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