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Research Methods: 1
M.Sc.
Physiotherapy/Podiatry/Pain
Correlation, Regression and Basic
Probability
Relationships Between Variables
Exploring relationships between
variables
What happens to one variable as
another changes
Relationships Between Variables
• Correlation;the strength of the linear
relationship between two variables.
• Regression; the nature of that relationship,
in terms of a mathematical equation.
• In this module we are only concerned with
linear relationships between variables.
Correlation
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Weight
Relationship between Height and Weight
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10
9
8
7
150
155
160
165
170
175 Height180
Correlation
Correlation Coefficient = r
-1  r  1
-1 perfect
negative relationship
0
+1 perfect
positive relationship
Correlation; r = +1,
perfect positive correlation
Correlation; r = -1,
perfect negative correlation
Correlation; 0 < r < 1,
positive correlation
Correlation; -1 < r < 0,
negative correlation
Correlation; r  0,
no linear relationship
Correlation; r  0,
no linear relationship
Correlation
• Closer to  1 the stronger
• Relationships do not necessarily mean what
you think, i.e. non-causal relationships
Spurious Correlation
• Coincidental Correlation; chance
relationships
• Indirect Correlation; related through some
third variable
• .
Putting a value on the Linear
Relationship
• Pearson’s Product Moment Correlation
Coefficient (PPM)
• Parametric data - Quantitative data where
it can be assumed both variables are
normally distributed, = r
Putting a value on the Linear
Relationship
• Spearmans Rank Correlation Coefficient
• Non parametric - Ordinal data or
quantitative data where one (or both)
variables are not normally distributed.
Calculated from the ranked data =  (rho)
Regression
• Identify the nature of the relationship
• Predict one variable from the other
• The independent variable (plotted on the xaxis) determines the dependant variable
(plotted on the y-axis)
The Regression line
The Method of Least Squares
(the smallest sum of the squared distances)
The Regression equation
y = 0.836x +
2.800
Y = bX + a
Y = the y-axis value
X = the x-axis value
b = the gradient (slope) of the line
a = the intercept point with the y - axis
The Regression prediction ?
• Residuals
• Coefficient of Determination
• Coefficient of Determination * 100 =
R squared (R2)
• How good a fit the equation (and the line) is
to the data
Basic Probability
Chance, possibilities and luck!
• How likely is anything to happen, is one
outcome more likely than any other?
• What are the odds of one particular outcome
occurring?
• If you toss a dice what is the probability of
getting a six ?
Probability
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p(Six) + p(Not Six) = 1
p(Six)1/6 + p(NotSix) 5/6 =1
If an event is certain to occur
p=1
If an event is impossible
p=0
All other events fall somewhere between
0 and 1, 0 < p < 1
Probability
• p(Event) =
trials of interest
total number of trials
• Theoretical and Relative frequencies
Probability
• 100 students attend a statistics lecture and
40 of them fall asleep within 10 minutes.
• What is the probability that one of the
students chosen at random will fall asleep
within 10 minutes?
• Pr (sleep) = 40/100 = 0.4
p = 0.4/40%
Probability
• 100 students attend a statistics lecture and
40 of them fall asleep within 10 minutes.
• What is the probability that one of the
students chosen at random will not fall
asleep within 10 minutes?
• Pr (not sleep) = 60/100 = 0.6 p = 0.6/60%
Probability
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Complimentary rule of Probability
Pr (not event) = 1 - Pr (event)
Addition rule of Probability
Pr(A or B) = Pr(A) + Pr(B) where A and B
are mutually exclusive events