; PPP19 Multiple and Logistic Regression
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# PPP19 Multiple and Logistic Regression

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```									 SIMPLE LINEAR REGRESSION
►Linear regression determines the
strength & direction of a relationship
between two variables measured at
the interval or ratio level.

►Linear regression formalizes the
relationship by designating (X) as
the independent variable and (Y) as
the dependent variable.
MORE ON LINEAR REGRESSION..
►Regression   develops an equation
that allows us to predict the value of
our “outcome” variable Y, on the
basis of a specified value of our
“predictor” variable X.
►The simple linear regression formula
is:

Y = a + bX
►Statisticalprediction using linear
regression is widely used in “real
world” research. Insurance firms
use predictor variables to determine
tests to predict job performance.
Professional schools do the same with
(e.g., LSAT, MSAT, GRE, GMAT,
etc.,).
►If two variables are perfectly correlated,
knowing the value of X, allows us to
perfectly predict the value of Y.
►As long as 2 variables are significantly
correlated, we can use scores on X to
predict scores on Y. For example, the
strong correlation between social support
(X) and mental health (Y) implies that if
we know a person’s social support level,
we can accurately predict their mental
health level (Y).
SIMPLE LINEAR REGRESSION EXAMPLE
►           Y = a + bx
►   Y the sum of: (1) the average
duration of employment denoted
by the intercept a, and (2) an
employment duration due to a
counseling session denoted by the
slope b.
►The   intercept a is the value Y takes
when X equals 0. It is the average
duration of post-release employment
if a young offender attends no
counseling sessions at all.
►The slope b (the regression
coefficient) is the amount of change
in Y (up or down), that is caused by
a one unit increase in X. Here it is
the average increase in employment
duration caused by attending each
► Differencebetween observed & predicted
values of Y is the error in the regression. The
higher the correlation, the more accurate our
predictions of Y using X .
►r²is improvement in our predictions of Y
using X….square Pearson’s r = .85, we
obtain r² =.72. This is the coefficient of
determination and means we improve our
predictions of Y by 72% using X.
Coefficient of non-determination is 1- r²,
and it is the percentage of variability in Y
not explained by X.
Multiple Linear Regression:
►Is a logical and mathematical
extension of simple linear
regression to situations where
we have one interval-level
dependent variable, and two
or more interval-level
independent (predictor
variables).
Y = a + b1X1 + b2X2
Y … respondent score on the dependent
variable.
a … the intercept.
b1…the regression coefficient for the first
predictor (X1).
b2…the regression coefficient for the
second predictor (X2).
X1…respondent score on first predictor
variable.
X2…respondent score on second predictor
variable.
Logistic Regression

►Logistic   regression predicts
the probability of a dependent
variable Y (measured as a
nominal/ordinal dichotomous
outcome) using a predictor
variable measured at the
interval level.
Logistic Regression 2
►Used in medical research to
determine variables that predict
whether a tumor is likely to be
cancerous or benign, what variables
predict probability of a heart attack or
stroke, etc.,
►In social science, used to predict the
odds that convicts will recidivate or
not; which variables predict if couples
will get divorced or not, etc.,
►In linear regression, we predict
values of Y using a combination of
each predictor variable multiplied by
its respective regression coefficient
as illustrated in the formula:

Y = a + b1X1
z
P(Y) = 1/1+ e

where z = a + b1X1

P(Y)…probability of Y occurring.
e…base of natural logarithms
z…sum of simple linear
regression coefficients (intercept
and slope).

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