Transcript Correlation and Regression

February 2013
 Study
& Abstract
 Graphic presentation of data.
 Statistical Analyses
Assume: You have two articles that provide conflicting evidence
regarding the imposition of formulary controls for mental health
drugs. One says prior authorization causes “gaps” in therapy.
The other says when you impose restrictions drug costs go down
as well as other services such as hospital, ER and doctor visits.
 Conceptualize
a Scatter Diagram
 Pearson Correlation Coefficient
 Linear regression analysis
 Hypothesis testing of relationship
between x and y
 Correlation
and regression are the two most
common methods to describe the relationship
between two quantitative variables (x and y)
 Correlation coefficient measures the strength
of the relationship between the two variables,
provided that the relationship is linear
 Regression analysis is an equation that allows
the estimation of the value of y for any given x.
A
good way to understand the
relationship between two variables is by
graphing them
 The
correlation coefficient r measures
how strong the linear relationship is
 The
relationship between two variables
are not necessarily linear
 When the relationship is not linear, r may not be an
appropriate description of the relationship
 The
value of r can range from 0 (no linear
association between x and y), and 1
(perfectly linear association)
 The value of r can also be positive or
negative
 The closer the value of r to +1 and -1, the
stronger the relationship is and the more
nearly it approximates a straight line.
• Also referred to as the “r-squared”
• It is always between 0 and 1. At the extreme
value of 0, the regression line is horizontal
• r2 measures the percent of the variance
explained by your mathematical model (i.e.
regression model)
• The closer r2 is to 1, the better the fit of the
regression line.
• r2’s are evaluated using 2 sample t-tests with n-2
d.f.
 Although
correlation tells us the strength
of the relationship between x and y, it
doesn’t allow us to predict the value of y
for a given value of x
 Instead of using r, we can use a linear
equation to represent the relationship
between x and y
yˆ = a + bx
 Always
pass through the means of x and
y
 The sum of deviations about the line is
equal to zero
 Sum of residuals above the line equals
the sum of those below the line
 The sum of the squared deviations about
the line is a minimum so it is the best fit
• This is where “least squares” comes from
 H0: β1
= 0 (i.e., there is no relationship
between x and y)
• H1: β1 ≠ 0
 Determine α
 Calculate t statistic with n-2 df
 Compare to critical value
 Reject H0 if t is greater than c.v.
null hypothesis: β1=0 cannot be rejected, it
means that a straight line model in x is not the
best model to use and doesn’t provide much
help for predicting y
 If null hypothesis: β1=0 is rejected we would
conclude that x provides significant information
for predicting y
 If
• BUT there still might be a better model out there
Graphs a and b are
examples when
H0: β1=0 cannot be
rejected
Graphs c and d are
examples when
H0: β1=0 is
rejected
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Common to see this used in biomedical research.
It is a special case of multiple regression where the dependent
variable (the one being studied) is a discrete, nominal variable,
for example 0 or 1, representing the presence or absence of some
characteristic.
For example, you could do a study of long term statin use where
the dependent variable (left side of the equation) is all cause
mortality and the independent variable (right side) is length of
statin use.
So the dependent variable could be 0 v. 1 for dead or alive.
The reference category would be patients who were alive at the
end of the study and the regression result will be expressed as an
“odds ratio.” In other words, is there a greater or less chance of
being alive at some point in the future as a result of using the
statin?
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So, if the odds ratio were .78 you could interpret this as
a 22% less chance of remaining alive as a result of
statin use.
Conversely, if the odds ratio were 1.22 you could
conclude there was a 22% greater chance of dying as a
result of statin use.
Odds ratio’s are assessed for significance using chisquare analysis, and p values are reported.
Note: Which way the percents go depends on which variable, life or death
you set up as the reference variable.
 Correlations
and regression are methods
to measure the relationship between two
variables
 Regression analysis can also study the
relationship of more than two variables,
such as in the example we will walk
through in class