BIOL 300 Chapter Notes - Chapter 4i, 8: Invertebrate, Standard Deviation, Sampling Distribution
May 30th, 2018 BIOL 300
Week 3 Reading
Interleaf 4: Correlation does not require causation
- An association (or correlation) between variables is a powerful clue that there may be a
casual relationship between those variables
- The problem is that two variables can be correlated without one being the cause of the
other
- A correlation between two variables can result instead from a common cause
- A confounding variable is an unmeasurable variable that changes in tandem with one or
more of the measured variables; this gives the false appearance of a causal relationship
between the measured variables
- Even more confusingly, a correlation between two variables may actually result from
reverse causation
o The variable identified as the effect by the researcher may actually be the cause
- The main purpose of experimental studies is to disentangle such effects
o In an experiment, the researcher is able to assign participants randomly to
different treatment groups
o Random assignment breaks the association between the confounding variable
and the explanatory variable, allowing the casual relationship between
treatment and response variables to be assessed
- Finding correlations and associations between variables is the first step in developing a
scientific view of the world
- The next step is determining whether these relationships are casual or coincidental
o This requires careful experimentation
Chapter 8: Fitting probability models to frequency data
- The ioial test is a eaple of a goodess-of-fit test
o A goodness-of-fit test is a method for comparing an observed frequency
distribution with the frequency distribution that would be expected under a
simple probability model governing the occurrence of different outcomes
- The binomial test, however, is limited to categorical variables with only two possible
outcomes
8.1 Example of a probability model: the proportional model
- The proportional model is a simple probability model in which the frequency of
occurrence of events is proportional to the number of opportunities
- Example: no weekend getaway
o Under the proportional model, we would expect that babies should be born at
the same frequency on all seven days of the week. But is this true?
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o The data show a lot of variation in the number of births from one day to the next
during the week
o Under the proportional model, which will be the null
hypothesis, the number of births on Monday should be
proportional to the numbers of Monday in 1999,
except for chance differences. The same should be true
for the other days of the week
o Does the variation among days in the table to the right
represent only chance variation?
▪ We can test the fit of the proportional model to
the data with a X2 goodness-of-fit test
8.2 X2 goodness-of-fit test
- The goodness-of-fit test uses a test statistic called to measure the discrepancy
between an observed frequency distribution and the frequencies expected under a
simple probability model serving as the null hypothesis
o is the Geek lette hi, pooued ke
o The simple model is rejected if the discrepancy, is too large
o The goodness-of-fit test compares frequency data to a probability model
stated by the null hypothesis
- Under the proportional model, each day of the week should have the same probability
of a birth, that is 1/7. This is the simplest model so it is our null hypothesis
o H0: The probability of birth is the same on every day of the week
o HA: The probability of birth is not the
same on every day of the week
- Because the proportional model is the null
hypothesis, we use it to generate the expected
frequency of births on each day of the week
o We expect the accumulated number of
births on each day of the week to
reflect the number of times each day of
the week occurred in 1999
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- We can now use these proportions to calculate the expected frequencies of births for
each day of the week under the proportional model
o E.g there were 350 total births in the data set, and under H0 the fraction 52/365
of them should have occurred on Sundays. The expected frequency of births for
Sunday is therefore
o Note: expected frequencies can have fractional components. The expected
frequencies are the average values expected with the null model
o The sum of the expected values should be the same as the sum of the observed
values
- The statistic measures the discrepancy between the observed frequencies from the
data and expected frequencies from the null hypothesis. It is calculated by the following
sum:
o is the frequency of individuals observed in the ith category, and
is the frequency expected in the category under the null hypothesis
o The numerator of this quantity is a difference between the data and what was
expected, which is squared so that positive and negative deviations are treated
equally
o calculations use the absolute frequencies (i.e. counts) for the observed and
expected frequencies, not proportions or relative frequencies
o To determine , we must calculate
for each day of the
week
▪ Sunday = 1, Monday = 2, etc.
▪ Repeating this calculation for the rest of the days, we obtain the values
shown in the last column of Table 8.2-2
o Adding these up, we get
- The statistic is the test statistic for the
goodness-of-fit test, the quantity measuring the
level of agreement between the data and the null
hypothesis
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Document Summary
An association (or correlation) between variables is a powerful clue that there may be a casual relationship between those variables. The problem is that two variables can be correlated without one being the cause of the other. A correlation between two variables can result instead from a common cause. A confounding variable is an unmeasurable variable that changes in tandem with one or more of the measured variables; this gives the false appearance of a causal relationship between the measured variables. Even more confusingly, a correlation between two variables may actually result from reverse causation: the variable identified as the effect by the researcher may actually be the cause. The main purpose of experimental studies is to disentangle such effects. Finding correlations and associations between variables is the first step in developing a scientific view of the world. The next step is determining whether these relationships are casual or coincidental: this requires careful experimentation.
