; PPP19 Multiple and Logistic Regression
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PPP19 Multiple and Logistic Regression

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  • pg 1
									 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
 auto insurance premiums. Employers
 administer aptitude & personality
 tests to predict job performance.
 Professional schools do the same with
 undergraduate marks & aptitude tests
 (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
 additional average amount of
 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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