Download GUIDELINES FOR BASIC STATISTICS(Va)

GUIDELINES FOR BASIC STATISTICS
(These guidelines are modified from materials developed by C. Caceras for IB 449
Limnology).
Introduction
We will be using some basic statistics this semester. Because a stats class is not required
for this course, we are going to use early lectures and labs to go over some very basic
statistical concepts. You will use these concepts repeatedly in your Homework and for
your Student-Driven Projects.
Asking questions in a scientifically meaningfully way
Statistics are of little use unless you first have a clear hypothesis and a specific
quantitative prediction to test. Consider a simple example. You suspect that herbivores
affect the amount of algae present in lake water. You make that a bit more specific and
develop a general hypothesis: The presence of herbivores reduces the abundance of
algae in lake water. Now you have something you can work with, but it’s still a pretty
vague notion. What you need to do is apply this hypothesis to a specific system. You
know from surveys of local lakes that some lakes have Daphnia (large herbivorous
zooplankton) whereas others do not. So, based on your hypothesis, you can now make a
specific, testable prediction: The mean abundance of algae will be greater in lakes
without Daphnia than in lakes where Daphnia are present. The hypothesis/prediction
can be stated formally as follows:
If herbivores affect the amount of algae in lake water,
then the mean abundance of algae will be greater in lakes without Daphnia than in
lakes with Daphnia.
All predictions are built upon underlying assumptions.
An assumption is a condition upon which the prediction stands, but which is not directly
tested. It is a fact taken for granted in an experiment. For example, you might ask
whether you have to control for nutrient differences between the lakes with and without
Daphnia. Answer: No, you will not be controlling for nutrients in your experimental
design. So you assume that the nutrients are either the same in the lakes or that nutrients
are not a factor that affects algal abundance.
If you get a significant effect of Daphnia on algal abundance, then at least one of your
assumptions must have been valid. HOWEVER, when you get a non-significant result,
then you examine your assumptions. Perhaps they were not valid. Perhaps no difference
arose because the nutrients were not the same and nutrients were a factor that affects
algal abundance. The next phase of the research would be to explicitly test the
assumption (turning it into an explicit hypothesis to test next).
So try to think of any underlying factors/arguments that may be present and underpinning
your prediction, but not being directly tested. Assumptions are really a part of the logical
argument you are building when constructing a hypothesis/prediction.
Once you have a quantitative prediction firmly in mind for your experiment or study
system, you can proceed to the next step – forming statistical hypotheses for testing.
Statistical hypotheses are possible quantitative relationships among means or associations
among variables. The null hypothesis states that there is no association between
variables or no difference among means. For our example, it would be stated as Ho: algal
abundance in Daphnia-absent lakes = algal abundance in Daphnia-present lakes (Ho: A =
B). The alternative hypothesis states the pattern in the data that is expected if your
predictions hold true. For our example, HA: algal abundance in Daphnia-absent lakes >
algal abundance in Daphnia-present lakes (HA: A>B).
Answering your questions
Now you have to go out, gather data, and evaluate whether or not the data support your
hypothesis. Our problem, essentially, is that we can’t know the mean abundance of algae
in ALL lakes with and without Daphnia. Instead, we have to pick a handful of each type
of lake (a sample) and measure algae in each. We then calculate mean (average) algal
abundance within each group of lakes and compare the two means. Inevitably one
group’s mean WILL be higher than the other’s, just by random chance. Repeat the last
sentence 10 times. If you don’t understand why we asked you to repeat that sentence,
ASK, this is a key aspect of the entire course.
After you have collected your data and calculated your two means (Daphnia-absent,
Daphnia-present), you notice that the mean for the Daphnia-absent group appears to be
higher than the mean for the Daphnia-present group. That’s nice, but it doesn’t tell us
much. As we just got through saying 10 times over, one or the other group’s mean was
going to be greater even if only by random chance. Our problem is to decide, based on
the data, whether the difference we observed was likely to be caused by chance alone or
whether we can conclude that there probably is some real difference between the two
groups.
Statistics help us make this decision by assigning p-values to tests for differences
between means. P-values range between zero (impossible for difference to be due to
chance) and 1 (certain that difference is due to chance). They can be thought of as the
probability that the differences among means observed were caused by chance variation.
If the p-value is very small (traditionally, and for this class, < 0.05), you conclude that
there is some systematic difference among means. You have measured a difference that
was probably NOT caused by chance. “Something else” (maybe grazing by Daphnia?) is
going on. You reject the null hypothesis (Ho A = B) and claim evidence for the
alternative hypothesis: (HA A>B); algal abundance was significantly greater in the
Daphnia-absent lakes than in the Daphnia-present lakes. If, on the other hand, the pvalue is > 0.05 there is a good chance that, if you tried the experiment again with two
new samples of lakes, you would have gotten the opposite result. You have to accept the
null hypothesis. You really have no strong evidence for a difference in algal abundance
between the two groups of lakes.
Types of variables
There are three different types of variables: discrete, continuous and categorical.
Discrete variables are those that cannot be subdivided. They must be a whole number.
E.g. cats would be discrete data because there cannot be a fraction of a cat. We count or
tally these data. Continuous variables are those values that cover an uninterrupted and
large range of scales. A good example of a continuous variable is pH. It can be 2.4, 7.0,
8.5, etc. - any of a wide range of values between 0 and 14. Categorical variables are
arbitrarily assigned by the experimenter. An example of a categorical variable is in an
experiment with two treatments: you supplement a certain number of plants with nitrogen
(treatment 1), and another number of plants get no nitrogen (treatment 2). So your
categorical variable has two levels: nitrogen and no nitrogen. Note that categorical
variables have distinct, mutually exclusive categories that lack quantitative intermediaries
(e.g. present vs. absent; blonde vs. brown vs. red hair.)
Descriptive statistics
There are several commonly used descriptive statistics to use on discrete and
continuous variables. These include the mean (average), variance, standard deviation,
and standard error. A brief description of each follows below (Adapted from Hampton
1994).
Mean: The mean is the sum of all the data, divided by the number of data points. In the
list 10, 30, 20, the mean is 20.
x  mean 
 xi
n
Variance: the average of the squared deviations in a sample or population. The variance
considers how far each point differs from the mean. The variance for a population is
denoted by 2 and is calculated as:
population SS
2 
N
WHERE:
2
SS  Sum of Squares  x i  x 
The best estimate of the population variance is the sample variance, s2:
s2 
sample SS
n -1
Standard deviation: A measure of the average amount by which each observation in a
series of observations differs from the mean.
  standard deviation  variance
Standard error: A measure of the uncertainty of the estimate of the true mean of a sample
that takes into account the sample size.
SE = standard deviation /
N
Analytical (Inferential) Statistics.
We will be using four basic statistical tools for this class: χ2 (chi-square) tests, t-tests,

analysis of variance (ANOVA), and correlation
(and regressions). Chi-square tests
determine whether a relationship exists between two categorical variables. t-tests are
used to make comparisons of mean values of 2 groups, ANOVA is used to make
comparisons of mean values of >2 groups, while correlations look for associations
between two continuous variables. Details about each and examples follow below.
X2test (chi-square test)(contingency test for independence)
A chi-square (χ2) test determines whether a relationship exists between two categorical
variables. All other statistics (see below) apply to dependent variables that are
continuous, i.e. they can take on many different values with an obvious ordering to the
values (e.g. a height of 60 is greater than one of 50). Examples are: pH, height, chemical
concentration).
Categorical data fall into distinct, mutually exclusive categories that lack quantitative
intermediates: (present vs. absent; male vs. female; brown vs. blue vs. green eyes).
A p-value of less than 0.05 indicates that the variables are not independent; i.e. two
categorical variables are related.
An example:
We observe people to see if hair color and eye color are related. Do people with
blond hair have blue eyes, etc? We develop a two-way (contingency) table to
categorize each person into one group for each of the two categorical variables (e.g.
brown eyes and brown hair; green eyes and red hair, etc.) We then study the table to
see which combinations of variables have more or less observations than would be
expected if the two variables were independent. To calculate this test, you must first
find the expected value for the number of observations for each combination of the
two variables based on the null hypothesis that the two variables are independent.
Then you compare the differences in distribution of values for the expected and
observed values.
t-tests (Read MSLS: 38)
A t-test is, most simply, a test to determine whether there is a significant difference
between two treatment means. A p-value of less than 0.05 is usually considered to
indicate a significant difference between the treatment means.
An example:
Let’s say we want to see how the presence of a predator affects the amount of food
consumed by a dragonfly larva. We could set up an experiment with 10 tanks with
just the dragonfly and 10 with both the dragonfly and predator (a total of two
treatments). We could then measure the amount of food in each tank and use a t-test
to see whether there was any difference in the mean amount of food in the 10 tanks
with just dragonflies (treatment 1) and in the mean amount of food in the 10 tanks
with predators (treatment 2).
ANOVA (Analysis of Variance)
ANOVA looks for differences among treatments (like a t-test), by examining the
variance around the mean. The main difference from a t-test is that ANOVA can
look for differences in more than two treatment means. In fact, a t-test is really just a
simple type of ANOVA. What you need to know is that if the ANOVA is significant,
then there is a difference among the treatments. However, if there is a significant
difference, ANOVA does not tell you anything about which treatments are
significantly different from the others, and you must use other tests, such as multiple
comparisons, to determine which are significantly different from the others.
An example:
Let’s modify the experiment above that had two treatments and used a t-test. Instead,
we have three treatments: dragonflies alone (treatment 1), dragonflies and predator 1
(treatment 2), and dragonflies and predator 2 (treatment 3). We would use an
ANOVA to see if there is a difference among the three treatments in the mean amount
of food.
Correlation
A t-test is used to test whether a difference between two group means is statistically
significant. Correlations require a different kind of data. Here you have a list of
subjects (individual animals, sites, ponds) and take measures of two different
continuous variables for each subject. Do subjects with a high level of variable X
also tend to have a high level of variable Y? These kinds of systematic associations
between two variables are described by correlation.
A correlation establishes whether there is some relationship between two continuous
variables - but does not determine the nature of that relationship, i.e. it cannot be
translated into cause and effect. The intensity of the relationship is measured by the
correlation coefficient or “r”. The value of r varies between -1 and 1. -1 indicates a
strong negative relationship between the two variables, 0 indicates no relationship,
and 1 a strong positive relationship.
An example:
You suspect that there is some relationship between degrees Fahrenheit and degrees
Celsius. To test this, you take an F and a C thermometer and measure water baths at
10 different temperatures. We then run a correlation using degrees F as your x
variable and degrees C as your y variable, or vice versa. (For a correlation, it doesn’t
matter which variable is on which axis.) We would probably find a strong positive
correlation - an r-value above 0.9, in this case.
Remember - correlations only tell you that there is some association between the two
variables, but not the nature of that association.
Regression
Regression is similar to correlation, but regressions do assume that changes in one
variable are predictive of changes in the other, resulting in an equation for a line. For
simple linear regression, this is the old y=mx + b line (m=slope, b= y-intercept).
Once this line is established, values of y can be found from values of x.
The strength of this relationship is measured by the coefficient of determination or
“R2” (the correlation coefficient squared). R2 ranges from 0 to 1.0, with 1.0 indicating
that the values of x perfectly predict y (this never happens in ecology).
An example:
We have data on light intensity and algal biomass for twenty locations within a reach
of stream. We suspect that higher light intensities may lead to more algal biomass.
So we could run a regression on light intensity and algal biomass, using light as our x
variable and algal biomass as our y variable. We end up with an equation for the line
and an R2. The R2 will probably be quite high. With a high R2 we can feel confident
in using the equation for predicting values of y, based on x. In this case, we would
say we could predict algal biomass based on light intensity.
It is very important to point out, however, that although there may be a statistical
relationship between the values, without an experiment, we cannot say for sure that
light causes the increase in algal biomass. In other words, correlation is not
causation! Although in this case, increased light most likely does result in increased
algal biomass, a manipulative experiment is necessary to determine the causal
relationship.
Reference:
Hampton, R.E. 1994. Introductory Biological Statistics. Wm. C. Brown Publishers.
Dubuque, Iowa.
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Summary: How to Choose Among Statistical Tests:
Chi-square (done on count data collected for two categorical variables. Do
distributions differ?)
Are eye color and hair color related? Or are they independent?
Is the distribution of eye color independent of the distribution of hair color?
t-test (done on discrete or continuous data collected for one categorical variable with only
two treatments. Do means of two treatments differ?)
Does the presence of a predator affect the amount of food consumed by a dragonfly
larva?
Does the mean amount of food eaten differ in the presence vs. absence of a
predator?
ANOVA (done on discrete or continuous data collected for one categorical variable with
more than two treatments. Do means of >2 treatments differ?)
Does the predator type affect the amount of food consumed by a dragonfly larva?
Does the mean amount of food eaten differ among three predator species?
Correlation (done on two continuous variables; no independent-dependent variables; Is
there a relationship, but no cause-effect)
Are these Fahrenheit temperatures related to these Celsius temperatures?
Regression (done on two continuous variables; clear independent-dependent variables;
Is there a relationship, and with cause-effect)
Do higher light intensities lead to greater algal biomass?
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