Biostats Exam 2

consequence of violating equal variance

MSE increases, so F decreases, so power decreases (β Type II error increases - less ability to reject false null hypothesis)

consequence of violating normality

MSE increases, so F decreases, so power decreases (β Type II error increases - less ability to reject false null hypothesis)

F =

MSa ÷ MSw

models of ANOVA

I. fixed effects II. random effects III. mixed effects

Tukey's HSD

"highly" or "honestly" significant difference; more conservative; finds fewer differences than actually exist; β slightly higher than desirable

rule of thumb: we would like 1-β to be greater than or equal to...

0.80

why do we replicate?

1. estimate random error 2. reduce standard error (improve precision) 3. increase power by increasing degrees of freedom 4. wider array of experimental units (better representation of variance)

deductions of ANOVA

1. if H₀ is true, then variance among replicates within groups estimates population variance 2. if H₀ is true, then variance among treatment groups estimates population variance 3. if H₀ is true, then within-groups variance and among-groups variance estimate the same thing (Sw² = Sa²; F = 1)

advantages of 2ⁿ factorial design

1. need a minimum number of measurements to see if a factor has a significant effect 2. two-way interactions are easier to interpret graphically 3. no need for MCTs (only 2 levels per factor)

two principles of effective experimental design

1. randomize 2. replicate

notes for multiway ANOVA

1. replication within cells is necessary 2. best to have a balanced design 3. can run MCTs if no interaction exists

how to conduct a randomized block ANOVA

1. set up blocks across some known or suspected gradient 2. randomize treatments within blocks 3. run ANOVA with fixed treatment effect and random block effect (model III)

two types of interaction

1. synergism 2. inhibition

for k = _____ groups, F-test is equivalent to t-test

2

Most experiments are designed with _____ in mind.

ANOVA

John Tukey

American (Princeton); invented the Tukey HSD unplanned MCT

statistical tests for homoscedasticity

Bartlett's, Levene's

for df1 = 1 and df2 = ∞...

F = t² = ℵ²

ANOVA assumptions

LINE 1. linear and additive 2. independence 3. normal distribution of errors 4. equal variance among cells

other MCTs

Scheffe's & Dunnett's (planned); Student-Newman-Keuls & Bonferroni (unplanned)

model I ANOVA

fixed effects; experiment; treatment levels deliberately chosen by investigator (fixed and can be manipulated); generally followed by MCT if significant ANOVA

With a significant interaction in a multiway ANOVA...

a specific statement about each factor has limited meaning; you can only say that "the effect of factor A varied depending on the level of factor B"

procedure-wise error rate

actual type I error = α × number of tests (multiple tests compound alpha)

one-way ANOVA alternate hypothesis

all means are not equal

pooled variance among treatment groups

also known as among-groups variance Sa² = n × ε (*x*i - grand mean)² / (k-1)

pooled variance among replicates within treatment groups

also known as within-groups variance or mean square error Sw² = ε ε (xij - *x*i)² / k(n-1)

variation in one-way ANOVAs

assumes equal variance (random variation in subjects equally dispersed among the treatments)

consequence of violating independence

botched experiment (results are misleading/invalid)

randomized block design

compensates for situations where known factors other than treatment group status (e.g. age, sex, agriculture plots, other gradients) are likely to affect what is being observed in the study; accounts for the fact that experimental units are not homogeneous; removes unwanted uncontrollable variation

Bonferroni correction

corrected α = desired α ÷ number of tests run all tests at corrected α

NPP will give a straight line if...

data are normally distributed

problems with the Bonferroni correction

decreasing α increases β, which greatly reduces the power of the statistical test

degrees of freedom 2

denominator degrees of freedom; degrees of freedom within

F distribution

distribution of repeated samplings and calculations of F; shape determined by degrees of freedom 1 & 2

example of one-way ANOVA

donut example; four different oils, determining if mean outcomes are different when various levels of ONE treatment are applied

why do we randomize?

eliminate bias (ensure independence)

balanced design

equal number of replicates per group

homoscedasticity

equal variances

within-groups variation is also known as...

error/residuals

Fisher's LSD

gives the smallest difference between means that allows you to state that they are significantly different; not very conservative; tends to find more differences than actually exist or than would exist by chance alone (α not well controlled)

unplanned comparisons

identify contrasts a posteriori to (after) ANOVA; Tukey's HSD; contrasts suggested by the data (normally do all of them)

planned comparisons

identify specific contrasts you want to do a priori to (before) ANOVA (e.g. comparing a control to several treatments); Fisher's LSD

to compensate for reduced power...

increase n (number of replicates)

which ANOVA assumption violation is fatal?

independence

replicate

individual unit/item

What is the problem with running multiple individual t-tests?

inflates alpha (increases type I error, likelihood of rejecting a true null hypothesis)

homogeneous subset

joined means (no significant difference by MCT)

k treatments would require how many t-tests?

k(k-1) / 2

Sir Ronald Fisher

knighted by Queen Elizabeth II; created ANOVA; statistician and geneticist

drawback of blocks

lowers degrees of freedom

inhibition

magnitude of response to one factor decreases with increasing value of the other factor

synergism

magnitude of response to one factor increases with increasing value of the other factor

consequence of violation of linear+additive

main effects not interpretable

if 1-β is low...

may want to redo study to reduce β (type II error; likelihood of accepting a false H₀)

grand mean

mean of all replicates (NOT mean of all group means)

model III ANOVA

mixed model (both fixed & random effects); requires at least two factors in the ANOVA (e.g. randomized block design); may combine experiments and observations

MCT

multiple comparisons test; used only AFTER getting a significant ANOVA (rejection of H₀); deal with problem of controlling effective α; can be planned or unplanned

multiway ANOVA is better than multiple one-way ANOVAs because...

multiple one-way ANOVAs would inflate α!

detection of violation of independence

none

detecting violation of normality

normal probability plots, shapiro-wilk test, frequency distribution

k (for multiway ANOVA)

number of cells

k

number of groups or treatments

degrees of freedom 1

numerator degrees of freedom; degrees of freedom among

two-level (2ⁿ) factorial design

only two levels of each factor n are tested

When do you run an MCT on a multiway ANOVA?

only when there is NO significant interaction (have to constrain it to one level of the other factor)

ANOVA

parametric statistical test; most useful statistical method; used to test whether two or more sample means come from the same population mean

advantages of randomized block

partitioning into blocks moves some error into blocks, reduces mean square error, increases F value (greater chance of revealing treatment effect)

normal probability plot

plot of normal (z) scores vs residuals (errors)

residuals plot

plot of residuals on the y axis versus fitted values on the x axis

model II ANOVA

random effects; observations; treatment levels randomly chosen and cannot be manipulated; specific levels are less important than just having different levels; significant ANOVA suggests other underlying mechanisms; usually not followed by MCT

how to ensure independence

randomization

R

replicates per cell

detecting violation of equal variance

residuals plot, variance versus mean, Bartlett's or Levene's tests

rectify violation of equal variance

run non-parametric test, transform data

rectify violation of normality

run non-parametric test, transform data

statistical testing is based on...

setting α (set α and hope 1-β is high)

detection of violation of linear+additive

significant interaction OR non-linear/non-additive response to treatment

multiway (factorial) ANOVA

simultaneously test for the effects of two or more factors; allows factor interactions to be examined

linear additive model

x(obs) = biological mechanism ± error

Shapiro-Wilk test

statistical test for normality; H₀ = normality

if null hypothesis in a one-way ANOVA is rejected...

the ANOVA does not tell you which means are different; must conduct a multiple contrast (comparisons) test

The effect of one factor may be due to...

the enhancing or inhibiting effect of the other factor.

ANOVA was developed to solve...

the problem of running multiple t-tests at weak power

as number of treatments increases, n increases proportional to...

the square of the treatments (reps ~ k²)

if we accept the null hypothesis in ANOVA, it could be because...

there are no significant differences OR because the power is too low to detect that difference (β too high)

increased degrees of freedom shifts the F distribution...

to the right

N

total size

rectify violation of linear+additive

transformation (interaction may become nonsignificant); if multi-way ANOVA, may do one-way ANOVA (perform at constant levels of one of the factors)

heteroscedasticity

unequal variances

How do we estimate type II error?

use power curves (∅, df1, df2, α)

it would be nice to know type II error in particular if...

we failed to reject H₀

one-way ANOVA null hypothesis

µ₁ = µ₂ = ... = µₙ

type I error

α; rejecting H₀ when it is true

type II error

β; failing to reject H₀ when it is false