Multivariate Analysis

Richard LeGates

URBS 492

The Elaboration Model

• History– Developed by Paul Lazarfeld at Columbia in 1946

– Based on Stouffers’ research explaining army morale in relation to reference groups and relative deprivation

• The Elaboration Pardigm– Identify a relationship between a dependent and an independent,

by e.g. Crosstab

– Divide into two subsets by introducing a third “control” variable

– Determine whether the test variable is antecedent (prior in time) or intervening between the other two variables

– Examine the relationships in the two partial tables

How the Model Clarifies The Relationship Among 3 Variables

• It is not possible to “prove” that one variable causes another.

• But one can:

– increase confidence that it one variable does cause another (through replication)

– Prove that one variable does not cause another by showing the relationship is spurious (explanation)

– Show that while there is a true relationship between two variables the apparent causal variable is mediated through a third variable (specification)

Replication, Explanation, Interpretation

• Replication– Partial relationships are essentially the same as in the original

– applies to both antecedent and intervening variables

• Explanation– Describes a spurious (false) relationship. E.g. Urban-ness explains the

relationship between storks and babies

– Test variable must be antecedent

– Partial relationships must be zero or significantly less than in the original.

• Interpretation– Test variable is intervening

– relationship diminishes. E.g. Education/acceptance of induction

– Original relationship genuine, but mediated through intervening variable

Descriptive Statistics

• Data Reduction capture central tendency in data.

• Proportional reduction of error (PRE) statistics– Nominal variables: lamda 0 to 1. 1.0 = perfect assocation | 0 = none

– Ordinal variables: gamma -1 (perfect negative) to + 1 (perfect positive)

– Ratio level variables: Pearson’s Product-moment correlation coefficient.• From -1 perfect negative association to + 1 perfect positive.

• Correlation matrices

• Regression analysis– Linear regression

– Multiple regression

– How to do it | What it means | Why it is useful

Measures of Association

1.000 -.166 -.472** .438**

-.166 1.000 .639** .139

-.472** .639** 1.000 -.298**

.438** .139 -.298** 1.000

. .149 .000 .000

.149 . .000 .228

.000 .000 . .009

.000 .228 .009 .

77 77 77 77

77 77 77 77

77 77 77 77

77 77 77 77

Dependency ratio,central city, 1990

Foreign-bornpopulation, centralcity, 1990

Median home value,cc, 1990

Percent of all personsliving in poverty,central city, 1990

Dependency ratio,central city, 1990

Foreign-bornpopulation, centralcity, 1990

Median home value,cc, 1990

Percent of all personsliving in poverty,central city, 1990

Dependency ratio,central city, 1990

Foreign-bornpopulation, centralcity, 1990

Median home value,cc, 1990

Percent of all personsliving in poverty,central city, 1990

PearsonCorrelation

Sig.(2-tailed)

N

Dependencyratio, centralcity, 1990

Foreign-bornpopulation,central city,

1990

Medianhome

value, cc,1990

Percent ofall

persons living inpoverty,central

city, 1990

Correlations

Correlation is significant at the 0.01 level (2-tailed).**.

.018 .073 .243 .808

709

GammaOrdinal by Ordinal

N of Valid Cases

ValueAsymp.

Std. Errora Approx. TbApprox.

Sig.

Symmetric Measures

Not assuming the null hypothesis.a.

Using the asymptotic standard error assuming the null hypothesis.b.

.081 .020 3.927 .000

.000 .000 .c .c

.218 .050 3.927 .000

.061 .011 .000d

.210 .030 .000d

SymmetricLambdaNominalby Nominal

ValueAsymp.

Std. Errora Approx. TbApprox.

Sig.

Directional Measures

Not assuming the null hypothesis.a.

Using the asymptotic standard error assuming the null hypothesis.b.

Cannot be computed because the asymptotic standard error equals zero.c.

Based on chi-square approximationd.

Bivariate, linear relationships Between Ratio Level Variables

• Often social scientists want to see if there is a relationship between two ratio level variables.

• A pearsons product-motion correlation coefficient ( r ) expresses this relationship.

• Correlation coefficients run from– -1 (perfect negative correlation); to

– + 1 (perfect positive correlation)

– An r of zero show no correlation at all

– numbers close to 0 like - .07 or + .14 show “weak” positive or negative associations

1.000 -.166 -.472** .438**

-.166 1.000 .639** .139

-.472** .639** 1.000 -.298**

.438** .139 -.298** 1.000

. .149 .000 .000

.149 . .000 .228

.000 .000 . .009

.000 .228 .009 .

77 77 77 77

77 77 77 77

77 77 77 77

77 77 77 77

Dependency ratio,central city, 1990

Foreign-bornpopulation, centralcity, 1990

Median home value,cc, 1990

Percent of all personsliving in poverty,central city, 1990

Dependency ratio,central city, 1990

Foreign-bornpopulation, centralcity, 1990

Median home value,cc, 1990

Percent of all personsliving in poverty,central city, 1990

Dependency ratio,central city, 1990

Foreign-bornpopulation, centralcity, 1990

Median home value,cc, 1990

Percent of all personsliving in poverty,central city, 1990

PearsonCorrelation

Sig.(2-tailed)

N

Dependencyratio, central

city, 1990

Foreign-bornpopulation,central city,

1990

Medianhome

value, cc,1990

Percent ofall

persons living inpoverty,central

city, 1990

Correlations

Correlation is significant at the 0.01 level (2-tailed).**.

Visual Representation of bivariate linear relationships

• Scattergrams show the association between two ratio level variables

• An imaginary line running through the points may slope up or down (or have no slope)

– A positive relationship has an upward slope

– A negative relationship has a downward slope)

• Points clustered close to the regression line show a strong relationship;

• Points dispersed around the regression line show a weak relationship

Clinton and Dole Votes 1996 Election

DOLE96

605040302010

CL

IN9

6

80

70

60

50

40

30

Percent White

100806040200

Pe

rce

nt

Ye

s V

ote

fo

r P

rop

20

9

50

40

30

20

10