Transcript Research in Business

Correlation Analysis
Correlation Analysis:
Introduction
• Management questions frequently revolve
around the study of relationships between two or
more variables. Thus a relational hypothesis is
necessary.
• Various Objectives are served with correlation
analysis.
• The strength, direction, shape and other
features of the relationship may be discovered.
• With correlation, one calculates an index to
measure the nature of the relationship between
variables
Bivariate Correlation Analysis
(BCA)
• The Pearson (product moment) correlation
coefficient varies over a range of +1
through to – 1.
• The designation r symbolizes the
coefficient’s estimate of linear association
based on sampling data.
• The coefficient p represents the population
correlation.
BCA contd.
• Correlation coefficient reveal the
magnitude and direction of relationships.
• The magnitude is the degree to which
variables move in unison or opposition.
• The size of a correlation of +.40 is the
same as one of -.40.
• The sign says nothing about size.
• The degree of correlation is modest.
BCA contd.
• Direction tells us whether two variables
move in the same direction, opposite
direction.
• When variables move in the same
direction, the two variables have a positive
relationship:
– As one increases, the other also increases
– Family income, e.g., is positively related to
household food expenditure.
• As income increases, food expenditures increase.
BCA contd.
• Some variables move in the opposite
direction, e.g., the prices of products/
services and their demand. These
variables are inversely related.
• The absence of a relationship is
expressed by a coefficient of
approximately zero.
Scatterplots for Exploring
Relationships
• Scattoerplot are essential for
understanding the relationships between
variables.
• They provide a means for visual inspection
of data that a list of values for two
variables cannot.
• Both the direction and the shape of a
relationship are conveyed in a plot.
• The magnitude of the relationship can also
be seen.
Simple Bivariate (i.e., two-variable)
plot:
Correlation Matrix
• The correlation is one of the most common
and most useful statistics.
• A correlation is a single number that
describes the degree of relationship
between two variables.
Correlation Example
• Let's assume that we want to look at the relationship
between two variables, height (in inches) and self
esteem.
• Our hypothesis is that height affects one's self esteem
(The direction of causality is not taken into account, i.e.,
it's not likely that self esteem causes one’s height).
• Data on the age and height of twenty individuals are
collected. We know that the average height differs for
males and females, so, to keep this example simple, the
example uses males only.
• Height is measured in inches. Self esteem is measured
based on the average of 10, 1-to-5, rating items (where
higher scores mean higher self esteem). Here's the
data for the 20 cases: Table 1 in WORD Format
Calculating the Correlation:
Example contd.
• We use the symbol r to stand for the
correlation. The value of r will always be
between -1.0 and +1.0. if the correlation is
negative, we have a negative relationship; if it's
positive, the relationship is positive.
• N
= 20
• ∑XY = 4937.6
• ∑X = 1308
• ∑Y = 75.1
• ∑X2 = 85912
• ∑Y2 = 285.45
Example contd.
• Plugging these values into the formula
given above, we get the value of r:
r = 0.73
• So, the correlation for our twenty cases
(.73) is shows a fairly strong positive
relationship.
• It seems that there is a relationship
between height and self esteem, at least in
this made up data!
Testing the Significance of a Correlation
• After having computed a correlation, we can
determine the probability that the observed
correlation occurred by chance.
• This can be done by conducting a significance
test.
• Most often we are interested in determining the
probability that the correlation is a real one and
not a chance occurrence.
• In this case, we are testing the mutually
exclusive hypotheses:
Testing the Significance
• Null Hypothesis: r = 0
• Alternative Hypothesis: r <> 0
• The easiest way to test this hypothesis is
to find a statistics book that has a table of
critical values of r.
• Most introductory statistics texts would
have a table like this. As in all hypotheses
testing, we need to first, determine the
significance level.
Testing the Significance contd.
• Here, we will use the common significance level
of alpha = .05.
– This means that we are conducting a test where
the odds that the correlation is a chance
occurrence are no more than 5 out of 100.
• Second, before we look up the critical value in a
table we also have to compute the degrees of
freedom (df).
– The df is simply equal to N-2 or, in this example, is
20-2 = 18.
Testing the Significance contd.
• Finally, we have to decide whether we are
doing a one-tailed or two-tailed test.
• In this example, since we have no strong
prior theory to suggest whether the
relationship between height and self
esteem would be positive or negative, we
will opt for the two-tailed test.
Testing the Significance contd.
• With the following three pieces of
information
– the significance level (alpha = .05)),
– degrees of freedom (df = 18), and
– type of test (two-tailed)
• we can now test the significance of the
correlation we have found.
• The critical value in the statistics book is
.4438.
Testing the Significance contd.
• This means that if the computed value of
correlation is greater than .4438 or less than .4438 (remember, this is a two-tailed test), we
can conclude that the odds are less than 5 out of
100 that this is a chance occurrence.
• Since our computed correlation 0f 0.73 is quite
higher, we conclude that it is not a chance
finding and that the correlation is "statistically
significant" (given the parameters of the
test). We can reject the null hypothesis and
accept the alternative.