Looking Ahead (and Behind)

• Week 4: Correlation

• Week 5: Chi-Square ( ${\chi }^2$ )

• Week 6: t-test

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

• This week: The Linear Model - Evaluating the Model

Announcements

• Lab Report Help

  • Lab Report Drop-ins

  • Results: review this week's practical on writing up lab report results

  • Discussion: review Academic Advising session

• Next week's practicals: The Linear Model (both lectures)

• Awards: SavioR nominations and Education Awards

• Feedback via the Suggestions Box

Objectives

After this lecture you will understand:

• The equation for a linear model with one predictor

  • b₀ (the intercept)

  • b₁ (the slope)

• The logic of NHST for b-values

  • Interpreting p and CIs

• How to assess model fit with $R^2$

General Model Equation

• 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: $y_i=b_0+b_1{x}_{1i}+e_i$

  • b₀: the intercept, or value of $y$ when $x$ is 0

  • b₁: the slope, or change in $y$ for every unit change in $x$

• The slope b₁ 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)

• 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? 🤔

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: $Congestio{n}_i=b_0+b_1\times Hug{s}_{1i}+e_i$

  • Use the lm() function to estimate b₀ and b₁

Having a Look

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)") +
  cowplot::theme_cowplot()

Having a Look

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()

• 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: $Congestio{n}_i=0.60-0.11\times Hug{s}_i$

The Story So Far

• Investigating whether hugs protect against colds

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

  • $Congestio{n}_i=0.60-0.11\times Hug{s}_i$

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

  • Captures a relationship that may in fact exist: significance and CIs of b₁

  • Explains the variance in the outcome: $R^2$ for the model

NHST for LM

• b₁ 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 b₁

Null Hypothesis of b₁

• b₁ 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 b₁

• $Congestio{n}_i=b_0+0\times Hug{s}_{1i}+e_i$

  • This is the null or intercept-only model

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") +
  geom_hline(yintercept = mean(cold_hugs$post_nasal_clear_log, na.rm = T), 
             linetype = "dashed", colour = "purple3")+
  cowplot::theme_cowplot()

Significance of b₁

• b₁ = 0 represents the null hypothesis

  • So, the alternative hypothesis is b₁ $\ne$ 0

• For our model, does b₁ = 0?

  • No! Here b₁ = -0.11

• Is our estimate of b₁ different enough from 0 to believe that it may actually not be 0 in the population?

  • Compare the estimate of b₁ to the variation in estimates of b₁

Significance of b₁

• Signal-to-noise ratio

  • Signal: the estimate of b₁

  • Noise: the standard error of b₁

• Scale b₁ by its standard error: $\frac{b_1}{S{E}_{b_1}}$

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

That's the t!

Significance of b₁

• $\frac{b_1}{S{E}_{b_1}}=t$

  • 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

Confidence Intervals for b₁

• 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 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 b₁ was 0

Confidence Intervals for b₁

• What can we conclude from these confidence intervals? 🤔

Interim Summary

• The key element of the linear model is b₁

  • Quantifies the relationship between the predictor and the outcome

  • Null hypothesis: b₁ = 0

  • Alternative hypothesis: b₁ $\ne$ 0

• Is b₁ different from 0?

  • Significance via t

  • Confidence intervals

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 $R^2$

• $R^2=\frac{Varianceexplainedbythemodel}{Totalvariance}$

  • 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? 🤔

Putting It All Together

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 b₁, which expresses the change in $y$ for each unit change in $x$

• Evaluating the model

  • Is it unlikely that b₁ isn't 0? Significance tests and CIs

  • How well does the model fit the data? $R^2$ and adjusted $R^2$