Biological Data Analysis

survivorship bias

only having data on individuals that lived through an incident

random sampling qualifications

1. each individual has an equal chance of being selected

2. selection is independent between individuals

precision

minimizing sampling error

accuracy

minimizing bias

categorical variables

qualitative groupings with no inherent magnitude on a numerical scale

nominal variables

categroical variables with no obvious order

ordinal variables

categorical varibles that have an intrinsic pattern

numeric (quantitative) variabled

have a magnitude on a numerical scale

continuous variables

numeric variables for data containing any real number within some bounds

discrete variables

numeric variables for data containing only whole numbers

ratio vs interval scale

ratio scales have a true zero point representing the absence of the variable, while interval scales do not.

frequency distribution

the number if times each value occurs in a sample

histograms

Descriptive Statistics

quantities that capture features of the sample data

arithmetic mean

regular old averaging

very impacted by outliers

median

middlemost measurement

somewhat resistent to outliers

geometric mean

used to summarize data when variables are multiplicative

standard deviation

square root of variance

s = sqrt(s²)

variance

spread of data

s²=(sum(observation - mean)²)/(n-1)

coefficient of variation

used for ratio scale variables (spread is expected to increase with mean)

CV=s/sample mean

IQR

range with middle 50% of the data

useful for skewed data

sampling distribution

distribution of values for an estimate that we might obtain with repeated sampling of a population

standard error

standard deviation of sampling distribution

SE(mean) = s/sqrt(n)

Confidence Interval

range of values that are likely to contain the target parameter

sample mean ± 1.96 * SE(mean) for normally distributed data

Probability

proportion of times event occurs if repeating random trial many times

value between 0 and 1, where probabilities of all possibilities must sum to 1

Probability Mass Function

probability of all possibilities for discrete data

Probability Density Function

probability of all possibilities for continuous data

Marginal Probability

the probability of a single event occurring, independent of other variables

ex. P(A)

Conditional Probability

measures the likelihood of an event occurring given that another event has already occurred

ex. P(A|B)

Union Probability

measures the likelihood that at least one of multiple events occurs

ex. P(A u B)

Compliment Probability

calculates the likelihood of an event not occurring

ex. P(A^c)

Intersection/ Joint Probability

the likelihood of two or more independent or dependent events occurring simultaneously

ex. P(A,B)

P(A and B)

P(A)*P(B)

P(A or B)

P(A) + P(B) - P(A,B)

Bayes Theorum

Bayes' Theorem helps us update probabilities based on prior knowledge and new evidence

𝑃(𝐴|𝐵)=(𝑃(𝐵|𝐴)×𝑃(𝐴))/𝑃(𝐵)

P(A given B)

P(A,B)/P(B)

probability distribution

the mathematical function that gives the probabilities of occurance of possible outcomes

Used to: Fit models to datam represent uncertainty in parameters, and portray prior information

Normal Distribution

continuous

symmetrical around its mean

unimodal

probability density is highest at the mean

normal distribution ranges

x (-inf, +inf)

u (-inf, +inf)

sigma > 0

Z-score

test statistic for a normal distribution

(X-u)/sigma

log normal distribution

continuous

positive only

positive right skew

Log normal distribution ranges

x (0, +inf)

u (-inf, +inf)

sigma > 0

central limit theorum

the sum or mean of measurements randomly sampled from ANY distribution is approximately normally distribution

location moment

mean

spread moment

variance

symmetry moment

skewness

heavy tailed-ness moment

kurtosis

t-distribution ranges

x (-inf, +inf)

u (-inf, +inf)

sigma > 0

v (nu) > 0

poisson distribution

discrete

positive only

only one parameter (lambda)

normal is a good approximator when lambda is large

often used for counts

poisson distribution ranges

x (positive whole numbers)

lambda > 0

lambda

= mean = variance (s²)

underdispersed population

variance < mean

overdispersed

variance > mean

binomial distribution

discrete

positive only

used for success/fail trials

binomial distribution ranges

x (# of successes) (whole numbers > 0)

n (# of trials) (whole numbers > 0)

p (probability of a success) [0,1]

bernoulli distribution

a special case of binomial distribution where n (number of trials) =1

multinomial distribution

generalization of the binomial, when there are >2 categories

ex. dice

gamma distribution

continuous

positive only

flexible

multiple common parameterizations

gamma distribution ranges

x [0, +inf)

alpha (shape parameter) > 0

theta (scale parameter) > 0

Beta distribution

continuous

bound between 0 and 1

used for proportions

Beta distribution ranges

x (0,1)

alpha >0

Beta > 0

Directed Acyclic Graphs (DAGs)

can use these to denote causal relationships between variables

DAG features

node

edge

direction

a/cyclic

confounding variables

influences both the dependent and independent variables, causing a spurious correlation

Fork example

sun intensity, sunburns, ice cream sales

pipe example

day of the year, temperature, how fast ice cream melts

the collider

ice cream sales, quality of ice cream, outside temperature

hypothesis testing

compares collected data to expectations under a null hypothesis to determine how unlikely the data are

p-value

probability of obtaining a value that is as or more extreme than the observed value, given the null hypothesis is true

test statistic

value calculated from the data that is used to eveluate how unlikely your data are, given the null hypothesis is true.

p-value < a

reject the null hypothesis

𝛼

probability of committing a type one error (false positive). Generally 0.05

𝛽

probability of committinga type two error (false negative)

p-value > 𝛼

The data are compatible or consistent with the null hypothesis

Confidence intervals

range of values that are likely to contain the target parameter

Permutation tests

generates a null distribution of the test statistic by repeatedly rearranging values

slightly less power than parametric tests (like t-tests) when sample sizes are small

performing a permutation test

calculate test statistic

randomly rearrange data into new groups

caclulate test statistic for permuted data

repeat 1000+ times to generate sampling distribution of test statistic under the null hypothesis

calculate p-value

Bootstrap

resampling data with replacement to approximate the sampling distribution of an estimate

useful for finding standard error or confidence interval for a parameter estimate

performing bootstrapping

sample with replacement from original sample

calculate estimated median from boootstrapped data

repeat 10000+ times

calculate SE

ways to reduce bias

control group - does not receive treatment but is exposed to the same conditions

randomization - random assignment of treatments

blinding

single blinding

hides treatment details from participants to prevent behavioral bias

double blinding

hides treatment details from participants and researchers to prevent expectancy effects

ways to reduce sampling error

replication - application of each treatment to multiple, independent units OR application of multiple, identical ttreatments to a single unit

balance - equal sample size in all groups

blocking - grouping of units that share properties (like location); within each block, treatments are randomly assigned

statistical power

probability that a random sample will lead to rejection of a false null hypothesis

1 - beta

Type 1 error

rejecting the null hypothesis when the null hypothesis is true

False +

alpha

Type 2 error

failing to reject the null hypothesis when the null hypothesis is false

False -

Beta

Statistical Power is most affected by

Sample Size and Variance of Data

chi squared goodness of fit test

compares frequency data to a probability model stated by the null hypothesis

common for hypothesis testing

chi squared test statistic (x²)

sum((observation - EV)²/EV)

chi squared distrubution

only affected by k (degrees of freedom)

k = df

k

# categories - 1 - # parameters estimated from the data

chi squared assumptions

random and independent sampling

expected frequencies >1

no more than 20% of categories should have expected frequencies < 5

one sample t-test

compares the mean of a sample with some “null mean”

t-distribution

continuous and symmetrical. Nu controls “heavy tailed-ness”

t-distribution test statistic

(sample mean - null mean) / SE of the sample mean

T-test df

n-1 = v

Paired t-test

compares the mean difference between two sample means to a null mean

controls for variation among plots that is otherwise difficult to control for

paired t-test test statistic

(mean difference - null mean)/SE of the mean difference

Unpaired t-test

compares the difference of one samle mean from another sample mean

Unpaired t-test test statistic

(sample 1 mean - sample 2 mean)/SE of sample 1 mean - sample 2 mean

Unpaired t-test df

v = n1 + n2 - 2