Download Statistics made simple

Statistics made simple
Dr. Jennifer Capers
Modified from Dr. Tammy Frank’s presentation, NOVA
Why do we need statistics?
• Example:
– Chemical may increase growth of animal
– Will be tested on housefly
– A colony of 20,000 houseflies are divided
into 2 groups
– Group 1 gets chemical in food
– Group 2 gets a placebo in same food
What comes next?
• 2 weeks later – take random sample of
25 house flies from each group,
measure wingspan
• What are the results?
Housefly results
• 25 houseflies from each group
• Group 1 (with chemical) – 7.5mm wingspan
• Group 2 (without) – 7.2mm wingspan
• What does this mean?
• Are group 1 flies really bigger?
• Some might say yes, some might say no
• Did you, by chance, happen to pick some larger
flies from group 2?
• Was there sampling error or bias?
• One way to be sure is to measure all
20,000 flies……not feasible
• So what do we do?
Statistics
• You say the flies are bigger, I say not
• Statistics provide rules to help us find out
• Statistics will help tell us if these are
significant (real) differences
• Is there bias? Where bigger ones in
group 2 picked by chance?
• Statistics will tell us what the chances are that
the results are due to sampling bias or random
chance
Significant Difference
• Real difference
• Due to chemical, not chance
• If test shows probability of getting
results by chance or random error is <5%,
we accept claim that chemical produced
larger fly
• If test shows that the probability of
getting results by chance or random error
is >5%, we reject claim that chemical
produced larger fly
• 5% is arbitrary cut-off point that is
generally accepted
• However, if the cost of making an incorrect
decision is very high, there will be higher cutoff like 1%
» such as research with cancer drugs, etc.
• Probability value is the p-value
• Measure of probability that the pattern we see
in our data is due to sampling error or random
chance
Scientific Method
• Remember that we cannot “prove”
anything. We can only accept or reject
a hypothesis
• A theory is the closest that a biologist
can come to “proving” a hypothesis
• Supported and validated by data and scientific
community
Null and Alternative Hypotheses
• For any experiment/survey/study, there must
be a null hypothesis and an alternative
hypothesis
• Set up so that one of them must be true, and one must be
false
• Null hypothesis (H0): = or ≤ or ≥
• Example:
– The average weight of hermit crab group A is the same as that
of hermit crab group B (=)
– OR
– The average weight of hermit crab group A is the same or
greater than that of hermit crab group B (≥)
– OR
– The average weight of hermit crab group A is the same or less
thank that of hermit crab group B (≤)
If null is true, then alternative
must be false
• Ho: average weight of hermit crab group A = average weight of
group B
• HA:
average weight of hermit crab group A ≠ average weight
of group B
Two-tailed hypotheses
• Use if you have no expectations
– You are trying to find out if weights are
different but have no reason for them to
be
• Ho :
average weight of hermit crab group A = average weight of
group B
• HA:
average weight of hermit crab group A ≠ average weight
of group B
One-tailed hypothesis
• Use if you have an expectation of the
outcome, based on previous studies or
information
• For example, previous studies have demonstrated
that Group A area has more hermit crab food that
Group B
• Ho: average weight of hermit crab group A ≤ average weight of
group B
• HA:
average weight of hermit crab group A › average weight of
group B
•
Alternative hypothesis corresponds to what you expect
• Always reject or accept the null
hypothesis, never reject the alternative
• If you accept or support the null, then don’t
mention the alternative
• If you reject the null, then accept or support
the alternative
• We never prove a hypothesis
• We just gain a measure of how confident we are
with our hypothesis
p-value
• The measure of the probability that the
pattern we see in our data is due to
random chance or sampling error
• 0.05 is the value most commonly used
• If p-value is ›0.05 (high p-value), accept null
» Weight is not significantly different
• If p-value is ≤0.05 (low p-value), reject null and
accept alternative
» Weight is significantly different
Important terms:
x = measurement value
∑ = sum of
n = sample size
df = degrees of freedom = n – 1
X = mean or average = ∑x/n
√ s2 = Standard deviation = average distance
from mean
• s2 = Variance = mean of sum of squares
•
•
•
•
•
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• ∑(x – X)2/df
• Tells you how much your values varied from mean
– Large variance means there is large spread in data, small
variance means data points are closer to mean
• What test do you use to get p?
• Depends on what type of data you are collecting
– Measurement variable or nominal variables?
• Measurement variables
• Something that can be counted or measured
• Involves numbers
• Examples: length, weight, quantity
• What are examples of tests that can be used?
t-test
• Used to determine if two sets of data
have the same mean
• Paired t-test – when measurements are
linked
• Patient before and after using drug
• The null would state there is no difference
• Unpaired t-test – when you have before
and after within 2 different groups
• Patients with drug (group 1) and patients
without drug (group 2)
What do you do when there are
more than 2 sets of data?
• ANOVA – analysis of variance
• Null would state that the means are equal
• Example would be if you had 5 groups of
patients taking drugs at different dosages
per group
• Single factor ANOVA
• Only vary one parameter – drug dosage
• Two factor ANOVA with or without
replication
• Vary dosage and time of day
• Nominal variables
• Usually involves categories
• A nominal variable is often a word or percentage
• Examples: color, sex, genotypes
• What are examples of tests that can be used?
• Goodness of fitness test
• Chi-square
• Graphing your results
– Using standard deviation bars versus
standard error bars
– Standard error (SE) = SD divided by square root of
sample size
http://mathbench.umd.edu/modules/prob-stat_bargraph/page01.htm
SD Bars or SE bars on your graph?
Standard Deviation – SD
Standard Error - SE
How far members of the population
deviate from the average
How far off is your estimate of the
mean?
Quantifies the population
Quantifies your experiment
Does NOT depend on sample size
DOES depend on sample size (a lot!!!!)
Use to characterize the population
Use to test your results – see next
row
Overlap doesn’t necessarily mean
insignificance
Overlap means insignificance (most of
the time)
Graphing Data on Excel
Enter data points
Highlight data set
Click on charts
Then click on which graph you want
This will be the result
Right click on one of the lines and then
Click on Format Data Series
Click on secondary axis
And there is your graph
Figure Title would go here