In this lecture we introduce linear models with two factors.
We look at the formulation of such models, and consider significance
tests for the factors.
Estimation for models with two (or more) factors can be performed by
the method of least squares in the usual way.
Fitted values and residuals are also defined as usual.
A Main Effects Model for Two Factors
Suppose that there are two factors, A and B, which
may be related to the response variable.
The most straightforward way of modelling these factors is to assume
that their effects are additive.
This leads to what can be termed the main effects two-way model.
The rationale for using the phrase main effects will become clear
when we look at interactions between factors.
Model for Two Factors as a Multiple Linear Regression
Written in the same way as was done for the one factor model:
\[Y_i = \mu + \alpha_2 z_{Ai2} + \ldots +
\alpha_K z_{AiK}
+ \beta_2 z_{Bi2} + \ldots + \beta_L z_{BiL} +
\varepsilon_i~~~~~(i=1,2,\ldots,n)\] where \[z_{Aij} =
\left \{
\begin{array}{ll}
1 & \mbox{unit } i \mbox{ observed at level } j \mbox{ of } A\\
0 & \mbox{otherwise}
\end{array}
\right .\] and \[z_{Bij} =
\left \{
\begin{array}{ll}
1 & \mbox{unit } i \mbox{ observed at level } j \mbox{ of } B\\
0 & \mbox{otherwise}
\end{array}
\right .\]
No \(\alpha_1 z_{Ai1}\) and \(\beta_1 z_{Bi1}\) as we assume treatment
constraints. That is, \(\alpha_1 = 0\)
and \(\beta_1 = 0\).
Tests for Main Effects in a Two-Way Model
In a two factor model we are usually interested in testing for the
statistical significance of both factors.
The importance of a factor may depend upon other factors in the
model, in the same way that the importance of a numerical predictor in a
multiple linear regression model depends upon the other explanatory
variables in the model.
The importance of the factors can be assessed by comparing
appropriate nested models using F tests.
Models Under Consideration
There are a number of possible main effects models based on at most
two factors A and B:
- Both A and B have an effect on the response: \[M_{AB}:~~Y_i = \mu + \alpha_2 z_{Ai2} + \ldots +
\alpha_K z_{AiK} + \beta_2 z_{Bi2} + \ldots + \beta_L z_{BiL} +
\varepsilon_i\]
- Only A has an effect on the response: \[M_A:~~Y_i = \mu + \alpha_2 z_{Ai2} + \ldots +
\alpha_K z_{AiK} + \varepsilon_i\]
- Only B has an effect on the response: \[M_B:~~Y_i = \mu + \beta_2 z_{Bi2} + \ldots +
\beta_L z_{BiL} + \varepsilon_i\]
- Neither A nor B have an effect on the response: \[M_0:~~Y_i = \mu + \varepsilon_i\]
Testing for the Importance of B adjusted for
A
This involves comparing the nested models \(M_{AB}\) and \(M_A\).
This can be done by testing \[H_0:~~\beta_2 = \beta_3 = \cdots = \beta_L =
0\] versus
\[H_1:~~\beta_2, \beta_3, \cdots, \beta_L
\mbox{ not all zero}\]
The appropriate F test statistic (on \(L-1, n-K-L+1\) degrees of freedom) is
\[F = \frac{(RSS_{A} -
RSS_{AB})/(L-1)}{RSS_{AB}/(n-K-L+1)}\]
As usual, large values of this F statistic provide evidence
against \(H_0\) (hence give small
p-values).
The relevant figures for this F test can be displayed in an
ANOVA table.
ANOVA Tables for Two Factor Main Effects Model
Factor A |
K-1 |
SSA |
MSA |
fA |
PA |
Factor B (adj. A) |
L-1 |
\(SSB|A\) |
\(MSB|A\) |
\(f_{B|A}\) |
\(P_{B|A}\) |
Residual |
n-K-L+1 |
RSS |
RMS |
|
|
Total |
n-1 |
TSS |
|
|
|
The residual row refers to residuals from model
MAB.
The row for factor A gives the F statistic for
testing the importance of A without adjusting for
B.
The second row gives the F statistic for testing the
importance of B adjusted for A as just discussed.
We can construct a new ANOVA table with first two rows swapped in
order to test for A adjusted for B.
Example: Foster Feeding Rats
In this example, we are looking at a dataset on baby rats fed by
foster mothers. Genotype (A, B, I or J) are recorded for both rat and
foster mother. The weight of baby rats is recorded at a given age. The
factors are genotypes of rat and mother. Does weight depend on
either?
who likes rats?
Analysis of the Data
Download ratgene.csv
## Rat <- read.csv(file = "ratgene.csv", header = TRUE)
head(Rat)
Rat Mother Weight
1 A A 61.5
2 A A 68.2
3 A A 64.0
4 A A 65.0
5 A A 59.7
6 A B 55.0
Let’s first fit a two-way model with Rat
variable first,
and Mother
variable second.
Rat.lm.1 <- lm(Weight ~ Rat + Mother, data = Rat)
anova(Rat.lm.1)
Analysis of Variance Table
Response: Weight
Df Sum Sq Mean Sq F value Pr(>F)
Rat 3 60.2 20.052 0.3317 0.802470
Mother 3 775.1 258.360 4.2732 0.008861 **
Residuals 54 3264.9 60.461
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Call:
lm(formula = Weight ~ Rat + Mother, data = Rat)
Residuals:
Min 1Q Median 3Q Max
-18.425 -5.584 2.499 5.416 13.745
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 56.909 2.478 22.964 <2e-16 ***
RatB -2.025 2.795 -0.725 0.4719
RatI -2.654 2.827 -0.939 0.3520
RatJ -2.021 2.757 -0.733 0.4668
MotherB 3.516 2.862 1.229 0.2246
MotherI -1.832 2.767 -0.662 0.5107
MotherJ -6.755 2.810 -2.404 0.0197 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 7.776 on 54 degrees of freedom
Multiple R-squared: 0.2037, Adjusted R-squared: 0.1152
F-statistic: 2.302 on 6 and 54 DF, p-value: 0.04732
This first model indicates that the genotype of the rat is
unimportant (P=0.802), but that the genotype of the foster
mother adjusted for the genotype of the rat is statistically significant
(P=0.00886).
Now we fit the model with Mother
variable first, and
Rat
variable second.
Rat.lm.2 <- lm(Weight ~ Mother + Rat, data = Rat)
anova(Rat.lm.2)
Analysis of Variance Table
Response: Weight
Df Sum Sq Mean Sq F value Pr(>F)
Mother 3 771.6 257.202 4.2540 0.009055 **
Rat 3 63.6 21.211 0.3508 0.788698
Residuals 54 3264.9 60.461
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Call:
lm(formula = Weight ~ Mother + Rat, data = Rat)
Residuals:
Min 1Q Median 3Q Max
-18.425 -5.584 2.499 5.416 13.745
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 56.909 2.478 22.964 <2e-16 ***
MotherB 3.516 2.862 1.229 0.2246
MotherI -1.832 2.767 -0.662 0.5107
MotherJ -6.755 2.810 -2.404 0.0197 *
RatB -2.025 2.795 -0.725 0.4719
RatI -2.654 2.827 -0.939 0.3520
RatJ -2.021 2.757 -0.733 0.4668
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 7.776 on 54 degrees of freedom
Multiple R-squared: 0.2037, Adjusted R-squared: 0.1152
F-statistic: 2.302 on 6 and 54 DF, p-value: 0.04732
There is little change in p-values when we alter the order
of the terms to get the second model. Note that zero change only occurs
in a special case (next lecture).
Fitted Values for the Rats Model
Calculate the fitted value for genotype A rat with genotype J
foster mother.
Calculate the fitted value for genotype J rat with genotype A
foster mother.
Calculate the fitted value for genotype B rat with genotype B
foster mother.
---
title: "Lecture 22: Introduction to the Two Factor Model"
subtitle: 161.251 Regression Modelling
author: "Presented by Jonathan Godfrey <a.j.godfrey@massey.ac.nz>"  
date: "Week 8 of Semester 2, `r lubridate::year(lubridate::now())`"
output:
  html_document:
    code_download: true
    theme: yeti
    highlight: tango
  html_notebook:
    code_download: true
    theme: yeti
    highlight: tango
  ioslides_presentation:
    widescreen: true
    smaller: true
  word_document: default
  slidy_presentation: 
    theme: yeti
    highlight: tango
  pdf_document: default
---





[View the latest recording of this lecture](https://R-Resources.massey.ac.nz/videos/251L22.mp4)
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https://r-resources.massey.ac.nz/data/161251/
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opts_chunk$set(comment="", fig.align="center", tidy=TRUE)
options(knitr.kable.NA = '')
library(tidyverse)
library(broom)
```


<!--- Do not edit anything above this line. --->
<!--- could switch data source to MASS::genotype --->

In this lecture we introduce linear models with two factors.

We look at the formulation of such models, and consider significance
    tests for the factors.

Estimation for models with two (or more) factors can be performed by
    the method of least squares in the usual way.

Fitted values and residuals are also defined as usual.

## A Main Effects Model for Two Factors

Suppose that there are two factors, *A* and *B*, which may be
    related to the response variable.

The most straightforward way of modelling these factors is to assume
    that their effects are additive.

This leads to what can be termed the main effects
    two-way model.

The rationale for using the phrase main effects will
    become clear when we look at interactions between factors.

### Model for Two Factors as a Multiple Linear Regression

Written in the same way as was done for the one factor model:

$$Y_i = \mu + \alpha_2 z_{Ai2} + \ldots + \alpha_K z_{AiK} 
 + \beta_2 z_{Bi2} + \ldots + \beta_L z_{BiL} + \varepsilon_i~~~~~(i=1,2,\ldots,n)$$
where $$z_{Aij} = 
\left \{ 
\begin{array}{ll}
1 & \mbox{unit } i \mbox{ observed at level } j \mbox{ of } A\\
0 & \mbox{otherwise}
\end{array}
\right .$$ and $$z_{Bij} = 
\left \{ 
\begin{array}{ll}
1 & \mbox{unit } i \mbox{ observed at level } j \mbox{ of } B\\
0 & \mbox{otherwise}
\end{array}
\right .$$

No $\alpha_1 z_{Ai1}$ and $\beta_1 z_{Bi1}$ as we assume treatment
constraints. That is, $\alpha_1 = 0$ and $\beta_1 = 0$.

### Tests for Main Effects in a Two-Way Model

In a two factor model we are usually interested in testing for the
    statistical significance of both factors.

The importance of a factor may depend upon other factors in the
    model, in the same way that the importance of a numerical predictor
    in a multiple linear regression model depends upon the other
    explanatory variables in the model.

The importance of the factors can be assessed by comparing
    appropriate nested models using *F* tests.

### Models Under Consideration

There are a number of possible main effects models based on at most two
factors A and B:

- Both A and B have an effect on the response: $$M_{AB}:~~Y_i = \mu + \alpha_2 z_{Ai2} + \ldots + \alpha_K z_{AiK} + \beta_2 z_{Bi2} + \ldots + \beta_L z_{BiL} + \varepsilon_i$$
- Only A has an effect on the response: $$M_A:~~Y_i = \mu + \alpha_2 z_{Ai2} + \ldots + \alpha_K z_{AiK} + \varepsilon_i$$
- Only B has an effect on the response: $$M_B:~~Y_i = \mu + \beta_2 z_{Bi2} + \ldots + \beta_L z_{BiL} + \varepsilon_i$$
- Neither A nor B have an effect on the response: $$M_0:~~Y_i = \mu + \varepsilon_i$$

### Testing for the Importance of *B* adjusted for *A*

This involves comparing the nested models $M_{AB}$ and $M_A$.

This can be done by testing
$$H_0:~~\beta_2 = \beta_3 = \cdots = \beta_L = 0$$
versus

$$H_1:~~\beta_2, \beta_3, \cdots, \beta_L \mbox{ not all zero}$$

The appropriate *F* test statistic (on $L-1, n-K-L+1$ degrees of
    freedom) is

$$F = \frac{(RSS_{A} - RSS_{AB})/(L-1)}{RSS_{AB}/(n-K-L+1)}$$

As usual, large values of this *F* statistic provide evidence
    against $H_0$ (hence give small *p*-values).

The relevant figures for this *F* test can be displayed in an ANOVA
    table.

### ANOVA Tables for Two Factor Main Effects Model

|                           |     Df      |    Sum Sq |   Mean Sq |     F value |     P value |
| :------------------------ | :---------: | --------: | --------: | ----------: | ----------: |
| Factor *A*              |   *K-1*   |   *SSA* |   *MSA* |     *f~A~* |     *P~A~* |
| Factor *B* (adj. *A*) |   *L-1*   | $SSB|A$ | $MSB|A$ | $f_{B|A}$ | $P_{B|A}$ |
| Residual                  | *n-K-L+1* |   *RSS* |   *RMS* |             |             |
| Total                     |   *n-1*   |   *TSS* |           |             |             |

The residual row refers to residuals from model *M~AB~*.

The row for factor *A* gives the *F* statistic for testing the
    importance of *A* without adjusting for *B*.

The second row gives the *F* statistic for testing the importance
    of *B* adjusted for *A* as just discussed.

We can construct a new ANOVA table with first two rows swapped in order
    to test for *A* adjusted for *B*.

## Example: Foster Feeding Rats



In this example, we are looking at a dataset on baby rats fed by foster mothers.
Genotype (A, B, I or J) are recorded for both rat and foster mother.
The weight of baby rats is recorded at a given age.
The factors are genotypes of rat and mother. Does weight depend on either?

![who likes rats?](../graphics/rats.jpg)




### Analysis of the Data

`r xfun::embed_file("../../data/ratgene.csv")`

```{r Rat.data, echo=-1, eval=-2}
Rat <- read.csv(file="../../data/ratgene.csv", header=TRUE)
Rat <- read.csv(file="ratgene.csv", header=TRUE)
head(Rat)
```

Let's first fit a two-way model with `Rat` variable first, and `Mother` variable second. 

```{r Rat.lm.1}
Rat.lm.1 <- lm(Weight ~   Rat + Mother, data=Rat)
anova(Rat.lm.1)
summary(Rat.lm.1)
```

This first model indicates that the genotype of the rat is unimportant
    (*P=0.802*), but that the genotype of the foster mother adjusted for
    the genotype of the rat is statistically significant (*P=0.00886*).

Now we fit the model with `Mother` variable first, and `Rat` variable second. 


```{r Rat.lm.2}
Rat.lm.2 <- lm(Weight ~  Mother + Rat  , data=Rat)
anova(Rat.lm.2)
summary(Rat.lm.2)
```

There is little change in *p*-values when we alter the order of the terms to get the second model. Note that zero change only occurs in a special case (next lecture).


### Comments and Conclusions

Overall, it seems clear that weight is not associated with genotype
    of the rat, so whether we adjust for this term is pretty much
    immaterial.

Weight is clearly associated with genotype of the foster mother.

Genotype A is reference level (baseline) for both `Mother` and `Rat` genotypes.

The `summary()` command (which gives parameter estimates) suggests that
    genotype J makes for poor foster mothers.

### Fitted Values for the Rats Model

1.  Calculate the fitted value for genotype A rat with genotype J foster
    mother.

2.  Calculate the fitted value for genotype J rat with genotype A foster
    mother.

3.  Calculate the fitted value for genotype B rat with genotype B foster
    mother.


Comments and Conclusions
Overall, it seems clear that weight is not associated with genotype of the rat, so whether we adjust for this term is pretty much immaterial.
Weight is clearly associated with genotype of the foster mother.
Genotype A is reference level (baseline) for both
Mother
andRat
genotypes.The
summary()
command (which gives parameter estimates) suggests that genotype J makes for poor foster mothers.