Correlation

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. Let's work through an example to show you how this
statistic is computed.

Correlation Example



Let's assume that we want to look at the relationship between two variables,
height (in inches) and self esteem. Perhaps we have a hypothesis that how tall
you are effects your self esteem (incidentally, I don't think we have to worry
about the direction of causality here -- it's not likely that self esteem
causes your height!). Let's say we collect some information on twenty
individuals (all male -- we know that the average height differs for males and
females so, to keep this example simple we'll just use males). 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 (don't take this too seriously -- I made this data up to
illustrate what a correlation is):


Now, let's take a quick look at the histogram for each variable:







And, here are the descriptive statistics:


Finally, we'll look at the simple bivariate (i.e., two-variable) plot:




You should immediately see in the bivariate plot that the relationship
between the variables is a positive one (if you can't see that, review the
section on types
of relationships) because if you were to fit a single straight line through
the dots it would have a positive slope or move up from left to right. Since
the correlation is nothing more than a quantitative estimate of the
relationship, we would expect a positive correlation.

What does a "positive relationship" mean in this context? It means
that, in general, higher scores on one variable tend to be paired with higher
scores on the other and that lower scores on one variable tend to be paired
with lower scores on the other. You should confirm visually that this is
generally true in the plot above.

Calculating the Correlation



Now we're ready to compute the correlation value. The formula for the
correlation is:





See Figure No.01
https://i.servimg.com/u/f40/11/10/02/04/no_0110.jpg


We use the symbol r to stand for the correlation. Through
the magic of mathematics it turns out that 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. You don't need to know how we came up
with this formula unless you want to be a statistician. But you probably will
need to know how the formula relates to real data -- how you can use the
formula to compute the correlation. Let's look at the data we need for the
formula. Here's the original data with the other necessary columns:

Person	Height (x)	Self Esteem (y)	x*y	x*x	y*y
1	68	4.1	278.8	4624	16.81
2	71	4.6	326.6	5041	21.16
3	62	3.8	235.6	3844	14.44
4	75	4.4	330	5625	19.36
5	58	3.2	185.6	3364	10.24
6	60	3.1	186	3600	9.61
7	67	3.8	254.6	4489	14.44
8	68	4.1	278.8	4624	16.81
9	71	4.3	305.3	5041	18.49
10	69	3.7	255.3	4761	13.69
11	68	3.5	238	4624	12.25
12	67	3.2	214.4	4489	10.24
13	63	3.7	233.1	3969	13.69
14	62	3.3	204.6	3844	10.89
15	60	3.4	204	3600	11.56
16	63	4	252	3969	16
17	65	4.1	266.5	4225	16.81
18	67	3.8	254.6	4489	14.44
19	63	3.4	214.2	3969	11.56
20	61	3.6	219.6	3721	12.96
Sum =	1308	75.1	4937.6	85912	285.45

The first three columns are the same as in the table above. The next three
columns are simple computations based on the height and self esteem data. The
bottom row consists of the sum of each column. This is all the information we
need to compute the correlation. Here are the values from the bottom row of the
table (where N is 20 people) as they are related to the symbols in the formula:





See Figure 1
https://i.servimg.com/u/f40/11/10/02/04/no_110.jpg
Now, when we plug these values into the formula given above, we get the
following (I show it here tediously, one step at a time):
See Figure 2
https://i.servimg.com/u/f40/11/10/02/04/no_210.jpg




So, the correlation for our twenty cases is .73, which is a fairly strong
positive relationship. I guess there is a relationship between height and self
esteem, at least in this made up data!

Testing the Significance of a Correlation



Once you've computed a correlation, you can determine the probability that
the observed correlation occurred by chance. That is, you can conduct a
significance test. Most often you are interested in determining the probability
that the correlation is a real one and not a chance occurrence. In this case,
you are testing the mutually exclusive hypotheses:



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 hypothesis testing, you need to first
determine the significance
level. Here, I'll use the common significance level of alpha = .05. This
means that I am conducting a test where the odds that the correlation is a chance
occurrence is no more than 5 out of 100. Before I look up the critical value in
a table I also have to compute the degrees of freedom or df. The df is simply
equal to N-2 or, in this example, is 20-2 = 18. Finally, I have to decide
whether I am doing a one-tailed or two-tailed
test. In this example, since I have no strong prior theory to suggest whether
the relationship between height and self esteem would be positive or negative,
I'll opt for the two-tailed test. With these three pieces of information -- the
significance level (alpha = .05)), degrees of freedom (df = 18), and type of
test (two-tailed) -- I can now test the significance of the correlation I
found. When I look up this value in the handy little table at the back of my
statistics book I find that the critical value is .4438. This means that if my
correlation is greater than .4438 or less than -.4438 (remember, this is a
two-tailed test) I can conclude that the odds are less than 5 out of 100 that
this is a chance occurrence. Since my correlation 0f .73 is actually quite a
bit higher, I conclude that it is not a chance finding and that the correlation
is "statistically significant" (given the parameters of the test). I
can reject the null hypothesis and accept the alternative.

The Correlation Matrix



All I've shown you so far is how to compute a correlation between two
variables. In most studies we have considerably more than two variables. Let's
say we have a study with 10 interval-level variables and we want to estimate
the relationships among all of them (i.e., between all possible pairs of
variables). In this instance, we have 45 unique correlations to estimate (more
later on how I knew that!). We could do the above computations 45 times to
obtain the correlations. Or we could use just about any statistics program to
automatically compute all 45 with a simple click of the mouse.

I used a simple statistics program to generate random data for 10 variables
with 20 cases (i.e., persons) for each variable. Then, I told the program to
compute the correlations among these variables. Here's the result:

See Figure
https://i.servimg.com/u/f40/11/10/02/04/result10.jpg

This type of table is called a correlation matrix. It lists the
variable names (C1-C10) down the first column and across the first row. The
diagonal of a correlation matrix (i.e., the numbers that go from the upper left
corner to the lower right) always consists of ones. That's because these are
the correlations between each variable and itself (and a variable is always
perfectly correlated with itself). This statistical program only shows the
lower triangle of the correlation matrix. In every correlation matrix there are
two triangles that are the values below and to the left of the diagonal (lower
triangle) and above and to the right of the diagonal (upper triangle). There is
no reason to print both triangles because the two triangles of a correlation
matrix are always mirror images of each other (the correlation of variable x
with variable y is always equal to the correlation of variable y with variable
x). When a matrix has this mirror-image quality above and below the diagonal we
refer to it as a symmetric matrix. A correlation matrix is always a
symmetric matrix.

To locate the correlation for any pair of variables, find the value in the
table for the row and column intersection for those two variables. For
instance, to find the correlation between variables C5 and C2, I look for where
row C2 and column C5 is (in this case it's blank because it falls in the upper
triangle area) and where row C5 and column C2 is and, in the second case, I
find that the correlation is -.166.

OK, so how did I know that there are 45 unique correlations when we have 10
variables? There's a handy simple little formula that tells how many pairs
(e.g., correlations) there are for any number of variables:

See Figure 3
https://i.servimg.com/u/f40/11/10/02/04/no_310.jpg

where N is the number of variables. In the example, I had 10 variables, so I
know I have (10 * 9)/2 = 90/2 = 45 pairs.

Dear Sir,

If have a time please

upload the power point lecuture

on correlation and Regression.

With regards,

amin khan

Dear Sir,

We have all the lecuture

notes on correlation and

regression.

So now no need to upload

the powe point lecuture.

Regards,

amin khan

Here is the presentation, Ameen Khan. Enjoy


Correlation and Regression Analysis

Dear Sir,

Thank you for uploading

this nice presentation.

Best regards,

ameen khan