biostats final

how do you decide to do a one sided test

problem tells you key info about hypothesis

• med to reduce blood pressure

use your own prior knowledge

what has more power 1 or 2 sided tests

1 sided

why can you not say that since ybar1>ybar1 then Hi: u1>u2

you should decide on the null hypothesis THEN look at y bars

How does our alternative hypothesis change for a one sided test

it is now directional

• greater/less than

What extra step do you need to do before calculating your test statistic in one sided tests?

Make sure your data agree with your alternative hypothesis!

How do you pick the table value for a one sided test

use the one sided bar on the table instead of the 2 sided indicators

What are your conclusions for a one sided test

same as for a 2 sided

t*>t table = reject

U* > table = rejct

p < alpha = reject

Why does a one sided test have more power?

a one-sided test concentrates all of its statistical power on detecting effects in that specific direction

instead of splitting the tails it is all in one

What is categorical data?

can be divided into distinct categories that do not have a natural order or ranking

• direct counts

• colors

χ 2 goodness of fit test.

• What is your null hypothesis?

• Your alternative hypothesis?

Ho: Pr(x) = %

Hi: Pr (x) is not % or one of the probabilities in the Ho is wrong

How do you calculate χ 2*?

sum ((obs-exp)²/exp)

How do you calculate your expected values for a x²?

Total * % given

(rt*ct ) / gt

What are your degrees of freedom in x²

n-1

What is your comparison for x²

chi square table

What is your decision for x²

X²* > table = reject

When can you use a one-sided alternative for x²

where there are only 2 samples

What are the assumptions of a χ 2 goodness of fit test?

data is random

smallest expected value must be greater than or equal to 5

How does sample size help with the “smallest expected value must be greater than or equal to 5” assumption of the x²

larger sample sizes make this assumption easier to achieve

you need to have a large enough sample that is assumption is not violated

Ho for contingency tables

Comparing proportions OR Determining independence

Ho: p1 = p2

or

Ho: 1 and 2 are independent

How do you calculate χ²* for contingency tables

sum ((obs-exp)²/exp)

degrees of freedom for χ²* for contingency tables

(row -1) (column -1)

How do you calculate your expected values for χ²* for contingency tables

( row total * column total ) / grand total

If you reject, know how to figure out which direction things are going in (particularly for tests of independence)

if you reject the Ho: a/b are independent it would mean that a/b are dependent

• but does that mean A likes or dislikes B

do conditional probabilities to find this out

• prob A + B present / total

• prob only A present / total

• if prob of only A is higher then they dislike eachother

what is a “r x k” tables

contingency table that is more than 2×2

relative risk = RR hat

p hat 1 / p hat 2

for a contingency table what size table will always result in a degrees of freedom =1

2×2 table

x² contingency test assumptions

random

smallest expected data is greater or equal to 5

why do we use a correlation

see if there is a relationship between x and y

What do we mean by comparing a continuous variable with a continuous variable?

numerical data vs numerical data

• height vs weight

continuous data

numerical variable that can take on any value within a given range

What is r, and what does it estimate?

r= pearsons correlation coefficent

estimates p (true unkown correlation coefficent)

has a value of -1 ≤ r ≤ 1

what does a r= 1 mean

what does a r=0 mean

r equal to (+) or (-) 1 means there is a good correlation

r equal to 0 means there is no correlation

What determines if r is positive? negative?

r is positive when there's a tendency for both variables to increase or decrease together

r is negative when one variable tends to increase as the other decreases.

What is the range for r?

-1 ≤ r ≤ 1

What is SScp?

• what does it tell you

• formula

tells you if the data goes up or down

used to represent variance

sum(xi-xbar)(yi-ybar)

If we're doing a correlation hypothesis test, what is our H0? our H1?

Ho: p = 0

H1: p ≠ 0 or p > 0

p = true correlation coefficient estimated by r

What is our t * for correlation

r * sqrt ( (n-2) / (1 - r²) )

degrees of freedom in correlation test

n-2

Why does a significant correlation not imply that one variable causes another?

correlation simply indicates a relationship exists, but doesn't prove which variable, if either, is influencing the other

there can be hidden variables

correlation assumptions

random data

relationship is approx straight/linear

x/y are both approx normal

how do you check that relationship is approx straight/linear and x/y are both approx normal for the correlation test

scatter plots

how is regression different from correlation?

Correlation measures the strength and direction of the relationship between two variables

regression analyzes how one variable affects another

Beta 1

• what is it

• formula

slope

SScp / SSx

Beta 0

• what is it

• formula

intercept

y bar - beta1(xbar)

What is a least squares lines, and how do we fit one to our data?

the straight line that best represents a set of data points in a scatter plot

• etermined by minimizing the sum of the squared differences between the actual data points and the corresponding points on the line

What is your equation for the least squares lines

y hat = beta0 + beta1 x

error or residual

the difference between the observed value and the predicted value from a regression model.

• distance from the line

• account for variability not represented by linear relationship

yi= beta0 + beta1 x + ei

What is the difference between Yhati and Yi ?

yhat 1 represents the predicted value of the dependent variable for the ith observation,

yi represents the actual observed value of the dependent variable for the ith observation.

What are we estimating with b0 and b1?

b0 = true unk population intercept

b1= true unk pop slope

How do we figure out if our slope is significant in a reegression

hypothesis test

test statistic for regression

b1 / (S residuals / SSx)

What is our H0? our H1? for regression

Ho: B1 = 0

hi: B1 is not 0 or b1 >0

What is SSresid?

sum ( yi - yhat) ²

• represents the unexplained variation in a dataset

What is SEb1?

sqrt (SS residuals / n-1) / SSx

• standard error of the slope coefficient

What does Y^ represent

the expected value from the least of squares line

regression assumptions

• how do you check these assumptions

1. random data

2. relationship between x and y must be straight

   • look at residual plot

3. residuals of each level of x are normally distributed

   • QQ plot of residuals

4. variance of the residuals is constant

   • residual plot

5. each residual is independent of every other residual

   • residual plot

how do you make a residual plot

graph y-yhat

what does it mean if your residuals show you have a curve?

violates linear assumption of regression

what should you see in a good residual plot

totally random data in a scatter plot

How do residual plots help you in assessing linearity and constant variance?

residual plots highlight problems in the data

• easier to see curves / funnels

How do q-q plots help you in validation the regression assumptions

comparing data distributions across different datasets for normality

What is R² ?

measure how how good x is at explaining y

R² formula

R² = (r= pearsons correlation coefficient)²

{ SScp / sqrt (SSx * SSy) } ²

or SS regression / SSy

outline for doing regressions

1. calc least of squares

2. hypothesis

3. check assumptions

4. if everything is ok → do hypothesis test

5. is R² significant?

what do you do if you data does not agree with you alternative hypothesis

you automatically Fail to reject Ho