PSYC2010 Final Exam

ANOVA

Analysis of variance is a test used to compare mean differences across two or more groups

One way ANOVA

A test of whether one independent variable affects one dependent variable

Independent groups ANOVA

A one way ANOVA where different participants are in each group or condition

Between subjects design

A design where each participant appears in only one condition

Repeated measures ANOVA

A one way ANOVA where the same participants provide scores in every condition

Factorial ANOVA

A form of ANOVA used when there are two or more independent variables

Two way ANOVA

A factorial ANOVA with two independent variables

Three way ANOVA

A factorial ANOVA with three independent variables

Factor in ANOVA

Another name for a categorical independent variable

Level in ANOVA

One condition or category within a factor

Dependent variable in ANOVA

The outcome measured to see whether it changes across levels of the independent variable

Omnibus test

A test that checks whether any group means differ but does not identify which means differ

Why ANOVA is used instead of many t tests

Multiple t tests inflate the chance of a Type I error

Type I error

Rejecting a true null hypothesis

Familywise error rate

The probability of at least one Type I error across a family of comparisons

Familywise error formula

FWER = 1 - (1 - alpha)^c

Symbol c in multiple comparisons

The number of comparisons in a family of tests

Symbol alpha

The chosen probability threshold for rejecting the null hypothesis

ANOVA null hypothesis

All population means are equal

ANOVA alternative hypothesis

At least one population mean differs from another population mean

ANOVA H0 with three groups

mu1 = mu2 = mu3

ANOVA H1 with three groups

At least one mu is different from another mu

What a significant ANOVA means

There is evidence that at least one group mean differs

What a significant ANOVA does not tell you

It does not tell you exactly which groups differ

What a non significant ANOVA means

There is not enough evidence that the group means differ

When follow up tests are needed

Follow up tests are used after a significant omnibus ANOVA to locate differences

Partitioning

Dividing total variability into separate sources of variability

Total variability

All variability in the scores before it is split into sources

Treatment variability

Variability due to differences between group means

Error variability

Variability within groups that is not explained by treatment

SStotal

Total sum of squares across all scores

SStreatment

Sum of squares due to treatment or between group differences

SSerror

Sum of squares due to within group error

SStotal formula

SStotal = sum of (X - grand mean)^2

SStreatment formula

SStreatment = sum of nk(Mk - grand mean)^2

SSerror formula

SSerror = sum of (Xik - Mk)^2

Partitioning formula for independent groups ANOVA

SStotal = SStreatment + SSerror

Alternative SSerror formula

SSerror = SStotal - SStreatment

Mean square

A variance estimate formed by dividing SS by its df

MStreatment

Treatment mean square or between groups variance estimate

MSerror

Error mean square or within groups variance estimate

MStreatment formula

MStreatment = SStreatment / dftreatment

MSerror formula

MSerror = SSerror / dferror

F ratio formula

F = MStreatment / MSerror

How to interpret F

Large F values suggest between group variance is large relative to within group variance

Small F value

The group means differ little compared with within group variability

Large F value

The group means differ more than expected from within group variability

Under H0 in ANOVA

MStreatment and MSerror should be similar

When MStreatment equals MSerror approximately

When the null hypothesis is true and group means differ only by sampling error

When MStreatment is larger than MSerror

When treatment effects may be present

Degrees of freedom in F

F has one df for treatment and one df for error

Why F has two df values

The numerator and denominator are separate variance estimates

ANOVA dftreatment formula

dftreatment = k - 1

ANOVA dferror formula

dferror = N - k

ANOVA dftotal formula

dftotal = N - 1

Relationship among ANOVA dfs

dftotal = dftreatment + dferror

Symbol N

Total number of participants across all groups

Symbol k

Number of groups or treatment conditions

Symbol nk

Number of participants in group k

Symbol Xi

The ith raw score

Symbol Xik

The ith raw score in group k

Symbol Mk

The mean of group k

Symbol grand mean

The mean across all participants and groups

Prime symbol

A mark showing a different group or population

How to use the F table

Compare Fobt to Fcrit using treatment df and error df

Decision rule for F test

Reject H0 when Fobt is greater than Fcrit

Report format for ANOVA

Report the test type and significance and DV and IV and F statistic and p value

Example ANOVA notation

Use F(2 27) = 8.60 and p < .05

What p less than .05 means

The obtained result is unlikely under the null hypothesis at the .05 criterion

Effect size in ANOVA

A measure of how important the independent variable is in explaining DV variability

Why effect size matters

A significant test can still have a small practical effect

Eta squared

The proportion of sample variance explained by the treatment

Eta squared formula

eta squared = SStreatment / SStotal

Omega squared

An estimate of the proportion of population variance explained by the factor

Omega squared formula

omega squared = (SStreatment - dftreatment x MSerror) / (SStotal + MSerror)

Symbol omega squared

An ANOVA effect size estimating population variance explained

Symbol eta squared

An ANOVA effect size estimating sample variance explained

Small omega squared

.01 is often treated as a small effect

Medium omega squared

.06 is often treated as a medium effect

Large omega squared

.15 is often treated as a large effect

Interpreting omega squared .56

About 56 percent of population variance is attributed to the factor

What remains when omega squared is .56

About 44 percent of variance is linked to other factors and error

Structural model in independent groups ANOVA

Each score equals the population mean plus the treatment effect plus error

Independent groups ANOVA model formula

Xik = mu + tauk + epsilonik

Symbol mu in ANOVA model

The population mean or grand mean component

Symbol tauk

The treatment effect for condition k

Symbol epsilonik

The error for participant i in condition k

Known systematic variance

Treatment effect explained by the model

Unknown unsystematic variance

Residual error not explained by the model

SS model

Another name for treatment sum of squares

SS residual

Another name for error sum of squares

ANOVA versus t test

ANOVA compares variance estimates while t tests compare mean differences

How ANOVA is like a t test

Both test whether group differences are larger than expected by error

How ANOVA differs from t test

ANOVA can test more than two groups in one omnibus test

Relationship between F and t with two groups

F = t^2 when the IV has only two levels

F from two group ANOVA

The F statistic equals the squared independent groups t statistic

One way ANOVA with two levels

It gives the same significance decision as the matching t test

Planned comparison

A follow up test chosen before looking at the data

A priori comparison

A planned comparison based on a theory or hypothesis

Post hoc comparison

A follow up test chosen after looking at the data