AN INTERVAL / RATIO MEASURE OF ASSOCIATION

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AN INTERVAL / RATIO MEASURE OF ASSOCIATION

• To measure the strength direction of an
  association between 2 interval/ratio variables,
  use Pearsons Correlation Coefficient (Pearsons
  r).
• With interval or ratio-level measurement, we
  obtain enough information to order using constant
  units or equal intervals between points on a
  scale.

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MORE ON INTERVAL/RATIO MEASURMENT.

• With interval or ratio measurement we can say
  exactly how much more or less of something a
  respondent has.
• (e.g., someone with 6 siblings has twice as many
  as a person with 3 siblings).

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MORE ON PEARSONS r.

• Pearsons r varies from -1.00 to 1.00.
• r -1.00 indicates a perfect negative or inverse
  association between X and Y. As X increases, Y
  decreases.
• r of 1.00 indicates a perfect positive
  association between X and Y. As X values
  increases, Y also increases.

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• The hours a child spends watching television is
  inversely (negatively) correlated with the hours
  a child spends playing outside.
• The hours a child spends reading is positively
  correlated with their vocabulary.

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MORE ON PEARSONS r.

• Very few relationships between social variables
  are perfect! Accordingly, the rules of thumb for
  interpreting Pearsons r
• -.60 to -.99 Strong negative association
• -.30 to -.59 Moderate negative association
• -.10 to -.29 Weak negative association
• 0 No association
• .10 to .29 Weak positive association
• .30 to .50 Moderate positive association
• .60 to .99 Strong positive association

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• Strong positive correlation between social
  support and happiness. As social support
  increases so does happiness.
• We can generalize our finding to the entire
  Nipissing Student population with 99 confidence.

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• Hypthesis testing (t-tests, ANOVA, chi-square,
  median test) tell us little about any possible
  association or relationship between variables.
• To document a relationship between 2 variables
  measured at the interval or ratio level we use
  the Pearsons Correlation Coefficient (Pearsons
  r).

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• Do not confuse correaltion with causation!
• Correlation means that variation in the scores on
  one variable correspond with variation in the
  scores of another variable.
• Causation means that variation in the scores of
  one variable CAUSE or CREATE variation in the
  scores of another variable.
• Correlations need to be combined with theory,
  logic, other evidence to determine causality.

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SIMPLE LINEAR REGRESSION

• Linear regression determines the strength
  direction of a relationship between 2
  interval/ratio variables.
• Linear regression formalizes the relationship by
  designating (X) as the independent variable and
  (Y) as the dependent variable.

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MORE ON LINEAR REGRESSION..

• Regression develops an equation that allows us to
  predict the value of outcome variable Y, on
  the basis of a specified value of our predictor
  variable X.
• The simple linear regression formula is
• Y a bX

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MORE ON LINEAR REGRESSION..

• Statistical prediction using linear regression is
  widely used in real world research. Insurance
  firms use predictor variables to determine
  auto, home, or health insurance premiums.
  Employers administer aptitude personality tests
  to applicants to predict future job performance.
  Professional schools do the same with
  undergraduate marks aptitude tests (e.g., LSAT,
  MSAT, GRE, GMAT, etc.,).

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MORE ON LINEAR REGRESSION..

• If two variables are perfectly correlated,
  knowing the value of X, allows us to perfectly
  predict the value of Y. Perfect correlations
  are rare!!
• As long as 2 variables are correlated, we can use
  scores on X to predict scores on Y. (e.g.,
  the strong correlation between social support (X)
  and mental health (Y) implies that if we know a
  persons social support level, we can accurately
  predict their mental health level (Y).

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SIMPLE LINEAR REGRESSION EXAMPLE

• Simple linear regression formula
• Y a bx
• Y is the sum of (1) the base line (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.

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A SIMPLE LINEAR REGRESSING EXAMPLE

• The intercept a is the value Y takes when X
  0. It is the average duration of employment if
  a young offender attends no counseling sessions.
• The slope b (the regression coefficient) is the
  amount of change in Y caused by a one unit
  increase in X. (e.g., average increase in
  employment caused by attending each counseling
  session.

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CORRELATION REGRESSION

• Difference in observed predicted values of Y
  is the regression error . A higher correlation
  implies more accurate predictions of Y using X .
• r² measures improvement in predictions of Y using
  X.if we square r .85, we obtain r² .72.
  This is the coefficient of determination so if we
  use X to predict Y we improve our accuracy by
  72. The coefficient of non-determination is 1-
  r², and it is the percentage of variability in Y
  not explained by X).

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• While correlation and linear regression both shed
  light on the relationship between 2
  interval/ratio level variables, regression allows
  us to make predictions and provides a stronger
  intuitive grasp of the relation between the 2
  variables.

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• Y a bx
• Y predicted value of Y (dependent variable).
• b unstandardized regression coefficient or
  slope.
• a intercept
• X a given or specified value of X
  (independent variable).
• Linear regression assumes the 2 variables
  (XY) are actually related in a straight-line
  mannerso that every time there is an increase of
  a given size in the value of X, a corresponding
  increase (or decreas) of a given size occurs in
  Y.

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• Ordinary least squares is based on the idea that
  of all the possible straight lines that could
  bisect the data points on a scatterplot, only one
  line will minimize the distance between each data
  point and that line.
• Linear regression calculates the straight line
  with the smallest sum of squared deviations and
  is known as the line of least squares. It is the
  regression line that minimizes the differences
  between the predicted and actual values of Y.

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• Y a bx
• Y predicted value of Y (dependent variable).
• b unstandardized regression coefficient or
  slope.
• a intercept
• X a given or specified value of X
  (independent variable).
• b .61 a -3.77 Y -3.77 .61X

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• Y a bx
• Y predicted value of Y (dependent variable).
• b unstandardized regression coefficient or
  slope.
• a intercept
• X a given or specified value of X
  (independent variable).
• b .61 a -3.77 Y -3.77 .61X

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• For every unit increase in X, there is a
  corresponding predicted increase of .61 units in
  Y. So for every additional year of education, we
  predict an increase of .61(1000) or 610 in
  monthly income.
• For someone with 9 years of education, we would
  predict Y -3.77 .61(9) -3.77 5.59 1.82
  or 1.82(1000) or 1,820 per month income.
• For someone with 25 years of education, we
  predict Y -3.77 .61(25) 11.48(1000) or
  11,480 per month income.

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Multiple Linear Regression

• Is a logical 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).

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• 1. Multiple Regression combines several
  independent variables (Xs) which provides more
  accurate predictions of (Y).
• 2. MR isolates the effect of each independent
  variable.

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• INCOME 10000 (1500 x YEARS OF EDUCATION)
• A person with 0 years of education is predicted
  to have 10,000 income. Since each additional
  year of education increases predicted income by
  1500, we predict that a person with 10 years of
  education would have an income of 25,000.
• 25000 10000 (1500 x 10)

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• INCOME 7000 (1200 x YEARS OF EDUCATION)
  (600 x AGE)
• A person with 0 years of education age is
  predicted to have 7,000 income. Since each
  additional year of education increases predicted
  income by 1200, and each additional year of age
  increases predicted income by 600, we predict a
  40 year old with 15 years of education would have
  an income of 49,000.
• 49000 7000 (1200 x 15) (600 x
  40)
• 49000 7000 (18000) (24000)

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• Y a b1X1 b2X2
• Y respondent score on the dependent variable.
• a the intercept.
• b1the regression coefficient for the first
  predictor (X1).
• b2the regression coefficient for the second
  predictor (X2).
• X1respondent score on first predictor variable.
• X2respondent score on second predictor variable.

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• Y a b1X1 b2X2
• (a 14 b1 3 b2 4.2)
• Y 14 3(X1) 4.2(X2)
• 14 3(10) 4.2(12) 10 counselling
  sessions and 12 years of
  education.
• 14 30 50.4
• 94.4 months of employment after release.