Comparing means adjusted for other predictors (analysis of covariance)

# Comparing means adjusted for other predictors (analysis of covariance)
## Professor Andy Field

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???
Music: Primus: Too many puppies

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---
class: center

![:scale 83%](media/spine_content_map.png)
???

We've seen this map of the process of fitting models before

---
class: center

![:scale 83%](media/spine_lec_04.png)

???

Today we focus back on the model itself to look at the form of the model we're fitting. The faded stuff still applies though - we'll look at bias, robust models, and of course samples and estimation and so on. But they are the same as for other models, what is different is the form of the model we're fitting.

---
#  Learning outcomes

* Explain how to compare means adjusting for other predictors using a linear model
  + Linear model with a categorical and continuous predictor
  + a.k.a. analysis of covariance (ANCOVA)

* Type I vs. Type III sums of squares

* Interpreting the model
  + Main effects
  + Covariates

---
# When and why
## Generally

* To test for differences between group means when we know that an extraneous variable affects the outcome variable

* Used to adjust the means for extraneous and confounding variables

## In experimental research

* Reduce error variance (sometimes)
  + By explaining some of the unexplained variance (SS<sub>R</sub>) the error variance in the model can be reduced

* Greater experimental control
  + By adjusting for known confounds, we can gain greater insight into the effect of the predictor variable(s)

---
background-image: url("media/milton_bed_crop_2018.jpg")
background-size: cover
class: middle

* A puppy therapy RCT
  + A no puppies control group 
  + 15 minutes of puppy therapy
  + 30 minutes of puppy therapy

* Outcome variable
  + Happiness (0 = unhappy to 10 = happy).

* Covariate
  + Love of puppies (0 = no love, 7 = all the love)
]

---
class: center
# The statistical model

![:scale 75%](media/ancova_statistical_model_01.png)

---
class: center
# The statistical model

![:scale 75%](media/ancova_statistical_model_02.png)

<br>

.ong[.center[.eq_lrge[
$$
`\begin{aligned}
\text{Happiness}_i &= \hat{b}_0 + \hat{b}_1\text{Long}_i + \hat{b}_2\text{Short}_i + e_i
\end{aligned}`
$$
]]]

---
class: center
# The statistical model

![:scale 75%](media/ancova_statistical_model_03.png)

<br>

.ong[.center[.eq_lrge[
$$
`\begin{aligned}
\text{Happiness}_i &= \hat{b}_0 + \hat{b}_1\text{Long}_i + \hat{b}_2\text{Short}_i + \hat{b}_3\text{Puppy love}_i + e_i
\end{aligned}`
$$
]]]

---
class: center
# Partitioning variance

![:scale 75%](media/dsr2_fig_12_03_independence_of_cov_and_treatment_01.png)
---
class: center
# Partitioning variance

![:scale 75%](media/dsr2_fig_12_03_independence_of_cov_and_treatment_02.png)
---
class: center
# Partitioning variance

![:scale 75%](media/dsr2_fig_12_03_independence_of_cov_and_treatment_03.png)
---
background-image: url("media/milton_20180308_202134.JPG")
background-size: cover
class: inverse

# The data

<div class="datatables html-widget html-fill-item-overflow-hidden html-fill-item" id="htmlwidget-08e9b5c542228cb86ba1" style="width:100%;height:auto;"></div>
<script type="application/json" data-for="htmlwidget-08e9b5c542228cb86ba1">{"x":{"filter":"none","vertical":false,"caption":"<caption>Table 1: Data for the puppy therapy example<\/caption>","data":[["1","2","3","4","5","6","7","8","9","10","11","12","13","14","15","16","17","18","19","20","21","22","23","24","25","26","27","28","29","30"],["72kfu0","2l47nl","uev2x2","fbnym1","i3d8d1","lg6i36","22o7c9","5223ie","v596m4","w36156","rdqvx6","vm94r0","nd241u","8i423t","mxjl12","wnjf54","26nx21","8486wm","46ijm4","2i965f","wrl5w6","764i23","6qgv7q","41kl44","rtm226","u3g6b2","60yb73","o4j585","genuvm","4sx8wg"],["No puppies","No puppies","No puppies","No puppies","No puppies","No puppies","No puppies","No puppies","No puppies","15 mins","15 mins","15 mins","15 mins","15 mins","15 mins","15 mins","15 mins","30 mins","30 mins","30 mins","30 mins","30 mins","30 mins","30 mins","30 mins","30 mins","30 mins","30 mins","30 mins","30 mins"],[3,2,5,2,2,2,7,2,4,7,5,3,4,4,7,5,4,9,2,6,3,4,4,4,6,4,6,2,8,5],[4,1,5,1,2,2,7,4,5,5,3,1,2,2,6,4,2,1,3,5,4,3,3,2,0,1,3,0,1,0]],"container":"<table class=\"display\">\n  <thead>\n    <tr>\n      <th> <\/th>\n      <th>id<\/th>\n      <th>dose<\/th>\n      <th>happiness<\/th>\n      <th>puppy_love<\/th>\n    <\/tr>\n  <\/thead>\n<\/table>","options":{"dom":"tp","columnDefs":[{"className":"dt-center","targets":[1,2,3]},{"className":"dt-right","targets":4},{"orderable":false,"targets":0}],"pageLength":10,"order":[],"autoWidth":false,"orderClasses":false}},"evals":[],"jsHooks":[]}</script>

---
# Data summary
## By group

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> Therapy group </th>
   <th style="text-align:center;"> Mean (happiness) </th>
   <th style="text-align:center;"> SD (happiness) </th>
   <th style="text-align:center;"> Mean (puppy love) </th>
   <th style="text-align:center;"> SD (puppy love) </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> No puppies </td>
   <td style="text-align:center;"> 3.22 </td>
   <td style="text-align:center;"> 1.79 </td>
   <td style="text-align:center;"> 3.44 </td>
   <td style="text-align:center;"> 2.07 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 15 mins </td>
   <td style="text-align:center;"> 4.88 </td>
   <td style="text-align:center;"> 1.46 </td>
   <td style="text-align:center;"> 3.12 </td>
   <td style="text-align:center;"> 1.73 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 30 mins </td>
   <td style="text-align:center;"> 4.85 </td>
   <td style="text-align:center;"> 2.12 </td>
   <td style="text-align:center;"> 2.00 </td>
   <td style="text-align:center;"> 1.63 </td>
  </tr>
</tbody>
</table>

## Overall

<table>
 <thead>
  <tr>
   <th style="text-align:center;"> Mean (happiness) </th>
   <th style="text-align:center;"> SD (happiness) </th>
   <th style="text-align:center;"> Mean (puppy love) </th>
   <th style="text-align:center;"> SD (puppy love) </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:center;"> 4.37 </td>
   <td style="text-align:center;"> 1.96 </td>
   <td style="text-align:center;"> 2.73 </td>
   <td style="text-align:center;"> 1.86 </td>
  </tr>
</tbody>
</table>

---
background-image: none
background-color: #000000
class: no-scribble

---
background-image: url("media/milton_20180606_113722_crop.jpg")
background-size: cover

# The model

.whitebox[.center[.eq_lrge[
$$
`\begin{aligned}
\hat{\text{Happiness}}_i &= \hat{b}_0 + \hat{b}_1\text{Long}_i + \hat{b}_2\text{Short}_i +  \hat{b}_3\text{Puppy love}_i
\end{aligned}`
$$
]]]

<br>
--

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> Therapy group </th>
   <th style="text-align:center;"> Long (30 mins vs. no puppies) </th>
   <th style="text-align:center;"> Short 1 (15 mins vs. no puppies) </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;background-color: white !important;"> No Puppies </td>
   <td style="text-align:center;background-color: white !important;"> 0 </td>
   <td style="text-align:center;background-color: white !important;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;background-color: white !important;"> 15 mins </td>
   <td style="text-align:center;background-color: white !important;"> 0 </td>
   <td style="text-align:center;background-color: white !important;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;background-color: white !important;"> 30 mins </td>
   <td style="text-align:center;background-color: white !important;"> 1 </td>
   <td style="text-align:center;background-color: white !important;"> 0 </td>
  </tr>
</tbody>
</table>

---
# What you'd expect the dummy variables to represent

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> Therapy group </th>
   <th style="text-align:center;"> Mean (happiness) </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> No puppies </td>
   <td style="text-align:center;"> 3.22 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 15 mins </td>
   <td style="text-align:center;"> 4.88 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 30 mins </td>
   <td style="text-align:center;"> 4.85 </td>
  </tr>
</tbody>
</table>

--
<br>
.ong[.eq_lrge[
$$
`\begin{aligned}
\hat{b}_0 &= \bar{X}_\text{No puppies} = 3.22 \\
\hat{b}_1 &= 4.85-3.22 = 1.63 \\
\hat{b}_2 &= 4.88-3.22 = 1.66 
\end{aligned}`
$$
]]

---
# What you actually get

```r
pupluv_lm <- lm(happiness ~ puppy_love + dose, data = pupluv_tib)
broom::tidy(pupluv_lm, conf.int = TRUE)
```

<br>
--

.center[
<table>
 <thead>
  <tr>
   <th style="text-align:left;"> term </th>
   <th style="text-align:right;"> estimate </th>
   <th style="text-align:right;"> std.error </th>
   <th style="text-align:right;"> statistic </th>
   <th style="text-align:right;"> p.value </th>
   <th style="text-align:right;"> conf.low </th>
   <th style="text-align:right;"> conf.high </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> (Intercept) </td>
   <td style="text-align:right;background-color: yellow !important;"> 1.789 </td>
   <td style="text-align:right;"> 0.867 </td>
   <td style="text-align:right;"> 2.063 </td>
   <td style="text-align:right;"> 0.049 </td>
   <td style="text-align:right;"> 0.007 </td>
   <td style="text-align:right;"> 3.572 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> puppy_love </td>
   <td style="text-align:right;background-color: yellow !important;"> 0.416 </td>
   <td style="text-align:right;"> 0.187 </td>
   <td style="text-align:right;"> 2.227 </td>
   <td style="text-align:right;"> 0.035 </td>
   <td style="text-align:right;"> 0.032 </td>
   <td style="text-align:right;"> 0.800 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> dose15 mins </td>
   <td style="text-align:right;background-color: yellow !important;"> 1.786 </td>
   <td style="text-align:right;"> 0.849 </td>
   <td style="text-align:right;"> 2.102 </td>
   <td style="text-align:right;"> 0.045 </td>
   <td style="text-align:right;"> 0.040 </td>
   <td style="text-align:right;"> 3.532 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> dose30 mins </td>
   <td style="text-align:right;background-color: yellow !important;"> 2.225 </td>
   <td style="text-align:right;"> 0.803 </td>
   <td style="text-align:right;"> 2.771 </td>
   <td style="text-align:right;"> 0.010 </td>
   <td style="text-align:right;"> 0.575 </td>
   <td style="text-align:right;"> 3.875 </td>
  </tr>
</tbody>
</table>
]

--
<br>
.center[.ong[
$$
`\begin{aligned}
\hat{\text{Happiness}}_i &= 1.789 + 2.225\text{ Long}_i + 1.786\text{ Short}_i + 0.416\text{ Puppy love}_i
\end{aligned}`
$$
]]

---
# Adjusting means

.center[.ong[
$$
`\begin{aligned}
\hat{\text{Happiness}}_i &= 1.789 + 2.225\text{ Long}_i + 1.786\text{ Short}_i + 0.416\text{ Puppy love}_i
\end{aligned}`
$$
]]

<br>

## Control group

.center[.ong[
$$
`\begin{aligned}
\hat{\text{Happiness}}_i &= 1.789 + 2.225\text{ Long}_i + 1.786\text{ Short}_i +  0.416\text{ Puppy love}_i \\
&= 1.789 + (2.225\times0) + (1.786\times0) +  (0.416\text{ Puppy love}_i)  \\
&= 1.789 +  (0.416\times\bar{X}_\text{Puppy love})  \\
&= 1.789 +  (0.416\times2.73)  \\
&= 2.925
\end{aligned}`
$$
]]

---
# Adjusting means

## 15 minute group

.center[.ong[
$$
`\begin{aligned}
\hat{\text{Happiness}}_i &= 1.789 + 2.225\text{ Long}_i + 1.786\text{ Short}_i +  0.416\text{ Puppy love}_i \\
&= 1.789 + (2.225\times0) + (1.786\times1) +  (0.416\text{ Puppy love}_i) \\
&= 1.789 + 1.786 + (0.416\times2.73) \\
&= 4.71
\end{aligned}`
$$
]]

## 30 minute group

.center[.ong[
$$
`\begin{aligned}
\hat{\text{Happiness}}_i &= 1.789 + 2.225\text{ Long}_i + 1.786\text{ Short}_i +  0.416\text{ Puppy love}_i \\
&= 1.789 + (2.225\times1) + (1.786\times0) +  (0.416\text{ Puppy love}_i) \\
&= 1.789 + 2.225 + (0.416\times2.73) \\
&= 5.15
\end{aligned}`
$$
]]

---

# The unadjusted model

.ong[.center[
$$
`\begin{aligned}
\hat{\text{Happiness}}_i &= \hat{b}_0 + \hat{b}_1\text{Long}_i + \hat{b}_2\text{Short}_i
\end{aligned}`
$$
]]

![](adjusting_means_files/figure-html/unnamed-chunk-40-1.png)

---
class: center

# The unadjusted model

.ong[.center[
$$
`\begin{aligned}
\hat{\text{Happiness}}_i &= \hat{b}_0 + \hat{b}_1\text{Long}_i + \hat{b}_2\text{Short}_i
\end{aligned}`
$$
]]

![](adjusting_means_files/figure-html/unnamed-chunk-41-1.png)

---
class: center

# The unadjusted model

.ong[.center[
$$
`\begin{aligned}
\hat{\text{Happiness}}_i &= \hat{b}_0 + \hat{b}_1\text{Long}_i + \hat{b}_2\text{Short}_i
\end{aligned}`
$$
]]

![](adjusting_means_files/figure-html/unnamed-chunk-42-1.png)

---
class: center
# The unadjusted model

.ong[.center[
$$
`\begin{aligned}
\hat{\text{Happiness}}_i &= \hat{b}_0 + \hat{b}_1\text{Long}_i + \hat{b}_2\text{Short}_i
\end{aligned}`
$$
]]

![](adjusting_means_files/figure-html/unnamed-chunk-43-1.png)

---
class: center
# The adjusted model

.ong[.center[
$$
`\begin{aligned}
\hat{\text{Happiness}}_i &= \hat{b}_0 + \hat{b}_1\text{Long}_i + \hat{b}_2\text{Short}_i +  \hat{b}_3\text{Puppy love}_i
\end{aligned}`
$$
]]

![](adjusting_means_files/figure-html/unnamed-chunk-44-1.png)

---
class: center
# The adjusted model

.ong[.center[
$$
`\begin{aligned}
\hat{\text{Happiness}}_i &= \hat{b}_0 + \hat{b}_1\text{Long}_i + \hat{b}_2\text{Short}_i +  \hat{b}_3\text{Puppy love}_i
\end{aligned}`
$$
]]

![](adjusting_means_files/figure-html/unnamed-chunk-45-1.png)
---
# Model parameters

.pull-left[
<table>
 <thead>
  <tr>
   <th style="text-align:left;"> Therapy group </th>
   <th style="text-align:center;"> Mean (happiness) </th>
   <th style="text-align:center;"> Adjusted Mean (happiness) </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> No puppies </td>
   <td style="text-align:center;"> 3.22 </td>
   <td style="text-align:center;"> 2.93 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 15 mins </td>
   <td style="text-align:center;"> 4.88 </td>
   <td style="text-align:center;"> 4.71 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 30 mins </td>
   <td style="text-align:center;"> 4.85 </td>
   <td style="text-align:center;"> 5.15 </td>
  </tr>
</tbody>
</table>
]

.pull-right[
.ong[.eq_lrge[
$$
`\begin{aligned}
\hat{b}_1 &= 5.15-2.93 = 2.22 \\
\hat{b}_2 &= 4.71-2.93 = 1.78 
\end{aligned}`
$$
]]
]

<br>

.center[
<table>
 <thead>
  <tr>
   <th style="text-align:left;"> term </th>
   <th style="text-align:right;"> estimate </th>
   <th style="text-align:right;"> std.error </th>
   <th style="text-align:right;"> statistic </th>
   <th style="text-align:right;"> p.value </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;background-color: white !important;"> (Intercept) </td>
   <td style="text-align:right;background-color: yellow !important;background-color: white !important;"> 1.789 </td>
   <td style="text-align:right;background-color: white !important;"> 0.867 </td>
   <td style="text-align:right;background-color: white !important;"> 2.063 </td>
   <td style="text-align:right;background-color: white !important;"> 0.049 </td>
  </tr>
  <tr>
   <td style="text-align:left;background-color: white !important;"> puppy_love </td>
   <td style="text-align:right;background-color: yellow !important;background-color: white !important;"> 0.416 </td>
   <td style="text-align:right;background-color: white !important;"> 0.187 </td>
   <td style="text-align:right;background-color: white !important;"> 2.227 </td>
   <td style="text-align:right;background-color: white !important;"> 0.035 </td>
  </tr>
  <tr>
   <td style="text-align:left;background-color: rgba(19, 108, 185, 0.4) !important;"> dose15 mins </td>
   <td style="text-align:right;background-color: rgba(19, 108, 185, 0.4) !important;background-color: yellow !important;"> 1.786 </td>
   <td style="text-align:right;background-color: rgba(19, 108, 185, 0.4) !important;"> 0.849 </td>
   <td style="text-align:right;background-color: rgba(19, 108, 185, 0.4) !important;"> 2.102 </td>
   <td style="text-align:right;background-color: rgba(19, 108, 185, 0.4) !important;"> 0.045 </td>
  </tr>
  <tr>
   <td style="text-align:left;background-color: rgba(19, 108, 185, 0.4) !important;"> dose30 mins </td>
   <td style="text-align:right;background-color: rgba(19, 108, 185, 0.4) !important;background-color: yellow !important;"> 2.225 </td>
   <td style="text-align:right;background-color: rgba(19, 108, 185, 0.4) !important;"> 0.803 </td>
   <td style="text-align:right;background-color: rgba(19, 108, 185, 0.4) !important;"> 2.771 </td>
   <td style="text-align:right;background-color: rgba(19, 108, 185, 0.4) !important;"> 0.010 </td>
  </tr>
</tbody>
</table>
]

---
background-image: none
background-color: #000000
class: no-scribble

<video width="100%" height="100%" controls id="my_video">
    <source src="media/milton_even_prettier_as_a_puppy.mp4" type="video/mp4">
</video>
---
# The *F*-statistic with multiple predictors

* The *F*-statistic is calculated using sums of squares

* Type I (sequential)
  + The default in R
  + Each predictor is evaluated taking account of previous predictors
  + **The order of predictors matters!**

* Type III
  + Each predictor is evaluated taking account of all other predictors
  + The order of predictors doesn’t matter

* Type II and IV
  + These exist too but let's not confuse things ...

---
background-image: url("media/milton_20180721_195630_crop.jpg")
background-size: cover
# .center[Type I sums of squares]

```r
lm(happiness ~ `puppy_love + dose`, 
   data = pupluv_tib) |> 
  anova()
```
.whitebox[
<table>
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:right;"> Df </th>
   <th style="text-align:right;"> Sum Sq </th>
   <th style="text-align:right;"> Mean Sq </th>
   <th style="text-align:right;"> F value </th>
   <th style="text-align:right;"> Pr(&gt;F) </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> puppy_love </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 6.73 </td>
   <td style="text-align:right;"> 6.73 </td>
   <td style="text-align:right;"> 2.22 </td>
   <td style="text-align:right;"> 0.15 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> dose </td>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 25.19 </td>
   <td style="text-align:right;"> 12.59 </td>
   <td style="text-align:right;"> 4.14 </td>
   <td style="text-align:right;"> 0.03 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Residuals </td>
   <td style="text-align:right;"> 26 </td>
   <td style="text-align:right;"> 79.05 </td>
   <td style="text-align:right;"> 3.04 </td>
   <td style="text-align:right;"> NA </td>
   <td style="text-align:right;"> NA </td>
  </tr>
</tbody>
</table>
]]
--
.pull-right[

```r
lm(happiness ~ `dose + puppy_love`,
   data = pupluv_tib) |> 
  anova()
```
.whitebox[
<table>
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:right;"> Df </th>
   <th style="text-align:right;"> Sum Sq </th>
   <th style="text-align:right;"> Mean Sq </th>
   <th style="text-align:right;"> F value </th>
   <th style="text-align:right;"> Pr(&gt;F) </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> dose </td>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 16.84 </td>
   <td style="text-align:right;"> 8.42 </td>
   <td style="text-align:right;"> 2.77 </td>
   <td style="text-align:right;"> 0.08 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> puppy_love </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 15.08 </td>
   <td style="text-align:right;"> 15.08 </td>
   <td style="text-align:right;"> 4.96 </td>
   <td style="text-align:right;"> 0.03 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Residuals </td>
   <td style="text-align:right;"> 26 </td>
   <td style="text-align:right;"> 79.05 </td>
   <td style="text-align:right;"> 3.04 </td>
   <td style="text-align:right;"> NA </td>
   <td style="text-align:right;"> NA </td>
  </tr>
</tbody>
</table>
]]

<br>
.whitebox[
.warning[
.txt_lrge[<svg style="height: 1em; top:.04em; position: relative; fill: #CA3E34;" viewBox="0 0 576 512"><path d="M192,320h32V224H192Zm160,0h32V224H352ZM544,112H512a32.03165,32.03165,0,0,0-32,32v16H416V128h32a32.03165,32.03165,0,0,0,32-32V64a32.03165,32.03165,0,0,0-32-32H416a32.03165,32.03165,0,0,0-32,32H352a32.03165,32.03165,0,0,0-32,32v32H256V96a32.03165,32.03165,0,0,0-32-32H192a32.03165,32.03165,0,0,0-32-32H128A32.03165,32.03165,0,0,0,96,64V96a32.03165,32.03165,0,0,0,32,32h32v32H96V144a32.03165,32.03165,0,0,0-32-32H32A32.03165,32.03165,0,0,0,0,144V288a32.03165,32.03165,0,0,0,32,32H64v32a32.03165,32.03165,0,0,0,32,32h32v64a32.03165,32.03165,0,0,0,32,32h80a32.03165,32.03165,0,0,0,32-32V416a32.03165,32.03165,0,0,0-32-32h96a32.03165,32.03165,0,0,0-32,32v32a32.03165,32.03165,0,0,0,32,32h80a32.03165,32.03165,0,0,0,32-32V384h32a32.03165,32.03165,0,0,0,32-32V320h32a32.03165,32.03165,0,0,0,32-32V144A32.03165,32.03165,0,0,0,544,112ZM416,64h32V96H416ZM128,96V64h32V96ZM240,448H160V384h32v32h48Zm176,0H336V416h48V384h32ZM544,288H480v64H96V288H32V144H64V256H96V192h96V96h32v64H352V96h32v96h96v64h32V144h32Z"/></svg> WARNING!
**Type I sums of squares: variable order matters!**]
]]
---
background-image: url("media/milton_20180720_100533_crop.jpg")
background-size: cover

# Type III sums of squares

--
.pull-left[

```r
lm(happiness ~ `puppy_love + dose`,
   data = pupluv_tib) |> 
car::Anova(type = 3)
```
.whitebox[
<table>
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:right;"> Sum Sq </th>
   <th style="text-align:right;"> Df </th>
   <th style="text-align:right;"> F value </th>
   <th style="text-align:right;"> Pr(&gt;F) </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> (Intercept) </td>
   <td style="text-align:right;"> 12.94 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 4.26 </td>
   <td style="text-align:right;"> 0.05 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> puppy_love </td>
   <td style="text-align:right;"> 15.08 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 4.96 </td>
   <td style="text-align:right;"> 0.03 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> dose </td>
   <td style="text-align:right;"> 25.19 </td>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 4.14 </td>
   <td style="text-align:right;"> 0.03 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Residuals </td>
   <td style="text-align:right;"> 79.05 </td>
   <td style="text-align:right;"> 26 </td>
   <td style="text-align:right;"> NA </td>
   <td style="text-align:right;"> NA </td>
  </tr>
</tbody>
</table>
]
]
--
.pull-right[

```r
lm(happiness ~ `dose + puppy_love`,
   data = pupluv_tib) |> 
car::Anova(type = 3)
```
.whitebox[
<table>
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:right;"> Sum Sq </th>
   <th style="text-align:right;"> Df </th>
   <th style="text-align:right;"> F value </th>
   <th style="text-align:right;"> Pr(&gt;F) </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> (Intercept) </td>
   <td style="text-align:right;"> 12.94 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 4.26 </td>
   <td style="text-align:right;"> 0.05 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> dose </td>
   <td style="text-align:right;"> 25.19 </td>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 4.14 </td>
   <td style="text-align:right;"> 0.03 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> puppy_love </td>
   <td style="text-align:right;"> 15.08 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 4.96 </td>
   <td style="text-align:right;"> 0.03 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Residuals </td>
   <td style="text-align:right;"> 79.05 </td>
   <td style="text-align:right;"> 26 </td>
   <td style="text-align:right;"> NA </td>
   <td style="text-align:right;"> NA </td>
  </tr>
</tbody>
</table>
]]

<br>
.whitebox[
.tip[
.txt_lrge[<svg style="height:1.5em; top:.04em; position: relative; fill: #2C5577;" viewBox="0 0 640 512"><path d="M512,176a16,16,0,1,0-16-16A15.9908,15.9908,0,0,0,512,176ZM576,32.72461V32l-.46094.3457C548.81445,12.30469,515.97461,0,480,0s-68.81445,12.30469-95.53906,32.3457L384,32v.72461C345.35156,61.93164,320,107.82422,320,160c0,.38086.10938.73242.11133,1.11328A272.01015,272.01015,0,0,0,96,304.26562V176A80.08413,80.08413,0,0,0,16,96a16,16,0,0,0,0,32,48.05249,48.05249,0,0,1,48,48V432a80.08413,80.08413,0,0,0,80,80H352a32.03165,32.03165,0,0,0,32-32,64.0956,64.0956,0,0,0-57.375-63.65625L416,376.625V480a32.03165,32.03165,0,0,0,32,32h32a32.03165,32.03165,0,0,0,32-32V316.77539A160.036,160.036,0,0,0,640,160C640,107.82422,614.64844,61.93164,576,32.72461ZM480,32a126.94015,126.94015,0,0,1,68.78906,20.4082L512,80H448L411.21094,52.4082A126.94015,126.94015,0,0,1,480,32Zm64,64v64a64,64,0,0,1-128,0V96l21.334,16h85.332ZM480,480H448V351.99609A15.99929,15.99929,0,0,0,425.5,337.377L303.1875,391.75a100.1169,100.1169,0,0,0-67.25-84.89062,7.96929,7.96929,0,0,0-10.09375,5.76562l-3.875,15.5625a8.16346,8.16346,0,0,0,5.375,9.5625C252,346.875,272,375.625,272,401.90625V448h48a32.03165,32.03165,0,0,1,32,32H144c-26.94531,0-48.13086-22.27344-47.99609-49.21875.63671-127.52734,101.31054-231.53516,227.36914-238.14063A160.02931,160.02931,0,0,0,480,320Zm0-192A128.14414,128.14414,0,0,1,352,160c0-32.16992,12.334-61.25391,32-83.76367V160a96,96,0,0,0,192,0V76.23633C595.666,98.74609,608,127.83008,608,160A128.14414,128.14414,0,0,1,480,288ZM432,160a16,16,0,1,0,16-16A15.9908,15.9908,0,0,0,432,160ZM162.94531,68.76953l39.71094,16.56055,16.5625,39.71094a5.32345,5.32345,0,0,0,9.53906,0l16.5586-39.71094,39.71484-16.56055a5.336,5.336,0,0,0,0-9.541l-39.71484-16.5586L228.75781,2.957a5.325,5.325,0,0,0-9.53906,0l-16.5625,39.71289-39.71094,16.5586a5.336,5.336,0,0,0,0,9.541Z"/></svg> TIP!
If you want *F*-statistics and have several predictors, use Type III sums of squares]
]]
---
# Bias in the *F*-statistic

.pull-left[
* For the significance of *F*-statistics to be accurate we assume that the relationship between the covariate and outcome is similar across groups.

* Known as **homogeneity of regression slopes**

* When the assumption is met the resulting *F*-statistic can be assumed to follow the *F*-distribution and the corresponding *p*-value is accurate.

* When the assumption is not met the *F*-statistic might not follow the *F*-distribution and the corresponding *p*-value is inaccurate

* Only relevant for the *F*-statistic
]

???
When the assumption of homogeneity of regression slopes is met the resulting F-statistic can be assumed to have the corresponding F-distribution; however, when the assumption is not met it can’t, meaning that the resulting F-statistic is being evaluated against a distribution different than the one that it actually has. Consequently, the Type I error rate of the test is inflated and the power to detect effects is not maximized (Hollingsworth, 1980). This is especially true when group sizes are unequal (Hamilton, 1977) and when the standardized regression slopes differ by more than 0.4 (Wu, 1984). 
---
## Homogeneity of regression slopes

.pull-left[
* For the significance of *F*-statistics to be accurate we assume that the relationship between the covariate and outcome is similar across groups.

* Known as **homogeneity of regression slopes**

* When the assumption is met the resulting *F*-statistic can be assumed to follow the *F*-distribution and the corresponding *p*-value is accurate.

* When the assumption is not met the *F*-statistic might not follow the *F*-distribution and the corresponding *p*-value is inaccurate

* Only relevant for the *F*-statistic
]

---
## Heterogeneity of regression slopes

.pull-left[
* For the significance of *F*-statistics to be accurate we assume that the relationship between the covariate and outcome is similar across groups.

* Known as **homogeneity of regression slopes**

* When the assumption is met the resulting *F*-statistic can be assumed to follow the *F*-distribution and the corresponding *p*-value is accurate.

* When the assumption is not met the *F*-statistic might not follow the *F*-distribution and the corresponding *p*-value is inaccurate

* Only relevant for the *F*-statistic
]

![](adjusting_means_files/figure-html/unnamed-chunk-23-1.png)
]

---
# Fitting the model
## Overall fit of each predictor

```r
pupluv_lm <- lm(happiness ~ dose + puppy_love, data = pupluv_tib) 
car::Anova(pupluv_lm, type = 3)
```

.center[
<table>
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:right;"> Sum Sq </th>
   <th style="text-align:right;"> Df </th>
   <th style="text-align:right;"> F value </th>
   <th style="text-align:right;"> Pr(&gt;F) </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> (Intercept) </td>
   <td style="text-align:right;"> 12.943 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 4.257 </td>
   <td style="text-align:right;"> 0.049 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> puppy_love </td>
   <td style="text-align:right;"> 15.076 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 4.959 </td>
   <td style="text-align:right;"> 0.035 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> dose </td>
   <td style="text-align:right;"> 25.185 </td>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 4.142 </td>
   <td style="text-align:right;"> 0.027 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Residuals </td>
   <td style="text-align:right;"> 79.047 </td>
   <td style="text-align:right;"> 26 </td>
   <td style="text-align:right;"> NA </td>
   <td style="text-align:right;"> NA </td>
  </tr>
</tbody>
</table>
]

--
<br>

.infobox[
<svg style="height: 1em; top:.04em; position: relative; fill: fill;" viewBox="0 0 512 512"><path d="M493.255 56.236l-37.49-37.49c-24.993-24.993-65.515-24.994-90.51 0L12.838 371.162.151 485.346c-1.698 15.286 11.22 28.203 26.504 26.504l114.184-12.687 352.417-352.417c24.992-24.994 24.992-65.517-.001-90.51zM164.686 347.313c6.249 6.249 16.379 6.248 22.627 0L368 166.627l30.059 30.059L174 420.745V386h-48v-48H91.255l224.059-224.059L345.373 144 164.686 324.687c-6.249 6.248-6.249 16.378 0 22.626zm-38.539 121.285l-58.995 6.555-30.305-30.305 6.555-58.995L63.255 366H98v48h48v34.745l-19.853 19.853zm344.48-344.48l-49.941 49.941-82.745-82.745 49.941-49.941c12.505-12.505 32.748-12.507 45.255 0l37.49 37.49c12.506 12.506 12.507 32.747 0 45.255z"/></svg> Love of puppies significantly predicted happiness *F*(1, 26) = 4.96, *p* = 0.035.

```r
broom::tidy(pupluv_lm, conf.int = TRUE)
```

.center[
<table>
 <thead>
  <tr>
   <th style="text-align:left;"> term </th>
   <th style="text-align:right;"> estimate </th>
   <th style="text-align:right;"> std.error </th>
   <th style="text-align:right;"> statistic </th>
   <th style="text-align:right;"> p.value </th>
   <th style="text-align:right;"> conf.low </th>
   <th style="text-align:right;"> conf.high </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> (Intercept) </td>
   <td style="text-align:right;"> 1.789 </td>
   <td style="text-align:right;"> 0.867 </td>
   <td style="text-align:right;"> 2.063 </td>
   <td style="text-align:right;"> 0.049 </td>
   <td style="text-align:right;"> 0.007 </td>
   <td style="text-align:right;"> 3.572 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> puppy_love </td>
   <td style="text-align:right;"> 0.416 </td>
   <td style="text-align:right;"> 0.187 </td>
   <td style="text-align:right;"> 2.227 </td>
   <td style="text-align:right;"> 0.035 </td>
   <td style="text-align:right;"> 0.032 </td>
   <td style="text-align:right;"> 0.800 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> dose15 mins </td>
   <td style="text-align:right;"> 1.786 </td>
   <td style="text-align:right;"> 0.849 </td>
   <td style="text-align:right;"> 2.102 </td>
   <td style="text-align:right;"> 0.045 </td>
   <td style="text-align:right;"> 0.040 </td>
   <td style="text-align:right;"> 3.532 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> dose30 mins </td>
   <td style="text-align:right;"> 2.225 </td>
   <td style="text-align:right;"> 0.803 </td>
   <td style="text-align:right;"> 2.771 </td>
   <td style="text-align:right;"> 0.010 </td>
   <td style="text-align:right;"> 0.575 </td>
   <td style="text-align:right;"> 3.875 </td>
  </tr>
</tbody>
</table>
]

--
<br>

---
## Adjusted means

```r
modelbased::estimate_means(pupluv_lm, fixed = "puppy_love")
```

.center[
<table>
 <thead>
  <tr>
   <th style="text-align:left;"> dose </th>
   <th style="text-align:right;"> puppy_love </th>
   <th style="text-align:right;"> Mean </th>
   <th style="text-align:right;"> SE </th>
   <th style="text-align:right;"> CI_low </th>
   <th style="text-align:right;"> CI_high </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> No puppies </td>
   <td style="text-align:right;"> 2.73 </td>
   <td style="text-align:right;"> 2.93 </td>
   <td style="text-align:right;"> 0.60 </td>
   <td style="text-align:right;"> 1.70 </td>
   <td style="text-align:right;"> 4.15 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 15 mins </td>
   <td style="text-align:right;"> 2.73 </td>
   <td style="text-align:right;"> 4.71 </td>
   <td style="text-align:right;"> 0.62 </td>
   <td style="text-align:right;"> 3.44 </td>
   <td style="text-align:right;"> 5.99 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 30 mins </td>
   <td style="text-align:right;"> 2.73 </td>
   <td style="text-align:right;"> 5.15 </td>
   <td style="text-align:right;"> 0.50 </td>
   <td style="text-align:right;"> 4.12 </td>
   <td style="text-align:right;"> 6.18 </td>
  </tr>
</tbody>
</table>
]

--
<br>

.infobox[
<svg style="height: 1em; top:.04em; position: relative; fill: fill;" viewBox="0 0 512 512"><path d="M493.255 56.236l-37.49-37.49c-24.993-24.993-65.515-24.994-90.51 0L12.838 371.162.151 485.346c-1.698 15.286 11.22 28.203 26.504 26.504l114.184-12.687 352.417-352.417c24.992-24.994 24.992-65.517-.001-90.51zM164.686 347.313c6.249 6.249 16.379 6.248 22.627 0L368 166.627l30.059 30.059L174 420.745V386h-48v-48H91.255l224.059-224.059L345.373 144 164.686 324.687c-6.249 6.248-6.249 16.378 0 22.626zm-38.539 121.285l-58.995 6.555-30.305-30.305 6.555-58.995L63.255 366H98v48h48v34.745l-19.853 19.853zm344.48-344.48l-49.941 49.941-82.745-82.745 49.941-49.941c12.505-12.505 32.748-12.507 45.255 0l37.49 37.49c12.506 12.506 12.507 32.747 0 45.255z"/></svg> At average levels of love of puppies, the mean happiness in the no puppy control group was *M* = 2.93 [1.70, 4.15] compared to *M* = 4.71 [3.44, 5.99] in the 15-minute group and *M* = 5.15 [4.12, 6.18] in the 30-minute group.
]

---
background-image: none
background-color: #000000
class: no-scribble

---
class: inverse
background-image: url("media/milton_20180906_163838_crop.jpg")
background-size: cover

# Testing assumptions

```r
library(ggfortify)
ggplot2::autoplot(pupluv_lm,
                  which = c(1, 3, 2, 4),
                  colour = "#5c97bf",
                  alpha = 0.5,
                  size = 1) +
  theme_minimal()
```
]

---
# Homogeneity of regression slopes

```r
hors_lm <- lm(happiness ~ `puppy_love*dose`, data = pupluv_tib)
car::Anova(hors_lm, type = 3)
```

<table>
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:right;"> Sum Sq </th>
   <th style="text-align:right;"> Df </th>
   <th style="text-align:right;"> F value </th>
   <th style="text-align:right;"> Pr(&gt;F) </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> (Intercept) </td>
   <td style="text-align:right;"> 0.771 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 0.316 </td>
   <td style="text-align:right;"> 0.579 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> puppy_love </td>
   <td style="text-align:right;"> 19.922 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 8.157 </td>
   <td style="text-align:right;"> 0.009 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> dose </td>
   <td style="text-align:right;"> 36.558 </td>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 7.484 </td>
   <td style="text-align:right;"> 0.003 </td>
  </tr>
  <tr>
   <td style="text-align:left;background-color: yellow !important;"> puppy_love:dose </td>
   <td style="text-align:right;background-color: yellow !important;"> 20.427 </td>
   <td style="text-align:right;background-color: yellow !important;"> 2 </td>
   <td style="text-align:right;background-color: yellow !important;"> 4.181 </td>
   <td style="text-align:right;background-color: yellow !important;"> 0.028 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Residuals </td>
   <td style="text-align:right;"> 58.621 </td>
   <td style="text-align:right;"> 24 </td>
   <td style="text-align:right;"> NA </td>
   <td style="text-align:right;"> NA </td>
  </tr>
</tbody>
</table>

.tip[
<svg style="height:1.5em; top:.04em; position: relative; fill: #2C5577;" viewBox="0 0 640 512"><path d="M512,176a16,16,0,1,0-16-16A15.9908,15.9908,0,0,0,512,176ZM576,32.72461V32l-.46094.3457C548.81445,12.30469,515.97461,0,480,0s-68.81445,12.30469-95.53906,32.3457L384,32v.72461C345.35156,61.93164,320,107.82422,320,160c0,.38086.10938.73242.11133,1.11328A272.01015,272.01015,0,0,0,96,304.26562V176A80.08413,80.08413,0,0,0,16,96a16,16,0,0,0,0,32,48.05249,48.05249,0,0,1,48,48V432a80.08413,80.08413,0,0,0,80,80H352a32.03165,32.03165,0,0,0,32-32,64.0956,64.0956,0,0,0-57.375-63.65625L416,376.625V480a32.03165,32.03165,0,0,0,32,32h32a32.03165,32.03165,0,0,0,32-32V316.77539A160.036,160.036,0,0,0,640,160C640,107.82422,614.64844,61.93164,576,32.72461ZM480,32a126.94015,126.94015,0,0,1,68.78906,20.4082L512,80H448L411.21094,52.4082A126.94015,126.94015,0,0,1,480,32Zm64,64v64a64,64,0,0,1-128,0V96l21.334,16h85.332ZM480,480H448V351.99609A15.99929,15.99929,0,0,0,425.5,337.377L303.1875,391.75a100.1169,100.1169,0,0,0-67.25-84.89062,7.96929,7.96929,0,0,0-10.09375,5.76562l-3.875,15.5625a8.16346,8.16346,0,0,0,5.375,9.5625C252,346.875,272,375.625,272,401.90625V448h48a32.03165,32.03165,0,0,1,32,32H144c-26.94531,0-48.13086-22.27344-47.99609-49.21875.63671-127.52734,101.31054-231.53516,227.36914-238.14063A160.02931,160.02931,0,0,0,480,320Zm0-192A128.14414,128.14414,0,0,1,352,160c0-32.16992,12.334-61.25391,32-83.76367V160a96,96,0,0,0,192,0V76.23633C595.666,98.74609,608,127.83008,608,160A128.14414,128.14414,0,0,1,480,288ZM432,160a16,16,0,1,0,16-16A15.9908,15.9908,0,0,0,432,160ZM162.94531,68.76953l39.71094,16.56055,16.5625,39.71094a5.32345,5.32345,0,0,0,9.53906,0l16.5586-39.71094,39.71484-16.56055a5.336,5.336,0,0,0,0-9.541l-39.71484-16.5586L228.75781,2.957a5.325,5.325,0,0,0-9.53906,0l-16.5625,39.71289-39.71094,16.5586a5.336,5.336,0,0,0,0,9.541Z"/></svg> Homogeneity of regression slopes **cannot be assumed** because:

1. The dose*puppy_love interaction effect is significant
2. The plots (see earlier) show that the relationship between puppy love and happiness is different in the 30 minute group to the other two groups.
]

---
background-image: url("media/milton_20190627_072357_crop.jpg")
background-size: cover

# Robust parameter estimates

```r
pupluv_rob <- robust::lmRob(happiness ~ puppy_love + dose, data = pupluv_tib)
summary(pupluv_rob)
```

.whitebox[.center[
<table>
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:right;"> Estimate </th>
   <th style="text-align:right;"> Std. Error </th>
   <th style="text-align:right;"> t value </th>
   <th style="text-align:right;"> Pr(&gt;|t|) </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> (Intercept) </td>
   <td style="text-align:right;"> 1.041 </td>
   <td style="text-align:right;"> 2.103 </td>
   <td style="text-align:right;"> 0.495 </td>
   <td style="text-align:right;"> 0.625 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> puppy_love </td>
   <td style="text-align:right;"> 0.633 </td>
   <td style="text-align:right;"> 0.556 </td>
   <td style="text-align:right;"> 1.139 </td>
   <td style="text-align:right;"> 0.265 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> dose15 mins </td>
   <td style="text-align:right;"> 1.855 </td>
   <td style="text-align:right;"> 2.642 </td>
   <td style="text-align:right;"> 0.702 </td>
   <td style="text-align:right;"> 0.489 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> dose30 mins </td>
   <td style="text-align:right;"> 1.400 </td>
   <td style="text-align:right;"> 1.750 </td>
   <td style="text-align:right;"> 0.800 </td>
   <td style="text-align:right;"> 0.431 </td>
  </tr>
</tbody>
</table>
]]

--
<br>

.infobox[
<svg style="height: 1em; top:.04em; position: relative; fill: fill;" viewBox="0 0 512 512"><path d="M493.255 56.236l-37.49-37.49c-24.993-24.993-65.515-24.994-90.51 0L12.838 371.162.151 485.346c-1.698 15.286 11.22 28.203 26.504 26.504l114.184-12.687 352.417-352.417c24.992-24.994 24.992-65.517-.001-90.51zM164.686 347.313c6.249 6.249 16.379 6.248 22.627 0L368 166.627l30.059 30.059L174 420.745V386h-48v-48H91.255l224.059-224.059L345.373 144 164.686 324.687c-6.249 6.248-6.249 16.378 0 22.626zm-38.539 121.285l-58.995 6.555-30.305-30.305 6.555-58.995L63.255 366H98v48h48v34.745l-19.853 19.853zm344.48-344.48l-49.941 49.941-82.745-82.745 49.941-49.941c12.505-12.505 32.748-12.507 45.255 0l37.49 37.49c12.506 12.506 12.507 32.747 0 45.255z"/></svg> Robust estimates showed that love of puppies **did not** have a significant effect on happiness, *b* = 0.63, *t* = 1.14, *p* = 0.265. For every unit increase in puppy love (on the 0-7 scale), predicted happiness (on the 0-10 scale) increased by 0.63 units.

# Heteroscedasticity consistent standard errors

```r
parameters::model_parameters(pupluv_lm,  vcov = "HC4")
```

.whitebox[.center[
<table>
 <thead>
  <tr>
   <th style="text-align:left;"> Parameter </th>
   <th style="text-align:right;"> Coefficient </th>
   <th style="text-align:right;"> SE </th>
   <th style="text-align:right;"> CI </th>
   <th style="text-align:right;"> CI_low </th>
   <th style="text-align:right;"> CI_high </th>
   <th style="text-align:right;"> t </th>
   <th style="text-align:right;"> df_error </th>
   <th style="text-align:right;"> p </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> (Intercept) </td>
   <td style="text-align:right;"> 1.789 </td>
   <td style="text-align:right;"> 0.544 </td>
   <td style="text-align:right;"> 0.95 </td>
   <td style="text-align:right;"> 0.671 </td>
   <td style="text-align:right;"> 2.908 </td>
   <td style="text-align:right;"> 3.288 </td>
   <td style="text-align:right;"> 26 </td>
   <td style="text-align:right;"> 0.003 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> puppy_love </td>
   <td style="text-align:right;"> 0.416 </td>
   <td style="text-align:right;"> 0.190 </td>
   <td style="text-align:right;"> 0.95 </td>
   <td style="text-align:right;"> 0.025 </td>
   <td style="text-align:right;"> 0.807 </td>
   <td style="text-align:right;"> 2.187 </td>
   <td style="text-align:right;"> 26 </td>
   <td style="text-align:right;"> 0.038 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> dose15 mins </td>
   <td style="text-align:right;"> 1.786 </td>
   <td style="text-align:right;"> 0.493 </td>
   <td style="text-align:right;"> 0.95 </td>
   <td style="text-align:right;"> 0.772 </td>
   <td style="text-align:right;"> 2.800 </td>
   <td style="text-align:right;"> 3.619 </td>
   <td style="text-align:right;"> 26 </td>
   <td style="text-align:right;"> 0.001 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> dose30 mins </td>
   <td style="text-align:right;"> 2.225 </td>
   <td style="text-align:right;"> 0.690 </td>
   <td style="text-align:right;"> 0.95 </td>
   <td style="text-align:right;"> 0.807 </td>
   <td style="text-align:right;"> 3.642 </td>
   <td style="text-align:right;"> 3.226 </td>
   <td style="text-align:right;"> 26 </td>
   <td style="text-align:right;"> 0.003 </td>
  </tr>
</tbody>
</table>
]]

--
<br>

* When we include both a categorical and continuous predictor, the categorical predictor compares means adjusted for the effect of the continuous predictor.
  + The effect of the categorical variable at average levels of the continuous predictor

* Test the overall effect of categorical predictors using the *F*-statistic
  + Use Type III sums of squares (other things being equal)
  + Test for homogeneity of regression slopes

* Break down the effects of categorical predictors using parameter estimates and their associated tests
  + Interpret in the same way as in previous lectures
  
* Test the usual assumptions for the linear model

* Apply a robust test if necessary