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Prediction intervals

1 / 11
by Dr. Lucy D'Agostino McGowan

confidence intervals

If we use the same sampling method to select different samples and computed an interval estimate for each sample, we would expect the true population parameter ( β1 ) to fall within the interval estimates 95% of the time.

2 / 11
by Dr. Lucy D'Agostino McGowan

Confidence interval for β1

How do we calculate the confidence interval for the slope?

3 / 11
by Dr. Lucy D'Agostino McGowan

Confidence interval for β1

How do we calculate the confidence interval for the slope?

ˆβ1±t∗SEˆβ1

3 / 11
by Dr. Lucy D'Agostino McGowan

How do we calculate it in R?

  • using the broom package
lm(Weight ~ WingLength, Sparrows) %>%
  tidy(conf.int = TRUE)
## # A tibble: 2 x 7
##   term        estimate std.error statistic  p.value conf.low conf.high
##   <chr>          <dbl>     <dbl>     <dbl>    <dbl>    <dbl>     <dbl>
## 1 (Intercept)    1.37     0.957       1.43 1.56e- 1   -0.531     3.26 
## 2 WingLength     0.467    0.0347     13.5  2.62e-25    0.399     0.536
4 / 11
by Dr. Lucy D'Agostino McGowan

How do we calculate it in R?

  • using the broom package
lm(Weight ~ WingLength, Sparrows) %>%
  tidy(conf.int = TRUE)
## # A tibble: 2 x 7
##   term        estimate std.error statistic  p.value conf.low conf.high
##   <chr>          <dbl>     <dbl>     <dbl>    <dbl>    <dbl>     <dbl>
## 1 (Intercept)    1.37     0.957       1.43 1.56e- 1   -0.531     3.26 
## 2 WingLength     0.467    0.0347     13.5  2.62e-25    0.399     0.536
  • "by hand"
t_star <- qt(0.025, df = 116 - 2, lower.tail = FALSE)
# or
t_star <- qt(0.975, df = 116 - 2)
0.467 - t_star * 0.0347
## [1] 0.3982596
0.467 + t_star * 0.0347
## [1] 0.5357404
4 / 11
by Dr. Lucy D'Agostino McGowan

Confidence intervals

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

5 / 11
by Dr. Lucy D'Agostino McGowan

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
5 / 11
by Dr. Lucy D'Agostino McGowan

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
5 / 11
by Dr. Lucy D'Agostino McGowan

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: confidence interval for μy
  • The confidence interval for an individual response y for a given x∗ value: prediction interval
6 / 11
by Dr. Lucy D'Agostino McGowan

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: confidence interval for μy
  • The confidence interval for an individual response y for a given x∗ value: prediction interval
  • Why are these different? Which do you think is easier to estimate?
6 / 11
by Dr. Lucy D'Agostino McGowan

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: confidence interval for μy
  • The confidence interval for an individual response y for a given x∗ value: prediction interval
  • 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?
6 / 11
by Dr. Lucy D'Agostino McGowan

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: confidence interval for μy
  • The confidence interval for an individual response y for a given x∗ value: prediction interval
  • 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
6 / 11
by Dr. Lucy D'Agostino McGowan

Confidence intervals

confidence interval for μy and prediction interval

ˆy±t∗SE

  • ˆy is the predicted y for a given x∗
  • t∗ is the critical value for the tn−2 density curve
  • SE takes ✌️ different values depending on which interval you're interested in
7 / 11
by Dr. Lucy D'Agostino McGowan

Confidence intervals

confidence interval for μy and prediction interval

ˆy±t∗SE

  • ˆy is the predicted y for a given x∗
  • t∗ is the critical value for the tn−2 density curve
  • SE takes ✌️ different values depending on which interval you're interested in
  • SEˆμ
7 / 11
by Dr. Lucy D'Agostino McGowan

Confidence intervals

confidence interval for μy and prediction interval

ˆy±t∗SE

  • ˆy is the predicted y for a given x∗
  • t∗ is the critical value for the tn−2 density curve
  • SE takes ✌️ different values depending on which interval you're interested in
  • SEˆμ
  • SEˆy
7 / 11
by Dr. Lucy D'Agostino McGowan

Confidence intervals

confidence interval for μy and prediction interval

ˆy±t∗SE

  • ˆy is the predicted y for a given x∗
  • t∗ is the critical value for the tn−2 density curve
  • SE takes ✌️ different values depending on which interval you're interested in
  • SEˆμ
  • SEˆy
  • Which will be larger?
7 / 11
by Dr. Lucy D'Agostino McGowan

Confidence intervals

confidence interval for μy and prediction interval

ˆy±t∗SE

  • ˆy is the predicted y for a given x∗
  • t∗ is the critical value for the tn−2 density curve
  • SE takes ✌️ different values depending on which interval you're interested in
  • SEˆμ=ˆσϵ√1n+(x∗−ˉx)2Σ(x−ˉx)2
  • SEˆy=ˆσϵ√1+1n+(x∗−ˉx)2Σ(x−ˉx)2
  • Which will be larger?
8 / 11
by Dr. Lucy D'Agostino McGowan

Confidence intervals

confidence interval for μy and prediction interval

ˆy±t∗SE

  • ˆy is the predicted y for a given x∗
  • t∗ is the critical value for the tn−2 density curve
  • SE takes ✌️ different values depending on which interval you're interested in
  • SEˆμ=ˆσϵ√1n+(x∗−ˉx)2Σ(x−ˉx)2
  • SEˆy=ˆσϵ√1+1n+(x∗−ˉx)2Σ(x−ˉx)2
  • Which will be larger?
  • What is the difference between these two equations?
8 / 11
by Dr. Lucy D'Agostino McGowan

Confidence intervals

confidence interval for μy and prediction interval

ˆy±t∗SE

  • ˆy is the predicted y for a given x∗
  • t∗ is the critical value for the tn−2 density curve
  • SE takes ✌️ different values depending on which interval you're interested in
  • SEˆμ=ˆσϵ√1n+(x∗−ˉx)2Σ(x−ˉx)2
  • SEˆy=ˆσϵ√1+1n+(x∗−ˉx)2Σ(x−ˉx)2
9 / 11
by Dr. Lucy D'Agostino McGowan

Confidence intervals

confidence interval for μy and prediction interval

ˆy±t∗SE

  • ˆy is the predicted y for a given x∗
  • t∗ is the critical value for the tn−2 density curve
  • SE takes ✌️ different values depending on which interval you're interested in
  • SEˆμ=ˆσϵ√1n+(x∗−ˉx)2Σ(x−ˉx)2
  • SEˆy=ˆσϵ√1+1n+(x∗−ˉx)2Σ(x−ˉx)2
  • an individual response will vary from the mean response μy with a standard deviation of σϵ
9 / 11
by Dr. Lucy D'Agostino McGowan

Let's do it in R!

lm(Weight ~ WingLength, data = Sparrows) %>% 
  predict()
##        1        2        3 
## 14.92020 15.85501 13.05059
10 / 11
by Dr. Lucy D'Agostino McGowan

Let's do it in R!

lm(Weight ~ WingLength, data = Sparrows) %>% 
  predict()
##        1        2        3 
## 14.92020 15.85501 13.05059
lm(Weight ~ WingLength, data = Sparrows) %>% 
  predict(interval = "confidence")
##        fit      lwr      upr
## 1 14.92020 14.63801 15.20240
## 2 15.85501 15.49396 16.21607
## 3 13.05059 12.74776 13.35342
10 / 11
by Dr. Lucy D'Agostino McGowan

Let's do it in R!

lm(Weight ~ WingLength, data = Sparrows) %>% 
  predict()
##        1        2        3 
## 14.92020 15.85501 13.05059
lm(Weight ~ WingLength, data = Sparrows) %>% 
  predict(interval = "confidence")
##        fit      lwr      upr
## 1 14.92020 14.63801 15.20240
## 2 15.85501 15.49396 16.21607
## 3 13.05059 12.74776 13.35342
lm(Weight ~ WingLength, data = Sparrows) %>%
  predict(interval = "prediction")

## WARNING predictions on current data refer to _future_ responses

##        fit      lwr      upr
## 1 14.92020 12.13329 17.70712
## 2 15.85501 13.05902 18.65101
## 3 13.05059 10.26151 15.83966
10 / 11
by Dr. Lucy D'Agostino McGowan

Let's do it in R!

What if we have new data?

new_sparrows <- data.frame(
  WingLength = c(30, 28, 25)
)
new_sparrows
##   WingLength
## 1         30
## 2         28
## 3         25
11 / 11
by Dr. Lucy D'Agostino McGowan

Let's do it in R!

What if we have new data?

new_sparrows <- data.frame(
  WingLength = c(30, 28, 25)
)
new_sparrows
##   WingLength
## 1         30
## 2         28
## 3         25
lm(Weight ~ WingLength, data = Sparrows) %>%
  predict(newdata = new_sparrows, interval = "prediction")
##        fit      lwr      upr
## 1 15.38761 12.59700 18.17822
## 2 14.45280 11.66790 17.23771
## 3 13.05059 10.26151 15.83966
11 / 11
by Dr. Lucy D'Agostino McGowan

confidence intervals

If we use the same sampling method to select different samples and computed an interval estimate for each sample, we would expect the true population parameter ( β1 ) to fall within the interval estimates 95% of the time.

2 / 11
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