The Linear Model 2: *R2* Strikes Back

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# The Linear Model 2: <em>R<sup>2</sup></em> Strikes Back
### Dr Jennifer Mankin
### 14 March 2022

---

## Looking Ahead (and Behind)

- Previously

- Samples, distributions, and *t*-tests
  
  - The take-away paper
  
--

- Last week: The Linear Model - Equation of a Line

- This week: The Linear Model - Evaluating the Model

---

## TAP Set Analysis

- View the Set Analysis document on Canvas > Take-away paper Information

- You **must use the Set Analysis results** to complete your Psychobiology report

- If you don't have exactly these results, DON'T PANIC

- Marking is based on everything you submitted, not just getting one right answer!

---

## Objectives

After this lecture you will understand:

- The equation for a linear model with one predictor

- *b*<sub>0</sub> (the intercept)
  
  - *b*<sub>1</sub> (the slope)
  
- The logic of NHST for *b*-values

- Interpreting *p* and CIs

- How to assess model fit with `\(R^2\)`

---

## General Model Equation
<br><br>

<center><font size = "18">
`\(outcome = model + error\)`
</font></center><p>

* We can use models to **predict** the outcome for a particular case

* This is always subject to some degree of **error**

---

## The Linear Model

- The linear model predicts the outcome `\(y\)` based on a predictor `\(x\)`

- General form: `\(y_{i} = b_{0} + b_{1}x_{1i} + e_{i}\)`

- *b*<sub>0</sub>: the **intercept**, or value of `\(y\)` when `\(x\)` is 0
  
  - *b*<sub>1</sub>: the **slope**, or change in `\(y\)` for every unit change in `\(x\)`

- The slope *b*<sub>1</sub> represents the relationship between the predictor and the outcome

---

## Today's Example

- "Does Hugging Provide Stress-Buffering Social Support? A Study of Susceptibility to Upper Respiratory Infection and Illness" ([Cohen et al., 2015](https://journals.sagepub.com/doi/full/10.1177/0956797614559284))

- Participants completed questionnaires and phone interviews over 14 days

- Including whether they had been hugged each day

--
  
- Then exposed to a cold virus! 🤒

- Measures of infection: amount of mucus, nasal clearing time
  
--
  
- Does receipt of hugs have a relationship with infection?

- What kind of relationship might we predict? 🤔

![GIF of Olaf from Frozen saying, "And I like warm hugs."](data:image/png;base64,#../../../static/images/warm_hugs.gif)

---

## Operationalisation

- Predictor: Percentage of days in which participants were hugged

- Higher percentage = more hugs

- Outcome: Nasal clearing time

- A measure of congestion
  
  - Longer time = more congestion (= worse cold)
  
--
  
- Model: `\(Congestion_{i} = b_{0} + b_{1}\times{Hugs_{1i}} + e_{i}\)`

- Use the `lm()` function to estimate *b*<sub>0</sub> and *b*<sub>1</sub>
  
---

## Having a Look

![](data:image/png;base64,#slides_files/figure-html/unnamed-chunk-1-1.png)]

---

## Having a Look

```r
cold_hugs %>% 
  mutate(pct_hugs = pct_hugs*100) %>% 
  ggplot(aes(x = pct_hugs, y = post_nasal_clear_log)) +
  geom_point(position = "jitter", alpha = .4) +
  scale_x_continuous(name = "Percentage of Days with Hugs") +
  scale_y_continuous("Congestion (Log)") +
  geom_smooth(method = "lm") +
  cowplot::theme_cowplot()
```

![](data:image/png;base64,#slides_files/figure-html/unnamed-chunk-2-1.png)]

- Very slight *negative* relationship

- How can we interpret this relationship?

---

## Creating the Model

```
## 
## Call:
## lm(formula = post_nasal_clear_log ~ pct_hugs, data = cold_hugs)
## 
## Coefficients:
## (Intercept)     pct_hugs  
##      0.5952      -0.1077
```
  
- For every *unit increase* in hugs, congestion changes by -0.11

- Here, "unit increase" = 1%
  
  - So, congestion goes down by 0.11 for every 1% increase in hugs
  
Model: `\(Congestion_i = 0.60 - 0.11\times{Hugs_i}\)`

---

## The Story So Far

- Investigating whether hugs protect against colds

- Linear model shows that more hugs are associated with less congestion (infection)

- `\(Congestion_i = 0.60 - 0.11\times{Hugs_i}\)`

- Is this model any good? What do we mean by "good"?

- Captures a relationship that may in fact exist: significance and CIs of *b*<sub>1</sub>
  
--
  
  - Explains the variance in the outcome: `\(R^2\)` for the model

---

## NHST for LM

- *b*<sub>1</sub> quantifies the relationship between the predictor and the outcome

- The *effect* of interest and the key part of the linear model!
  
  - Our estimate of the true relationship in the population (the model parameter)

- So...is this value **significant**?

---

## NHST for LM

- Our recipe for significance testing is:

- Data

- A test statistic

- The distribution of that test statistic under the null hypothesis

- The probability *p* of finding a test statistic as large as the one we have (or larger) if the null hypothesis is true
  
- First, we need to sort out the null hypothesis of *b*<sub>1</sub>

---

## Null Hypothesis of *b*<sub>1</sub>

- *b*<sub>1</sub> captures the relationship between variables

- How much the outcome *y* changes for each unit change in *x*
  
  - Null hypothesis: the outcome *y* **does not change** when *x* changes
  
- What would this look like in terms of the linear model? 🤔

---

## Null Hypothesis of *b*<sub>1</sub>

- `\(Congestion_{i} = b_{0} + 0 \times Hugs_{1i} + e_{i}\)`

- This is the **null** or **intercept-only model**

![](data:image/png;base64,#slides_files/figure-html/unnamed-chunk-4-1.png)]

---

## Significance of *b*<sub>1</sub>

- *b*<sub>1</sub> = 0 represents the null hypothesis

- So, the alternative hypothesis is *b*<sub>1</sub> `\(\ne\)` 0

- For our model, does *b*<sub>1</sub> = 0?

- No! Here *b*<sub>1</sub> = -0.11

- Is our estimate of *b*<sub>1</sub> different **enough** from 0 to believe that it may actually not be 0 in the population?

- Compare the estimate of *b*<sub>1</sub> to the variation in estimates of *b*<sub>1</sub>

---

## Significance of *b*<sub>1</sub>

- Signal-to-noise ratio

- Signal: the estimate of *b*<sub>1</sub>
  
  - Noise: the standard error of *b*<sub>1</sub>
  
- Scale *b*<sub>1</sub> by its standard error: `\(\frac{b_{1}}{SE_{b_1}}\)`

- What do you get when you divide a normally distributed value by its standard error...???? 🤔

---

## That's the *t*!

![GIF meme of Kermit the Frog drinking tea](data:image/png;base64,#../../../static/images/kermit.gif)

---

## Significance of *b*<sub>1</sub>

- `\(\frac{b_{1}}{SE_{b_1}} = t\)`

- Compare our value of *t* to the *t*-distribution to get *p*, just as we've seen before
  
  - If *p* is smaller than our chosen alpha level, our predictor is considered to be significant
  
--
  
<table class="table" style="margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:left;"> Term </th>
   <th style="text-align:right;"> <em>b</em> </th>
   <th style="text-align:right;"> <em>SE</em><sub>b</sub> </th>
   <th style="text-align:right;"> <em>t</em> </th>
   <th style="text-align:left;"> <em>p</em> </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Intercept </td>
   <td style="text-align:right;"> 0.60 </td>
   <td style="text-align:right;"> 0.04 </td>
   <td style="text-align:right;"> 16.04 </td>
   <td style="text-align:left;"> &lt; .001 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Percentage of Days with Hugs </td>
   <td style="text-align:right;"> -0.11 </td>
   <td style="text-align:right;"> 0.05 </td>
   <td style="text-align:right;"> -2.05 </td>
   <td style="text-align:left;"> .041 </td>
  </tr>
</tbody>
</table>

---

## Confidence Intervals for *b*<sub>1</sub>

- Give us the range of likely sample estimates of `\(\beta_1\)` from other samples

- Only if our interval is one of the 95% of intervals that does in fact contain the population value!
  
  - Review [Lecture 2](https://and.netlify.app/lectures/#lecture-2) for more on CIs

--
  
- Key info: does the confidence interval cross or include 0?

- If yes, it's likely that we could have gathered a sample where *b*<sub>1</sub> was 0
  
---

## Confidence Intervals for *b*<sub>1</sub>

- What can we conclude from these confidence intervals? 🤔

<table class="table" style="margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:left;"> Term </th>
   <th style="text-align:right;"> <em>b</em> </th>
   <th style="text-align:right;"> <em>SE</em><sub>b</sub> </th>
   <th style="text-align:right;"> <em>t</em> </th>
   <th style="text-align:left;"> <em>p</em> </th>
   <th style="text-align:right;"> CI<sub>upper</sub> </th>
   <th style="text-align:right;"> CI<sub>lower</sub> </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Intercept </td>
   <td style="text-align:right;"> 0.60 </td>
   <td style="text-align:right;"> 0.04 </td>
   <td style="text-align:right;"> 16.04 </td>
   <td style="text-align:left;"> &lt; .001 </td>
   <td style="text-align:right;"> 0.522 </td>
   <td style="text-align:right;"> 0.668 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Percentage of Days with Hugs </td>
   <td style="text-align:right;"> -0.11 </td>
   <td style="text-align:right;"> 0.05 </td>
   <td style="text-align:right;"> -2.05 </td>
   <td style="text-align:left;"> .041 </td>
   <td style="text-align:right;"> -0.211 </td>
   <td style="text-align:right;"> -0.005 </td>
  </tr>
</tbody>
</table>

---

## Interim Summary

- The key element of the linear model is *b*<sub>1</sub>

- Quantifies the relationship between the predictor and the outcome
  
  - Null hypothesis: *b*<sub>1</sub> = 0
  
  - Alternative hypothesis: *b*<sub>1</sub> `\(\ne\)` 0

- Is *b*<sub>1</sub> different from 0?

- Significance via *t*
  
  - Confidence intervals
  
---

.pollEv[
<iframe src="https://embed.polleverywhere.com/discourses/IXL29dEI4tgTYG0zH1Sef?controls=none&short_poll=true" width="800px" height="600px"></iframe>
]

---

## A Good Model

- Captures a relationship that does in fact exist

- Isn't just noise (random variation)
  
  - Quantified with significance/CIs
  
--
  
- Is useful for understanding the outcome variable

- Explains variance in the outcome

- Quantified with `\(R^2\)`
  
---

## Explaining Variance

- We want to explain variance, particularly in the outcome

- Goodness of Fit: How well does the model fit the data?

- Better fit = model is better able to explain the outcome
  
- So, how do we quantify model fit?

---

## Goodness of Fit with `\(R^2\)`

---

## Goodness of Fit with `\(R^2\)`

- `\(R^2 = \frac{Variance\ explained\ by\ the\ model}{Total\ variance}\)`

- Interpret as a percentage of variance explained
  
  - Applies to our sample only
  
  - Larger value means better fit

- Adjusted `\(R^2\)`: estimate of `\(R^2\)` **in the population**

- How does this value look? 🤔
  
<br><br>

<table class="table" style="margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:right;"> R<sup>2</sup> </th>
   <th style="text-align:right;"> Adjusted R<sup>2</sup> </th>
   <th style="text-align:right;"> <em>F</em> </th>
   <th style="text-align:left;"> <em>p</em> </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:right;"> 0.01 </td>
   <td style="text-align:right;"> 0.01 </td>
   <td style="text-align:right;"> 4.22 </td>
   <td style="text-align:left;"> .041 </td>
  </tr>
</tbody>
</table>

---

## Putting It All Together

```r
hugs_lm %>% summary()
```

```
## 
## Call:
## lm(formula = post_nasal_clear_log ~ pct_hugs, data = cold_hugs)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.59522 -0.27880  0.01766  0.25219  0.79262 
## 
## Coefficients:
##             Estimate Std. Error t value            Pr(>|t|)    
## (Intercept)  0.59522    0.03710  16.043 <0.0000000000000002 ***
## pct_hugs    -0.10773    0.05243  -2.055              0.0405 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.3572 on 403 degrees of freedom
##   (1 observation deleted due to missingness)
## Multiple R-squared:  0.01037,	Adjusted R-squared:  0.007912 
## F-statistic: 4.222 on 1 and 403 DF,  p-value: 0.04055
```

---

## Summary

- The linear model (LM) expresses the relationship between at least one predictor, `\(x\)`, and an outcome, `\(\hat{y}\)`

- Linear model equation: `\(y_{i} = b_{0} + b_{1}x_{1i} + e_{i}\)`
  
  - Most important result is the parameter *b*<sub>1</sub>, which expresses the change in `\(y\)` for each unit change in `\(x\)`

- Evaluating the model

- Is it unlikely that *b*<sub>1</sub> isn't 0? Significance tests and CIs
  
  - How well does the model fit the data? `\(R^2\)` and adjusted `\(R^2\)`

---

.pollEv[
<iframe src="https://embed.polleverywhere.com/discourses/IXL29dEI4tgTYG0zH1Sef?controls=none&short_poll=true" width="800px" height="600px"></iframe>
]

---

## More Strikes??

- Have a look at the impact of pension changes: http://uss-pension-model.com/

- Staff working conditions are your learning conditions!

---