Factorial ANOVA

The linear model

Two-factor

	Low N	Medium N	High N
Low temp.	XXX	XXX	XXX
High temp	XXX	XXX	XXX

\[ y_{ijk} = \mu + \alpha_i + \beta_j + \alpha_i\beta_j + \varepsilon_{ijk} \]

\(\alpha_i\) is the effect of the \(i_{th}\) temperature
\(\beta_j\) is the effect of the \(j_{th}\) nitrogen level
\(\alpha_i\beta_{j}\) is the effect of the \(ij_{th}\) interaction.

The linear model

Two-factor

	Low N	Medium N	High N
Low temp.	XXX	XXX	XXX
High temp	XXX	XXX	XXX



 Temp   Nitrogen 
------ ----------
 Low      Low    
 Low      Low    
 Low      Low    
 Low     Medium  
 Low     Medium  
 Low     Medium  
 Low      High   
 Low      High   
 Low      High   
 High     Low    
 High     Low    
 High     Low    
 High    Medium  
 High    Medium  
 High    Medium  
 High     High   
 High     High   
 High     High

\[ y_{i} = \beta_{0i} + \beta_{1i} + \beta_{2i} + \beta_{3i} + \beta_{4i} + \beta_{5i} + \beta_{6i} + \varepsilon{i} \]

The linear model

Two-factor



 T     N     NA   (Intercept)   THigh   NMedium   NHigh   THigh:NMedium   THigh:NHigh 
---- ------ ---- ------------- ------- --------- ------- --------------- -------------
Low   Low    NA        1          0        0        0           0              0      
Low   Low    NA        1          0        0        0           0              0      
Low   Low    NA        1          0        0        0           0              0      
Low  Medium  NA        1          0        1        0           0              0      
Low  Medium  NA        1          0        1        0           0              0      
Low  Medium  NA        1          0        1        0           0              0      
Low   High   NA        1          0        0        1           0              0      
Low   High   NA        1          0        0        1           0              0      
Low   High   NA        1          0        0        1           0              0      
High  Low    NA        1          1        0        0           0              0      
High  Low    NA        1          1        0        0           0              0      
High  Low    NA        1          1        0        0           0              0      
High Medium  NA        1          1        1        0           1              0      
High Medium  NA        1          1        1        0           1              0      
High Medium  NA        1          1        1        0           1              0      
High  High   NA        1          1        0        1           0              1      
High  High   NA        1          1        0        1           0              1      
High  High   NA        1          1        0        1           0              1

The linear model

Two-factor

	Low N	Medium N	High N
Low temp.	XXX	XXX	XXX
High temp	XXX	XXX	XXX

\[ y_{i} = \beta_{0i} + \beta_{1i} + \beta_{2i} + \beta_{3i} + \beta_{4i} + \beta_{5i} + \beta_{6i} + \varepsilon{i} \]

\(\beta_0\) is the mean of the \(T_{L}:N_{L}\) group
\(\beta_1\) is the difference between \(T_{H}:N_{L}\) and \(T_{L}:N_{L}\)
\(\beta_2\) is the difference between \(T_{L}:N_{M}\) and \(T_{L}:N_{L}\)

The linear model

Two-factor

	Low N	Medium N	High N
Low temp.	XXX	XXX	XXX
High temp	XXX	XXX	XXX

\[ y_{ijk} = \mu + \alpha_i + \beta_j + \alpha_i\beta_j + \varepsilon_{ijk} \]

\(\alpha_i\) is the effect of the \(i_{th}\) temperature at the base level of \(\beta\)

…

Factorial ANOVA

Factor	MS	F-ratio (both fixed)	F-ratio (A fixed, B random)	F-ratio (both random)
A	\(MS_A\)	\(MS_A/MS_{Resid}\)	\(MS_A/MS_{A:B}\)	\(MS_A/MS_{A:B}\)
B	\(MS_B\)	\(MS_B/MS_{Resid}\)	\(MS_B/MS_{Resid}\)	\(MS_B/MS_{A:B}\)
A:B	\(MS_{A:B}\)	\(MS_{A:B}/MS_{Resid}\)	\(MS_{A:B}/MS_{Resid}\)	\(MS_{A:B}/MS_{Resid}\)

Balance

When balanced

\[SS_{TOTAL} = SS_{A} + SS_{B} + SS_{A:B} + SS_{Resid}\]

Factoral ANOVA

Design balance

When balanced
\[SS_{TOTAL} = SS_{A} + SS_{B} + SS_{A:B} + SS_{Resid}\]
When not balanced
\[SS_{TOTAL} \ne SS_{A} + SS_{B} + SS_{A:B} + SS_{Resid}\]

Factorial ANOVA

Factoral ANOVA

Design balance

When balanced
\[SS_{TOTAL} = SS_{A} + SS_{B} + SS_{A:B} + SS_{Resid}\]
When not balanced
\[SS_{TOTAL} \ne SS_{A} + SS_{B} + SS_{A:B} + SS_{Resid}\]

can’t use sequential SS (Type I SS)
should use either
- hierarchical (Type II SS)
- marginal (Type III SS)

Factorial ANOVA

Assumptions

Normality
Homogeneity of variance
Independence
Considerations for Balance

Worked examples

Format of starling.csv data files

SITUATION	MONTH	MASS	GROUP
S1	November	78	S1Nov
..	..	..	..
S2	November	78	S2Nov
..	..	..	..
S3	November	79	S3Nov
..	..	..	..
S4	November	77	S4Nov
..	..	..	..
S1	January	85	S1Jan
..	..	..	..

SITUATION	Categorical listing of roosting situations
MONTH	Categorical listing of the month of sampling.
MASS	Mass (g) of starlings.
GROUP	Categorical listing of situation/month combinations - used for checking ANOVA assumptions

> starling <- read.csv('../data/starling.csv',strip.white=T)

Error in file(file, "rt"): cannot open the connection

> head(starling)

Error in head(starling): object 'starling' not found

Worked Examples

Question: what effects do roosting situations and season have on the mass of starlings

Linear model:

\[ Mass_{ijk} = \mu + \alpha_i + \beta_j + \alpha_i\beta_j +\varepsilon_{ijk} \hspace{1cm} \varepsilon \sim{} \mathcal{N}(0, \sigma^2) \]

Worked Examples

Format of stehman.csv data files

PH	HEALTH	GROUP	BRATING
3	D	D3	0.0
..	..	..	..
3	H	H3	0.8
..	..	..	..
5.5	D	D5.5	0.0
..	..	..	..
5.5	H	H5.5	0.0
..	..	..	..
7	D	D7	0.2
..	..	..	..

PH	Categorical listing of pH (not however that the levels are numbers and thus by default the variable is treated as a numeric variable rather than a factor - we need to correct for this)
HEALTH	Categorical listing of the health status of the seedlings, D = diseased, H = healthy
GROUP	Categorical listing of pH/health combinations - used for checking ANOVA assumptions
BRATING	Average bud emergence rating per seedling

> #Worked examples
> stehman <- read.csv('../data/stehman.csv', strip.white=T)

Error in file(file, "rt"): cannot open the connection

> head(stehman)

Error in head(stehman): object 'stehman' not found

Worked Examples

Question: what effects do pH and health have on the bud emergence rating of spruce seedlings

Linear model:

\[ Buds_{ijk} = \mu + \alpha_i + \beta_j + \alpha_i\beta_j +\varepsilon_{ijk} \hspace{1cm} \varepsilon \sim{} \mathcal{N}(0, \sigma^2) \]

Worked Examples

Format of quinn.csv data files

SEASON	DENSITY	RECRUITS	SQRTRECRUITS	GROUP
Spring	Low	15	3.87	SpringLow
..	..	..	..	..
Spring	High	11	3.32	SpringHigh
..	..	..	..	..
Summer	Low	21	4.58	SummerLow
..	..	..	..	..
Summer	High	34	5.83	SummerHigh
..	..	..	..	..
Autumn	Low	14	3.74	AutumnLow
..	..	..	..	..

SEASON	Categorical listing of Season in which mussel clumps were collected independent variable
DENSITY	Categorical listing of the density of mussels within mussel clump independent variable
RECRUITS	The number of mussel recruits response variable
SQRTRECRUITS	Square root transformation of RECRUITS - needed to meet the test assumptions
GROUPS	Categorical listing of Season/Density combinations - used for checking ANOVA assumptions

Error in file(file, "rt"): cannot open the connection

Error in head(quinn): object 'quinn' not found

Worked Examples

Question: what effects do season and density have on barnacle recruitment

Linear model:

\[ Recruits_{ijk} = \mu + \alpha_i + \beta_j + \alpha_i\beta_j +\varepsilon_{ijk} \hspace{1cm} \varepsilon \sim{} \mathcal{N}(0, \sigma^2) \]

Workshop 7.6a: Factorial ANOVA

Murray Logan

19 Jul 2017

Background

Factorial ANOVA

Factorial ANOVA

The linear model

The linear model

The linear model

The linear model

The linear model

Factorial ANOVA

Design Balance

Balance

Factoral ANOVA

Design balance

Factorial ANOVA

Factoral ANOVA

Design balance

Factorial ANOVA

Factorial ANOVA

Assumptions

Worked examples

Worked examples

Worked Examples

Worked Examples

Worked Examples

Worked Examples

Worked Examples