[ Home ] PSYC 220 (Research Methods), Fall 2006 S. J. Gilbert, SUNY-Oneonta
A CORRELATION/SIGNIFICANCE-TESTING/
LESSON
Let’s
clear up some confusion concerning HYPOTHESIS and NULL-HYPOTHESIS. In a Correlational study – the type you are
considering in Assignment 8 – the NULL HYPOTHESIS is the assumption that we
always start with, that there is NO RELATIONSHIP between the two measures in
question. NO RELATIONSHIP means that
where people stand on one measure, is UNRELATED to where they stand on the
other. If
we measure HAPPINESS and INCOME of 50 people, the NULL HYPOTHESIS is that
they are not related (i.e., that how high a person’s score is on our
happiness measure is UNRELATED to how high the person’s score is on our
income measure). The ALTERNATE (or
RESEARCH) HYPOTHESIS, in contrast, is a claim that the two measures ARE RELATED
– that how high a person’s score is on one measure IS RELATED to how high the
person’s score is on the other. We
can state two kinds of ALTERNATE HYPOTHESES: DIRECTIONAL and
NON-DIRECTIONAL. A DIRECTIONAL
HYPOTHESIS states the DIRECTION (positive vs. negative) of the expected
relationship. For example: “The
hypothesis is that higher happiness scores are associated with higher income
scores.” This is the kind we usually
state, because we usually have an idea concerning how two variables are
likely to be related. A
NON-DIRECTIONAL hypothesis is noncommittal concerning the direction of the
relationship. For example: “The
hypothesis is that happiness is related, in some fashion, to income.” When
we ask SPSS to calculate the correlation coefficient for two variables (like
HAPPINESS and INCOME), SPSS gives us an r statistic (e.g., r = +.45), and a p
(probability) statistic (e.g., p = .02).
The r statistic tells us how strong a correlation is (1.0 is the
strongest it can be, 0 is the least strong it can be), and the direction of
the relationship (+ or -). What
about the p statistic? We will be
seeing it all semester, and you will be learning, deeply, what it means. Here’s the short course. How
do we find out whether or not two variables are related in nature? We cannot measure all of nature to find
out; we can only sample nature
(e.g., choose 50 of the 6 billion people in the world, and give them our two
measures). Due to chance factors, two
variables that are in fact unrelated in nature, are
unlikely to yield a perfect-zero correlation coefficient when we measure any
specific group of 50 people (just like a perfectly balanced coin is unlikely
to yield exactly 25 heads in 50 tosses).
So, the question becomes: how
high does a correlation coefficient obtained from a particular sample of
people have to be -- how far from 0 -- before we conclude that it is so high, that we can reject the idea
that the two variables are unrelated in nature. Is +.27 high enough? +.41?
+.73? The
p statistic tells us the probability that a correlation as high (or higher)
than we received from our sample of subjects, could have happened if the two
variables were truly UNCORRELATED in nature.
The LOWER the p, the less likely it is that we could have attained
such a correlation in our sample, if those variables are NOT ACTUALLY RELATED
in the world. For
example, with 50 subjects, here are some possible correlation coefficients,
and the p statistics associated with them.
Notice
that as the correlation in our sample of 50 people gets bigger, the
probability associated with it gets smaller.
So, in a world in which HAPPINESS and INCOME are, in fact, NOT
RELATED, we would expect to get a correlation as high as +.20 from a sample
of 50 people about 35% of the time.
Now, 35% is not all that rare.
So, a correlation of +.20 would NOT be high enough to cause us to
REJECT the NULL HYPOTHESIS (that happiness and income are unrelated) and
instead, accept the ALTERNATE HYPOTHESIS that they must be positively
related. But
what if we got a correlation in our sample of 50 people, of r = .50? The table tells us that in a world in
which HAPPINESS and INCOME are, in fact, NOT RELATED, we would expect to get
a correlation as high as +.50 only 1% of the time (1 in 100). Two interpretations are possible. One is that HAPPINESS and INCOME are, in
fact, unrelated, and that our study was the 1 in 100 that would, by chance,
yield a correlation between them as high as +.50. If we believed that, we would RETAIN (keep
believing) the NULL HYPOTHESIS, despite the +.50 correlation that we obtained
in our study. The
other possibility is that HAPPINESS and INCOME are, in fact, positively related
in nature (in reality, in the world), and the +.50 correlation we got in our
study attests to that fact. Well,
we’re back to our original question, which, if you remember, was “how high does a correlation coefficient have to be -- how far from
0 -- before we conclude that it is so
high, that we can reject the idea that the two variables are unrelated in
nature?” Psychology has come to accept
a particular convention. If the p
associated with a particular correlation coefficient is equal to or less than
5% -- the famous p < .05 level -- then we reject the NULL HYPOTHESIS (that
the variables are UNRELATED in nature) and accept, instead, the ALTERNATE
HYPOTHESIS (that the variables must be RELATED in nature). Thus, our table tells us that if we
measured the HAPPINESS and INCOME of 50 people, and obtained a correlation of
+.30 (p = .08), we would have to RETAIN the NULL HYPOTHESIS (of no
relationship existing in nature), and would NOT ACCEPT the ALTERNATE
HYPOTHESIS (nature produces a positive relationship between HAPPINESS and
INCOME). But, if we, instead, obtained
a correlation of +.40 (p < .02), we would REJECT the NULL HYPOTHESIS, and
instead, ACCEPT the ALTERNATE HYPOTHESIS.
We would, in effect, decide that it is so unlikely (2 chances in 100)
that a world in which HAPPINESS and INCOME are unrelated would produce a
correlation this high (+.40), that we will adopt the belief that HAPPINESS
and INCOME ARE ACTUALLY POSITIVELY RELATED in the world, and that our 50
subjects’ +.40 correlation reflects this relationship! If
you get this, you’re in great shape.
If you don’t get it, print it out, put it aside, and then reread it
when you mind and attitude permit. SJG |
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