Prediction intervals

How do we calculate it in R?

• In with the confint function:

mod <- lm(frequency_score ~ group, data)
summary(mod)

Call:
lm(formula = frequency_score ~ group, data = data)

Residuals:
   Min     1Q Median     3Q    Max 
-38.80 -22.55 -11.80  30.20  95.20 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)   39.800      7.757   5.131 8.82e-06 ***
groupsquare   -8.000     10.970  -0.729     0.47    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 34.69 on 38 degrees of freedom
Multiple R-squared:  0.0138,    Adjusted R-squared:  -0.01215 
F-statistic: 0.5318 on 1 and 38 DF,  p-value: 0.4703

confint(mod)

                2.5 %   97.5 %
(Intercept)  24.09636 55.50364
groupsquare -30.20830 14.20830

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)

-8 - t_star * 10.97

[1] -30.2076

-8 + t_star * 10.97

[1] 14.2076

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(frequency_score ~ group, data = data)
predict(mod) 

   1    2    3 
39.8 39.8 39.8 

mod <- lm(frequency_score ~ group, data = data)
predict(mod, interval = "confidence") 

   fit      lwr      upr
1 39.8 24.09636 55.50364
2 39.8 24.09636 55.50364
3 39.8 24.09636 55.50364

mod <- lm(frequency_score ~ group, data = data)
predict(mod, interval = "prediction") 

   fit       lwr      upr
1 39.8 -32.16311 111.7631
2 39.8 -32.16311 111.7631
3 39.8 -32.16311 111.7631

Let’s do it in R!

What if we have new data?

new_data <- data.frame(
  group = c("square", "circle")
)
new_data

   group
1 square
2 circle

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

   fit       lwr      upr
1 31.8 -40.16311 103.7631
2 39.8 -32.16311 111.7631