Biostats final

Truth

Something we can never know with certainty

Elements of a good scientific research question

• Clarifies your primary goal

• Identifies the scope of inference

• Interesting to others

• Can be answered with data

• Grounded in theory

• Feasible

Goals of scientific research

• Description: numerically summarize a quantity of interest

• Prediction: forecast some future event based on historical or current data

• Explanation: understand whether a particular variable is cause of some other variable

Explanatory variable

variable inducing a causal effect

Response variable

variable receiving causal effects

Quantitative

numeric: continuous or discrete

Qualitative

categorical, factor: nominal or ordinal

Continuous variable

scale down to any number of decimal places (ex. height, weight, temp)

Discrete variable

can only take on whole numbers (ex. number of students in the classroom)

Nominal variable

categories do not have an order

Ordinal

categories have an order (ex. lifespan)

Sample size

• effects the shape of distribution

• can lead to misleading patterns or conclusions if it is too small

• can increase uncertainty if too small

DAGs

• used to visualize the causal assumptions of a hypothesis

• communicates your causal assumptions

Cause

refers to a variable or event that produces a change in another variable

Counterfactual

refers to a hypothetical scenario

The fork

• Confounder

• Can lead to spurious relationships

• Can mask real causal relationships between explanatory and response

• directed, open

The pipe

• Mediator

• Can cause post-treatment bias with the total effect

• directed, closed

The inverted fork

• Collider

• directed, closed

Total effect

indirect effects + direct effects

Observational Study

• sensitive to confounding variables

• inferring causation requires formal causal inference approaches for statistical analysis

Experimental Study

• randomization breaks association between treatments and other variables

• causal inference approaches can be necessary to avoid post-treatment bias

Randomization

The defining feature of an experiment

Populations

entire group of individuals of units you are interested in studying

Samples

a subset of the population that is observed or measured

Parameter

Quantities with unknown values

Statistic

a numerical summary calculated from a sample

estimate

the value of a statistic used as an approximation of a parameter

estimand

parameter of interest

Sampling distribution

distribution of possible outcomes of an estimate based on our sampling process

Precision

• measure of how consistent samples should be when we repeatedly sample from a population

• describes sampling error

Accuracy

• describes bias

How to maximize precision

replication

How to maximize accuracy

doing random sampling

Random trial

each test is a random process

outcome

end results of each trial

sample space

set of all possible outcomes

Frequentist

probability is the proportional of trials n where we observe the event of interest, X

Bayesian

Probability is as a strength of beleif

Kolmogorov’s Axioms of probability

• Rule 1: the probability if any event X is non-negative

  • P(X) > 0

• Rule 2: the probability of any possible outcome in the same sample space is certain (1)

  • P(Ω)=1

• Rule 3: Addition rule

  • P(A or B)=P(A)+P(B)

Joint probablity

probability of the first and second event

Marginal probability

to get the total probability

Independent events

• two events are independent if the occurrence does not affect the probability of the other

• (A and B)=P(A)*P(B)

Mutually exclusive

Example: an individual cannot test positive and negative for the infection at the same time

Conditional probability

• defined as the probability of event A given that know event B is true

• P(A|B)= P(A and B)/P(B)

Random variable

where the term random implies an element of chance in terms of how we observe the variable

Probability distribution

probability of observing each mutually exclusive outcome

Probability density

how probability is distributed across the possible value of a continuous random variable

Probability mass

can be quantified as area under the probability density function for any interval of interest

Empirical Rule

• 68% of observations are within 1 standard deviation of the mean

• 95% of the observations are within 2 standards deviation of the mean

Parameter estimation

assumes a single trait true parameter value

point estimates

represents the single best estimate of the parameter of interest

sampling distribution

• the probability used to describe a sample estimate

• an illustration of uncertainty about the estimates taken from samples

The mean in a sampling distribution

the true parameter value

Standard error

standard deviation of the sampling distribution

Law of large numbers

as the sample size N increases, the point estimate ultimately converges on the true parameter value

Central Limits Theorem

the distribution of a sample estimate will be approximately normal at large sample sizes regardless of the shape of the probability distribution for the underlying probability distribution

Confidence Interval

Use info from the standard error to quantity at a range of possible values for the true proportion parameter at a given level of confidence

Steps of null hypothesis significance testing

1) Specify the null and alternative hypothesis

2) Determine the test statistic and significance value

3) Compute the sampling distribution for the null hypothesis

4) Compute p-value

5) Make a decision

Null hypothesis

Ho, represents hypothesis with “no effect”

Alternative hypothesis

Ha, represents the opposite of the null hypothesis

Null distribution

sampling distribution for the null hypothesis

p-value

probability of getting an estimate as or more extreme than our sample estimate, assuming the null is true

Type I error

We reject a null hypothesis that is actually true

Type II error

• Failing to reject a null hypothesis that false

Factors that affect Type II errors

• low sample size and high variability

• effect size is small

• low significance value

One-tailed test

the alternative hypothesis is directional

significance value (α)

refers to the probability of observations in the tails

Two-tailed test

does not specify directionality of a potential effect

prior probability

P(Hypothesis), quantitative statement of your degree of belief about the hypothesis prior to collecting new data

Likelihood

P(Data|Hypothesis), probability of new data you observe given that the hypothesis is true

Marginal Likelihood

P(Data), overall probability of the data integrated across all possible hypotheses

Posterior Probability

P(Hypothesis|Data), our updated probability given the new data

Point estimates

represent the single best estimate of the parameters of interest

Steps of bayesian

1) Specify the prior distribution

2) Quantify the likelihood'

3) Quantify the marginal likelihood'

4) Quantify the posterior distribution

Binomial distribution

probability distribution for a binary variable where the outcome of that binary variable is examined across n trials

Statistical models

• quantitative representations of our scientific models

• consist of an equation, or a set of equations, that describe single variables, and more often in causal inference, the relationship between variables.