biostats final

one-sided hypotheses

H0: µ1 = µ2 (there is no difference between the two groups) 

H1: µ1 > µ2 or µ1 < µ2 (difference exists in a specific direction based on expectation or interest)

why does one-sided alternative hypothesis change

More powerful for detecting difference in a specific direction and ignores potential differences in the opposite direction 

how is alternative hypothesis for one-sided tests chosen

when study suggest change in one direction

one-sided test assumptions

sample mean follows same direction of alt hypothesis (µ1 < µ2, then we expect y1 < y2)

critical value one-sided test

Significance level (a) is concentrated in one tail, increases the test's ability to detect a difference 

categorical data

frequencies, how often each category occurs

null hypothesis of goodness of fit test

The observed frequencies match the expected frequencies

alternative hypothesis of goodness of fit test

at least one of the expected proportions is incorrect

how is chi-square different from two-sample test

chi-square uses categorical variables and two-sample uses continuous numerical data

assumptions of chi-square test

data is random, smallest expected value is >= 5

sample size and chi square test

• Larger samples increase likelihood of detecting small, insignificant differences 

• Smaller samples may fail to meet assume of expected cell frequencies are greater than 5 = inaccurate p-values 

X²* = 0

perfect fit between observed and expected frequencies, FTR, variables are completely independent

contingency table

comparison of two categorical factors, each with two or more levels

usage of contingency tables

comparing proportions and testing association/independence

comparing proportions hypotheses

• H0: The population proportions are equal across groups (p1 = p2) 

• H1: The proportions differ (p1=/ p2, p1 > p2, p1 < p2) 

testing independence hypotheses

• H0: The variables are independent (there is no relationship or association) 

• H1: The variables are not independent (there is a relationship or association) 

assumptions of contingency tables

data is random, all expected values are >= 5

parameter of X²*

degrees of freedom, as d.f increases, distribution becomes more spread out and less concentrated near 0

proportion

shows how much something is in relation to total

parameter for proportions and what estimates it

p = part/whole, estimated by sample proportion

X²*

measures the total difference between the observed and expected frequencies across all categories, large value = reject null

one-sided alternative conditions for chi-square

if d.f = 1

correlation

compares two continuous variables, answers direction and strength

sample correlation coefficient

describes direction and strength, estimates population correlation coefficient

positive correlation coefficient

As one variable tends to increase/decrease, the other variable also tends to increase/decrease - same direction 

negative correlation coefficient

As one variable tends to increase, the other tends to decrease and vice versa – opposite direction 

0 correlation coefficient

no linear relationship

strength of relationship

close to +1 or -1, strong, closer to 0, weak

correlation coefficient range

+1 to -1

null hypothesis correlation test

H0: p=0 (there is no relationship between X and Y) 

alternative hypothesis correlation test

• H1: p=/0 (there is a relationship between X and Y 

• IF one-sided: p> 0 (X and Y have a positive relationship or p < 0 (X and Y have a negative relationship) 

assumptions of correlation test

random data, X and Y are linear, for one-sided (H1: p < 0 then r < 0)

scatterplots

show correlations on a graph

does order of variables matter

No, order of the formula gives the exact same value regardless of data point on x or y axis 

Why is neither variable in correlation tests considered independent or dependent? 

We are finding association, variables may effect one another or not 

regression test

compares two continuous variables where one variable is used to predict the other

least squares line

relationship between two continuous variables represented using a straight line, provides a model to predict Y from X

center of least squares line

(xbar , ybar)

residuals

Distance between each observed value and its predicted value on the regression line, observed value - predicted value

b1

slope, estimates direction and rate of change

b0

y-intercept, reflects average relationship between x and y

SSr/sum of squared residuals

Small SSr: line fits well (overall error is low) 

Large SSr: line fits poorly (overall error is high) 

coefficient of determination

how much variation in the dependent variable is explained by the least squares line using the independent variable

high r² (close to 1): least squares line explains large portion of the variation in Y

low r² (close to 0): most of the variation in Y is unexplained

range of R²

0 to 1

significance of slope tested

helps determine whether the observed relationship is statistically meaningful rather than due to random chance

null hypothesis regression test

no relationship between X and Y, slope is equal to zero (B1 = 0)

alternative hypothesis

there is a relationship between X and Y, slope is not equal to 0

assumptions of regression test

data must be random, X and Y are linear, normal distribution, residuals are independent, variance of residuals are constant, if one-sided, B1 < 0 and b1 < 0

residual plots

show predicted values/residuals, X values on x-axis, residuals on y-axis, reference line at y = 0  

normal residual plot

balanced below and above 0 line, no trend, if Q-Q plot is made, points would align closely

curved residual plots

U shape/inverted U, shows it is not linear

funnel shaped residual plot

widen or narrow in a funnel-like pattern (left to right), violates variance

when to use one-sample test

compare data to pre-set value

when to use two-sample test

compare two independent groups of data

when to use correlation test

used for associations

when to use regression test

to model/predict outcomes

when to use welchs test

independent groups, unequal variances, unequal sample sizes

when to use classic t-test

independent groups, equal variances, equal sample sizes

when to use paired t-test

dependent data

when to use mann whitney U test

when there are extreme outliers

when to use chi-squared test

when testing proportions or frequencies of categorical