Title: SPSS Chapter 8 Correlation and Linear Regression
1SPSS Chapter 8Correlation and Linear Regression
2Review Measures of Association
- The most important thing is to answer the
question How strong is the relationship between
the independent and the dependent variables? - This distinction is important because it is
possible that a change in the independent
variable can cause only slight changes in the
dependent variable. - The calculated value of most measures of
association ranges from 0 to 1 or 1. A value of
zero indicates no relationship, a value of 1
indicates perfect positive relationship, and a
value of 1 indicates perfect negative
relationship. The sign of the measure indicates
the direction of the relationship
3Review PRE Measures (nominal or ordinal data)
- Prediction based How much better can we predict
the DV by knowing the IV than by not knowing the
IV? - General rules about PRE Relationships
- Less than 0.1 Weak
- Greater than 0.1 but less than 0.2 Moderate
- Greater than 0.2 but less than 0.3 Moderately
strong - Greater than 0.3 strong
4Pearsons r
- Relationship between 2 interval level variables
- Not PRE measure
- -1 (perfect negative) to 1 (perfect positive)
- Used to get overall picture of relationships
between variables
5Regression
- More specialized than correlation
- Regression Coefficient
- Estimates the effect of an IV on a DV
- PRE Measure of association
- How completely the IV explains the DV
- Variables
- DV Interval Level
- IV nominal, ordinal or interval
- Used to model causal relationships between one or
more IVs and a DV
6Correlation Example
- Interest Gender composition of State
Legislatures - Using Descriptives finds 6 to 40
- Explanations
- of college grads
- Christian residents
7Correlation Matrix
8Scatterplot
- Graphic plot of all cases across 2 variables
- Horizontal Axis IV (christad)
- Vertical Axis DV (womleg)
- Interactive Scatterplot tailored to report BASIC
regression estimates - Can also use scatter but does not report
regression estimates
9Scatterplot Regression Estimates
- Y a b (X)
- Y Dependent Variable
- X Independent Variable
- a Y Intercept
- Estimated value of Y when X 0
- b Regression Coefficient
- Change in DV for each 1-unit change in IV
- R-Square Strength of Relationship
- proportion of the variation in DV explained by
the IV
10Scatterplot Regression Estimates
- Y a b (X)
- Women Legislators 3.01 0.74 (college)
- For each 1-unit increase in the percentage of
college graduates, there is a .74-unit increase
in the percentage of female legislators.
11Example of scatterplot
This scatter plot represents age vs. size of a
plant. As the plant ages, its size tends to
increase. If it seems that the points follow a
linear pattern then we say that there is a high
linear correlation, while if it seems that the
data do not follow a linear pattern, we say that
there is no linear correlation. If the data
somewhat follow a linear path, then we say that
there is a moderate linear correlation
12Scatterplot Regression Estimates
- Estimate of women legislators for a state
- Start with intercept (3.01)
- Add 0.74 for each percentage of states pop with
a college degree - If 10 Y 3.01 .74 10 10.4
13Regression
- y a bx
- Bivariate
- One IV to predict a DV
- Multiple
- More than one IV to predict a DV
14Regression Example
- Factors related to motor vehicle deaths
- Population density
- Sparsly populated
- Drive More
- Drive Faster
- Fewer People to witness/help
- Fewer Hospitals
- Younger drivers more likely to get in accidents
15Hypotheses
- In comparing states, those with lower population
density will have higher number of car crash
fatalities than more populated states - In comparing states, those with more people under
20 will have higher number of car crash
fatalities than states with fewer people under 20
- Variables
- Carfatal motor vehicle deaths per 100k
residents - Density state population per square mile
- Under20 of population 19 years and younger
16Regression
- Y a b (X)
- MV Fatalities 20.278 0.014density
- Alaska 1 per sq mile
- 20 per 100,000
- New Jersey 1,000 per sq mile
- 6 per 100,000
- b for each addition person per square mile, MV
Fatality drops by .014.
17Regression Null Hypothesis
- In the population no relationship exists between
the IV and the DV - In the population the true regression coefficient
is equal to 0 - Further, the regression coefficient of -.014 was
occurred by chance.
18Regression Interpretation
- Coefficients
- Constant Y Intercept
- Pop Per Sq Mile IV
- B values for constant and IV
- T value from analysis (get p-value from
t-table) - Generally look for t (absolute value) above 2
significant - Sig P-values for this test
- Ignore Constant (for most applications)
- Probability of obtaining the results if null is
correct - If over .05 observed results occur too frequently
by chance
19Regression Interpretation
- Model Summary
- R-Square
- Overall measure of how well IV explains DV
- Multiply by 100 represents of variance
explained - Of all variation among states in MV fatality
rates 38.3 is explained by population density
20R-Squared
- ? b (standard error of b)
21Regression Example What do we know
- Population density significant negative effect on
MV Fatality - Percentage of Younger people has significant
positive effect on MV Fatality - Correlation between density and of younger
people
22Regression Example What do we know
- Population density significant negative effect on
MV Fatality - Percentage of Younger people has significant
positive effect on MV Fatality - Small correlation between density and of
younger people (-.139) - Density up of Young Down
- Perhaps states with higher percentages of young
people have higher fatality rates because they
have lower population densities, not younger
drivers. - Relationship between under20 and carfatal may be
spurious
23Multiple Regression
- Designed to estimate the partial effect of an IV
on DV, controlling for effect of other IVs - Y a b(X1) b(X2)
24- MV Fatalities 7.535 -.012(density) 0.428
(under20)