Regression

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Regression

• Bivariate
• Multivariate

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Simple Regression line line of best fit Y
a bX The mean point is always on the line Y
is the criterion variable X is the predictor
variable (whenever this is appropriate)
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The line gives mean relationship The standard
error gives the variability We predict Y using
the equation and then Compare that to the
measurements of Y. This gives you the standard
error of the estimate.
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The regression line seeks to minimize the sum of
the squared errors of prediction. The square root
of the average squared error of prediction is
used as a measure of the accuracy of prediction.
This measure is called the standard error of the
estimate and is designated as sest. The formula
for the standard error of the estimate is
                     
where N is the number of pairs of (X,Y) points.
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An alternate formula for the standard error of
the estimate is                     .
where is sy is the population standard deviation
of Y and ? is the population correlation between
X and Y
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• Simple (bivariate) Univariate
• Y a bX
• Multivariate
• Y a b1X1 b2X2 ..

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• Multivariate one predicted variable influenced by
  2 or more other variables.
• Analyze all at once to find out how much of the
  relationship is explained by each of the factors.

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Four measurements were made of male Egyptian
skulls from five different time periods ranging
from 4000 B.C. to 150 A.D. We wish to analyze the
data to determine if there are any differences in
the skull sizes between the time periods and if
they show any changes with time. The researchers
theorize that a change in skull size over time is
evidence of the interbreeding of the Egyptians
with immigrant populations over the years.
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Basibregmatic height Bh
Max skull breadth Mb
Basioalveolar length Bl
Nasal height Nh
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The scatterplots of measurements vs. Year. Four
measurements of male Egyptian skulls from 5
different time periods. Thirty skulls are
measured from each time period. The
measurement MB appears to change more
significantly with year than NH. Both MB and NH
increase over time however, plots of BH and BL
vs. Year show that BH and BL decrease over time.
In this study the direction of the change is not
important since any change in skull size would be
evidence of interbreeding.
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Mb
Y -18575.5 0.371Mb
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Bh
Year 6167.843 0.181Bh
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Bl
Year 10716.279 -0.425Bl
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Nh
Year -6271.2 0.170nh
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• Individually the four measurements show change
• If add up the r2 of all individual correlations
  r2 0.355
• BUT they are interdependent.
• So this over-estimates the relationship

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• Because there are four different measurements
  that characterize skull size, we must use
  multivariate techniques that allow multiple
  dependent variables. Our dependent variables are
  the measurements MB, BH, BL, and NH. The
  predictor variable is Year.

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Year -3687.401 0.287Mb - 0.088Bh -0.356Bl
0.128nh
A multivariate regression of the data, treating
Year as a continuous predictor, shows that, when
all four measurements are taken together, the
measurements appear to change over the years
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More complex methodsOnly include variables that
are significantly contributing to the relationship
.307 if add up Bl Mb
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Residuals

• The difference between Y and the value of Y for
  each individual
• Can use this value to calculate R2
• The proportion of total variability of the Y
  scores that is accounted for by the regression
  equation

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• Can also calculate how much each variable
  contributes.
• Can use multiple regression to control for a
  third variable. Eliminate the influence of one
  variable and look at the remainder.

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• Factor Analysis
• Which variables go together?
• Identifies factors that group together.
• ANCOVA
• Anova with a covariate able to control for a
  variable and then analyze.

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Effect size
for experiments Cohens d mean1 mean 2
sd
For correlations r and r2 give effect size
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Can compare very different types of information
if you know the effect size

• Anxiety and depression
• Study 1 correlation 2 scales r .37
• Study 2 control and experimental reduced anxiety
  to see if impact on depression
• Change in depression Effect size d .8
• Same effect size