psyc3010

RM variable

________ treated as multiple DVs and combined /weighted to maximise difference between levels of other variables.

covariate

________ is continuous control variable known to be associated with DV.

ANOVA

________ cannot detect non- linear relationships, reduces power.

rho

________ or radj is population coefficient, compared to r.

moderator

________ enhances or attenuates relationship between IV /predictor and DV /criterion.

levels of WPF

Variance and covariance are the same at all ________ (often violated)

block

________ (trial) and group are both fixed factors; participant is not fixed.

error used for any effect in RM ANOVA is = to

the interaction between that effect and the effect of participants (applies to main and simple effects and their follow up tests)

DV

Interaction: effect of A on ________ changes over levels of B.

heteroscedasticity

________: variance of Y values are consistent across yhat values (homogeneity of variance)

Semipartial variation

________ (spr2)= proportion of variance in DV uniquely explained by IV.

participant reacts

Contrast- previous treatment sets standard to which ________ (more broad than sensitisation)

regression

Moderated ________ asks if XY interaction significantly contributes to prediction of Y.

Orthogonal contrasts

________: variance is partitioned without overlap, use different portions of variance to avoid inflating t1 error.

main effect

are the means of the population corresponding to each level of the factor different

variance

spread of scores around mean

MSerror

pooled within-cell variance (SSerror/DFerror)

E(MSerror)

long term average of variance in each sample = population variance

orthogonal contrasts

variance is partitioned without overlap, use different portions of variance to avoid inflating t1 error

block 1

predict DV from IV (IV entered)

contrast

previous treatment sets standard to which participant reacts (more broad than sensitisation)

adaptation

adjustment to previous treatment changes reaction to next (more broad than habituation)

direct carry-over

learn something in previous trials that is applied to latter ones

within participant variability

(variance of participant outcome over IV trials)

interaction

effect of A on DV changes over levels of B

regression intx

relationship between X and Y varies over values of Z (moderator)

error used for any effect in RM ANOVA is = to

the interaction between that effect and the  effect of participants (applies to main and simple effects and their follow up tests)

one-way factorial design

are the means from each level of the factor different from the grand mean (each other)

independent t test or one way ANOVA

two-way factorial design

combine 2 one-way designs, every factor is crossed

is there a main effect of factor 1 or 2, or an interaction

2 way ANOVA

structural model of one-way ANOVA

DV score= grand mean + effect of a treatment factor (j) + error for i person in treatment (j)

structural model of two-way ANOVA

DV score = grand mean + effect of treatment (j) at factor (A) + effect of treatment (k) at factor (B) + effect of differences in factor A treatments at different levels of factor B treatments + error for (i) person in j and k treatments

assumptions of ANOVA

population normally distributed with homogeneity of variance

samples are random and independent

interval/ratio data

eta squared (η2)

proportion of variance in sample DV accounted for by effect

easy to interpret, most common

biased estimate of true variance in population

sseffect (e.g. AxB) /sstotal

omega squared (Ω²)

proportion of variance in population accounted for by effect

less biased, more conservative

however usually same or similar to eta squared depending on sample size (if n = >20)

larger error variance, smaller sample size = bigger diff. between eta and omega sq

(sseffect-(dfeffectxMSerror))/sstotal+MSerror

partial eta squared (pη2)

proportion of total variance (residual variance) in sample accounted for by investigated effect

model removes (controls for) variance attributable to other effects + interaction

compare variable A to variance attributable to A + error

inflated, gives larger effect size

values for each effect are not comparable (and may add up to >1), cannot make meaningful comparisons

useful only when only have controls and 1 focal IV but often still report n2

simple simple comparisons

follow up significant simple simple effects with >2 levels

t test and linear contrasts (+- 1SD from mean)

2 way interaction

the effect of one factor changes depending on the level of another factor (averaging over levels of a third factor)

AKA simple effects not the same

3 way interaction

the two-way interaction between two factors changes depending on the level of the third factor

SALE - reduce error, increase power

increase **s**ample, increase ***a*** level, **l**arger effects, decrease **e**rror variance

4 ways to reduce error

improve operationalisation of variables

improve measurement of variables

improve design

improve methods of analyses

Cohen’s d

measures effect size (overlap of distributions between groups)

closely related to power

concomitant variable

control variable (interval) closely associated with DV, increases power by removing error variance

blocking design set up

divide people into groups (blocks) according to control variable (e.g. IQ) known to be associated with DV

people within blocks each randomly assigned to different levels of IV (stratified random assignment)

blocking design positives/drawbacks

* benefits
* may equate treatment groups better (assuming equal n for blocking levels)
* more power
* can check interactions of treatment and blocks
* limits
* more expensive
* loss of power if blocking variable is not correlated with DV (fewer df error, higher fcrit)
* artificial grouping due to arbitrary levels of blocking IV may result in loss of information (e.g. IQ hi, med, lo)

covariance

multiply together cross products of deviation scores

is scale dependant, need scale info to answer questions on association

pearson’s *r*

relationship between 2 variables in terms of stdevs (average cross product of standardised scores)

standardised but biased to sample

*rho* or radj

population coefficient, compared to *r*

*r*2

standardised, tends to be overly liberal (high) with small samples

regression

estimating scores on variable on the basis of scores of a predictor variable

“regress DV on IV”

linear regression equ.

y = ab+x

y = mx+c

Y^ = 𝑏1𝑥 + 𝑏0

Y^ = 𝑏X + 𝑎

standardised linear regression equ.

zY = β1𝑧X + β0

standardised Beta

unstandardised beta change from units to standard deviations

B = Zscore change in Y predicted by 1 SD increase in X

standard error of estimate

bigger correlation = smaller

how scores are expected to cluster relative to line

x percent of people expected to be within \~ x SDs of regression line

underestimated for small samples

ANCOVA design

adding an extra IV (post hoc) that is known to account for changes in DV in a study where there is a large amount of unexplained variance (reduce error)

covariance is tendency of two scores to vary together

covariate is continuous control variable known to be associated with DV

can be applied to any ANOVA design

difference between ANCOVA and blocking

blocking at design level

ANCOVA post-hoc error term adjustment

ANCOVA test conducted

effects of covariate are subtracted from error term, then treatment means are adjusted to account for (remove) differences in covariate

if there isn’t same mean on covariate, it is a confound and removed from data (partial out/control for effects of covariate from focal IV and error term)

issue with ANCOVA

only done on the assumption there should be no differences in groups due to random assignment

if covariate unrelated to DV, error term is reduced and power not increased (increases t2 error)

assumptions of ANCOVA

DV is normally distributed in population, homogeneity of variance

random sample

linear relationship between DV and covariate, overall and within each group

homogeneity of regression slopes

* no IV x covariate interaction
* relationship between DV and covariate is same in each group

strengths of ANCOVA

ability to analyse continuous variables

splitting into groups causes loss of info and increase in error

better than blocking if variables are continuous and is applied correctly

multivariate regression

variation as function of multiple predictors acting together

predictors are correlated and contribution overlaps

have to remove overlap to accurately understand effect of variance

chronbach’s A

indicator of internal consistency for 2 items on continuous scale

how well items hang together

scales should have A > .7 to reduce error

collinearities

intercorrelations among predictors

smaller is better, less error

to maximise R2 should have high validities, low collinearities

SMR

all variables entered simultaneously

each predictor evaluated in terms of what it uniquely adds to prediction (unique variance)

model r2 evaluated in 1 step

HMR

predictors are entered sequentially in prespecified order based on logic/theory (a priori, predictors can be entered singly or in blocks

SPSS outputs r2 and r2 change for each step

fuller model = variables added, reduced model = without added variables

*each* predictor evaluated on what it adds to prediction at point of entry, model r2 assessed in multiple steps

assumptions of multiple regression

population normally distributed

heteroscedasticity: variance of Y values are consistent across yhat values (homogeneity of variance)

no linear relationship between yhat and errors of prediction

independence of errors

moderator

enhances or attenuates relationship between IV/predictor and DV/criterion

mediator

indirect relationships between IV and DV via 3rd variable

accounts for the underlying mechanism or process between those variables

bootstrapping

take sample and assume it was taken from population without bias

create new samples based on that data

report results with larger set of data

uses CI, not p

assumptions of mediation

SMR shows IV is related to mediator (path A)

IV must be associated with DV (path C, though not necessary for indirect effect to be present, esp. with suppression models)

assumptions of WP ANOVA

sample is randomly drawn from population (often violated due to convenience samples)

DV scores are normally distributed in population

compound symmetry

sphericity

determines where variance and covariance is roughly equal – is sig. if assumptions are violated

often not sig. when assumptions are violated (not robust)

if sphericity is violated, F is positively biased (t1 error)

when does sphericity not matter

between-participants (as unrelated treatments)

when within-participants designs only have 2 levels

only 1 covariance

epsilon adjustment

change Fcrit by adjusting df'

e is no. by which Fcrit is multiplied

is 1 when sphericity is not violated

smaller = more conservative

greenhouse-geisser is most common e adjustment

MANOVA

creates linear composite of DVs

RM variable treated as multiple DVs and combined/weighted to maximise difference between levels of other variables

creates predicted DV score that maximises difference across levels of IV

uncommon, tends towards t1 error, very specific use case

advantages/disadvantages of MANOVA

advantages

* very powerful, more sensitive
* simplifies procedure (less people)

disadvantages of within-participants

* restrictive assumptions
* sequencing effects (counterbalancing required)

mixed model ANOVA (split-plot ANOVA)

best way to do WP ANOVA

has a between participants and within participants factor

can use non-experimental variables

observations are independent, can avoid carry-over and sequencing effects

assumptions of mixed model ANOVA

homogeneity of variance between groups and levels of factor

DV is normally distributed in population

homogeneity of interaction between RM factor and p factor at all levels of WPF (error for RM factor doesn’t change between groups)

variance-covariance matrix consistency (variance and covariance are the same at all levels of WPF (often violated), solved via epsilon adjustments)

pooled variance-covariance matrix has sphericity

mixed model ANOVA error terms

1 required for p within g factor (between factor)

1 for within participants factor and interaction (block x p within g)

all df terms add up to dftotal

following up sig. intx:

* between participants, use error term from between participants main effect (MS Ps within G)


* within participants, separate error term as in MANOVA (MS B \* Ps within G)

\

mixed model ANOVA simple effects

one-way ANOVA of WPF at BPF

repeat for each block of trials

each simple effect has own unique error term

average of simple effect error terms is same as that of MS bxps within g intx

or pooled error term (MS ps within cell)

ANOVA

as opposed to correlational design, can infer causality

due to random assignment to IV levels

interaction: effect of A on DV changes over levels of B

design methods for increasing power in ANOVA

blocking

improving measurement

remove individual differences (within ps or RM design)

include covariate (ANOVA)

design methods for increasing power in regression

use of covariate

improved measurement

one-way ANOVA error term: main effect of A

MSw

between-participants two-way ANOVA error term: main effect of A

MSw

between-participants two-way ANOVA error term: AxB interaction

MSw

between-participants two-way ANOVA error term: simple effect of A at B1

MSw

within-participants one-way ANOVA error term: main effect of A

MSa\*p

within-participants two-way ANOVA error term: main effect of A

MSa\*p

within-participants two-way ANOVA error term: interaction

MSa\*b\*p

within-participants two-way ANOVA error term: simple effect of A at B1

MSa\*b1\*p

mixed model two-way ANOVA error term: main effect of the BPF

MS(Ps within BPF Gs)

mixed model two-way ANOVA error term: main effect of the WPF

MS(WPF \* Ps within BPF Gs)

mixed model two-way ANOVA error term: WPF \* BPF interaction

MS(WPF \* Ps within BPF Gs)

mixed model two-way ANOVA error term: simple effect of WSF at BPF1

MS(WSF \* Ps within BPF1)

mixed model two-way ANOVA error term: simple effect of BPF at WSF1

MS(Ps within BPF at WSF1) or MS(within cell)