biostats quiz 4

multivariate analysis

used to analyze more than 2 variables at once —> use for concepts you know have are complex and influenced by multiple factors (such as self-esteem)

goal of multivariate analysis

find patterns/correlations b/w several variables simultaneously

multiple regression

multiple, numeric predictors that describe a response variable

method to find an equation that best describes a response variable as a function of >1 explanatory variables when all of our predictor variables are numeric

partial regression coefficient

effects of one predictor on response when all other explanatory variables are held constant

coefficient of multiple determination (R²)

proportion of variance in Y explained by all X

R² diff between linear and multiple regression

linear = corresponds to a single correlation coefficient

multiple = proportion of variance explained by all predictors combined

adjusted R²

only increases if new predictor improves model significantly

prevents overestimation of model’s explanatory power by adjusting for degrees of freedom

also explains ___% of variation in [insert response variable]

why multiple regression is useful

can determine which factor is more important when two predictors are both positively correlated to same response variable

ultimate goal of multiple regression

explain the most amount of variation using the fewest variables NOT which factors are significant

hypothesis testing

determines if findings of a study provide evidence to support a specific theory relevant to a larger population

model selection vs hypothesis testing

1) which set of variables = best model vs does this variable matter

2) uses adjusted R²/AIC vs uses p-values from t-test or F-stats to determine if coefficients are significantly different from 0

multiple comparisons problem

testing single predictor = probability of Type I error = 0.05, multiple predictors means each test carries its own alpha level so more predictors = increase in probability of Type I

why we don’t like adding more factors to explain more variance

trade off between model fit and complexity

increases likelihood of Type I errors

factors interact (many, many possible interaction terms)

could add infinite number of factors (where would we stop)

p-hacking

repeatedly testing diff variables, models, and data subsets until we get one with a p-value less than 0.05 (increases likelihood of Type I)

AIC

strives to address trade-off between model fit and complexity (penalty for adding more terms)

ideal AIC value

most negative value

problems with AIC

doesn’t tell you about quality of model, influenced by sample size, and only useful for model selection

collinearity

issue with two predictor variables overlapping too much

variance components and repeatability

proportion of variation in our response variable that is due to our random effect

variance components

VarCorr() function

intercept = variability among random groups

residual = variability within groups

repeatability

proportion of total variation that is explained by random effect

repeatability ← varAmong / (varAmong + varWithin)

comparing AIC numbers

every term added subtracts two from the value so for a complicated model to be justified, it has to more than 2 number smaller than the next, simpler model

ordination

summarize data in a reduced number of dimensions while accounting for as much of the variability in the original data set as possible

measure stress

solution to maintaining differences between points in ordination as you increase the number of points

stress

reflects how well ordination summarizes observed distances among the samples

eigenanalyses

decomposes square matrix into eigenvectors and eigenvalues

eigenvectors

directions of greatest variance in data, forming axes of ordination space (stay the same no matter how much data is distorted)

eigenvalues

indicate amount of variance explained by each eigenvector

why ordination is important when dealing with complex communities

reduces complexity by identifying the most important underlying gradients or axes that explain variation in community structure AND visualization that readily allows researchers to identify patterns and relationships

characteristics of communities that we use ordination for

have associations between membres, have a lot of zeroes in the matrix, many potential causal variables (but a small fraction explain most of variation), lots of noise/stochasticity

PCA

euclidean distances (can’t interpret 0s well)

PCoA, NMDS

non-Euclidean dissimilarities to calculate differences

NMDS

ranks pairwise distances between samples and preserves these rankings instead of actual data values

PCA and PCoA = eigenanalyses

both use eigenvectors as axes

autocorrelation

response variable can be at least partially predicted by itself at earlier time points or certain spacial distances

errors associated with autocorrelation

type I errors (false positives)

significance of degrees of freedom

crucial for determining the p-value and critical values to assess the statistical significance of your results, also helps you understand the precision and reliability of results

temporal autocorrelation

correlation between values of a variable at different points in time

autocorrelation function

ACF is a plot of autocorrelation between a variable and itself separated by specified lags

autocorrelation and residuals

use residuals because if a data wasn’t autocorrelated, the residuals would be random

pattern = residuals are autocorrelated

we don’t use data values because residuals are the part of the dataset that’s NOT explained by the model

durbin-watson test

use to test for autocorrelation

Ho = no correlation among the residuals

Ha = residuals are autocorrelated

want a p-value greater than 0.05 so we can fail to accept null!

lag function

add to model as a predictor —> lag(response)

unconstrained ordination

• finds patterns without considering external explanatory variables.

• It is data-driven, meaning it purely reflects variations in the response variables (e.g., species composition) without forcing them to align with specific predictors

  • PCA, NMDS, PCoA

    • exploratory and find natural variation patterns

constrained ordination

• forces the ordination to be guided by known explanatory variables (e.g., environmental factors like pH, temperature, or soil type).

• It maximizes variation explained by these predictors, making it hypothesis-driven rather than purely exploratory.

• explain variation using predictors

PERMANOVA

non-parametric statistical test used in multivariate analysis to compare groups based on a distance matrix. It is especially useful in ecological and community composition studies.

uses of multiple regression

prediction - predict or estimate a response variables affected by many explanatory variables

causation - is there a functional relationship between response variable and explanatory variables and which factor cause variation in response variable?

running out of df

df = statistical power, no df = no possibility of finding a significant result

rank deficiency

too many factors (main or random effects) relative to sample size