Download lecture

Determining the Independence
of Variables & Variates
■ Much quantitative
research compares
distributions (empirical/
theoretical)
■ But are the variables/
variates independent?
■ Importance: the kind of
hypothesis testing &
assessment of probability
related to independence
■ Comparing:
– Surface area of
skeletal elements
collected together at a
point along a river
– Heart rates of
individuals before/
after each climbs four
flights of stairs
– Opinions about a bill to
tax farm effluents
among adult males in
a small farming
community
Determining the Independence
of Variables & Variates
■ Key question to ask:
– How reasonable is the
assumption of
independence in light
of the null hypothesis?
■ Subsistence strategy and
post-marital residence
among societies that all
share a “recent” common
origin?
■ Opinions about an issue
among people who have
never associated
■ Change over a five year
period in the percentage
of people involved in
wage labour in several
communities in Mexico.
Determining the Independence
of Variables & Variates
■ Steps to take to figure this out:
– Think about what you need to know? Become familiar
with related literature surrounding an issue.
– State your variable definitions and assumptions in such
a way that both you and others can evaluate them
– Plan your choice of statistical tests around expectations
about independence or its absence*
* Yes, more on these tests later
Problem Statement
■ You now have:
– Determining relevant problem in the literature
– Articulating a clear hypothesis
– Considering variables, operational definition &
independence
– Statistical population, samples, and sampling methods
Probability
■ Descriptive & inferential
stats depend on
probability
– And probability based
on assumption of
empirical/ theoretical
regularities in world
Theoretical Determination of
Probability
■ Coin example:
–With a fair coin flip,
50% chance heads,
50% tails
■ P(H)=0.5, P(T)=0.5 with
summed probabilities= 1
■ Dice example:
– With fair dice, total of
36 combinations
– Only one way to role a
two, 1/36 = 0.03; two
ways to role a three,
2/36 = 0.05, etc.
– Determine all logical
possibilities
Empirical Determination of
Probability
■ Calculate probabilities by measuring actual trials
– A concern: are you characterizing a regularity that
applies to relevant space and time?
Complex Events &
Determination of Probability
■ Deck of card example:
– Individual card, 1/52,
e.g., P(2♣)=0.02
– P(♣)=13/52=0.25
■ Complex Events (more
than one) :
– P(♣Ŭ♠)=
13/52+13/52 = 0.50
■ Complex Events
(consecutive independent
successes)
– P(n events) =
(event freq. / trials)n
– P(♠twice in row)=
(13/52)x(13/52)=0.06
The Binomial Distribution &
Empirical Probabilities
■ Binomial distribution
formalizes calculation of
probabilities when two
possible outcomes, and
specific number of trials
– P(p+q)n
– For 3 trials:
(p+q)x(p+q)x(p+q)
p3 + 3p2q + 3pq2 + q3
■ Randomly grading the
labs, what is the
probability that you will
be the top scorer at least
once over the first three
labs?
– 1 of 24 students
– p = 0.04, q = 0.96
– Plug in p and q for
terms than include
at least one success
= 0.12
Probabilities for Events &
Probabilities for Distributions
■ Calculate probability of
events under
assumptions of regularity
and consistency
(theoretically or
empirically determined)
■ Determine how likely our
event is, and then
interpret it accordingly.
■ However, often want to
know how likely is a
distribution given
assumptions of
regularity?
■ Or, how likely is this
variate in a distribution of
variates?
■ Answering these kinds of
questions require a
theoretical foundation in
the normal distribution.