Biol 315

population

the set of all “subjects” relevant to the scientific hypothesis under examination

variables

characteristics that differ among individuals

parameters

quantities describing a population (denoted by Greek letters)

census

a collection of data where the population is examined

random sample

each and every member of the population has an equal chance of being selected and each member is selected independently of others

mean

denoted by a bar

standard deviation

denoted by an “s”

sample

the subset of cases selected from a statistical population that are actually examined during a particular study

sample statistics

calculated from the collected sample and used to estimate the population parameters (denoted by roman letters)

how to get a good sample

take a random sample, be unbiased an precise

to get a good sample

carefully define your statistical population and select a sample that is as representative of the population a possible, where each subject is selected randomly and measurements are precise

bad samples

volunteer sample or a sample of convenience

experimental study

assigning treatment randomly, creating groups, imposing change

observational study

relying on comparisons of already existing conditions

2 types of variables

numerical (quantitative) and categorical (qualitative)

2 types of numerical variable

interval (arbitrarty zero) and ratio (true zero)

2 types of categorical variables

nominal (no order) and ordinal (ordered)

frequency distribution

describes the number of times each value of a variable occurs

histograms

used for numerical data - x axis has a continuous scale, data are “binned” into continuous categories, the bins are touching

histograms y-axis can be

frequency (count of observations in each bin), proportion(of the total observations in each bin) and density(the proportion of the total observations per unit of the bin width)

location or central tendency

distributions with a different central measurement using the mean

spread or scale

distributions with a different spread measured using the standard deviation

shape or skew

distributions with a long tail on one side or the other

mean

arithmetic average

median

middle of the data

mode

most commonly occurring observations

scale

most basic (max - min), not very informative

variance

“expected” squared difference between an observation and the mean

standard deviation is

positive square root of the variance

what is meant by “estimation”

it’s using the sample data to learn about the popualtion

estimation

the process of inferring a population parameter from sample data

uncertainty

a situation in which something is not known; in statistics it is the error of an estimate

Sampling distribution

the distribution of all the values for an estimate that we might have obtained when we sampled a population

a 95% confidence interval is a

range of values, calculated from sample data, that would contain the true population parameter in 95 out of 100 samples if the sampling process were repeated

uncertainty

decreases an precision increases with sample size

hypothesis testing

to determine whether an estimate can be simply explained by chance or is it special

Null hypothesis

is a specific statement made about population for the sake of argument, forces us to take a skeptical view

null hypothesis is used to

create a null model, compare test statistic calculated from the sample to the model

H0 is rejected if

we are surprised by the test statistic

P means

probability of observing a test statistic as extreme as, or more more extreme than, the one observed, assuming H0 is true

significance level α

a probability used as a criterion for rejecting the null hypothesis

P-value > α

fail to reject H₀

P-value < α

reject H₀

two-tailed tests

deviation is either direction would reject null hypothesis

type I error (α)

rejecting a true null hypothesis (false posititve)

type II error (β)

Failing to reject a false null hypothesis (false negative)

Power

the probability of correctly rejecting a false H₀

Power depends on

how different the truth is from the null hypothesis, type I error rate, and sample size

things to consider when designing an experiment

reduce bias and decrease sampling error

reduce bias

have a control group, use randomization, use blinding

decrease sampling error

use replication, ensure balance, use blocking, implement extreme treatments

Control group

units that are similar to the treatment units except that they do not receive the treatment

random assignment

units that are otherwise “identical” are assigned to be controls or treatments

blinding

concealing information about whether a participant is in the control or treatment group (single blind) and sometimes researchers (double blind)

replication

application of treatment ti multiple, independent experimental subjects or units

balance

an equal number of units in the control and treatment minimizes the sampling error in both

blocking

divide experimental units into groups with known confounding variables

extreme treatments

a treatment you may add to an experiment to see if by doing more (or less) of a treatment will elicit more (or less) of an effect

probability distributions

the probability with which each possible observation of a variable occurs

normal distribution

continuous probability distribution, bell shaped curve, symmetrical around the mean

binomial distribution

discrete probability distribution, outcome of a number of bernouli trials

Bernoulli trials

only 2 possible outcomes, outcomes are indenpedent of each other

Poisson distribution

frequency distribution of events that occur rarely and randomly

Poisson conditions

the probability of 2 or more occurrences in a single sample subdivision is negligibly small, probability of one occurrence is proportional to the size of the subdivision, outcomes are independent, probability of an occurence is identical for all sample subdivisions

analysis of frequencies

compares the observed frequency of observations in different categories with the expected frequency under a null hypothesis

goodness-of-fit assumptions

no more than 20% of categories have expected counts <5, no category with expected count <1

how to calculate a G-test

calculate expected frequencies, calculate g based on the dissimilarities between observed and expected, compare G to the tabulated X² distribution

G test df

= k - p - 1, k= categories, p = estimated parameters

contigency analysis

asks whether 2 categorical variables are associated with one another

extrinsic hypothesis

derived from information other than the data you are analyzing

intrinsic hypothesis

expected frequencies are derived from the data you are analyzing

contingency df

k-p-1 or (r-1)(c-1), r = rows, c = columns

normal distribution defined by two parameters

mean and standard deviation

properties of a normal distribution

continuous distribution, probability measuted as the area under the curve, symmetrical, 2/3 of the are under the curve lies within on sd of the mean

central limit theorem

the mean of a large number of measurements randomly sampled from a non-normal (or normal) population is approximately normally distributed

Student’s t distribution

estimate the standard error or the mean use z-scores to estimate the probability of obtaining a particular sample mean given the population of means from which we are sampling

Z vs t

t distribution is more spread out because of uncertainty about the true population standard deviation

student’s t df

n - 1, n = sample size

single sample t - test

comapres the mean of a random sample to a population mean proposed in a null hypothesis

single-sample t-test assumptions

variable is normally distributed in the population, data is a random sample of the population

paired samples

individual observations in 2 samples are connected, i.e. an individual before and after treatment

paired t-test

difference between two paired observations, testing whether the mean difference between paired measurements equals some specified value (usually zero)

paired t-test assumptions

that the differences are normally distributed and the each pair is independent from the others

when can we ignore violations of assumptions

small deviations, large sample size, non-normality if deviations are small and size is large (central limit theory)

what to do when assumptions are violated

ignore violations, transformation, permutation/randomization/resampling methods

transformation

changes data to another scale of measurement to fit assumptions

data transformation by

applying the same mathematical formula to each observation, needs to be applied to each individual data point, must be backwards convertible to the original data

Anova

analysis of variance, compares and determines if there is significant difference between means of two or more unpaired groups

ANOVA assumptions

independence, normality, homogeneity of variance (homoscedasticity), random sampling

total sum of squares (SST)

measures the total variation in the dataset

between-group sum of squares (SSB or SSG)

this measures the variation between the group means and the grand mean

error sum of squares (SSE) or within-group SS (SSW)

this measures the variation within each group

R²

summarizes the contribution of the group difference to the total variation

Anova is a general linear model GLM

a mathematical representation of the relationship between a response variable and one or more explanatory variables

GLM fixed

categories of the explanatory variable are pre-determined (drug trial dosage)

GLM random

randomly sampled from a larger pool of groups, groups are not of particular interest

Tukey’s test

tests which treatments/groups differ from each other, compares all possible pairs of means to infer which one(s) differ(s) while controlling type I error, only do if anova rejected H0

balanced design

all treatment groups have the same n, tukey’s test

unbalanced design

not all treatments have the same n, tukey-kramer test

tukey’s df

= N - k, N = total sample size (sum for all groups), k = total number of groups in the anova