Biometry exam 2

peace love 2 way anova

2 Sample T-test

• requires samples with equal variance and normal distribution

• tests the null hypothesis of the two means of populations are equal

• non paired, independent samples

• If the CI of one includes the mean of another, they are not significant

Welch’s approximate t test

• compares mean of 2 normally distributed populations with unequal variances

Mann whitney U test

• non parametric t-test

• deals with ranked values

• assumes both samples are random and distributions are similar

• for tied ranks, take the numbers they cover and divide by n

Paired sample t

• t test for samples that are paired, where you are interested in the difference between them

• if the null hypoth is true, the difference between µ1 and µ2 is 0

• n = # of pairs

• groups must be balanced

the null hypoth is true, but you reject the null

• type 1 error

• prob = α (typically 0.05)

the null hypothesis is true, and you fail to reject the null

• correct

• prob = 1 - α

the null hypothesis is false, and you fail to reject the null

• type 2 error

• prob = β

  
  

the null hypothesis is false, and you reject the null

• correct

• prob = 1-β

statistical power

• the probability of not making a type 2 error

• 1 - β

Randomization / permutation

• under the assumption of the null hypoth being true

• mixes up real data randomly and makes new estimate from random data

• randomization without replacement

jack-knifing

• deterministic, always the same outcome

• data must be separable into independent subunits

• delete each observation 1 at a time and calculate pseudovalue for each observation

• a way to put a CI on something with no simple formula for doing so

Bootstrapping

• creates a random dataset by random sampling of the data with replacement of the original values

• non replicable, different outcome every time

F ratio

• ratio of variance estimates

• F = s²1 / s²2

• If null is true, F = 1 and will follow normal distribution

• 1 DF each for numerator and denominator (total 2 df’s)

• 2 sided test, compare against F.975 to get 5% confidence

types of factors

• fixed and random

Y_ij = µ + α_i + b_ij + e_ij

Y_ij = response

µ = mean

α_{i =} affect of i level of factor

b_ij = effect of j individual in i level of factor

e_ij = error / residual variance

Sum of Squares

• derived from the mean

• 3 types, SS total, SS within groups, and SS between groups (also referred to as SS group or SS factor)

• SS within + SS between = SS total

Total sum of squares

• sum of each individual against overall mean

within sum of squares

• sum of each group individual against group mean

factor / group / between sum of squares

• group mean against overall mean

ANOVA

• used to compare the means of 1 or more factors levels

• estimates α_i (and sublevels when included) to compare them to see if they differ

• assumes independent

Mean square =

SS / df

model1 anova

• fixed factors that do not sample all possibilities

• interested in the outcome of specific treatments/setups

• repeatable

model 2 anova

• random factors

• groups are sampled randomly from all possible groups/treatments

• not interested in one particular group or treatment, rather interested in making inferences about the population

Tukey-Kramer

• post hoc analysis for fixed effects only

• compares all group means to all other group means to determine which are different

• can tease apart which groups/levels differ from one another

natural log

• used when the data is right skewed

• or when the variance increases as the mean increases

arcsin square root

• used when the dependent variable is a proportion

Kruskall Wallace

• non parametric 1 way anova

• rank based

• more difficult to see a difference when one is present in rank based analyses

Nested anova

• multilevel, hierarchical, with multiple causal factors nested within one another

• every possible combination of factors is not tested

factorial anova

• includes interaction in sources of variation

• every possible combination of factors is tested

F ratios when both factors are fixed

Interaction: F = MS interaction/MS error

factor A: F = MS A / MS error

factor B: F = MS B / MS error

Pure model 1

• both factors fixed

• F value for all factors is calculated by dividing factor by MS error

Pure model 2

• both factors random

• F for factors is calculated by dividing individual factor MS by interaction MS

• F for interaction is calculated by dividing MS interaction by MS error

F value when both factors are random

factor a: F = MS A / MS AB

factor b: F = MS B / MS AB

interaction: MS AB / MS error

Mixed model

• 1 factor random, 1 factor fixed

F statistic calculation for mixed model

• fixed factor: MS fixed / MS interaction

• random factor: MS random / MS error

• interaction: MS interaction / MS error

• on minitab, calculate these by hand

What do you need to estimate an interaction

• multiple measurements for all combinations of factors

interaction

• the affect of factor a depends on factor b

AIC

• measure of goodness of fit, often a criteria for selecting best model

• AIC = 2k - 2 ln(L hat)

• lower value is better

Bonferroni

• controls for false positive, but reduces power, making type 2 errors more likely

• assumes all tets are independent

• test is significant only if P < α / M

• where α is significance level, typically .05, and M is # of tests

fishers combined probability test

• takes tests of the same null hypothesis and asks: based on results of all tests, should we reject the null?

• utilized for when multiple experiments have been performed

• natural log each p value and multiply by 2

• sum and compare to chi sq

Holms bonferroni

• increases the power of bonferroni through a sliding scale

• rank each p value smallest to largest

• P(k) < α / M + 1 - k

• where M is # of tests, α is significance level, and k is the rank

False Discovery / FDR

• rank p’s smallest to largest and calculate the cutoff

• if P value is smaller than the cutoff, it is significant

• cutoff = chosen FDR - rank / M

• where M is # of tests

FDR

• fraction of significant tests that are false

• determined by researcher, often 10, 5, etc, can be wrong

r² / Coefficient of determination

• proportion of variance in 1 variable that can be predicted from another

r =

Sxy / SxSy

Co variance / Sxy

Sxy = 1 / n-1 x Sum of (x-x̄)(y-ȳ)

1. For each “pair” of data points, multiply (x-x̄)(y-ȳ)

2. then sum all results and multiply by 1 / n-1

F value formulas for nested anova

Group: MS Group / MS Subgroup

Subgroup: MS Subgroup / MS Error

Homoskedasticity

• variance is equal amongst observations

• usually referring to variance of the error is constant across dataset

• homogeneity of variance