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Chapter 9
Analyzing Data
Multiple Variables
Basic Directions
Review page 180 for basic directions on
which way to proceed with your analysis
 Provides
statistical decision steps based
upon the level of measurement for your
independent and dependent variables
Elaboration ‘Models’
An association has been found to be
statistically significant
Consider controlling for variables that
would serve as plausible explanations
Run chi-square or other comparable tests
Partialling
When a control variable is introduced, that is
deemed first-order partialling

Should you add a second, 2nd order, and so on
The original bivariate relationship is called the
zero-order relationship
Good for replicating patterns
Can use minitab stat > tables and put in multiple
variables of interest

Don’t use too many – keep it clean
Spurious Relationships
If you introduce a third variable (a control)
and the relationship that existed in the
bivariate setting is now non-significant or
even less strong… then, the original
relationship is spurious
Consider the ice cream and murder example
Specification
Specification: when the control variable leads to
only ‘some’ of the values of the test variable to
become non-significant or weakened
It is called specification because there is a
determination of which relationship holds
Suppressing Relationships
If there is no relationship or a very weak one, introduce
control variable to see if the ‘weak’ relationship continues

Could be that the variables involved are suppressor
variables
Within this structure you can also identify the
intervening variables: the one that was keeping the
original relationship weak
Partial Correlations
When a correlation exists between two variables,
X and Y, the correlation may be explained by a
third variable that is correlated with both X and Y.
A partial correlation is used to control for the
effect of a third variable when examining the
correlation between X and Y.
If the correlation between X and Y is reduced, the
third variable is responsible for the effect.
Two-Way ANOVA
ANOVA can be used for factorial designs: ones
that employ more than one IV (or factor).
The factorial design is very popular in the social
sciences. The big advantage over single variable
designs is that it can provide some unique and
relevant information about how variables interact
or combine in the effect they have on the DV.
A two way factorial design tells us about two main
effects and the interaction.
Two-Way ANOVA
The effects
 Treatment Effect: a difference in population means

Main Effect: a difference in population means for a factor
collapsed over the levels of all other factors in the design

Interaction: occurs when the effect on one factor is not the
same at the levels of another
Select: Stat > ANOVA > Two-Way ANOVA
Multiple R
Multiple correlation finds the correlation coefficient (r) for
every pair of variables
The multiple correlation coefficient, R, is the correlation
coefficient between the observed values of Y and the
predicted values of Y.

The value of R will always be positive and will take on a
value between zero and one.

The direction of the multivariate relationship between the
independent and dependent variables can be observed in
the sign, positive or negative, of the regression weights.
Multiple R
The interpretation of R is similar to the
interpretation of the correlation coefficient, the
closer the value of R to one, the greater the linear
relationship between the independent variables
and the dependent variable.
Multiple Regression
Multiple regression finds the linear equation
that best predicts the value of one of the
variables (the dependent variable) from the
others.
Multiple Regression
Y = a + bX + cZ + e
The coefficients (a, b, and c) are chosen
so that the sum of squared errors is
minimized.

The estimation technique is then called
least squares or ordinary least squares
(OLS).
Multiple Regression
The predictors in a regression equation have no
order and one cannot be said to enter before the
other.
Generally in interpreting a regression equation, it
makes no scientific sense to speak of the variance
due to a given predictor.


Measures of variance depend on the order of
entry in step-wise regression and on the
correlation between the predictors.
The semi-partial correlation or unique variance
has little interpretative utility.
Multiple Regression
The standard test of a specified regression
coefficient is to determine if the multiple
correlation significantly declines when the
predictor variable is removed from the
equation and the other predictor variables
remain.

Test is given by the t or F next to the
coefficient.