stats y2

General Linear Model (GLM)

statistical framework that describes the relationship between a DV and one or more IVs

when to use a GLM

to test a hypothesis with a numeric outcome

e.g. regression, correlation, t test, anova

linear regression

tests whether there is a linear association between numeric variables, often continuous variables

works with categorical predictors as an alt to anova (dummy code predictor)

What does a significant association in linear regression indicate?

slope and intercept are significantly different from 0

What are the assumptions of multiple regression?

No multicollinearity, homoscedasticity, linear relationship between DV and IVs, normally distributed residuals.

centering values

mean removed from each datapoint in variable

intercept becomes mean DV value

=to make intercept more interpretable

df in multiple regression

dfm- number of variables in the model

dfe- number of ppts-number of variables-1

What does the F statistic in ANOVA represent?

The ratio of variance due to differences between groups to variance within groups. (F = MSbetween/MSwithin)

group effects

the deviation of each group mean from the grand mean

What is the significance of Mauchly's test in repeated measures ANOVA?

It tests the assumption of sphericity, which is the equality of variances of the differences between treatment levels.

What is a two way anova used for

when there are two categorical IVs

What is the null hypothesis for testing the interaction between two independent variables in a two way anova

no effect of interaction on DV

What is the formula for the individual score in the context of ANOVA?

Individual score (DV) = grand mean + main effect A + main effect B + interaction + error.

why use a Bonferroni correction in post hoc tests

reduce chance of type 1 error from multiple comparisons

What does the term 'homogeneity of variances' refer to in ANOVA?

The assumption that different groups have similar variances.

assumptions of anovas

observations are independent events and identically distributed

homogeneity of variances

normality of residuals

if assumptions for anova not met

do anova anyway (less power and risk of type 2 error

transform the dv

use kruskal wallis non parametric test

What is the difference between fixed effects and random effects in ANOVA?

Fixed effects are chosen levels of factors

random effects are factors selected randomly from a population.

What is the significance of the residuals in regression analysis?

Residuals are the differences between observed values and predicted values, indicating the model's accuracy.

What does 'dummy coding' refer to in regression analysis?

Transforming categorical predictors into numerical format for analysis. df loses 1

What is the main effect in the context of ANOVA?

The individual effect of one independent variable on the dependent variable.

grand mean

average score across all subjects no matter the condition

df in two way anova

DFmaineffect = k-1 for each factor

DFinteraction= DFaxDFb

sphericity

The assumption that the variances of the differences between treatment levels are equal between repeated measures deigns with 3+levels

mauchlys test

tests whether variances of differences between conditions are equal

if p<0.5, F test has too many false positives and no sphericity

correct df using greenhouse geisser correction or huyh-fedlt correction

What is ANCOVA?

An extension of ANOVA that includes a numeric covariate that may explain additional variance in DV

strengths of ancova

reduces within-group variance (error) as co variate explains some of the error

eliminates confounds by including them

contrast coding in linear models

A method to transform categorical predictors into numeric values for analysis, allowing for comparisons against a reference level.

dummy contrast coding

one level of a categorical variable is defined as the reference (0) and other levels are compared to it (1)

intercept shows mean of reference level and slope shows the difference between each other level and the reference level

successive differences coding

tests whether there are differences between levels sequentially, with the intercept representing the grand mean and slope showing mean differences between groups

deviation coding

A method used for factors with two levels, assigning one level -0.5 and one 0.5

intercept becomes grand mean and slope shows mean dif between groups

What are the five important assumptions in order in statistical modeling according to Gelman and Hill (2007)?

Validity, Linearity and additivity, Independence of errors, Homogeneity of variances, Normality of residuals.

multicollinearity

when predictors are strongly correlated with each other

=can inflate standard errors and complicate regression coefficient estimation. above 0.7 concern

What does the Variance Inflation Factor (VIF) measure?

The relationship between variables, with values greater than 5 indicating high collinearity, which is a concern in regression analysis.

Cook's distance

detects influential outliers in regression by investigating how much predicted values change if an observation is removed

higher value= affects data more

centering

Shifting a variable so its mean becomes 0, keeping the same shape size and relationship

z score transformation

rescaling variable so its mean is 0 and sd is 1

What is log transformation used for?

To smooth out tails of a distribution and make data more normally distributed.

Akaike Information Criterion (AIC)?

A measure used to compare different models, where a smaller AIC indicates a better fit.

Linear Mixed Models (LLMs)?

Models that account for random effects in data with nested or hierarchical structures, allowing for varying intercepts or slopes.

fixed effects in LLMs?

Explanatory variables hypothesized to affect the DV.

random effects in LLMs?

Categorical grouping variables considered random samples from a larger population, like participants or schools.

What is REML?

Restricted Maximum Likelihood, the default parameter estimation criterion for linear mixed models.

random effect variables

categorical, ideally 5+ levels, represent a sample from a broader population

What is the purpose of centering variables in LLMs?

To improve interpretability of the intercept and reduce multicollinearity issues.

What is the significance of including random slopes in a model?

To examine how predictors interact with random effects, though it can complicate the model and lead to overfitting.

nested random effects

a lower level grouping factor exists only within one specific level of a higher level factor

crossed random effects

a factor appears in multiple levels of another factor

What is the purpose of transformations in statistical analysis?

To meet assumptions of normality, linearity, and homogeneity of variance, improving model accuracy.

What does it mean if a model fails to converge?

It indicates that the model is too complex or improperly specified, often due to overfitting or insufficient data.

p-value

probability of observing a test statistic that is at least as extreme or more extreme than the one we observed if the null hypothesis is true and we repeat our experiment many times

C: doesnt tell you the null is false

alpha level

 threshold for declaring significance (5%)

C: levels should be different for different contexts (Fisher, 1935)

effect size

 tells us how practical the result is in the real world

‘small’ <0.2 cohens d hedges g, 0.1-0.3 correlation, 0.01 anova, cohens f 0.1

‘medium’ <0.5 cohens d hedges g, 0.3-0.5 correlation, 0.06 anova, 0.25 cohens f

‘large’ 0.8 cohens d hedges g, >0.5 correlation, 0.14 anova, 0.4 cohens f

eta squared n2

 effect size for main effects in anova that tells us the proportion of variance in the dv explained by the predictor

partial eta squared n2p

proportion of partial variance after accounting for the other predictors in the model that the predictor explains in the dv

generalised eta squared n2g

 estimates the effect size in a design where only the term of interest was manipulated, accounting for the fact that some terms cannot be manipulated

-formula depends on design

cohens f (partial)

a transformation of partial eta squared when the population means are equal and an indefinitely large number as the means are further and further apart

factors that affect power

sample size- larger gives larger power

expected effect size- larger gives larger power

type 1 error rate- as tolerance for type1 error in, power inc

reliability of measures- more reliable larger power

power

probability of correctly rejecting the null, if type 2 error rate is .2, power is .8

priori power analysis

what sample size do we need to have 80% power in detecting an effect size

sensitivity power analysis

what is the smallest effect size we can detect with the power, sample size, and alpha we have

meta analysis strengths

+higher power than individual studies

+overall effect across studies

+can identify potential publication bias

+can explore impact of design and analysis decisions

+more accurate estimation of effect size for future power analysis

fixed effect model for meta analyses

assumes all studies are estimating the same population effect size. any error is due to sampling

random effects model for meta analyses

allows the population effect to differ between studies. allows for differences in design, population, dosage

heterogeneity of meta analysis

the extent to which effect sizes vary between studies

heterogeneity measures

cochrans Q statistic- difference between observed effect sizes and overall effect size

I2- percentage of variability in the effect sizes not caused by sampling error

tau-squared- alternative measure of heterogeneity between study

funnel plot

plots effect size against precision of the study

• studies with high precision should cluster at the top

• studies with low precision should scatter widely at the bottom

• unsymmetrical could be sign of publication bias

error/residuals (anova)

difference between individual points and its group mean

sum of squares

summarises the total variance associated with each component of the model

SS= SSwithin+SSbetween

mean squares

we cant use ss to compare so use mean squares

MSwithin= SSwithin/DFbetween

MSbetween= SSbetween/DFbetween

homoscedasticity

similar amount of variation in y at each value of x

VIF

1/1-R2

calculate effect n2 for anova

SSeffect/SStotal

mu with a hat

sample estimate of the population mean value

if linear mixed effects model doesnt converge

change number of iterations of optimiser

remove correlations between random effects

dummy coding intercept and slope

intercept is mean of reference level

slope is mean difference between levels

calculate each groups mean in deviation coding

intercept +0.5xslope

anova equation

Individual score (dependent)= grand mean + group mean difference from grand mean + error (diff between group mean and individual score) 

df for ancova or multiple regression

number of observations- (k-1)

multiple regression formula

Dv= intercept + predictor slope 1x IV1+ predictor slope 2XIV2+error 

degrees of freedom

linear regression N-1

multiple regression N-(K-1)

anova N-1

ancova N-(K-1)

fixed effects lmm depends

random effects lmm number of random variables

meta analysis K-1

poisson distribution

count variable

variance = mean

when to use z score

when you want to compare distribution and coefficients across predictors with different scales