Regression on Faithful with Section 9.3 content

Transcription

1 Regression on Faithful with Section 9.3 content The faithful data frame contains 272 obervational units with variables waiting and eruptions measuring, in minutes, the amount of wait time between eruptions, and the length of eruptions. head(faithful) eruptions waiting We look at a scatterplot and include the regression line when waiting is treated as the response variable and eruptions is the explanatory one. gf_point(waiting ~ eruptions, data=faithful) %>% gf_lm() waiting eruptions It appears there is a positive association. The model utility test confirms this: faithful.lm.result <- lm(waiting ~ eruptions, data=faithful) summary(faithful.lm.result) Call: lm(formula = waiting ~ eruptions, data = faithful) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) <2e-16 *** eruptions <2e-16 *** 1

2 --- Signif. codes: 0 '***' '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Residual standard error: on 270 degrees of freedom Multiple R-squared: , Adjusted R-squared: F-statistic: 1162 on 1 and 270 DF, p-value: < 2.2e-16 The point estimate b 1 = of the true slope β 1 produces a standardized t-score t = b 1 0 = = SE b We count the number of rows in the faithful data frame nrow(faithful) [1] 272 and seeing n = 272, we use n 2 degrees of freedom to find the P -value pt(34.09, df=270, lower.tail=false) * 2 [1] e-100 This number, which usually appears next to the standardized t-score in the output above, was reported there as being less than We produce a histogram of residuals and a scatter plot of residuals vs. fitted values to verify model assumptions: gf_histogram( ~residuals(faithful.lm.result), color="black" ) count residuals(faithful.lm.result) gf_point( residuals(faithful.lm.result) ~ fitted(faithful.lm.result) ) 2

3 residuals(faithful.lm.result) fitted(faithful.lm.result) Remember that another way to obtain the same standardized t-score is by standardizing the correlation. Specifically, our data yields correlation cor(waiting ~ eruptions, data=faithful) [1] and the standard error for correlation is 1 r 2 1 (0.9008) n 2 = yielding standardized correlation. = , t = r 0 = = SE r practically the same as the t-value reported above (the difference between above and here is attributable to round-off error). Estimates at a given explanatory value (Section 9.3) We have b 0 and b 1 in the output above, giving us regression line waiting = (eruption). It is easier to pin down the average waiting time corresponding to a 2-minute eruption than to pin down the wait time on the next (or any individual) instance of a 2-minute eruption. But if we were to give a single number as estimate, we would use the same as an estimate of both the next wait time for a 2-minute eruption and the average wait time over all 2-minute eruptions. That number would be the one produced by the line (2). = Confidence interval for the mean response If we seek a 95% confidence interval for the mean waiting time over the population of eruptions lasting 2 minutes, we can do the following (so long as the mosaic package is loaded) 3

4 waittimeest = makefun(faithful.lm.result, interval="confidence") waittimeest(eruptions=2) You can modify this to ask simultaneously for confidence intervals of the mean response over multiple populations. Here, I do it focusing on the populations of eruptions lasting 1.8 minutes, 2 minutes, and 2.4 minutes. waittimeest(eruptions=c(1.8, 2, 2.4), interval="confidence") fit lwr upr You can use the level switch to change the level of confidence. To obtain a 90% confidence interval for the mean response (wait time) for eruptions lasting 2 minutes I do: waittimeest(eruptions=2, interval="confidence", level=.9) fit lwr upr Prediction intervals A prediction interval is built in similar fashion from the estimate that arises from the regression line, but has a larger margin of error than the same-level confidence interval for the mean response (discussed above). Having used the makefun() command to create the waittimeest() command, we can indicate we are after a prediction interval instead. This next command requests both 90% and 95% prediction intervals for the next response (wait time) when a 2-minute eruption arises. waittimeest(eruptions=2, interval="prediction", level=c(.9,.95)) fit lwr upr [1,] [2,] A plot showing confidence and prediction bands around the regression line This is a plot you aren t likely to produce on a whim. But it helps illustrate the concepts and make comparisons between prediction and confidence intervals for mean responses. temp_var <- predict(faithful.lm.result, interval="prediction") Warning in predict.lm(faithful.lm.result, interval = "prediction"): predictions on current data refer new_df <- cbind(faithful, temp_var) gf_point(waiting ~ eruptions, data=new_df) + geom_line(aes(y=lwr), color = "red", linetype = "dashed") + geom_line(aes(y=upr), color = "red", linetype = "dashed") + geom_smooth(method=lm, se=true) 4

5 waiting eruptions File creation date: Author: Thomas Scofield 5