1. Biol 500: basic statistics
Goals:
1) understand basics of experimental design
- controls
- replication
2) understand how to report quantitative data
3) be able to interpret the reporting of statistical results in a
journal article

2. Replication:
allows you to determine if the
difference between treatments
or groups of samples is greater
than the variation within a
treatment or group
Is there a difference in how
effective the 3 drugs are in
curing headaches?

3. Replication:
allows you to determine if the
difference between treatments
or groups of samples is greater
than the variation within a
treatment or group
Is there a difference in how
effective the 3 drugs are in
curing headaches?
Generally, overlapping error bars
indicate no significant difference
between the mean values that
are being graphed
Bars don’t overlap = probably
different ? no
yes

4. Controls:
From these data, could you tell if the least effective drug has
any effect at all?

5. Controls:
From these data, could you tell if the least effective drug has
any effect at all?
Including a control that
is the same in all respects
except the key variable
you are manipulating is
key to interpreting your
results

6. Controls:
Procedural controls allow you to diagnose problems in your
experiment, samples or technique
When we amplify DNA from unknown samples by PCR, we
include a positive control (a DNA sample that always works)
and a negative control (all the PCR reagents, but no DNA)
This allows us to interpret the results of our gels, and to
troubleshoot any problems

7. Do squirrels bury acorns?
My experiment: I remove all the squirrels from 3 clumps of trees
in one park, but leave the squirrels 3 control clumps of trees
in another park on the other side of town
Park A
Park B

8. Pseudoreplication
In this example the unit of replication is the park, not the clump
of trees – I have no actual replication
Park A
Park B
Any difference that I measure could be due to differences among
the two parks, and not due to my squirrel-removal treatment

9. Avoiding pseudoreplication
Correct design would be to have squirrel-removal and control
areas in each of several replicate parks
This lets you assess differences between treatment and control
areas, while simultaneously measuring variation among parks
Park C
Park A
Park B

10. n = 10
Did these two classes do differently on my 418 midterm?

11. n = 20

12. n = 44

13. n = 44 The statistical approach is to ask if the means of these
X = ± 29.7 SD
range:
X = ± 38.8 SD
range:
The statistical approach is to ask if the means of these
two populations are significantly different

14. n = 44 the standard deviation (SD) is what you should report if you
X = ± 29.7 SD
range:
X = ± 38.8 SD
range:
the standard deviation (SD) is what you should report if you
are actually interested in the variation – ie, for purposes of
deciding where to draw the line between grades

15. √ X = 133.9 ± 29.7 SD or ± 4.3 SE range: 59 - 183 n = 44
the standard error (SE, or SEM) is SD √ n
sample size

16. X = ± 29.7 SD or ± 4.3 SE
range:
n = 44
X = ± 38.8 SD or ± 5.8 SE
range:
the standard error is what you report when you want to
compare the means of different treatments or samples

17. unpaired two-tailed t-test: df = 86, t = 1.20, P = 0.23
X = ± 29.7 SD
X = ± 38.8 SD
unpaired two-tailed t-test: df = 86, t = 1.20, P = 0.23
a t-test compares 2 populations by calculating a test statistic
called t and determining the probability (P, or p) of getting
that value of t, with that sample size, by chance alone

18. unpaired two-tailed t-test: df = 86, t = 1.20, P = 0.23
X = ± 29.7 SD
X = ± 38.8 SD
unpaired two-tailed t-test: df = 86, t = 1.20, P = 0.23
paired would be, you compare the % scores on midterm
versus final for each student; most tests are unpaired

19. unpaired two-tailed t-test: df = 86, t = 1.20, P = 0.23
X = ± 29.7 SD
X = ± 38.8 SD
unpaired two-tailed t-test: df = 86, t = 1.20, P = 0.23
one-tailed if you have some reason to think, in advance,
that the 2009 scores will only be higher (or lower) than 2007
- cuts your P-value in half, but you need a reason to do this!

20. unpaired two-tailed t-test: df = 86, t = 1.20, P = 0.23
X = ± 29.7 SD
X = ± 38.8 SD
unpaired two-tailed t-test: df = 86, t = 1.20, P = 0.23
power of your test will depend on your degrees of freedom,
which is (sample size) – (number of groups)
- in this case: ( students) – (2 groups) = = 86

21. unpaired two-tailed t-test: df = 86, t = 1.20, P = 0.23
X = ± 29.7 SD
X = ± 38.8 SD
unpaired two-tailed t-test: df = 86, t = 1.20, P = 0.23
P values below 0.05 are accepted as significant, meaning there
is less than a 5% chance of getting a test statistic this large if
the groups are not really any different

22. 3 or more samples can be compared using a
df subscripted under F ratio
F2,129 = 7.12
P <0.001
overall P for 3-way
comparison of means
n = 44
n = 44
n = 44
3 or more samples can be compared using a
one-way Analysis of Variance, or ANOVA
instead of calculating a t statistic, ANOVA calculates an
F-ratio, which compares variation within groups (error bars)
to the differences in mean values among groups
2 degrees of freedom: 1st = (# of groups – 1)
2nd = (total sample size) – (# of groups)

23. If your overall P-value is significant, you can then do a post-hoc
Scheffe:
P =
0.002
Scheffe:
P =
0.050
n = 44
n = 44
n = 44
If your overall P-value is significant, you can then do a post-hoc
(“after the fact”) test to work out which specific means are
different from each other
Bonferroni - not too conservative; may see differences that aren’t real
Scheffe - very conservative; if it sees a difference, there really is one
Dunnett - compares each mean to a control; most powerful

24. 2-way ANOVA tests for interactions among 2 or more factors
factors: aspirin, yes/no
tylenol, yes/no

25. 2-way ANOVA tests for interactions among 2 or more factors
when the response to two treatments combined is not what
you would expect from adding their individual effects, this is
an interaction
interactions are usually the most biologically interesting result!

26. 2-way ANOVA tests for interactions among 2 or more factors
A B C D
NOT appropriate to do a 1-way ANOVA on these data,
because that requires that each treatment be independent of
the other treatments
- since 2 treatments involve aspirin, they are not independent
- also, you miss the interaction, which is the important result

27. Correlation analysis is appropriate when you think 2 variables
are related, but not in a cause-and-effect way
- arm length and leg length are related, but longer arms do not
cause you to have longer legs; both are due to your height
Regression analysis is when you believe a change in one
predictor variable (what you manipulate) causes a change
in the response variable (the thing you measure)
- adding more water makes
plants grow taller

28. Output of a regression
analysis includes:
1) ANOVA table
tells you if your model
explains a significant
amount of the variation
in the response

29. Output of a regression analysis includes: 1) ANOVA table
2) equation of the
best-fit line
summarizes the relationship
between predictor and response

30. Output of a regression analysis includes: 1) ANOVA table
2) equation of the
best-fit line
3) table testing the
effect of each
predictor
in multiple regression, you can test
many possible predictors that might
matter, and see which significantly
affect the response variable

31. Output of a regression
analysis includes:
1) ANOVA table
2) equation of the
best-fit line
3) table testing the
effect of each
predictor
4) r2
r2 is the % of variation in the
response that is due a change
in the predictor

32. More scatter = lower r2
You can have a low r2,
but still have a significant
slope

33. ANOVA and regression are both types of linear models,
which test the same basic equation:
response variable = model + error
variance in the
response that is
not explained by
the model
thing you
measure
predictors, and
coefficients that
tell you how they
affect the response
this is what a simple linear
regression model looks like

34. Does predictor X affect response?
test is to set the coefficient = 0,
which drops out the predictor,
and see if the model (now just the
residual error term) is really any
worse

35. Parametric versus non-parametric tests
All of the tests we have discussed are parametric tests
- they use the numerical values of your actual data
- however, they also have built-in assumptions that your
data, and the residual errors, fit a normal distribution
(bell curve)

36. Parametric versus non-parametric tests
% scores
If your data do not fit a normal
distribution, you can transform
the raw numbers to make them
more “normal” – put the data
through a mathematical function
arcsine(square-root(%)) is a
standard transformation for %’s
which stop at 100%, and are often
not normally distributed

37. Parametric versus non-parametric tests
Alternatively, there are non-parametric versions of most
common statistical tests that use ranked values instead of
the raw data
- are typically more conservative: if they see a difference,
it is real
- make no assumptions about the shape of the distribution
raw ranked (high to low)
3 5
2 6
6 3
4 4
9 2
1 7
12 1