# PSYC 300B Chapter Notes - Chapter 9: Helicopter Flight Controls, Sydney Trains T Set, Descriptive Statistics

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**preview**shows half of the first page. to view the full**2 pages of the document.**•multi-factorial research designs: involves 2 or more factors (IVs), each

with more than 1 level, & 1 DV in a single balanced design

-analyzed as a single experiment rather than a series of 1-way designs

•𝛼 remains at 0.05, thus type I error is not inﬂated

-types of factorial designs (determined by measurement of factors):

(i) independent/between-group design: all factors are

independent variables, & participants contribute data to only

one experimental condition

(ii) repeated-measures/within-group design: all factors are

repeated-measures variables, & participants contribute data to

every level of every factor

(iii) mixed factorial: has at least 1 independent & 1 RM factor

-naming the design is based on:

(a) # of factors

(b) # of levels of each factor

(c) how each factor is measured (ex. between-groups, RM, etc.)

•data matrix: a display that includes the factors, the levels of each factor, &

the cell means for each experimental group (DV)

-factor (independent variable): labeled on the outside of the data matrix

-levels (subgroups of that factor): a sub-heading, labeled outside of matrix

-cell means: represent data for each condition in the experiment

•cell mean is 1 level of each factor at 1 level of another factor

•# of cell means = multiplying design label

•only use a ﬁgure if you have at least 4 cell means (a 2 x 2 design) or a

signiﬁcant interaction

-marginal means: the averaged mean score across 1 level of 1 factor

•have both row & column marginal means

•# of marginal means = adding design label

-main effects: there is one main effect for each factor in the design, in

which each main effect is part of the total treatment effect

•# of main effects = # of factors

•you test the null for each factor, so a unique F(obs) is computed for

every main effect

•test main effects by using the marginal means (the effect of one factor

is averaged over the levels of another factor)

•some main effects can be predicted a priori from theory/past research

•column main effect — the same as applying a 1-way ANOVA to

all the means

-determines whether one factor affects the other

•row main effect — like running a t-test on the means

-determines whether there is an association between factors

•interaction: the extent to which the effect of 1 factor varies/changes across

levels of a 2nd factor

-whether or not the presence of second variable inﬂuence the behaviour

of participants in the study

-represents a test of the treatment effect & is calculated from cell means

•therefore requires a computed F(obs) value

-represents a signiﬁcant interaction — the effect of one factor changes

across levels of a second factor

-cannot predict the presence of an interaction a priori, thus they cannot be

analyzed with planned comparisons

-can only use post-hoc comparisons if we have a signiﬁcant interaction term

•generalizability of a factor: the extent to which 1 factor is the same/

consistent across levels of the other factor

-represents a non-signiﬁcant interaction — the behaviour is the same,

regardless of whether or not the factor is included

•advantages — combining 2 factors into just 1 analysis, rather than have to

carry out multiple individual 1-way designs, is better because it:

(i) is more economical — you can study 2 more factors with the same

set of participants, therefore needing less

(ii) reduces some unexplained variability — easier to control for

some confound variable, & there is less opportunity for experimenter

error or variability among participants

(iii) you have the ability to test for the an interaction or to examine

the generalizability of a factor — major advantage of the design

•logic of ANOVA — applied to a multi-factorial design:

-apply the random sampling model of hypothesis testing

-participants in each condition are randomly sampled from a unique

population, so that each population has the same parameters

•under the null, each population will remain the same

•under the alternative, at least one of the means will shift

-makes the assumptions:

(i) each condition is associated with its own population

(ii) the DV is normally distributed in each population

(iii) homogeneity of population variances for each cell

(iv) homogeneity of sample sizes (equal # of scores in each cell) !

(should not violate either homogeneity assumption)

(v) n ≥ 7 for each cell

-each condition has it’s own population distribution, sample distribution,

& sampling distribution of the means

•all the sampling distributions for each condition are then combined

in order to create a unique null hypothesis distribution for each

treatment effect

•F(obs) — because there is more than 1 null hypothesis to test (more than 1

sampling distribution), consequently need to compute more than 1 F-ratio

-# of treatment effects = # of F(obs) = # of unique sampling distributions

-each F-test is independent of each other & they do not inﬂuence one

another → does not inﬂate type I error

-there is a unique F(obs) for each effect in the design

(i) 1 for each main effect (individual variable)

(ii) 1 for each possible combination of main effects (interaction terms)!

[ex. for a 3 factor design — there are 3 main effects (A, B & C),

& 4 interaction terms (A x B, A x C, B x C, A x B x C)]

-compute F(obs) using the formula — F(obs) = (MSeffect)÷(MSerror)

•MSeffect = between-group variance estimate for the effect

-the main effect (interaction)

•MSerror = within-group variance estimate

-represents pooled population S2 estimates

-computed from scores in each cell

•extremely powerful because it considers the error from each

cell, rather than just one error estimate

-SST = SScells + SSerror

•SScells is further divided into row main effects, column main effects, &

the interaction — SScells = SSrows + SScolumns + SSrows x column

-compute an F-ratio for treatment effect

•when F(obs) = 1, then F(obs) = only error variance

•when F(obs) > 1, then F(obs) = error + treatment variance

•ANOVA source table — components:

-SST = total variability in the experiment — ∑(x - x̅GM)2

-SSerror = variability within cells — ∑SSi

-SScells = deviation of cell means from x̅GM — ∑ncells(x̅cells - x̅GM)2

•SSrows = main effect of the row factor — ∑nrows(x̅rows - x̅GM)2

•SScols = main effect of the column factor — ∑ncols(x̅cols - x̅GM)2

•SSrows x cols = interaction — SST - SSe - SSrows - SScols

-dfT = total variability in the experiment — nT - 1

-dferror = how much each score varies from its cell mean — nT - # of cells

-dfcells = # of cells - 1

•dfrows = deviation of row marginal means from x̅GM — # of rows - 1

•dfcol = deviation of column marginal means from x̅GM — # of col - 1

•dfrows x col = everything left over after considering the variability

explained by each main effect — (# of rows - 1) x (# of col - 1)

-MSrows = mean squared rows — SSrows ÷ dfrows

-MScol = mean squared columns — SScols ÷ dfcols

-MSrows x cols = mean squared interaction — SSrows x cols ÷ dfrows x cols

-η²rows = rows effect size — SSrows ÷ SST

-η²cols = columns effect size — SScols ÷ SST

-η²rows x cols = interaction effect size — SSrows x cols ÷ SST

-R²rows = rows effect size — SSrows ÷ (SST - SScols - SSrows x cols) or !

SSrows ÷ (SSe + SSrows)

-R²cols = columns effect size — SScols ÷ (SST - SSrows - SSrows x cols) or !

SScols ÷ (SSe + SScols)

-R²rows x cols = interaction effect size — SSrows x cols ÷ (SST - SSrows - SScols) or !

SSrows x cols ÷ (SSe + SSrows x cols)

PSYC 300B - Chapter 9: Multi-Factorial Research Designs

η² ≠ R² because:

-Multiple R² partials out variability from the total

variably that is accounted for by other effects

-instead just focuses on how much variability for a

speciﬁc treatment effect is accounted for

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