Prediction intervals

How do we calculate it in R?

• In with the confint function:

mod <- lm(battery_percent ~ screen_time, data)
summary(mod)

Call:
lm(formula = battery_percent ~ screen_time, data = data)

Residuals:
    Min      1Q  Median      3Q     Max 
-61.818 -17.353   2.546  19.108 115.720 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) 68.150787   2.503928  27.218  < 2e-16 ***
screen_time -0.022447   0.008347  -2.689  0.00735 ** 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 24.53 on 630 degrees of freedom
Multiple R-squared:  0.01135,   Adjusted R-squared:  0.009781 
F-statistic: 7.233 on 1 and 630 DF,  p-value: 0.007349

confint(mod)

                 2.5 %       97.5 %
(Intercept) 63.2337308 73.067842966
screen_time -0.0388383 -0.006056541

How do we calculate it in R?

• “by hand”

t_star <- qt(0.025, df = nrow(data) - 2, lower.tail = FALSE)
# or
t_star <- qt(0.975, df =  nrow(data) - 2)

-0.022447 - t_star * 0.008347

[1] -0.03883831

-0.022447 + t_star * 0.008347

[1] -0.00605569

Confidence intervals

There are ✌️ other types of confidence intervals we may want to calculate

• The confidence interval for the mean response in \(y\) for a given \(x^*\) value
• The confidence interval for an individual response \(y\) for a given \(x^*\) value
• Why are these different? Which do you think is easier to estimate? It is harder to predict one response than to predict a mean response. What does this mean in terms of the standard error?
• The SE of the prediction interval is going to be larger

Confidence intervals

confidence interval for \(\mu_y\) and prediction interval

\[ \hat{y}\pm t^* SE\]

• \(\hat{y}\) is the predicted \(y\) for a given \(x^*\)
• \(t^*\) is the critical value for the \(t_{n-2}\) density curve
• \(SE\) takes ✌️ different values depending on which interval you’re interested in

Confidence intervals

confidence interval for \(\mu_y\) and prediction interval

\[\hat{y}\pm t^* SE\]

• \(\hat{y}\) is the predicted \(y\) for a given \(x^*\)
• \(t^*\) is the critical value for the \(t_{n-2}\) density curve
• \(SE\) takes ✌️ different values depending on which interval you’re interested in
• \(SE_{\hat\mu} = \hat{\sigma}_\epsilon\sqrt{\frac{1}{n}+\frac{(x^*-\bar{x})^2}{\Sigma(x-\bar{x})^2}}\)
• \(SE_{\hat{y}}=\hat{\sigma}_\epsilon\sqrt{1 + \frac{1}{n}+\frac{(x^*-\bar{x})^2}{\Sigma(x-\bar{x})^2}}\)

Confidence intervals

confidence interval for \(\mu_y\) and prediction interval

\[\hat{y}\pm t^* SE\]

• \(\hat{y}\) is the predicted \(y\) for a given \(x^*\)
• \(t^*\) is the critical value for the \(t_{n-2}\) density curve
• \(SE\) takes ✌️ different values depending on which interval you’re interested in
• \(SE_{\hat\mu} = \hat{\sigma}_\epsilon\sqrt{\frac{1}{n}+\frac{(x^*-\bar{x})^2}{\Sigma(x-\bar{x})^2}}\)
• \(SE_{\hat{y}}=\hat{\sigma}_\epsilon\sqrt{\color{red}1 + \frac{1}{n}+\frac{(x^*-\bar{x})^2}{\Sigma(x-\bar{x})^2}}\)

Let’s do it in R!

mod <- lm(battery_percent ~ screen_time, data = data)
predict(mod) 

       1        2        3 
62.69606 61.25943 59.91258 

mod <- lm(battery_percent ~ screen_time, data = data)
predict(mod, interval = "confidence") 

       fit      lwr      upr
1 62.69606 60.70358 64.68855
2 61.25943 59.27788 63.24097
3 59.91258 57.48676 62.33840

mod <- lm(battery_percent ~ screen_time, data = data)
predict(mod, interval = "prediction") 

       fit      lwr      upr
1 62.69606 14.47649 110.9156
2 61.25943 13.04031 109.4785
3 59.91258 11.67317 108.1520

Let’s do it in R!

What if we have new data?

new_data <- data.frame(
  screen_time = c(300, 250, 523)
)
new_data

  screen_time
1         300
2         250
3         523

predict(
  mod, 
  newdata = new_data, 
  interval = "prediction")

       fit       lwr      upr
1 61.41656 13.198504 109.6346
2 62.53893 14.320523 110.7573
3 56.41079  8.024989 104.7966