STSCI 2150 Prelim 2

TUNA TEA

1. think up a research question

2. state the null hypothesis

3. state the alternative hypothesis

4. test

- test name

- test statistic formula

- test statistic value

5. evaluate extremeness

- null distribution

- p-value or critical value

- decision

- state conclusion in English

6. assess the next step

asks how unusual it is to get the data we got when the null hypothesis is true

hypothesis testing

asks how unusual it is to get the data we got when the null hypothesis is true

null hypothesis (H0)

a specific statement about a population parameter made for the purposes of argument, usually the simplest statement

alternative hypothesis

represents all other possible parameter values except that stated in the null hypothesis, usually the statement of greatest interest

null distribution

the probability distribution of possible values of the test statistic when the null hypothesis is true, chi-squared with df = _ OR binomial with N = __ and p = __

P-value

the probability of getting the given result, or something as extreme or more extreme, when the null hypothesis is true

significance level (alpha)

a probability used as a criterion for rejecting the null hypothesis; if the P-value for a test is less than or equal to alpha, then we reject the null hypothesis

test of a single proportion

TU: a population proportion equals a specific number?
H₀: p = #
H_A: p != # (or p > # or p < #)
T: test of a single proportion, find the number of successes, X
E: binomial distribution is null distribution

type I error

rejecting a true null hypothesis, false positive, probability of type I error is alpha

type II error

not rejecting a false null hypothesis, false negative

power

the ability of a test to reject a false null hypothesis, the probability of a correct decision when the null hypothesis is false. larger sample gives more power to reject a false null hypothesis

p

the population parameter for the true proportion of individuals in the "success" category

p hat

sample proportion, X/N = # subjects with attribute / total subjects

binomial distribution

describes the probability of a given number of "successes" from a fixed number of independent trials each with the same probability of "success" -- the null distribution for a test of a single proportion

binomial distribution assumptions

-the number of trials (N) is fixed
-individual trials are independent
-the probability of success (p) is the same for every trial

binomial probability formula

P(x)= (nCx) (p^x) (1-p)^n-x

x = number of "successes"

n = number of trials

p = probability of success

confidence interval

(sample estimate +/- 2*SE of sample estimate)

Agresti-Coull method

4th formula on the sheet

duality of confidence intervals and hypothesis testing

reject null, H_A favored = CI does not include hypothesized value

do not reject null, H₀ favored = CI includes hypothesized value

3 types of statistical inference

1. interval estimate (confidence interval)
2. point estimate (phat = X/N)
3. hypothesis test (H₀: p=#, H_A: p!=#)

test of association

are two categorical variables independent or not? shown on mosaic plot

independent ("no association")

two variables are independent if the probability of a particular outcome of one variable is about the same for both levels of the other variable

relative risk

ratio of probabilities for the event of two groups

pr(worse outcome in group 1)/pr(worse outcome in group 2)

The probability of a (focus event) is (RR) times greater for the (numerator group) vs. the (denominator group).

odds ratio

ratio of odds of the event for two groups

OR = O1/O2 = odds(event in group 1) / odds(event in group 2) = AD/BC

The odds of (event) are (OR) times greater for r for the (numerator group) vs. the (denominator group).

test statistic (X²)

sum (observed - expected)²/expected or just X

contingency analysis

estimates and tests for an association between two or more categorical variables

residuals

observed - expected

sample size requirements for Chi-squared test

- all cells must have expected counts greater than or equal to 1
- at least 80% of cells must have expected counts greater than or equal to 5

options to evaluate extremeness

compare P-value to alpha OR compare test statistic value to critical value.

P-value < alpha, reject null

X² > critical value, reject null

finding P-value in R

1 - pchisq(test stat, df)

degrees of freedom for test of association

df = (columns-1)(rows-1)

critical value

the value of the distribution at a set alpha value; the value beyond which the probability of such a value or greater is less than the set alpha level

finding critical value in R

qchisq(1-alpha, df)

Agresti-Caffo interval

95% CI for true difference of two proportions (p1-p2)

big difference = large association

uniform distribution

probability model in which the probability of each outcome category is the same, p1=p2=p3=…

proportional distribution

model comes from a large entity, the data should match the number of opportunities

goodness-of-fit test question

do the collected categorical data fit a hypothesized probability distribution? compares observed counts to expected counts according to a specified probability distribution

H0: the data have a ___ distribution (uniform, proportional, or poisson)

discrete distribution

probability distribution describing a discrete random variable

degrees of freedom for goodness-of-fit

df = # of categories - 1 (uniform, proportional)

df = # of categories - 2) (poisson)

poisson distribution

describes the probability that a certain number of events occur in a block of time or space, when those events happen independently of each other and occur with equal probability at every point in time or space. No upper bound

clumped events

lots of low and high counts, presence of one individual attracts others

dispersed events

lots of middle counts, presence of one individual repels others