PSY 503: Foundations of Statistical Methods in Psychological Science

class: center, middle, inverse, title-slide

.title[
# PSY 503: Foundations of Statistical Methods in Psychological Science
]
.subtitle[
## More LM: Categorical Predictors
]
.author[
### Jason Geller, Ph.D. (he/him/his)
]
.institute[
### Princeton University
]
.date[
### Updated:2022-11-02
]

---

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# Outline

- Check-in questions

- Effect size and power in MR

- Categorical predictors

- How linear modeling is related to t-tests/ANOVAs
  
  - Contrast coding for categorical predictors with two means 
    
  - Contrast coding for categorical predictors with three means or more
---
# Check-in Questions

- What the heck is heteroskedasticity?

---
# Heteroskedasticity

- Non-constant error (residual) variance

- Residuals and predictors are correlated

---
# Heteroskedasticity

- Consequences:

- Causes standard errors (SE) to be unreliable
  
    - Increased Type 1 and Type 2 Error

- Solution:

- Easiest would be to use robust methods for the SE     
---
.pull-left[

```r
library(performance)
model_parameters(model2) %>%
  flextable() %>%
  autofit()
```

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class="cl-fcd7be33">47.5059719</td><td class="cl-fcd7be28">62.82149992</td><td class="cl-fcd7be46">14.1843873</td><td class="cl-fcd7be32">262</td><td class="cl-fcd7be47">0.0000000000000000000000000000000002841237</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-fcd7be64">PIL_total</td><td class="cl-fcd7be5c">-0.38157775</td><td class="cl-fcd7be5a">0.03384062</td><td class="cl-fcd7be5b">0.95</td><td class="cl-fcd7be5a">-0.4482119</td><td class="cl-fcd7be5c">-0.31494356</td><td class="cl-fcd7be65">-11.2757330</td><td class="cl-fcd7be50">262</td><td class="cl-fcd7be6e">0.0000000000000000000000026622090508643975</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-fcd7be78">AUDIT_TOTAL_NEW</td><td class="cl-fcd7be79">-0.09210892</td><td class="cl-fcd7be8d">0.09391092</td><td class="cl-fcd7be6f">0.95</td><td class="cl-fcd7be8d">-0.2770251</td><td class="cl-fcd7be79">0.09280729</td><td class="cl-fcd7be82">-0.9808116</td><td class="cl-fcd7be8c">262</td><td 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.pull-right[

```r
model_parameters(model2, vcov = "HC3") %>% #get corrected SE# robust params
flextable() %>%
  autofit()# robust params
```

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class="cl-fcf379a6">46.87639900</td><td class="cl-fcf379a6">63.45107283</td><td class="cl-fcf379d9">13.106827</td><td class="cl-fcf379c4">262</td><td class="cl-fcf379e2">0.000000000000000000000000000001603377</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-fcf379f6">PIL_total</td><td class="cl-fcf379ec">-0.38157775</td><td class="cl-fcf37a01">0.03597990</td><td class="cl-fcf379ed">0.95</td><td class="cl-fcf379ec">-0.45242431</td><td class="cl-fcf379ec">-0.31073119</td><td class="cl-fcf379f7">-10.605305</td><td class="cl-fcf379e3">262</td><td class="cl-fcf37a00">0.000000000000000000000425863763393213</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-fcf37a0a">AUDIT_TOTAL_NEW</td><td class="cl-fcf37a0b">-0.09210892</td><td class="cl-fcf37a15">0.08647293</td><td class="cl-fcf37a02">0.95</td><td class="cl-fcf37a0b">-0.26237929</td><td class="cl-fcf37a0b">0.07816145</td><td class="cl-fcf37a0c">-1.065176</td><td class="cl-fcf37a14">262</td><td class="cl-fcf37a16">0.287776789141752753486969140794826671</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-fcf37a33">DAST_TOTAL_NEW</td><td class="cl-fcf37a1f">1.03415391</td><td class="cl-fcf37a3c">0.48095976</td><td class="cl-fcf37a1e">0.95</td><td class="cl-fcf37a1f">0.08711542</td><td class="cl-fcf37a1f">1.98119239</td><td class="cl-fcf37a32">2.150188</td><td class="cl-fcf37a28">262</td><td class="cl-fcf37a29">0.032455663970932233097776986596727511</td></tr></tbody></table></div></template>
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---
# Check-in Questions

- What the heck is heteroskedasticity?

- Assumptions of simple linear model

---
# Simple Linear Modeling Assumptions

- Normality of residuals

- Missingness

- Outliers

---
# Burning Questions

- What the heck is heteroskedasticity?

- Assumptions of simple linear model

- Difference between `$\epsilon$` and residual
---
# Errors

- Error is basically an unobservable  quantity in our model

- We use residuals as a proxy
---
# Outliers

<img src="out.png" width="70%" style="display: block; margin: auto;" />
---
# Outliers: Discrepancy

> A data point that is unusual in the context of the least squares model

`$$e_i^* = \dfrac{e_i}{S_{E(-i)}\sqrt{1-h_i}}$$`

<img src="hdl.png" width="100%" style="display: block; margin: auto;" />
---
# Outliers: Leverage

> Leverage measures how far a data point is from the mean

`$$h_i = \dfrac{1}{n} + \dfrac{\left(x_i -\overline{x}\right)^2}{\sum_{j=1}^{n}{\left(x_j -\overline{x}\right)^2}} \qquad \text{and}\qquad \overline{h} = \dfrac{k}{n} \qquad \text{and}\qquad \dfrac{1}{n} \leq h_i \leq 1.0$$`
<img src="hdl.png" width="100%" style="display: block; margin: auto;" />
---
# Cook's Distance

- A measure of how much of an effect a single data point has on the whole model 
 
- Often described as leverage + discrepancy (residuals)
 
`$$D_i = \dfrac{e_i^2}{k \times \frac{1}{n}\sum{e_i^2}} \times \dfrac{h_i}{1-h_i}$$`
`$$e_i^* = \dfrac{e_i}{S_{E(-i)}\sqrt{1-h_i}}$$`
- How do we calculate how much change is bad?

 - `$\frac{4}{N-K-1}$`
---
# Calculate in R

```r
augment(model) # in broom package
```

---
# The Impact of Individual Predictors on the Model: Effect Size

.pull-left[
- R is the multiple correlation
- `$R^2$` is the multiple correlation squared
- All overlap in Y, used for overall model
- `$A+B+C/(A+B+C+D)$`
]

.pull-right[
<img src="https://raw.githubusercontent.com/doomlab/learnSTATS/master/vignettes/pictures/regression/19.PNG" width="100%" style="display: block; margin: auto;" />
]
---
# Effect Size

.pull-left[

- sr is the semi-partial correlations

- Unique contribution of IV to `$R^2$` for those IVs

- Increase in proportion of explained Y variance when X is added to the equation

- `$A/(A+B+C+D)$`
]

.pull-right[
<img src="https://raw.githubusercontent.com/doomlab/learnSTATS/master/vignettes/pictures/regression/19.PNG" width="100%" style="display: block; margin: auto;" />
]
---
# Effect Size

.pull-left[

- pr is the partial correlation

- Partial correlation asks how much of the Y variance, which is not estimated by the other IVs, is estimated by this variable.

- `$A/(A+D)$`

- Removes the shared variance of the control variable (Say X2) from both Y and X1

- Pr > sr
]

.pull-right[

<img src="https://raw.githubusercontent.com/doomlab/learnSTATS/master/vignettes/pictures/regression/19.PNG" width="100%" style="display: block; margin: auto;" />
]
---
# Partial Correlations

- We would add these to our other reports:

- Meaning: `$b = -0.38$`, 95\% CI `$[-0.45, -0.31]$`, `$t(262) = -11.28$`, `$p < .001$`, `$pr^2 = .30$``
 - Alcohol: `$b = -0.09$`, 95\% CI `$[-0.28, 0.09]$`, `$t(262) = -0.98$`, `$p = .328$`, `$pr^2 < .01$``
 - Drugs: `$b = 1.03$`, 95\% CI `$[0.25, 1.82]$`, `$t(262) = 2.59$`, `$p = .010$`, `$pr^2 < .01$``

```r
library(ppcor)
partials <- pcor(master)
partials$estimate^2
#spcor.test (semi-partial)
```

---
# Multiple Regression: Power

- We can use the `pwr` library to calculate the required sample size for any particular effect size

- First, we need to convert the `$R^2$` value to `$f^2$`, which is a different effect size, not the ANOVA *F*

```r
library(pwr)
r2=glance(model2)

R2= r2$r.squared
f2 <- R2 / (1-R2)
R2
```

```
## [1] 0.3586693
```

```r
f2
```

```
## [1] 0.559258
```
---
# Multiple Regression: Power

- `u` is degrees of freedom for the model, first value in the F-statistic
- `v` is degrees of freedom for error, but we are trying to figure out sample size for each condition, so we leave this one blank. 
- `f2` is the converted effect size.
- `sig.level` is our `$\alpha$` value
- `power` is our power level
- The final sample size is *v + k + 1 where k is the predictors*

```r
#f2 is cohen f squared 
pwr.f2.test(u = model2$df[1], 
            v = NULL, f2 = f2, 
            sig.level = .05, power = .80)
```

---
class: middle
# Categorical Predictors
---
# Modeling Categorical Variables

- Today's Dataset

- Winter(2016)

- Are smell words (e.g., *rancid*) rated as more negative/unpleaseant than taste words (e.g., *sweet*)?
    
    - 1 to 9 rating scale

```r
library(tidyverse)
senses<- read_csv("data/winter_2016_senses_valence.csv")

senses_filt <- senses %>%
 filter(Modality=="Taste" | Modality=="Smell")
```
---
# Linear Model

`$$\red{Y_i} = \beta_0 + \beta_1 \blue{X_i} + \green{\varepsilon_i}$$`

- So far .blue[predictor variable] has been continuous

- We can also use linear modeling for categorical variables

---
# Categorical Variables

- Terminology

- *Factor*: a variable with a fix set of categories
  
  - *Levels*: The individual categories within a factor

- In our dataset, what is the factor and what are its levels?
---
# Linear Modeling and t-tests/ANOVAs

What do we do in linear modeling?

- Fit a line (least squares method)

---
# Linear Modeling and *t*-tests/ANOVAs

Within a t-test/ANOVA framework we want to know if means differ between groups

```r
t.test() # test mean differences
```

<img src="More_LM_categorical_files/figure-html/unnamed-chunk-21-1.png" width="60%" style="display: block; margin: auto;" />
---
# Linear Modeling and t-tests/ANOVAs

- First, calculate `$SS_{total}$`

.pull-left[
<img src="More_LM_categorical_files/figure-html/unnamed-chunk-22-1.png" width="100%" style="display: block; margin: auto;" />
]
.pull-left[
<img src="More_LM_categorical_files/figure-html/unnamed-chunk-23-1.png" width="100%" style="display: block; margin: auto;" />
]
---
# Fit a line

- Find line using least squares method

- Mean is best line fit

<img src="More_LM_categorical_files/figure-html/unnamed-chunk-24-1.png" width="100%" style="display: block; margin: auto;" />
---
# Fit a line

<img src="More_LM_categorical_files/figure-html/unnamed-chunk-25-1.png" width="100%" style="display: block; margin: auto;" />
---
# Single Line

- We have two equations

- `$y = b_0$` = intercept(mean of smell)
  
  - `$y = b_0$` = intercept(mean of touch)
  
- How do we get one linear equation?
---
class:middle
# Dummy Coding/Treatment Coding
---
# Dummy Coding/Treatment Coding

- R's default system

- 0’s and 1’s, with reference level at intercept

- R does this automatically (0 = whatever comes first alphabetically)

- Smell = 0 
  
       - Taste = 1

`$$\operatorname{Val} = \alpha + \beta_{1}(X_i) + \epsilon$$`

`$X_i$` = Indicator of group (0 or 1)
---
# Dummy Coding/Treatment Coding

- Prediction for Smell `$X_i =0$`

`$$\operatorname{Val} = \alpha + \beta_{1}(\operatorname{Modality}_{\operatorname{Taste}}0) + \epsilon$$`
$$
\operatorname{\bar{Y}} = \alpha(Smell_{mean})
$$
- Prediction for Taste `$X_i =1$`

`$$\operatorname{Val} = \alpha + \beta_{1}(\operatorname{Modality}_{\operatorname{Taste}}1) + \epsilon$$`
`$$\operatorname{\bar{Y}{taste}} = \alpha + \beta_{1}(\operatorname{Modality}_{\operatorname{Taste}}1) + \epsilon$$`

---
# Dummy Coding By Hand

.pull-left[

```r
senses_dum <- senses_filt %>%
 mutate(mod=ifelse(Modality=="Smell", 0, 1))
```
]

.pull-right[

<img src="More_LM_categorical_files/figure-html/unnamed-chunk-27-1.png" width="100%" style="display: block; margin: auto;" />
]
---
# Categorical Contrast Coding

`$$slope=\frac{\mu_{diff}}{run}$$`

.pull-left[
 
 
<img src="More_LM_categorical_files/figure-html/unnamed-chunk-28-1.png" width="80%" style="display: block; margin: auto;" />
]

.pull-right[

<img src="twomean.JPG" width="100%" style="display: block; margin: auto;" />
]
---
# Dummy Coded Regression

```r
lm(Val ~ Modality, data=senses_filt)
```

- Intercept: `$\bar{Y} = b_0$`(Smell) = 5.47

- Slope ($b_1$): Valance of taste words are .337 higher

- One unit increase (going from 0 to 1; from Smell to Taste) associated with .337 increase in valance scores (mean difference)

- Adding intercept and slope together will give us mean valence of taste words

---
# Change Reference Level
.pull-left[

```r
senses_dum <- senses_filt %>%
 mutate(mod=factor(Modality)) %>% 
 mutate(mod1=relevel(mod, ref="Taste")) # relevel the var 
contrasts(senses_dum$mod1)
```

```
##       Smell
## Taste     0
## Smell     1
```

```r
contrasts(senses_dum$mod)
```

```
##       Taste
## Smell     0
## Taste     1
```
]
.pull-right[

```r
lm(Val ~ mod, data=senses_dum)
```

]

---
class:middle

# Effects Coding/Sum Coding

---
# Effects Coding/Sum Coding

- So far the intercept at 0 has referred to a particular baseline or reference level

- Centering (subtracting mean from each value) changes the intercept to correspond to the overall mean

- While mostly done for continuous variables, you can apply centering to categorical variables
---
# Effects Coding/Sum Coding (.5, -.5)

- Mean of dummy coded variable is = .5

- Subtract .5 from (0, 1) and we get *+ 0.5* and *-0.5*

- Y intercept is now the grand mean

`$$\frac{\mu_1 + \mu_2}{2}$$`
- Slope is still the difference

```r
senses_filt_sum <- senses_filt %>%
 mutate(mod_sum=as.factor(Modality), mod_sum_r = as.factor(Modality)) # add a new var to sum code

contrasts(senses_filt_sum$mod_sum) <- c(0.5, -0.5) # change 0 - 1 to +.5 and -.5
```

---
# Effects Coding Results

.pull-left[

```r
# reg regression
lm(Val ~ Modality, data=senses_filt)
```

]

.pull-right[

```r
# regression with sum coding (.5 - .5)
lm(Val ~ mod_sum, data=senses_filt_sum)
```
]

- Intercept is grand mean (mean of means): `$\hat \beta_0=5.64$`

- Mean of Smell: `$\hat\beta_0+\hat\beta_1\times .5=5.64-0.33(0.5)=5.47$`

- Mean of Taste: `$\hat\beta_0+\hat\beta_1\times -.5=5.64 + (-0.33)*-0.5=5.80$`

---
# Default Sum Coding R behavior

- +1 and -1

```r
senses_filt_sum <- senses_filt_sum %>%
 mutate(mod_sum55=ifelse(mod_sum=="Taste", .5, -.5), mod_sum11 = ifelse(mod_sum=="Taste", 1, -1)) # add a new var to sum code (1, -1)
```

---
# Sum Coding (-1 + 1) Interpretation

.pull-left[

What does this do to our interpretation?

- Intercept is now centered at 0 (grand mean)

- Slope rise is still the same (difference between categories) but:

-  Stepping from one category to another (the run) results in overall change of 2

- Results are halved
]

.pull-right[
 
 
<img src="More_LM_categorical_files/figure-html/unnamed-chunk-37-1.png" width="100%" style="display: block; margin: auto;" />

]
---
# Sum Coding (+1, -1) Model Results

.pull-left[

```r
lm(Val ~ mod_sum55, data=senses_filt_sum)
```
]
.pull-right[

```r
lm(Val ~ mod_sum11, data=senses_filt_sum)
```

- Intercept is grand mean: `$\hat\beta_0$`=5.64

- Mean of Smell:

- Mean of Taste:
]
---
# Why -0.5 and +0.5?

<img src="contrastcodes.bmp" width="100%" style="display: block; margin: auto;" />
---
# The General Linear F-Test

- Can test overall influence for 2 or more levels of a factor

- We can think about the hypotheses for the overall test being:

- `$H_0$`: We cannot predict the dependent variable (over and above a restricted model (only an intercept))
    
  - `$H_1$`: We can predict the dependent variable (over and above a model with only an intercept)

---
# Restricted Model

`$$Y_{ij} = \mu + \epsilon$$`
.pull-left[

- Restricted model (Intercept-only): each score `$Y_{ij}$` is the result of a single population mean plus random error

`$$SS_{error}(R)=\sum(y_i-\bar{y})^2=SS_{total}$$`
where:

`$y_i$` = observed value
`$\bar{y}$` = mean value 
]

.pull-right[
<img src="More_LM_categorical_files/figure-html/unnamed-chunk-41-1.png" width="80%" style="display: block; margin: auto;" />
]
---
# Full Model

`$$Y_{ij} = \mu_j + \epsilon$$`
.pull-left[
- Full model (all predictors/levels): each score `$Y_ij$` is the result of a different group mean plus random error

`$$SSE(F)=\sum(y_{ij}-\hat{y}_{ij})^2=SSE$$`
where:

`$i$` = Person
`$j$` = Group
`$y_i$` = Observed value
`$\hat{y}$` = Value estimated by regression line
]

.pull-right[
<img src="More_LM_categorical_files/figure-html/unnamed-chunk-42-1.png" width="80%" style="display: block; margin: auto;" />
]
---
# F-ratio

- F-ratio is measure of signal to noise

- Tells us if overall model is significant fit to the data

- `$H_0$`: We cannot predict the dependent variable (over and above a model with only an intercept)
    
    - `$H_1$`: We can predict the dependent variable (over and above a model with only an intercept)

`$$\begin{aligned} &&df_{R} = N - 1\\  &&df_{F} = N - a\\ &&SSE(R)=SS_{total}\\&&SSE(F)=SS_{error}\end{aligned}$$`

---
# F-ratio

`$$F = \frac{SS_{R}-SS_{F}/{df_{R}-df_{F}} (p-1)}{SS_{F}/df_F(N-p)} = \frac{MS_{model}}{MS_{error}}$$`

- If Full =  Restricted , then F=1

- If Full > Restricted, F > 1

- If Full < Restricted, F < 1

*Degrees of freedom: F(a-1, n - a)*

---
# Plotting Categorical Effects

- Boxplot

<img src="More_LM_categorical_files/figure-html/unnamed-chunk-43-1.png" width="100%" style="display: block; margin: auto;" />
---
# Violin Plots

.pull-left[

```r
mod=lm(Val ~ Modality, data=senses_filt)

means=modelbased::estimate_means(mod)

d=ggplot(senses_filt, aes(x = Modality, y = Val)) +
  # Add base data
  geom_violin(aes(fill = Modality), color = "white") +
  geom_jitter2(width = 0.05, alpha = 0.5) +

# Add pointrange and line from means
  geom_line(data = means, aes(y = Mean, group = 1), size = 1) +
  geom_pointrange(
    data = means,
    aes(y = Mean, ymin = CI_low, ymax = CI_high),
    size = 1,
    color = "white"
  ) +
  # Improve colors
  scale_fill_material() +
  theme_modern()
```
]

.pull-right[

<img src="More_LM_categorical_files/figure-html/unnamed-chunk-45-1.png" width="100%" style="display: block; margin: auto;" />
]

---
# Activity

Mental Health and Drug Use

- CESD: Depressions scores
- unemp: 1=employed 0=unemployed

```r
d <- read_csv("https://raw.githubusercontent.com/ASKurz/Applied-Longitudinal-Data-Analysis-with-brms-and-the-tidyverse/master/data/unemployment_pp.csv")
```
---
# Activity

1. Change `unemp` variable to a factor with categorical labels

2. Dummy code the unemployment variable

3. Contrast code the unemployment variable

3. Run `lm` on the dummy coded variable

6. Interpret the output

7. Use output to extract the means for each group only using the output

8. Plot the results
---
# Linear Models with Multiple Levels

- So far we have only been looking at two levels

- We easily can extend linear modeling approach to multiple levels

- Let's go back to our sense data

- Before filtering it down to 2 senses it had 5 senses!

```r
senses<- read_csv("data/winter_2016_senses_valence.csv")

glimpse(senses)
```

```
## Rows: 405
## Columns: 3
## $ Word <chr> "abrasive", "absorbent", "aching", "acidic", "acrid", "adhesi…
## $ Modality <chr> "Touch", "Sight", "Touch", "Taste", "Smell", "Touch", "Taste"…
## $ Val <dbl> 5.398113, 5.876667, 5.233370, 5.539592, 5.173947, 5.240000, 5…
```
---
Treatment/Dummy Coding: Multilevel Factors

```r
lm(Val~Modality, data=senses) %>%
  tidy()
```

- What is going on here? There are only 4 levels, but we actually have 5 levels.

---
# Dummy Coding Extension

.pull-left[
1. Create one fewer dummy codes than levels (K (number of levels)-1)

2. Choose one of your levels as baseline and assign all zeros for this level across each dummy code

3. For first dummy code, assign 1 to first group and 0s for rest of levels

4. For the second dummy code, assign 1 to second group and 0s for rest of levels

5. For third dummy code, assign 1 to third group and 0s for rest of levels

6. For fourth dummy code, assign 1 to fourth group and 0s for rest of levels
]

.pull-right[

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---
# Linear Equation

`$$\operatorname{Val} = \alpha + \beta_{1}(\operatorname{Modality}_{\operatorname{Smell}}) + \beta_{2}(\operatorname{Modality}_{\operatorname{Sound}}) + \beta_{3}(\operatorname{Modality}_{\operatorname{Taste}}) + \beta_{4}(\operatorname{Modality}_{\operatorname{Touch}}) + \epsilon$$`

---
# Hello Again Sums of Squares

<img src="ss_aov.bmp" width="40%" style="display: block; margin: auto;" />
---
# The General Linear F-Test
 
- We can think about the hypotheses for the overall test being:

`$$H_0: b_1 = 0$$`
         
`$$H_0: \mu_1 = \mu_2 = \mu_3 = \mu_4 = \mu_5$$`

`$$H_1:  b_1  \neq b_2 \neq b_3 \neq b_4 \neq b_5$$`
`$$H_1:  \mu_1  \neq \mu_2 \neq \mu_3 \neq \mu_4 \neq \mu_5$$`
- *Analysis of Variance (ANOVA)*
---
# Restricted Model

`$$Y_{ij} = \mu + \epsilon$$`
.pull-left[

- Restricted model (Intercept-only): each score `$Y_{ij}$` is the result of a single population mean plus random error

`$$SS_{error}(R)=\sum(y_i-\bar{y})^2=SS_{total}$$`

where:

`$y_i$` = observed value
`$\bar{y}$` = mean value 
]

.pull-right[
<img src="More_LM_categorical_files/figure-html/unnamed-chunk-52-1.png" width="100%" style="display: block; margin: auto;" />
]
---
# Full Model

`$$Y_{ij} = \mu_j + \epsilon$$`
.pull-left[
- Full model (all predictors/levels): each score `$Y_ij$` is the result of a different group mean plus random error

`$$SSE(F)=\sum(y_{ij}-\hat{y}_{ij})^2=SSE$$`
where:

`$i$` = Person
`$j$` = Group
`$y_i$` = Observed value
`$\hat{y}$` = Value estimated by regression line
]

.pull-right[
<img src="More_LM_categorical_files/figure-html/unnamed-chunk-53-1.png" width="100%" style="display: block; margin: auto;" />
]
---
# F-ratio

- F-ratio is measure of signal to noise

`$$\begin{aligned} &&df_{R} = N - 1\\  &&df_{F} = N - p\\ &&SSE(R)=SS_{total}\\&&SSE(F)=SS_{error}\end{aligned}$$`

---
# F-ratio

`$$F = \frac{SS_{R}-SS_{F}/{df_{R}-df_{F}} (p-1)}{SS_{F}/df_F(N-p)} = \frac{MS_{model}}{MS_{error}}$$`

- If Full =  Restricted , then F=1

- If Full > Restricted, F > 1

- If Full < Restricted, F < 1

*Degrees of freedom: F(p-1, n - p)*

---
# ANOVA Table

```r
lm(Val~Modality, data=senses) %>%
  parameters::model_parameters()
```

<table class="huxtable" style="border-collapse: collapse; border: 0px; margin-bottom: 2em; margin-top: 2em; ; margin-left: auto; margin-right: auto; " id="tab:unnamed-chunk-54">
<col><col><col><col><col><col><col><col><col><tr>
<th style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0.4pt 0.4pt; padding: 6pt 6pt 6pt 6pt; font-weight: bold;">Parameter</th><th style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0.4pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: bold;">Coefficient</th><th style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0.4pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: bold;">SE</th><th style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0.4pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: bold;">CI</th><th style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0.4pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: bold;">CI_low</th><th style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0.4pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: bold;">CI_high</th><th style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0.4pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: bold;">t</th><th style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0.4pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: bold;">df_error</th><th style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0.4pt 0.4pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: bold;">p</th></tr>
<tr>
<td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0.4pt; padding: 6pt 6pt 6pt 6pt; background-color: rgb(242, 242, 242); font-weight: normal;">(Intercept)</td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt; padding: 6pt 6pt 6pt 6pt; background-color: rgb(242, 242, 242); font-weight: normal;">5.58  </td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt; padding: 6pt 6pt 6pt 6pt; background-color: rgb(242, 242, 242); font-weight: normal;">0.0189</td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt; padding: 6pt 6pt 6pt 6pt; background-color: rgb(242, 242, 242); font-weight: normal;">0.95</td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt; padding: 6pt 6pt 6pt 6pt; background-color: rgb(242, 242, 242); font-weight: normal;">5.54 </td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt; padding: 6pt 6pt 6pt 6pt; background-color: rgb(242, 242, 242); font-weight: normal;">5.62   </td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt; padding: 6pt 6pt 6pt 6pt; background-color: rgb(242, 242, 242); font-weight: normal;">295   </td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt; padding: 6pt 6pt 6pt 6pt; background-color: rgb(242, 242, 242); font-weight: normal;">400</td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0.4pt 0pt 0pt; padding: 6pt 6pt 6pt 6pt; background-color: rgb(242, 242, 242); font-weight: normal;">0       </td></tr>
<tr>
<td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0.4pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">ModalitySmell</td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">-0.109 </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.0564</td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.95</td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">-0.22 </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.00229</td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">-1.93</td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">400</td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0.4pt 0pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.0549  </td></tr>
<tr>
<td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0.4pt; padding: 6pt 6pt 6pt 6pt; background-color: rgb(242, 242, 242); font-weight: normal;">ModalitySound</td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; background-color: rgb(242, 242, 242); font-weight: normal;">-0.174 </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; background-color: rgb(242, 242, 242); font-weight: normal;">0.0376</td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; background-color: rgb(242, 242, 242); font-weight: normal;">0.95</td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; background-color: rgb(242, 242, 242); font-weight: normal;">-0.248</td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; background-color: rgb(242, 242, 242); font-weight: normal;">-0.101  </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; background-color: rgb(242, 242, 242); font-weight: normal;">-4.64</td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; background-color: rgb(242, 242, 242); font-weight: normal;">400</td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0.4pt 0pt 0pt; padding: 6pt 6pt 6pt 6pt; background-color: rgb(242, 242, 242); font-weight: normal;">4.66e-06</td></tr>
<tr>
<td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0.4pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">ModalityTaste</td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.228 </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.0431</td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.95</td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.144</td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.313  </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">5.3 </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">400</td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0.4pt 0pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">1.96e-07</td></tr>
<tr>
<td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0.4pt; padding: 6pt 6pt 6pt 6pt; background-color: rgb(242, 242, 242); font-weight: normal;">ModalityTouch</td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt; padding: 6pt 6pt 6pt 6pt; background-color: rgb(242, 242, 242); font-weight: normal;">-0.0452</td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt; padding: 6pt 6pt 6pt 6pt; background-color: rgb(242, 242, 242); font-weight: normal;">0.0374</td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt; padding: 6pt 6pt 6pt 6pt; background-color: rgb(242, 242, 242); font-weight: normal;">0.95</td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt; padding: 6pt 6pt 6pt 6pt; background-color: rgb(242, 242, 242); font-weight: normal;">-0.119</td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt; padding: 6pt 6pt 6pt 6pt; background-color: rgb(242, 242, 242); font-weight: normal;">0.0282 </td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt; padding: 6pt 6pt 6pt 6pt; background-color: rgb(242, 242, 242); font-weight: normal;">-1.21</td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt; padding: 6pt 6pt 6pt 6pt; background-color: rgb(242, 242, 242); font-weight: normal;">400</td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0.4pt 0.4pt 0pt; padding: 6pt 6pt 6pt 6pt; background-color: rgb(242, 242, 242); font-weight: normal;">0.227   </td></tr>
</table>

---
# ANOVA Table

```r
aov1<-aov(Val~ 1, data=senses)
aov2 <- aov(Val~Modality, data=senses)
#anova(aov1, aov2) compare two models 
aov(Val~Modality, data=senses) %>%
 parameters::model_parameters()
```

---
# Model Comparison Approach vs Traditional Approach to ANOVA

- Traditional formulation of ANOVA asks the same question in a different way:

- Is the variability between groups (variance due to differences between groups)
greater than expected on the basis of the within-group variability (the variability
within a group) observed, and random sampling of group members?

- Both use sum of squares
- Both use F-statistic
- F = `$\frac{MSR}{MSE}$` (Same Mean Squared Error on ANOVA table outputs)
---
# Post-Hoc Comparisons

```r
aov_em=aov(Val~Modality, data=senses) # fit ANOVA

aov_em %>%
parameters::model_parameters(.) # print out ANOVA table
```

- The Modality factor is significant. Now what?

---
# Post-Hoc Comparisons

- The best package ever created: `emmeans`

- Allows one to extract marginal means for the model and also test comparisons of interest

---
# Pairwise Tests

- Get means and pairwise comparisons

.pull-left[

```r
# get pairwise tests between all groups

as.data.frame(emmeans::emmeans(aov_em, specs = "Modality")) %>%
flextable()
```

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Smell</td><td class="cl-fed68e70">0.10865148</td><td class="cl-fed68e70">0.05643072</td><td class="cl-fed68e70">400</td><td class="cl-fed68e70">1.925396</td><td class="cl-fed68e70">0.3055011645361</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-fed68e7a">Sight - Sound</td><td class="cl-fed68e70">0.17447036</td><td class="cl-fed68e70">0.03757671</td><td class="cl-fed68e70">400</td><td class="cl-fed68e70">4.643046</td><td class="cl-fed68e70">0.0000456942086</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-fed68e7a">Sight - Taste</td><td class="cl-fed68e70">-0.22846083</td><td class="cl-fed68e70">0.04313872</td><td class="cl-fed68e70">400</td><td class="cl-fed68e70">-5.295957</td><td class="cl-fed68e70">0.0000019440745</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-fed68e7a">Sight - Touch</td><td class="cl-fed68e70">0.04522812</td><td class="cl-fed68e70">0.03736969</td><td class="cl-fed68e70">400</td><td class="cl-fed68e70">1.210289</td><td class="cl-fed68e70">0.7454337311622</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-fed68e7a">Smell - Sound</td><td class="cl-fed68e70">0.06581888</td><td class="cl-fed68e70">0.06230922</td><td class="cl-fed68e70">400</td><td class="cl-fed68e70">1.056327</td><td class="cl-fed68e70">0.8286561493660</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-fed68e7a">Smell - Taste</td><td class="cl-fed68e70">-0.33711231</td><td class="cl-fed68e70">0.06581321</td><td class="cl-fed68e70">400</td><td class="cl-fed68e70">-5.122259</td><td class="cl-fed68e70">0.0000046594174</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-fed68e7a">Smell - Touch</td><td class="cl-fed68e70">-0.06342336</td><td class="cl-fed68e70">0.06218459</td><td class="cl-fed68e70">400</td><td class="cl-fed68e70">-1.019921</td><td class="cl-fed68e70">0.8461335445944</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-fed68e7a">Sound - Taste</td><td class="cl-fed68e70">-0.40293120</td><td class="cl-fed68e70">0.05058618</td><td class="cl-fed68e70">400</td><td class="cl-fed68e70">-7.965243</td><td class="cl-fed68e70">0.0000000000000</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-fed68e7a">Sound - Touch</td><td class="cl-fed68e70">-0.12924225</td><td class="cl-fed68e70">0.04576577</td><td class="cl-fed68e70">400</td><td class="cl-fed68e70">-2.823993</td><td class="cl-fed68e70">0.0397092275599</td></tr><tr style="overflow-wrap:break-word;"><td class="cl-fed68e85">Taste - Touch</td><td class="cl-fed68e84">0.27368895</td><td class="cl-fed68e84">0.05043260</td><td class="cl-fed68e84">400</td><td class="cl-fed68e84">5.426827</td><td class="cl-fed68e84">0.0000009903166</td></tr></tbody></table></div></template>
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---
# `afex` ANOVA package

```r
library(afex)

one_fit <- aov_ez("Word", "Val", data = senses, between = c("Modality"))

afex_plot(one_fit, x="Modality")
```

<img src="More_LM_categorical_files/figure-html/unnamed-chunk-59-1.png" width="100%" style="display: block; margin: auto;" />
---
# Assumptions

- Within linear modeling framework, do normal assumptions checks

```r
check_model()
```

<img src="More_LM_categorical_files/figure-html/unnamed-chunk-61-1.png" width="100%" height="50%" style="display: block; margin: auto;" />
---
# Effect Sizes: eta

- `$\eta^2$`

> Interpretation: % of variance explained

`$$\eta^2 = \frac{SS_{model}}{SS_{total}}$$`

- `$\eta^2$` cannot easily be compared between studies, because the total variability in a study ($SS_{total}$) depends on the design of a study, and increases when additional variables are manipulated

- .01: Small
- .06: Medium
- .14: Large
---
# Effect Sizes:eta

- `$\eta_p^2$`

> Interpretation: % of variance explained for one effect (partailing out others)

`$$\eta_p^2 = \frac{SS_{model}}{SS_{model} + SS_{error}}$$`
---
# Effect Sizes: eta

- `$\eta^2_g$` - Generalized eta-sqaured

`$$\frac{SS_{model}}{SS_{model} + SS_{subject} + SS_{error}}$$`

---
# Less Biased Effect Size: Omega

- `$\omega^2$`

`$$\omega^2 = \frac{SS_{model} - df_{model}\cdot MS_{error}}{SS_{total} + MS_{error}}$$`
- `$\omega_p^2$`

`$$\frac{df_{model} \times (MS_{model} - MS_{error})}{SS_{model} + (N - df_{model}) \times MS_{error}}$$`

- .01: Small
- .06: Medium
- .14: Large
---
# Effect Size: Cohen's f

`$$\text{Cohen's} f_p =
\sqrt{\frac{\eta^2_p}{1-\eta^2_p}} =
\sqrt{\frac{SS_{effect}}{SS_{error}}}$$`

- .14: Small
- .39: Medium
- .59: Large

https://imaging.mrc-cbu.cam.ac.uk/statswiki/FAQ/effectSize
---
# Caculate ANOVA Effect Size in R

```r
cohens_f(aov_em)
```

```r
eta_squared(aov_em)
```

```r
omega_squared(aov_em)
```

---
# Power

- We can also run power analyses for omnibus tests (e.g., number of participants needed  to find a sufficiently powered main effect or interaction)

- May not sufficiently power one for the smallest desired effect size of interest
  
    - Recommendation: Perform on smallest desired effect size (e.g., mean comparison  while controlling for multiple corrections)
  
- Often complex tests cannot be performed analytically and you must use numerical  methods

- Same approach we have already done!

---
# Power Analysis: ANOVA

- You are planning a reaction-time study involving three groups (k = 3)
- Pilot research & data from literature suggest effect size is medium `$f$` = .39
- Suppose you want a power of 0.9
- How many subjects do you need in each sample group?
---
# Power Analysis: ANOVA

```r
library(pwr)

pwr.anova.test(k=3,n=NULL,f=.39,sig.level=0.05,power=0.9)
```

```
## 
##      Balanced one-way analysis of variance power calculation 
## 
##               k = 3
##               n = 28.75626
##               f = 0.39
##       sig.level = 0.05
##           power = 0.9
## 
## NOTE: n is number in each group
```

```r
#k = groups
#n= sample size
#es = cohen's f
```

---
# Superpower

- Same RT study

- Pilot research & data from literature suggest population means might be 400, 450  and 500 ms with a sample within-group standard deviation of 100 ms

- Suppose you want a power of 0.9

- How many subjects do you need in each sample group?

---
# Non-Parametric

.pull-left[
- Kruskal Wallis Test

- Can be used if assumptions are not met

- Extension of Mann-Whitney test
      
    
    ```r
    kruskal.test(Val ~ Modality, data=senses)
    ```
    
    ```
    ## 
    ## 	Kruskal-Wallis rank sum test
    ## 
    ## data:  Val by Modality
    ## Kruskal-Wallis chi-squared = 57.815, df = 4, p-value = 8.344e-12
    ```
]

.pull-right[
- Welch's F test (W-test)

```r
library(onewaytests)

welch.test(Val ~ Modality, data = senses)
```

```
## 
##   Welch's Heteroscedastic F Test (alpha = 0.05) 
## ------------------------------------------------------------- 
##   data : Val and Modality 
## 
##   statistic  : 13.01914 
##   num df     : 4 
##   denom df   : 101.1231 
##   p.value    : 1.374323e-08 
## 
##   Result     : Difference is statistically significant. 
## -------------------------------------------------------------
```
]