The Linear Model 3: Return of the *yi*

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# The Linear Model 3: Return of the <em>y<sub>i</sub></em>
### Dr Milan Valášek
### 28 March 2022

---

- **LM2:** &nbsp; <i>R</i><sup>2</sup> Strikes Back

- **LM3:** &nbsp; .secondary[Return of the <i>y<sub>i</sub></i>]

]
]

---

# Stats wars

- **LM1:** &nbsp; A New Equation

- **LM2:** &nbsp; <i>R</i><sup>2</sup> strikes back

- **LM3:** &nbsp; .secondary[Return of the <i>y<sub>i</sub></i>]

---

## Today

- Extending the linear model

- Multiple predictors

- Transforming variables in a model

- Mean-centring
  
  - Scaling
  
  - *z*-transforming

---

## Basic linear model

`$$\text{outcome}_{\text{obs}} = \text{intercept} + \text{slope} \times \text{predictor}_{\text{obs}} + \text{residual}_{\text{obs}}$$`

- The model is a line through the scatter of data

- The line shows what the value of outcome for a given value of predictor *should* be according to the model

- Residual is the difference between prediction and observation

---

## Mean as linear model

- The simplest linear model is **the mean**
- `$y_i = b_0 + e_i$`
- `$b_0 = Mean(y)$`
- That's *literally* the same as `$y_i = b_0 + 0\times x_{1_i} + e_i$`
- Mean is the *intercept-only* model: a linear model where all `$b$` coefficients other than `$b_0$` have been set (fixed) to zero

<br>

---

## Other coefficients?

- Just like we can fix `$b_1$` to zero in `$y_i = b_0 + 0\times x_{1_i} + e_i$`, we can fix any other `$b$` coefficient as well

- We can think of the basic single-predictor linear model as

`$$y_i = b_0 + b_1 \times x_{1_i} + 0 \times x_{2_i} + 0 \times x_{3_i} + \dots + 0 \times x_{n_i} + e_i$$`

- We're just ignoring all but one of the infinity possible predictors we could put in the model

- Not including a predictor in a model is **the same as saying that there is no relationship between that variable and the outcome**

- It's just said *implicitly* rather than aloud

- We can include them in the model if we wish to so that their associated `$b$` coefficient gets estimated, rather than set to 0

---

## Variables are dimensions

- We've been representing the mean as a line on a plot of 2 variables
- It can also be represented as a point on the number line
- Every predictor *adds a dimension*

<br>

---

## More complex models

- Including more than one predictor allows us to model the outcome variable in a more sophisticated way

- Every slope ( `$b_n$` coefficient, for `$n>0$`) expresses the relationship between a given predictor and the outcome *after the relationship of all other predictors has been accounted for*

- A relationship &ndash; causal or not &ndash; between two variables can drastically change when another variable is taken into account

- It's important to consider all variables with a known effect when modelling a relationship (especially in observational research)
  - Say we find a relationship between home environment and mental health
  - However, mental health has a strong genetic component
  - Parental predisposition to worse mental health is also linked to home environment
  - Can we *really* claim a relationship between environment and mental health if we don't consider genetics?
  
---

## Breast is best but is it smartest?

- Lot of ink has been spilled over the claim that breastfeeding leads to increase in child IQ ([BBC](https://www.bbc.co.uk/news/health-31925449), [The Guardian](https://www.theguardian.com/science/brain-flapping/2015/mar/18/breastfeeding-raises-iq-worrying-questions),
[The New York Times](https://www.nytimes.com/2018/05/09/well/family/breast-feeding-has-no-impact-on-iq-by-age-16.html),
[FiveThirtyEigth](https://fivethirtyeight.com/features/everybody-calm-down-about-breastfeeding/))

- When assessed at face value breastfed children have higher IQ

- Whether or not a person breastfeeds their child is also linked to things like socio&#8209;economic status or the person's IQ

- When these effects are adjusted for, the effect shrinks substantially &ndash; 3 IQ points difference is a [generous estimate](https://onlinelibrary.wiley.com/doi/full/10.1111/apa.13139) and even that has been [contested](https://www.pure.ed.ac.uk/ws/portalfiles/portal/30394557/Ritchie_Breastfeeding_letter_21_07_2016.pdf)

<br>

**The linear model allows us to build these more nuanced models and get closer to the Truth about the Universe<sup>TM</sup>**

---

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]

---

## Mutiple predictors in practice

- Today's example focuses on data about babies' birth weights and parental characteristics ([source](https://www.sheffield.ac.uk/mash/statistics/datasets))

<div data-pagedtable="false">
  <script data-pagedtable-source type="application/json">
{"columns":[{"label":["ID"],"name":[1],"type":["dbl"],"align":["right"]},{"label":["Length"],"name":[2],"type":["dbl"],"align":["right"]},{"label":["Birthweight"],"name":[3],"type":["dbl"],"align":["right"]},{"label":["Headcirc"],"name":[4],"type":["dbl"],"align":["right"]},{"label":["Gestation"],"name":[5],"type":["dbl"],"align":["right"]},{"label":["smoker"],"name":[6],"type":["dbl"],"align":["right"]},{"label":["mage"],"name":[7],"type":["dbl"],"align":["right"]},{"label":["mnocig"],"name":[8],"type":["dbl"],"align":["right"]},{"label":["mheight"],"name":[9],"type":["dbl"],"align":["right"]},{"label":["mppwt"],"name":[10],"type":["dbl"],"align":["right"]},{"label":["fage"],"name":[11],"type":["dbl"],"align":["right"]},{"label":["fedyrs"],"name":[12],"type":["dbl"],"align":["right"]},{"label":["fnocig"],"name":[13],"type":["dbl"],"align":["right"]},{"label":["fheight"],"name":[14],"type":["dbl"],"align":["right"]},{"label":["lowbwt"],"name":[15],"type":["dbl"],"align":["right"]},{"label":["mage35"],"name":[16],"type":["dbl"],"align":["right"]}],"data":[{"1":"1360","2":"56","3":"4.55","4":"34","5":"44","6":"0","7":"20","8":"0","9":"162","10":"57","11":"23","12":"10","13":"35","14":"179","15":"0","16":"0"},{"1":"1016","2":"53","3":"4.32","4":"36","5":"40","6":"0","7":"19","8":"0","9":"171","10":"62","11":"19","12":"12","13":"0","14":"183","15":"0","16":"0"},{"1":"462","2":"58","3":"4.10","4":"39","5":"41","6":"0","7":"35","8":"0","9":"172","10":"58","11":"31","12":"16","13":"25","14":"185","15":"0","16":"1"},{"1":"1187","2":"53","3":"4.07","4":"38","5":"44","6":"0","7":"20","8":"0","9":"174","10":"68","11":"26","12":"14","13":"25","14":"189","15":"0","16":"0"},{"1":"553","2":"54","3":"3.94","4":"37","5":"42","6":"0","7":"24","8":"0","9":"175","10":"66","11":"30","12":"12","13":"0","14":"184","15":"0","16":"0"},{"1":"1636","2":"51","3":"3.93","4":"38","5":"38","6":"0","7":"29","8":"0","9":"165","10":"61","11":"31","12":"16","13":"0","14":"180","15":"0","16":"0"},{"1":"820","2":"52","3":"3.77","4":"34","5":"40","6":"0","7":"24","8":"0","9":"157","10":"50","11":"31","12":"16","13":"0","14":"173","15":"0","16":"0"},{"1":"1191","2":"53","3":"3.65","4":"33","5":"42","6":"0","7":"21","8":"0","9":"165","10":"61","11":"21","12":"10","13":"25","14":"185","15":"0","16":"0"},{"1":"1081","2":"54","3":"3.63","4":"38","5":"38","6":"0","7":"18","8":"0","9":"172","10":"50","11":"20","12":"12","13":"7","14":"172","15":"0","16":"0"},{"1":"822","2":"50","3":"3.42","4":"35","5":"38","6":"0","7":"20","8":"0","9":"157","10":"48","11":"22","12":"14","13":"0","14":"179","15":"0","16":"0"},{"1":"1683","2":"53","3":"3.35","4":"33","5":"41","6":"0","7":"27","8":"0","9":"164","10":"62","11":"37","12":"14","13":"0","14":"170","15":"0","16":"0"},{"1":"1088","2":"51","3":"3.27","4":"36","5":"40","6":"0","7":"24","8":"0","9":"168","10":"53","11":"29","12":"16","13":"0","14":"181","15":"0","16":"0"},{"1":"1107","2":"52","3":"3.23","4":"36","5":"38","6":"0","7":"31","8":"0","9":"164","10":"57","11":"35","12":"16","13":"0","14":"183","15":"0","16":"0"},{"1":"755","2":"53","3":"3.20","4":"33","5":"41","6":"0","7":"21","8":"0","9":"155","10":"55","11":"25","12":"14","13":"25","14":"183","15":"0","16":"0"},{"1":"1058","2":"53","3":"3.15","4":"34","5":"40","6":"0","7":"29","8":"0","9":"167","10":"60","11":"30","12":"16","13":"25","14":"182","15":"0","16":"0"},{"1":"321","2":"48","3":"3.11","4":"33","5":"37","6":"0","7":"28","8":"0","9":"158","10":"54","11":"39","12":"10","13":"0","14":"171","15":"0","16":"0"},{"1":"697","2":"48","3":"3.03","4":"35","5":"39","6":"0","7":"27","8":"0","9":"162","10":"62","11":"27","12":"14","13":"0","14":"178","15":"0","16":"0"},{"1":"808","2":"48","3":"2.92","4":"33","5":"34","6":"0","7":"26","8":"0","9":"167","10":"64","11":"25","12":"12","13":"25","14":"175","15":"0","16":"0"},{"1":"1600","2":"53","3":"2.90","4":"34","5":"39","6":"0","7":"19","8":"0","9":"165","10":"57","11":"23","12":"14","13":"2","14":"193","15":"0","16":"0"},{"1":"1313","2":"43","3":"2.65","4":"32","5":"33","6":"0","7":"24","8":"0","9":"149","10":"45","11":"26","12":"16","13":"0","14":"169","15":"1","16":"0"},{"1":"792","2":"53","3":"3.64","4":"38","5":"40","6":"1","7":"20","8":"2","9":"170","10":"59","11":"24","12":"12","13":"12","14":"185","15":"0","16":"0"},{"1":"1388","2":"51","3":"3.14","4":"33","5":"41","6":"1","7":"22","8":"7","9":"160","10":"53","11":"24","12":"16","13":"12","14":"176","15":"0","16":"0"},{"1":"575","2":"50","3":"2.78","4":"30","5":"37","6":"1","7":"19","8":"7","9":"165","10":"60","11":"20","12":"14","13":"0","14":"183","15":"0","16":"0"},{"1":"569","2":"50","3":"2.51","4":"35","5":"39","6":"1","7":"22","8":"7","9":"159","10":"52","11":"23","12":"14","13":"25","14":"200","15":"1","16":"0"},{"1":"1363","2":"48","3":"2.37","4":"30","5":"37","6":"1","7":"20","8":"7","9":"163","10":"47","11":"20","12":"10","13":"35","14":"185","15":"1","16":"0"},{"1":"300","2":"46","3":"2.05","4":"32","5":"35","6":"1","7":"41","8":"7","9":"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</div>

---

## Birth weight, mother's age, and gestation time

```r
p1 <- bweight %>%
  ggplot(aes(mage, Birthweight)) +
  geom_point(size = 3, alpha = .4) +
  geom_smooth(method = "lm", color = theme_col, fill = second_col) +
  labs(x = "Mother's age at birth", y = "Birth weight (lbs)")

p2 <- bweight %>%
  ggplot(aes(Gestation, Birthweight)) +
  geom_point(size = 3, alpha = .4) +
  geom_smooth(method = "lm", color = second_col, fill = theme_col) +
  labs(x = "Gestation duration (weeks)", y = "")

cowplot::plot_grid(p1, p2)
```

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

---

## Fit model using `lm()`

```r
## Intercept-only model
m_null <- lm(Birthweight ~ 1, bweight)

## Add mother's age as predictor
m_age <- lm(Birthweight ~ mage, bweight)
# alternatively update(m_null, ~ . + mage)

## Add gestation duration as predictor
m_gest <- lm(Birthweight ~ mage + Gestation, bweight)
# same as update(m_age, ~ . + Gestation)
```

---

## Results - null model

```r
summary(m_null)
```

```
## 
## Call:
## lm(formula = Birthweight ~ 1, data = bweight)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.39286 -0.37286 -0.01786  0.33464  1.25714 
## 
## Coefficients:
##             Estimate Std. Error t value            Pr(>|t|)    
## (Intercept)  3.31286    0.09318   35.55 <0.0000000000000002 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.6039 on 41 degrees of freedom
```

---

## Results - Mother's age as predictor

```r
summary(m_age)
```

```
## 
## Call:
## lm(formula = Birthweight ~ mage, data = bweight)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.39275 -0.37288 -0.01786  0.33473  1.25702 
## 
## Coefficients:
##               Estimate Std. Error t value      Pr(>|t|)    
## (Intercept) 3.31238583 0.44072153   7.516 0.00000000362 ***
## mage        0.00001845 0.01685112   0.001         0.999    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.6114 on 40 degrees of freedom
## Multiple R-squared:  2.996e-08,	Adjusted R-squared:  -0.025 
## F-statistic: 1.199e-06 on 1 and 40 DF,  p-value: 0.9991
```

---

## Results - M's age and gestation time

```r
summary(m_gest)
```

```
## 
## Call:
## lm(formula = Birthweight ~ mage + Gestation, data = bweight)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.77485 -0.35861 -0.00236  0.26948  0.96943 
## 
## Coefficients:
##               Estimate Std. Error t value    Pr(>|t|)    
## (Intercept) -3.0092887  1.0567990  -2.848     0.00699 ** 
## mage        -0.0007953  0.0120469  -0.066     0.94770    
## Gestation    0.1618369  0.0258242   6.267 0.000000221 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.4371 on 39 degrees of freedom
## Multiple R-squared:  0.5017,	Adjusted R-squared:  0.4762 
## F-statistic: 19.64 on 2 and 39 DF,  p-value: 0.00000126
```

---

## Model prediction

- Linear model can tell us the expected value of outcome for any combination of predictor values

- According to our model, expected birth weight for a baby whose mother is 29 years old and whose gestation period was 38 weeks is:

`$$\begin{aligned}\hat{y}&=-3.01 + 0 \times \text{age} + 0.16 \times \text{gestation}\\&=-3.01 + 0 \times 29 + 0.16 \times 38\\&=-3.01 + 0 + 6.08\\&=3.07\end{aligned}$$`

- Let's compare to observations in sample

```r
bweight %>% filter(mage == 29 & Gestation == 38) %>%
  rmarkdown::paged_table()
```

---

## Negative intercept?!

- The intercept always tells us the value of the outcome when all predictors are 0

- Not always sensible (instantaneous childbirth in women aged 0 is not a common occurrence)
  
.codePanel[

```r
bweight %>%
  ggplot(aes(Gestation, Birthweight)) +
  geom_point(size = 3, alpha = .4) +
  geom_vline(xintercept = 0, lty=2) +
  geom_hline(yintercept = coefs[1], lty=2) +
  geom_point(data = tibble(x = 0, y = coefs[1]),
             mapping = aes(x, y), colour = "#fdfdfd", size = 4) +
  geom_abline(intercept = coefs[1], slope = coefs[3], color = second_col, size = 1) +
  geom_point(data = tibble(x = 0, y = coefs[1]),
             mapping = aes(x, y), pch=21, size = 4) +
  xlim(c(0, 45)) +
  ylim(c(-4, 5)) +
  labs(x = "Gestation duration (weeks)", y = "Birth weight (lbs)")
```

![](data:image/png;base64,#slides_files/figure-html/unnamed-chunk-9-1.png)]

---

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]

---

## Transforming variables in the model

- We can apply various transformations to variables in the model

- Centring, scaling, standardising
  
  - Non-linear transformations are also possible (<i>e.g.,</i> log-transform)
  
- Transforming variables **changes the interpretation of the coefficients**

---

## Centring

- Centring *predictors* changes the interpretation of the intercept

```r
# untransformed predictor
lm(Birthweight ~ Gestation, bweight)
```

```
## 
## Call:
## lm(formula = Birthweight ~ Gestation, data = bweight)
## 
## Coefficients:
## (Intercept)    Gestation  
##     -3.0289       0.1618
```

```r
# centred predictor
bweight <- bweight %>%
  mutate(gest_cntrd = Gestation - mean(Gestation, na.rm=TRUE))

lm(Birthweight ~ gest_cntrd, bweight)
```

```
## 
## Call:
## lm(formula = Birthweight ~ gest_cntrd, data = bweight)
## 
## Coefficients:
## (Intercept)   gest_cntrd  
##      3.3129       0.1618
```
]

---

## Centring

- What's the weight of a baby born to a "typical" mother in terms of age and pregnancy duration

```r
# centre mother's age
bweight <- bweight %>%
  mutate(age_cntrd = mage - mean(mage, na.rm=TRUE))

lm(Birthweight ~ age_cntrd + gest_cntrd, bweight) %>% summary()
```

```
## 
## Call:
## lm(formula = Birthweight ~ age_cntrd + gest_cntrd, data = bweight)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.77485 -0.35861 -0.00236  0.26948  0.96943 
## 
## Coefficients:
##               Estimate Std. Error t value             Pr(>|t|)    
## (Intercept)  3.3128571  0.0674405  49.123 < 0.0000000000000002 ***
## age_cntrd   -0.0007953  0.0120469  -0.066                0.948    
## gest_cntrd   0.1618369  0.0258242   6.267          0.000000221 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.4371 on 39 degrees of freedom
## Multiple R-squared:  0.5017,	Adjusted R-squared:  0.4762 
## F-statistic: 19.64 on 2 and 39 DF,  p-value: 0.00000126
```
]

---

## Scaling

- Scaling *predictors* or *outcome* changes the interpretation of the slopes

```r
# untransformed outcome
lm(Birthweight ~ gest_cntrd, bweight)
```

```
## 
## Call:
## lm(formula = Birthweight ~ gest_cntrd, data = bweight)
## 
## Coefficients:
## (Intercept)   gest_cntrd  
##      3.3129       0.1618
```

```r
# scaled outcome
bweight <- bweight %>%
  mutate(bweight_g = Birthweight / 2.205 * 1000) # 2.205 lbs in kg

lm(bweight_g ~ gest_cntrd, bweight)
```

```
## 
## Call:
## lm(formula = bweight_g ~ gest_cntrd, data = bweight)
## 
## Coefficients:
## (Intercept)   gest_cntrd  
##     1502.43        73.39
```
]

---

## Standardising

- Sometimes it's useful to talk about change in outcome associated with a 1 *SD* change in predictors

```r
# untransformed predictor
lm(Birthweight ~ Gestation, bweight)
```

```
## 
## Call:
## lm(formula = Birthweight ~ Gestation, data = bweight)
## 
## Coefficients:
## (Intercept)    Gestation  
##     -3.0289       0.1618
```

```r
# standardised predictor

bweight <- bweight %>%
  mutate(gest_z = scale(Gestation))
lm(bweight_g ~ gest_z, bweight)
```

```
## 
## Call:
## lm(formula = bweight_g ~ gest_z, data = bweight)
## 
## Coefficients:
## (Intercept)       gest_z  
##        1502          194
```
]

---

## It's all the same model!

```r
p1 <- bweight %>%
  ggplot(aes(Gestation, Birthweight)) +
  geom_point(size = 3, alpha = .4) +
  geom_smooth(method = "lm", color = second_col, fill = theme_col) +
  labs(x = "Gestation time (weeks)", y = "Birth weight (lbs)")

p2 <- bweight %>%
  ggplot(aes(gest_cntrd, Birthweight)) +
  geom_point(size = 3, alpha = .4) +
  geom_smooth(method = "lm", color = second_col, fill = theme_col) +
  labs(x = "Gestation time (weeks from mean)", y = "")

p3 <- bweight %>%
  ggplot(aes(Gestation, bweight_g)) +
  geom_point(size = 3, alpha = .4) +
  geom_smooth(method = "lm", color = second_col, fill = theme_col) +
  labs(x = "Gestation time (weeks)", y = "Birth weight (g)")

p4 <- bweight %>%
  ggplot(aes(gest_z, bweight_g)) +
  geom_point(size = 3, alpha = .4) +
  geom_smooth(method = "lm", color = second_col, fill = theme_col) +
  labs(x = "Gestation time (z-score)", y = "")

cowplot::plot_grid(p1, p2, p3, p4, nrow = 2)
```

![](data:image/png;base64,#slides_files/figure-html/unnamed-chunk-14-1.png)]

---

## Standardised coefficients

- Standardised coefficients are equivalent to `$b$` coefficients in a model **where _both_ the predictors and the outcome have been *z*-transformed**

- We'll call them `$B$` to distinguish them from "raw" coefficients `$b$` but there is a lot of [confusion in literature about the notation](https://www.theanalysisfactor.com/confusing-statistical-terms-1-alpha-and-beta/) (you may see `$b$`, `$B$`, `$\beta$`, or `$Beta$` used to mean either of the two)

- `$B$` expresses the change in outcome in terms of number of *SD* as a result of 1 *SD* change in predictor

---

## Standardised coefficients

- Handy function &ndash; `QuantPsyc::lm.beta()`

- Only gives `$B$` for slopes, not intercept!

```r
m_gest <- lm(Birthweight ~ mage + Gestation, bweight)

# raw coefficients (b)
m_gest %>% coef()
```

```
##   (Intercept)          mage     Gestation 
## -3.0092887340 -0.0007952874  0.1618368592
```

```r
# standardised coefficeints (B)
m_gest %>% QuantPsyc::lm.beta()
```

```
##         mage    Gestation 
## -0.007462176  0.708383324
```

```r
# same as if we z-transform everything ourselves
lm(scale(Birthweight) ~ scale(mage) + scale(Gestation), bweight) %>% coef() %>% round(9)
```

```
##      (Intercept)      scale(mage) scale(Gestation) 
##      0.000000000     -0.007462176      0.708383324
```
]

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---

## Take-home message

- Linear model can be easily extended to more than one predictor

- Each predictor entered into the model *adds an extra dimension* to the space in which the model exists

- Each `$b$` coefficient (except for `$b_0$`) is a slope of the regression plane in its dimension

- Both *including* and *omitting* a variable is a claim about its relationship with the outcome

- A `$b$` coefficient for a predictor tells us about the relationship between the predictor and the outcome **after accounting for** the relationship between all other predictors and the outcome

- Intercept may not be a sensible value if variables are not transformed

- Transforming variables *changes the interpretation* of the coefficients

- Standardised coefficients, `$B$`, express the change in outcome in terms of number of *SD* as a result of 1 *SD* change in predictor

---

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