party by ideology Statistic DF Value Prob ------------------------------------------------------ Chi-Square 4 342.6161 <.0001 Likelihood Ratio Chi-Square 4 360.6052 <.0001 Mantel-Haenszel Chi-Square 1 313.3983 <.0001 Phi Coefficient 0.5242 Contingency Coefficient 0.4643 Cramer's V 0.3706 Statistic Value ASE ------------------------------------------------------ Gamma 0.6489 0.0253 Kendall's Tau-b 0.4561 0.0209 Stuart's Tau-c 0.4470 0.0205 Somers' D C|R 0.4512 0.0209 Somers' D R|C 0.4610 0.0211 Pearson Correlation 0.5015 0.0222 Spearman Correlation 0.5025 0.0226 Lambda Asymmetric C|R 0.1829 0.0321 Lambda Asymmetric R|C 0.2844 0.0229 Lambda Symmetric 0.2362 0.0220 Uncertainty Coefficient C|R 0.1352 0.0129 Uncertainty Coefficient R|C 0.1327 0.0127 Uncertainty Coefficient Symmetric 0.1340 0.0128

An Example:Simpson’s Paradox I

Simpson’s Paradox II

Same Example:Spurious Relationship I

Spurious Relationship II

Causality

“Correlation is not causation”Null hypothesis can be rejected, not “proven”

or “accepted”Causality can be rejected, not “proven” or

“accepted”Two variables may show a relationship or

association. But does one cause the other?

Suggesting Causality

Covariation or Association Demonstrated

Time Order Shown

Causal Linkage Explained

Alternative Explanations Eliminated