LECTURE 7: linear models part 2: categorical predictor > 2 levels
FANR 6750 (Experimental design)

Fall 2024
1 / 37

outline

1) Motivating example

2 / 37

outline

1) Motivating example

2) Extending the linear model

2 / 37

outline

1) Motivating example

2) Extending the linear model

3) Dummy variables with > 2 levels

2 / 37

outline

1) Motivating example

2) Extending the linear model

3) Dummy variables with > 2 levels

4) F-tests

2 / 37

outline

1) Motivating example

2) Extending the linear model

3) Dummy variables with > 2 levels

4) F-tests

5) Comparing linear model and ANOVA approaches

2 / 37

categorical predictorsIn lecture 4, we learned about fitting models with categorical predictors with 1 (measurement error) or 2 (frog) levels3 / 37

categorical predictorsIn lecture 4, we learned about fitting models with categorical predictors with 1 (measurement error) or 2 (frog) levelsLinear models with these categorical predictors correspond to one-sample or two-sample t-tests
3 / 37

categorical predictors

In lecture 4, we learned about fitting models with categorical predictors with 1 (measurement error) or 2 (frog) levels

Linear models with these categorical predictors correspond to one-sample or two-sample t-tests
Binary categorical predictors are quite common in ecological studies (e.g., male vs female, juvenile vs adult, disturbed vs undisturbed)

3 / 37

categorical predictors

In lecture 4, we learned about fitting models with categorical predictors with 1 (measurement error) or 2 (frog) levels

Linear models with these categorical predictors correspond to one-sample or two-sample t-tests
Binary categorical predictors are quite common in ecological studies (e.g., male vs female, juvenile vs adult, disturbed vs undisturbed)

However, you will often encounter categorical predictors with >2 levels

What are some examples?

3 / 37

motivating example

Foresters are studying the effect of 4 different fertilizers (treatments) on the growth of loblolly pine, which are grown on 3 plots (replicates) receiving each treatment. Data are average height per plot after 5 years:

	Treatment
Replicate	A	B	C	D
1	11	9	7	5
2	10	8	6	4
3	9	7	5	3

4 / 37

motivating example

	Treatment
Replicate	A	B	C	D
1	11	9	7	5
2	10	8	6	4
3	9	7	5	3

Notation

The number of groups (treatments) is $\large a=4$
The number of observations within each group (replicates) is $\large n=3$
$\large y_{ij}$ denotes the $\large j$ th observation from the $\large i$ th group

4 / 37

a brief tangentWhat counts as an observation?5 / 37

a brief tangent

What counts as an observation?

Experimental unit

the physical unit that receives a particular treatment

5 / 37

a brief tangent

What counts as an observation?

Experimental unit

the physical unit that receives a particular treatment

Observational unit

the physical unit on which measurements are taken

5 / 37

a brief tangent

What counts as an observation?

Experimental unit

the physical unit that receives a particular treatment

Observational unit

the physical unit on which measurements are taken

These are not always the same!

5 / 37

a brief tangent

What counts as an observation?

Experimental unit

the physical unit that receives a particular treatment

Observational unit

the physical unit on which measurements are taken

These are not always the same!

Examples

Agricultural fields given different fertilizer, crop yield measured
Rats given different diets, disease state measured
Microcosm given different predator abundance, tadpole growth measured

5 / 37

motivating example

Question: Is there a difference in growth among the four treatment groups?

6 / 37

motivating example

Question: Is there a difference in growth among the four treatment groups?

6 / 37

motivating example

Hypotheses

$\large H_0 : \mu_A = \mu_B = \mu_C = \mu_D$
$\large H_a :$ At least one inequality

7 / 37

motivating example

Hypotheses

$\large H_0 : \mu_A = \mu_B = \mu_C = \mu_D$
$\large H_a :$ At least one inequality

How should we test the null?

7 / 37

motivating example

Hypotheses

$\large H_0 : \mu_A = \mu_B = \mu_C = \mu_D$
$\large H_a :$ At least one inequality

How should we test the null?

We could do this using 6 t-tests

7 / 37

motivating example

Hypotheses

$\large H_0 : \mu_A = \mu_B = \mu_C = \mu_D$
$\large H_a :$ At least one inequality

How should we test the null?

We could do this using 6 t-tests

But this would alter the overall (experiment-wise) $\large \alpha$ level because each individual test has a chance (usually $\large \alpha = 0.05$ ) of incorrectly rejecting a true null hypothesis, and this is multiplied when multiple tests are used

7 / 37

extending the linear model

Fortunately, extending the previous linear model to include additional predictor levels is "straightforward"

The model looks familiar

$\large y_{ij} = \beta_0 + \beta_ix_{ij} + \epsilon_{ij}$

$\large \epsilon_{ij} \sim normal(0, \sigma)$

8 / 37

extending the linear model

Fortunately, extending the previous linear model to include additional predictor levels is "straightforward"

The model looks familiar

$\large y_{ij} = \beta_0 + \beta_ix_{ij} + \epsilon_{ij}$

$\large \epsilon_{ij} \sim normal(0, \sigma)$

What's tricky is how we code $\large x_{ij}$ when it is more than just 0 or 1

8 / 37

extending the linear model

Fortunately, extending the previous linear model to include additional predictor levels is "straightforward"

The model looks familiar

$\large y_{ij} = \beta_0 + \beta_ix_{ij} + \epsilon_{ij}$

$\large \epsilon_{ij} \sim normal(0, \sigma)$

What's tricky is how we code $\large x_{ij}$ when it is more than just 0 or 1
Fortunately, what we learned about how R codes dummy variables will help

8 / 37

dummy variable coding

data(loblollydata)
loblollydata[c(1,4,7,10),]

##    Treatment Replicate Height
## 1          A         1     11
## 4          B         1      9
## 7          C         1      7
## 10         D         1      5

fit1 <- lm(Height ~ Treatment, data = loblollydata)
model.matrix(fit1)[c(1,4,7,10),]

##    (Intercept) TreatmentB TreatmentC TreatmentD
## 1            1          0          0          0
## 4            1          1          0          0
## 7            1          0          1          0
## 10           1          0          0          1

9 / 37

dummy variable coding

model.matrix(fit1)[c(1,4,7,10),]

##    (Intercept) TreatmentB TreatmentC TreatmentD
## 1            1          0          0          0
## 4            1          1          0          0
## 7            1          0          1          0
## 10           1          0          0          1

Remember that the model matrix has one column for every parameter in the model

How many parameters is this model estimating?

10 / 37

dummy variable coding

model.matrix(fit1)[c(1,4,7,10),]

##    (Intercept) TreatmentB TreatmentC TreatmentD
## 1            1          0          0          0
## 4            1          1          0          0
## 7            1          0          1          0
## 10           1          0          0          1

Remember that the model matrix has one column for every parameter in the model

How many parameters is this model estimating?

$\large E[y_{ij}] = \beta_0 + \beta_1 I(B)_{ij} + \beta_2 I(C)_{ij} + \beta_3 I(D)_{ij}$ where $I(B/C/D)_{ij}$ are dummy variables (0/1 depending on treatment)

10 / 37

dummy variable coding

model.matrix(fit1)[c(1,4,7,10),]

##    (Intercept) TreatmentB TreatmentC TreatmentD
## 1            1          0          0          0
## 4            1          1          0          0
## 7            1          0          1          0
## 10           1          0          0          1

Remember that the model matrix has one column for every parameter in the model

How many parameters is this model estimating?

$\large E[y_{ij}] = \beta_0 + \beta_1 I(B)_{ij} + \beta_2 I(C)_{ij} + \beta_3 I(D)_{ij}$ where $I(B/C/D)_{ij}$ are dummy variables (0/1 depending on treatment)

What is the interpretation of each parameter?

10 / 37

motivating example

broom::tidy(fit1)

## # A tibble: 4 × 5
##   term        estimate std.error statistic     p.value
##   <chr>          <dbl>     <dbl>     <dbl>       <dbl>
## 1 (Intercept)    10.0      0.577     17.3  0.000000126
## 2 TreatmentB     -2.00     0.816     -2.45 0.0400     
## 3 TreatmentC     -4.00     0.816     -4.90 0.00120    
## 4 TreatmentD     -6        0.816     -7.35 0.0000801

11 / 37

motivating example

broom::tidy(fit1)

## # A tibble: 4 × 5
##   term        estimate std.error statistic     p.value
##   <chr>          <dbl>     <dbl>     <dbl>       <dbl>
## 1 (Intercept)    10.0      0.577     17.3  0.000000126
## 2 TreatmentB     -2.00     0.816     -2.45 0.0400     
## 3 TreatmentC     -4.00     0.816     -4.90 0.00120    
## 4 TreatmentD     -6        0.816     -7.35 0.0000801

Be sure you understand what each parameter means!

11 / 37

motivating example

broom::tidy(fit1)

## # A tibble: 4 × 5
##   term        estimate std.error statistic     p.value
##   <chr>          <dbl>     <dbl>     <dbl>       <dbl>
## 1 (Intercept)    10.0      0.577     17.3  0.000000126
## 2 TreatmentB     -2.00     0.816     -2.45 0.0400     
## 3 TreatmentC     -4.00     0.816     -4.90 0.00120    
## 4 TreatmentD     -6        0.816     -7.35 0.0000801

Be sure you understand what each parameter means!
t-statistics and p-values are calculated and interpreted in the same way as the frog example

11 / 37

motivating example

broom::tidy(fit1)

## # A tibble: 4 × 5
##   term        estimate std.error statistic     p.value
##   <chr>          <dbl>     <dbl>     <dbl>       <dbl>
## 1 (Intercept)    10.0      0.577     17.3  0.000000126
## 2 TreatmentB     -2.00     0.816     -2.45 0.0400     
## 3 TreatmentC     -4.00     0.816     -4.90 0.00120    
## 4 TreatmentD     -6        0.816     -7.35 0.0000801

Be sure you understand what each parameter means!
t-statistics and p-values are calculated and interpreted in the same way as the frog example
But wait, the null hypothesis was $\mu_A = \mu_B = \mu_C = \mu_D$

11 / 37

regression F-statisticsummary(fit1)

## 
## Call:
## lm(formula = Height ~ Treatment, data = loblollydata)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
##     -1     -1      0      1      1 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   10.000      0.577   17.32  1.3e-07 ***
## TreatmentB    -2.000      0.816   -2.45   0.0400 *  
## TreatmentC    -4.000      0.816   -4.90   0.0012 ** 
## TreatmentD    -6.000      0.816   -7.35  8.0e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1 on 8 degrees of freedom
## Multiple R-squared:  0.882,    Adjusted R-squared:  0.838 
## F-statistic:   20 on 3 and 8 DF,  p-value: 0.000449
12 / 37

regression F-statistic

The F-statistic tests whether all regression coefficients (other than the intercept) are all 0

$H_0$ : All regression coefficients are 0
$H_A$ : At least one regression coefficient is not 0

13 / 37

regression F-statistic

The F-statistic tests whether all regression coefficients (other than the intercept) are all 0

$H_0$ : All regression coefficients are 0
$H_A$ : At least one regression coefficient is not 0

F-statistics are the ratio of sample variances

$F = \frac{s^2_A}{s^2_B}$

Null hypothesis is that population variances are equal ( $\sigma^2_A = \sigma^2_B$ )
Always positive (variances cannot be negative)
Usually > 1 (by putting larger variance in the numerator)

13 / 37

regression F-statistic

The F-statistic tests whether all regression coefficients (other than the intercept) are all 0

$H_0$ : All regression coefficients are 0
$H_A$ : At least one regression coefficient is not 0

F-statistics are the ratio of sample variances

$F = \frac{s^2_A}{s^2_B}$

Null hypothesis is that population variances are equal ( $\sigma^2_A = \sigma^2_B$ )
Always positive (variances cannot be negative)
Usually > 1 (by putting larger variance in the numerator)

Where do the variances come from in the linear model and what do they tell us about differences among groups?

13 / 37

regression F-statisticTo understand why the test is based on variance, it is helpful to consider several types of means:14 / 37

regression F-statistic

To understand why the test is based on variance, it is helpful to consider several types of means:

Grand mean

$\large \bar{y}. = \frac{\sum_i \sum_jy_{ij}}{a \times n}$

14 / 37

regression F-statistic

To understand why the test is based on variance, it is helpful to consider several types of means:

Grand mean

$\large \bar{y}. = \frac{\sum_i \sum_jy_{ij}}{a \times n}$

Group means

$\large \bar{y}_i = \frac{\sum_j y_{ij}}{n}$

15 / 37

regression F-statistic

To understand why the test is based on variance, it is helpful to consider several types of means:

Grand mean

$\large \bar{y}. = \frac{\sum_i \sum_jy_{ij}}{a \times n}$

Group means

$\large \bar{y}_i = \frac{\sum_j y_{ij}}{n}$

We can now decompose the observations as

$\large y_{ij} = \color{#446E9B}{\bar{y}.} + \color{#D47500}{(\bar{y}_i - \bar{y}.)} + \color{#3CB521}{(y_{ij} - \bar{y}_i)}$

16 / 37

the additive model

The decomposition

$\Large y_{ij} = \color{#446E9B}{\bar{y}.} + \color{#D47500}{(\bar{y}_i - \bar{y}.)} + \color{#3CB521}{(y_{ij} - \bar{y}_i)}$

17 / 37

the additive model

The decomposition

$\Large y_{ij} = \color{#446E9B}{\bar{y}.} + \color{#D47500}{(\bar{y}_i - \bar{y}.)} + \color{#3CB521}{(y_{ij} - \bar{y}_i)}$

The additive model

$\Large y_{ij} = \color{#446E9B}{\mu} + \color{#D47500}{\alpha_i} + \color{#3CB521}{\epsilon_{ij}}$

17 / 37

the additive model

The decomposition

$\Large y_{ij} = \color{#446E9B}{\bar{y}.} + \color{#D47500}{(\bar{y}_i - \bar{y}.)} + \color{#3CB521}{(y_{ij} - \bar{y}_i)}$

The additive model

$\Large y_{ij} = \color{#446E9B}{\mu} + \color{#D47500}{\alpha_i} + \color{#3CB521}{\epsilon_{ij}}$

where

$\Large \epsilon_{ij} \sim normal(0, \sigma^2)$

17 / 37

the additive model

$\large y_{ij} = \mu + \alpha_i + \epsilon_{ij}$

$\large \epsilon_{ij} \sim normal(0, \sigma^2)$

Notes

$\large \mu$ is the grand mean of the population, estimated by $\large \bar{y}.$

18 / 37

the additive model

$\large y_{ij} = \mu + \alpha_i + \epsilon_{ij}$

$\large \epsilon_{ij} \sim normal(0, \sigma^2)$

Notes

$\large \mu$ is the grand mean of the population, estimated by $\large \bar{y}.$
$\large \alpha_i$ is the effect of treatment i, estimated by $\large\bar{y}_i - \bar{y}.$

18 / 37

the additive model

$\large y_{ij} = \mu + \alpha_i + \epsilon_{ij}$

$\large \epsilon_{ij} \sim normal(0, \sigma^2)$

Notes

$\large \mu$ is the grand mean of the population, estimated by $\large \bar{y}.$
$\large \alpha_i$ is the effect of treatment i, estimated by $\large\bar{y}_i - \bar{y}.$
- It is the deviation of the group mean from the grand mean
- If all $\large \alpha_i = 0$ , there is no treatment effect
- Thus, we can re-write the null hypothesis $H_0 : \alpha_1 = \alpha_2=... =\alpha_a = 0$

18 / 37

the additive model

$\large y_{ij} = \mu + \alpha_i + \epsilon_{ij}$

$\large \epsilon_{ij} \sim normal(0, \sigma^2)$

Notes

$\large \mu$ is the grand mean of the population, estimated by $\large \bar{y}.$
$\large \alpha_i$ is the effect of treatment i, estimated by $\large\bar{y}_i - \bar{y}.$
- It is the deviation of the group mean from the grand mean
- If all $\large \alpha_i = 0$ , there is no treatment effect
- Thus, we can re-write the null hypothesis $H_0 : \alpha_1 = \alpha_2=... =\alpha_a = 0$
$\large \epsilon_{ij}$ is the residual error, estimated by $\large y_{ij} - \bar{y}_i$
- It is the unexplained (random) deviation of the observation from the group mean

18 / 37

sums of squares

Variation among groups

$\Large SS_A = n \sum_i (\bar{y}_i - \bar{y}.)^2$

19 / 37

motivating example

Question: Is there a difference in growth among the four treatment groups?

20 / 37

sums of squares

Variation among groups

$\Large SS_A = n \sum_i (\bar{y}_i - \bar{y}.)^2$

Variation within groups

$\Large SS_W = \sum_i \sum_j (y_{ij} - \bar{y}_i)^2$

21 / 37

motivating example

Question: Is there a difference in growth among the four treatment groups?

22 / 37

sums of squares

Variation among groups

$\Large SS_A = n \sum_i (\bar{y}_i - \bar{y}.)^2$

Variation within groups

$\Large SS_W = \sum_i \sum_j (y_{ij} - \bar{y}_i)^2$

Total variation

$\Large SS_T = SS_A + SS_W = \sum_i \sum_j (y_{ij} - \bar{y}.)^2$

23 / 37

motivating example

Question: Is there a difference in growth among the four treatment groups?

24 / 37

mean squaresTo covert the sums of squares to variances, divide by the degrees of freedom25 / 37

mean squares

To covert the sums of squares to variances, divide by the degrees of freedom

Mean squares among

$\Large MS_A = \frac{SS_A}{a-1}$

25 / 37

mean squares

To covert the sums of squares to variances, divide by the degrees of freedom

Mean squares among

$\Large MS_A = \frac{SS_A}{a-1}$

Mean squares within

$\Large MS_W = \frac{SS_W}{a(n-1)}$

25 / 37

F-statistic

$\LARGE F = \frac{MS_A}{MS_W}$

26 / 37

F-statistic

$\LARGE F = \frac{MS_A}{MS_W}$

Note:

26 / 37

F-statistic

$\LARGE F = \frac{MS_A}{MS_W}$

Note:

F is a ratio measuring the variance among groups to the variance within groups

26 / 37

F-statistic

$\LARGE F = \frac{MS_A}{MS_W}$

Note:

F is a ratio measuring the variance among groups to the variance within groups
If there is a large treatment effect, what happens to $MS_A$ ?

26 / 37

F-statistic

$\LARGE F = \frac{MS_A}{MS_W}$

Note:

F is a ratio measuring the variance among groups to the variance within groups
If there is a large treatment effect, what happens to $MS_A$ ?
If there is little residual variation, what happens to $MS_W$ ?

26 / 37

F-statistic

$\LARGE F = \frac{MS_A}{MS_W}$

Note:

F is a ratio measuring the variance among groups to the variance within groups
If there is a large treatment effect, what happens to $MS_A$ ?
If there is little residual variation, what happens to $MS_W$ ?
Large values of $F$ indicate treatment effects are large relative to residual variation, but can we conclude there is a treatment effect?

26 / 37

the F-distribution

If the null hypothesis is true ( $\sigma^2_A = \sigma^2_B$ ), the ratio of sample variances follows an F-distribution

Properties:

$F > 0$
$F$ -distribution is not symmetrical
Shape of distribution depends on an ordered pair of degrees of freedom, $df_A$ and $df_B$

27 / 37

f-distribution

Just like the t-distribution, we can use the F-distribution to calculate p-values and test the null hypothesis that the variances are equal

28 / 37

analysis of varianceUsing the F-statistic to test whether there is a treatment effect is commonly referred to as Analysis of Variance (ANOVA)29 / 37

analysis of varianceUsing the F-statistic to test whether there is a treatment effect is commonly referred to as Analysis of Variance (ANOVA)Especially in experimental settings, ANOVA is very commonly used  
29 / 37

analysis of variance

Using the F-statistic to test whether there is a treatment effect is commonly referred to as Analysis of Variance (ANOVA)

Especially in experimental settings, ANOVA is very commonly used
The linear model approach is increasingly common, especially in observational settings

29 / 37

analysis of variance

Using the F-statistic to test whether there is a treatment effect is commonly referred to as Analysis of Variance (ANOVA)

Especially in experimental settings, ANOVA is very commonly used
The linear model approach is increasingly common, especially in observational settings
From a practical standpoint, the linear model and ANOVA approaches provide exactly the same information, just presented in different ways (just like the linear model vs t-test we saw previously)

29 / 37

worked example

Suppose we are interested in the effect of elevation on the abundance of Canada Warblers

30 / 37

worked example

Suppose we are interested in the effect of elevation on the abundance of Canada Warblers

	Elevation
Replicate	Low	Medium	High
1	1	2	4
2	3	0	7
3	0	4	5
4	2	3	5

30 / 37

Image courtesy of William H. Majoros via Wikicommons

worked exampledata(warblerdata)
fit.lm <- lm(Count ~ Elevation, data = warblerdata)
summary(fit.lm)

## 
## Call:
## lm(formula = Count ~ Elevation, data = warblerdata)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -2.250 -0.688 -0.250  0.938  1.750 
## 
## Coefficients:
##                 Estimate Std. Error t value Pr(>|t|)    
## (Intercept)        5.250      0.717    7.32  4.5e-05 ***
## ElevationLow      -3.750      1.014   -3.70   0.0049 ** 
## ElevationMedium   -3.000      1.014   -2.96   0.0160 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.43 on 9 degrees of freedom
## Multiple R-squared:  0.63,    Adjusted R-squared:  0.548 
## F-statistic: 7.66 on 2 and 9 DF,  p-value: 0.0114
31 / 37

worked example

data(warblerdata)
fit.aov <- aov(Count ~ Elevation, data = warblerdata)
summary(fit.aov)

##             Df Sum Sq Mean Sq F value Pr(>F)  
## Elevation    2   31.5   15.75    7.66  0.011 *
## Residuals    9   18.5    2.06                 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

32 / 37

anova table from `lm`

In case you want more detail about the relationship between lm and aov output:

lm() function also returns residuals (e.g., $y_i - E[y_i]$ )

fit.lm$residual

##     1     2     3     4     5     6     7     8     9    10    11    12 
## -0.50  1.50 -1.50  0.50 -0.25 -2.25  1.75  0.75 -1.25  1.75 -0.25 -0.25

Residual sum of squares

sum(fit.lm$residuals^2)

## [1] 18.5

Residual mean square

sum(fit.lm$residuals^2)/9

## [1] 2.056

33 / 37

anova table from `lm`

In case you want more detail about the relationship between lm and aov output:

What about among group variation?

fit.lm$fitted.values

##    1    2    3    4    5    6    7    8    9   10   11   12 
## 1.50 1.50 1.50 1.50 2.25 2.25 2.25 2.25 5.25 5.25 5.25 5.25

Treatment sum of squares

sum((fit.lm$fitted.values - mean(fit.lm$fitted.values))^2)

## [1] 31.5

Treatment mean square

sum((fit.lm$fitted.values - mean(fit.lm$fitted.values))^2)/2

## [1] 15.75

34 / 37

anova table from `lm`

In case you want more detail about the relationship between lm and aov output:

F-statistic and p-value

MSa <- sum((fit.lm$fitted.values - mean(fit.lm$fitted.values))^2)/2
MSe <- sum(fit.lm$residuals^2)/9
(F <- MSa/MSe)

## [1] 7.662

(p <- pf(F, 2, 9, lower.tail = FALSE))

## [1] 0.0114

35 / 37

recap

1) Models with categorical variables can be fit using either lm() or aov()

36 / 37

recap

1) Models with categorical variables can be fit using either lm() or aov()

2) The test traditionally known as ANOVA is just an extension of the linear model used analyze a binary predictor variable

36 / 37

recap

1) Models with categorical variables can be fit using either lm() or aov()

2) The test traditionally known as ANOVA is just an extension of the linear model used analyze a binary predictor variable

3) R codes categorical predictors using dummy variables and lm coefficients correspond to the difference between the reference level and the other treatment levels

36 / 37

recap

1) Models with categorical variables can be fit using either lm() or aov()

2) The test traditionally known as ANOVA is just an extension of the linear model used analyze a binary predictor variable

3) R codes categorical predictors using dummy variables and lm coefficients correspond to the difference between the reference level and the other treatment levels

4) lm() or aov() provide the exact same information, just presented in different ways

36 / 37

recap

1) Models with categorical variables can be fit using either lm() or aov()

2) The test traditionally known as ANOVA is just an extension of the linear model used analyze a binary predictor variable

3) R codes categorical predictors using dummy variables and lm coefficients correspond to the difference between the reference level and the other treatment levels

4) lm() or aov() provide the exact same information, just presented in different ways

5) ANOVA output commonly used for manipulative experiments, linear model output for observational studies

36 / 37

recap

1) Models with categorical variables can be fit using either lm() or aov()

2) The test traditionally known as ANOVA is just an extension of the linear model used analyze a binary predictor variable

3) R codes categorical predictors using dummy variables and lm coefficients correspond to the difference between the reference level and the other treatment levels

4) lm() or aov() provide the exact same information, just presented in different ways

5) ANOVA output commonly used for manipulative experiments, linear model output for observational studies

6) But how do we tell which groups differ from each other (aside from just the reference level)...

↑, ←, Pg Up, k	Go to previous slide
↓, →, Pg Dn, Space, j	Go to next slide
Home	Go to first slide
End	Go to last slide
Number + Return	Go to specific slide
b / m / f	Toggle blackout / mirrored / fullscreen mode
c	Clone slideshow
p	Toggle presenter mode
t	Restart the presentation timer
?, h	Toggle this help