D. Results
Results are hard to write
There are often multiple independent and
dependent variables:
D. Results
Results are hard to write
There are often multiple independent and
dependent variables:
- effect of density on D. putrida number
- effect of density on D. putrida mean mass
- effect of intra and interspecific comp on D.
putrida.
- effects on D tripunctata…
Pick a species or a dependent variable?
D. Results
Results are hard to write
There are often multiple independent and
dependent variables:
Organization of the text might be driven by
the best way to present the data visually –
what information is in Table 1?
1) Make a declarative statement about the significance or strength of a pattern:
“The mean number of Drosophila putrida progeny emerging from low density (N=10)
and high density (N=20) treatments were significantly different (t-test, t = 6.54, p =
0.05).”
1) Make a declarative statement about the significance or strength of a pattern:
“The mean number of Drosophila putrida progeny emerging from low density (N=10)
and high density (N=20) treatments were significantly different (t-test, t = 6.54, p =
0.05).”
2) Describe the pattern:
“The mean number of Drosophila putrida progeny emerging from low density (N=10)
and high density (N=20) treatments were significantly different (t-test, t = 6.54, p =
0.05).” More flies emerged from high density treatments than from low density
treatments (Table 1).
3) Elaborate. Maybe you do the correlation with size, or another test.
“The mean number of Drosophila putrida progeny emerging from low density (N=10)
and high density (N=20) treatments were significantly different (t-test, t = 6.54, p =
0.05).” More flies emerged from high density treatments than from low density
treatments (Table 1). However, the mean number of flies emerging from high
density treatments was not twice as much as the low density treatments; there was
only a 25% increase in progeny number with a doubling of initail flies added (20 vs.
10).
4) Now you might move on to the effects on number for D. tripunctata, OR another
dependent variable for D. putrida. I think it might be best to keep going with effects
on D. putrida… because the effects might reinforce a general pattern that would be
more obvious if the pieces were laid consecutively:
3) Elaborate. Maybe you do the correlation with size, or another test.
“The mean number of Drosophila putrida progeny emerging from low density (N=10)
and high density (N=20) treatments were significantly different (t = 6.54, p = 0.05).”
More flies emerged from high density treatments than from low density treatments
(Table 1). However, the mean number of flies emerging from high density
treatments was not twice as much as the low density treatments; there was only a
25% increase in progeny number with a doubling of initial flies added (20 vs. 10).
4) Now you might move on to the effects on number for D. tripunctata, OR another
dependent variable for D. putrida. I think it might be best to keep going with effects
on D. putrida… because the effects might reinforce a general pattern that would be
more obvious if the pieces were laid consecutively:
The mean mass of D. putrida metamorphs also differed between density treatments (t =
8.2, p = 0.01); flies emerging from low density treatments were larger, on average,
than flies emerging from high density treatments (Table 1).
4) Now you might move on to the effects on number for D. tripunctata, OR another
dependent variable for D. putrida. I think it might be best to keep going with effects
on D. putrida… because the effects might reinforce a general pattern that would be
more obvious if the pieces were laid consecutively:
The mean mass of D. putrida metamorphs also differed between density treatments (t =
8.2, p = 0.01); flies emerging from low density treatments were larger, on average,
than flies emerging from high density treatments (Table 1).
And…
There was also a significant decline in mean mass of D. putrida over time (ANOVA, F =
16.2, df = 3, 25, p = 0.001); flies that emerged early were larger than flies that
emerged later (Fig 1).
So, in this way, we can see the effect of density (10 vs. 20) on all D. putrida responses.
This makes it easier for US to make sense of these patterns, as well as helping the
reader.
5) At the end of the results (or at the beginning of the discussion), we might recap the
effect of our independent variables ON the response variables. In the first or second
sentence of the discussion, we’d say:
Intraspecific competition had a significant negative impact on D. putrida metamoph
abundance, size, and developmental rate. Metamorphs in the low density
treatment were more abundant, larger, and had a faster development time than D.
putrida emerging from high density treatments.
Distributions and Transformations
I. Probability Distributions
A. Uses
- a probability distribution often frames a particular
question; it generates a hypothetical distribution based on
random chance. These distributions are often used as
“standards” – or expected values – to compare observed
distributions against.
- to test the hypothesis of chance
- to test whether the assumptions for a test are met (normal).
I. Probability Distributions
A. Uses
B. Probability
1. classical probability – based on theory, alone. (chance of a “head” is
0.50, chance of a 5 on a 6-sided die is 1/6, chance of a homozygote recessive
from a cross between two heterozygotes is 0.25… etc.)
2. Empirical probability – based on prior knowledge of the population
of interest. Known frequencies of disease, sex ratio, etc.).
I. Probability Distributions
A. Uses
B. Probability
1. classical probability – based on theory, alone. (chance of a “head” is
0.50, chance of a 5 on a 6-sided die is 1/6, chance of a homozygote recessive
from a cross between two heterozygotes is 0.25… etc.)
2. Empirical probability – based on prior knowledge of the population
of interest. Known frequencies of disease, sex ratio, etc.).
Calculating the probability of these events is done using the
“Rule of Division”:
number of ways an event can occur divided by
the total number of events that can occur.
I. Probability Distributions
A. Uses
B. Probability
C. Discrete Probability Distributions
– for discrete or categorical variables
1. Binomial Distribution: for variables with two mutually exclusive and
discrete values (“p” and “q”): gametes A or a, sex male or female, heads or
tails; and the events occur repeatedly “k” times.
- assumptions:
- the probability of event p and event q are the same each trial
- the trials are independent
- the number of times a value occurs in k trials = x
- the binomial describes the probabilities of each x, p(x), for each
value of x from 0 to k.
I. Probability Distributions
A. Uses
B. Probability
C. Discrete Probability Distributions
– for discrete or categorical variables
1. Binomial Distribution: for variables with two mutually exclusive and
discrete values (“p” and “q”): gametes A or a, sex male or female, heads or
tails; and the events occur repeatedly “k” times.
- distribution = (p + q)k = 1
- Product Rule: the probability of independent events co-occurring is the
product of their independent probabilities.
Flip three coins, both with p(heads) = q(tails) = 0.5.
They are independent (don’t affect one another), so probability
of 3 heads = (p) x (p) x (p) = p3
(Product/And Rule) 1st a head AND the second AND the third
Probability of 2 heads and a tail:
(p) x (p) x (q) = 0.125
OR
(p) x (q) x (p) = 0.125
OR
(q) x (p) x (p) = 0.125
SUM THESE = 0.375 = 3p2q
Flip a coin with p(heads) = 0.5, 3 times
(or three coins once… 3 events).
Distribution = (p + q)3 = 1
1p3 + 3p2q + 3q2p + 1 q2 = 1
(3 heads, 2 heads:1 tail, 2 tails:1 head, 3 tails)
Generic Formula for a given outcome (x) =
p(x) = k!
pxq(k-x)
x! (k-x)!
Probability of 3 heads = p(3 heads) = 3*2*1 0.53(0.5)(3-3)= 1 x 0.53
3! (0)!
Distribution = “binomial expansion” = (p + q)3 = 1
1p3 + 3p2q + 3q2p + 1q3 = 1
(3 heads, 2 heads:1 tail, 2 tails:1 head, 3 tails)
red = probability of an outcome
green = number of outcomes that satisfy those conditions
Pascal’s Triangle: These are the constants for terms in a
binomial expansion: 1p3 + 3p2q + 3q2p + 1 q3 = 1
K=1
K=2
K=3
K=4
I. Probability Distributions
A. Uses
B. Probability
C. Discrete Probability Distributions
– for discrete or categorical variables
1. Binomial Distribution: for variables with two mutually exclusive and
discrete values (“p” and “q”): gametes A or a, sex male or female, heads or
tails; and the events occur repeatedly “k” times.
The number of combinations increases with k…. To the point where with 30
events, the distribution approaches a normal distribution.
I. Probability Distributions
A. Uses
B. Probability
C. Discrete Probability Distributions
– for discrete or categorical variables
1. Binomial Distribution: for variables with two mutually exclusive and
discrete values (“p” and “q”): gametes A or a, sex male or female, heads or
tails; and the events occur repeatedly “k” times.
The number of combinations increases with k…. To the point where with 30
events, the distribution approaches a normal distribution.
Many multiple outcomes can reduced to a binomial… roll a dice… reduce
prob to ‘1’ or ‘not 1’.
Utility:
Testing sex ratio in litters of multiple offspring:
# males
#females
P(x)
Freq per 100
6
0
0.0156
1.56
5
1
0.0937
9.37
4
2
0.2344
23.44
3
3
0.3125
31.25
2
4
0.2344
23.44
1
5
0.0937
9.37
0
6
0.0156
1.56
These become expected values for frequency distribution tests;
it tells us what to expect due to chance.
I. Probability Distributions
A. Uses
B. Probability
C. Discrete Probability Distributions
– for discrete or categorical variables
1. Binomial Distribution:
2. Poisson Distribution: The Poisson distribution predicts the distribution
of rare, INDEPENDENT events, usually across time or space. If the real
distribution deviates from the Poisson, then the events are probably NOT
INDEPENDENT.
“RARE” in the sense that N is less than what COULD occur.
Utility:
# of maple seeds/quadrat
Frequency
0
35
1
28
2
15
3
10
4
7
5
5
6
0
RARE, in the sense that a m2 quadrat could contain MANY MORE
than just 5 seeds.
Formula:
p(x) = Xxe-x
x!
Formula:
p(1) = 1.411e-1.41
1!
= 0.344
# of maple seeds/quadrat
Frequency
Exp p(x)
xN (100)
0
35
0.244
24.4
1
28
0.344
34.4
2
15
0.243
24.3
3
10
0.114
11.4
4
7
0.04
4
5
5
0.011
1.1
6
0
0.003
3
I. Probability Distributions
A. Uses
B. Probability
C. Discrete Probability Distributions
– for discrete or categorical variables
1. Binomial Distribution:
2. Poisson Distribution:
D. Normal Distribution
- often approximated by continuous variables
I. Probability Distributions
D. Normal Distribution
- often approximated by continuous variables
1. Properties:
- defined by the mean and s.d.
- centered on mean; symmetrical
- spread defined by s.d.:
- 68.26% of the values are within 1 s.d. of the mean
- 95.46% of the values are within 2 s.d. of the mean
- 99.93% of the values are within 3 s.d. of the mean
I. Probability Distributions
D. Normal Distribution
- often approximated by continuous variables
1. Properties:
2. The Z score:
- In a large, normally distributed population, 95% of the population is within
1.96 standard deviations of the mean.
- we can determine how many deviational units separate a point from the
mean with this equation:
z = (x – X)/sd
- the smaller the sample size, the more deviational units we must include to
be sure to include 95% of the points.
I. Probability Distributions
D. Normal Distribution
- often approximated by continuous variables
1. Properties:
2. The Z score:
3. Normal Approximation of the Binomial:
- in a reasonably large sample (N > 30), where neither p or q is close
to 1, the binomial has a reasonably normal distribution.
how large? K x p and k x q are both greater than 5
mean approximated by = kp
sd approximated by = kpq